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==Resumen== | ==Resumen== | ||
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Desde principios del presente siglo se han incrementado notablemente los esfuerzos por coordinar y sincronizar grupos de robots que forman redes para el trabajo colaborativo. Estas ''Redes de Robots'' (RR´s) se justifican en misiones que no son posibles llevarlas a cabo con robots actuando de forma individual por razones de su propia capacidad. Entre los aspectos de estudio de mayor interés destaca la propuesta de ''leyes de control'' que gobiernen el comportamiento colectivo. Para muchas misiones de estas RR´s, la etapa primaria es su ''formación'' con un patrón geométrico o una distribución determinada [1]. La formación consiste en un posicionamiento relativo espacial con respecto a una o más referencias, fijas o móviles, para cada uno de los robots que forman parte de la red [2]. Para este problema de formación se han utilizado distintas metodologías [2-8]. La mayor parte de estas metodologías se fundamenta en la teoría de grafos cuyos inicios están documentados desde el siglo XVIII y que en la década de 1960 se extiende para aplicaciones en redes complejas [9-15]. Mediante la teoría de grafos es posible expresar interacciones entre sistemas dinámicos de manera general y organizada, facilitando la formulación de leyes de control para gobernar el comportamiento entero de una red. | Desde principios del presente siglo se han incrementado notablemente los esfuerzos por coordinar y sincronizar grupos de robots que forman redes para el trabajo colaborativo. Estas ''Redes de Robots'' (RR´s) se justifican en misiones que no son posibles llevarlas a cabo con robots actuando de forma individual por razones de su propia capacidad. Entre los aspectos de estudio de mayor interés destaca la propuesta de ''leyes de control'' que gobiernen el comportamiento colectivo. Para muchas misiones de estas RR´s, la etapa primaria es su ''formación'' con un patrón geométrico o una distribución determinada [1]. La formación consiste en un posicionamiento relativo espacial con respecto a una o más referencias, fijas o móviles, para cada uno de los robots que forman parte de la red [2]. Para este problema de formación se han utilizado distintas metodologías [2-8]. La mayor parte de estas metodologías se fundamenta en la teoría de grafos cuyos inicios están documentados desde el siglo XVIII y que en la década de 1960 se extiende para aplicaciones en redes complejas [9-15]. Mediante la teoría de grafos es posible expresar interacciones entre sistemas dinámicos de manera general y organizada, facilitando la formulación de leyes de control para gobernar el comportamiento entero de una red. | ||
| − | Dentro del campo de la teoría de grafos y redes complejas, el ''acoplamiento difusivo estático'' es una estrategia ampliamente utilizada para realizar la interconexión entre los nodos (robots) de la red [16,17]. El principio de acoplamiento difusivo estático es un esquema de retroalimentación constante [17] entre los robots que forman parte de la red, pudiendo ser esta conectividad global o parcial (respectivamente: todos los robots se conectan con el resto de robots o sólo parte de ellos). La retroalimentación consiste en insertar como parte de la entrada del sistema dinámico del robot una señal que es la diferencia entre las variables de salida del robot de interés y las variables de salida del resto de robots en la red [18]. De este tipo de acoplamiento se han derivado distintas metodologías con el propósito de forzar una respuesta deseada en los robots, denominada estado de ''sincronía''[14]; ó bien, una respuesta de sincronía natural, denominada ''consenso'' [19-21]. Una modalidad de estrategia de control que destaca para redes de gran dimensión es el control ''pinning ''(líderes referentes), que propone una acción de control en sólo un reducido número de sistemas dinámicos de la red y que inducen a la red entera hacia un estado de sincronía [11,14-15,22]. Estas estrategias se pueden llevar a cabo tanto mediante el control centralizado como con el control distribuido [23]. Con el control centralizado el procesamiento reside en un procesador que se encarga de monitorear las variables de estado de cada robot y de comunicar las acciones de control necesarias para alcanzar el objetivo de movimiento. Con el control distribuido el procesamiento reside en dos o más procesadores con acceso total o parcial a la información de variables de estado de la red; típicamente los robots llevan sus | + | Dentro del campo de la teoría de grafos y redes complejas, el ''acoplamiento difusivo estático'' es una estrategia ampliamente utilizada para realizar la interconexión entre los nodos (robots) de la red [16,17]. El principio de acoplamiento difusivo estático es un esquema de retroalimentación constante [17] entre los robots que forman parte de la red, pudiendo ser esta conectividad global o parcial (respectivamente: todos los robots se conectan con el resto de robots o sólo parte de ellos). La retroalimentación consiste en insertar como parte de la entrada del sistema dinámico del robot una señal que es la diferencia entre las variables de salida del robot de interés y las variables de salida del resto de robots en la red [18]. De este tipo de acoplamiento se han derivado distintas metodologías con el propósito de forzar una respuesta deseada en los robots, denominada estado de ''sincronía''[14]; ó bien, una respuesta de sincronía natural, denominada ''consenso'' [19-21]. Una modalidad de estrategia de control que destaca para redes de gran dimensión es el control ''pinning ''(líderes referentes), que propone una acción de control en sólo un reducido número de sistemas dinámicos de la red y que inducen a la red entera hacia un estado de sincronía [11,14-15,22]. Estas estrategias se pueden llevar a cabo tanto mediante el control centralizado como con el control distribuido [23]. Con el control centralizado el procesamiento reside en un procesador que se encarga de monitorear las variables de estado de cada robot y de comunicar las acciones de control necesarias para alcanzar el objetivo de movimiento. Con el control distribuido el procesamiento reside en dos o más procesadores con acceso total o parcial a la información de variables de estado de la red; típicamente los robots llevan sus procesadores a bordo. |
| − | La formación de robots es aún un problema abierto [3-5], especialmente para RR´s de gran dimensión y RR´s heterogéneas. Las redes heterogéneas de robots están integradas por robots con distintos modelos dinámicos, con distintas dimensiones de vectores de variables de estado, con distinta dimensión de variables de salida o una combinación de las condiciones anteriores [16,17]. Las soluciones propuestas aún son parciales y requieren una integración para resolver problemas íntimamente relacionados. Tales problemas, además de los ya mencionados, son la evasión de obstáculos, la no colisión entre robots, las saturaciones en los accionamientos para el control, la diversidad de escenarios y las limitaciones en los enlaces de comunicación para el control[14-15]. | + | La formación de robots es aún un problema abierto [3-5], especialmente para RR´s de gran dimensión y RR´s heterogéneas. Las redes heterogéneas de robots están integradas por robots con distintos modelos dinámicos, con distintas dimensiones de vectores de variables de estado, con distinta dimensión de variables de salida o una combinación de las condiciones anteriores [16,17]. Las soluciones propuestas aún son parciales y requieren una integración para resolver problemas íntimamente relacionados. Tales problemas, además de los ya mencionados, son la evasión de obstáculos, la no colisión entre robots, las saturaciones en los accionamientos para el control, la diversidad de escenarios y las limitaciones en los enlaces de comunicación para el control [14-15]. |
Las contribuciones del presente trabajo se aplican en la ''formación ''de una RR´s utilizando herramientas derivadas de la teoría de grafos. Estas contribuciones son: (i) un tutorial para la formulación de RR´s mediante el acoplamiento difusivo estático en el que se describen de manera detallada los elementos y variables que intervienen en la red, (ii) una ley de control para una formación deseada de la RR´s bajo esta formulación y (iii) un procedimiento para la no colisión entre los robots que forman la red utilizando la misma topología de la RR´s. La metodología desarrollada se ejemplifica con simulaciones y experimentos mediante la construcción de una red de ''Robots Móviles Terrestres a base de Ruedas ''(RMTR’s). | Las contribuciones del presente trabajo se aplican en la ''formación ''de una RR´s utilizando herramientas derivadas de la teoría de grafos. Estas contribuciones son: (i) un tutorial para la formulación de RR´s mediante el acoplamiento difusivo estático en el que se describen de manera detallada los elementos y variables que intervienen en la red, (ii) una ley de control para una formación deseada de la RR´s bajo esta formulación y (iii) un procedimiento para la no colisión entre los robots que forman la red utilizando la misma topología de la RR´s. La metodología desarrollada se ejemplifica con simulaciones y experimentos mediante la construcción de una red de ''Robots Móviles Terrestres a base de Ruedas ''(RMTR’s). | ||
| − | El resto de este documento está organizado como sigue. En la sección 2 se expone un tutorial que describe paso a paso | + | El resto de este documento está organizado como sigue. En la sección 2 se expone un tutorial que describe paso a paso la formulación de una RR´s basado en el modelo para cada <math display="inline">i</math>-ésimo robot que forma parte de la red y la estrategia de acoplamiento difusivo estático. En la sección 3 se propone la ley de control para la formación de una red de robots construida con robots tipo uniciclo. La efectividad de la ley de control se demuestra utilizando la teoría de estabilidad de Lyapunov. En la sección 4 se presentan los resultados de simulación y experimentación. En la sección 5 se describe un procedimiento para evitar las colisiones entre robots de la red ejemplificada en la sección 4 y se presentan resultados de simulación y experimentación. Finalmente, en la sección 6 se presentan las conclusiones generales. |
==2. Formulación de una red de robots== | ==2. Formulación de una red de robots== | ||
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| [[Image:draft_Aparicio Nogué_723966060-image1.jpeg|300px]] [[Image:draft_Aparicio Nogué_723966060-image2.jpeg|96px]] | | [[Image:draft_Aparicio Nogué_723966060-image1.jpeg|300px]] [[Image:draft_Aparicio Nogué_723966060-image2.jpeg|96px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 1 | + | | colspan="1" | '''Figura 1.''' (a) Diagrama a bloques de un robot individual, (b) Representación simplificada |
|} | |} | ||
| − | ===2.2 Modelo del | + | ===2.2 Modelo del i-ésimo robot en una red de robots=== |
Para un robot <math display="inline">{R}_{i}</math> que pertenece a una red se puede proponer un nuevo vector de entrada <math display="inline">{\mathit{\boldsymbol{\vartheta }}}_{i}\mathit{\boldsymbol{\in \, }}{\mathit{\mathbb{R}}}^{{p}_{{\vartheta }_{i}}}</math>, de manera que su modelo quedaría como | Para un robot <math display="inline">{R}_{i}</math> que pertenece a una red se puede proponer un nuevo vector de entrada <math display="inline">{\mathit{\boldsymbol{\vartheta }}}_{i}\mathit{\boldsymbol{\in \, }}{\mathit{\mathbb{R}}}^{{p}_{{\vartheta }_{i}}}</math>, de manera que su modelo quedaría como | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math>{\overset{\cdot}{\mathit{\boldsymbol{x}}}}_{i}={\mathit{\boldsymbol{f}}}_{i}(t,{\mathit{\boldsymbol{x}}}_{i},{\mathit{\boldsymbol{u}}}_{i}) | + | | style="text-align: center;" | <math>{\overset{\cdot}{\mathit{\boldsymbol{x}}}}_{i}={\mathit{\boldsymbol{f}}}_{i}(t,{\mathit{\boldsymbol{x}}}_{i},{\mathit{\boldsymbol{u}}}_{i})</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
|- | |- | ||
| − | | style="text-align: center;" | <math>{\mathit{\boldsymbol{y}}}_{i}={\mathit{\boldsymbol{h}}}_{i}(t,{\mathit{\boldsymbol{x}}}_{i},{\mathit{\boldsymbol{u}}}_{i}) | + | | style="text-align: center;" | <math>{\mathit{\boldsymbol{y}}}_{i}={\mathit{\boldsymbol{h}}}_{i}(t,{\mathit{\boldsymbol{x}}}_{i},{\mathit{\boldsymbol{u}}}_{i})</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
|- | |- | ||
| − | | style="text-align: center;" | <math>{\mathit{\boldsymbol{u}}}_{i}={\mathit{\boldsymbol{\Upsilon }}}_{i}\left( t,\mathit{\boldsymbol{y}},{\mathit{\boldsymbol{\vartheta }}}_{i}\right) | + | | style="text-align: center;" | <math>{\mathit{\boldsymbol{u}}}_{i}={\mathit{\boldsymbol{\Upsilon }}}_{i}\left( t,\mathit{\boldsymbol{y}},{\mathit{\boldsymbol{\vartheta }}}_{i}\right)</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(2) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(2) | ||
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image3.jpeg|150px]] | | [[Image:draft_Aparicio Nogué_723966060-image3.jpeg|150px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 2 | + | | colspan="1" | '''Figura 2.''' Modelo de un <math display="inline">i</math>-ésimo robot que pertenece a una red. |
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image4.jpeg|306px]] | | [[Image:draft_Aparicio Nogué_723966060-image4.jpeg|306px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 3 | + | | colspan="1" | '''Figura 3.''' Diagrama a bloques de una red de robots. |
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">G=(R,\varepsilon )</math>, | + | | style="text-align: center;" | <math display="inline">\mathit{\cal G}=(R,\varepsilon)</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" |(3) | | style="width: 5px;text-align: right;white-space: nowrap;" |(3) | ||
|} | |} | ||
|} | |} | ||
| − | donde <math display="inline">R=</math><math>\left\{ {R}_{i}:i=1,2,\ldots ,N\right\}</math> es el conjunto de <math display="inline">N</math> robots y <math display="inline">\varepsilon=</math><math>\left\{ {\varepsilon}_{ij}({\mathit{\boldsymbol{y}}}_{j}):i,j=1,2,\ldots ,N\right\}</math> es el conjunto de <math display="inline"> | + | donde <math display="inline">R=</math><math>\left\{ {R}_{i}:i=1,2,\ldots ,N\right\}</math> es el conjunto de <math display="inline">N</math> robots y <math display="inline">\varepsilon=</math><math>\left\{ {\varepsilon}_{ij}({\mathit{\boldsymbol{y}}}_{j}):i,j=1,2,\ldots ,N\right\}</math> es el conjunto de <math display="inline">N\times N</math> conexiones dirigidas de la red. Una conexión dirigida <math display="inline">\, {\varepsilon}_{ij}({\mathit{\boldsymbol{y}}}_{j})\in {\mathit{\mathbb{R}}}^{{p}_{i}}</math> es una función que corresponde a un par de robots <math display="inline">({R}_{i},{R}_{j})</math> siendo la salida <math display="inline">{\mathit{\boldsymbol{y}}}_{j}\in {\mathit{\mathbb{R}}}^{{s}_{j}}</math> (salida del robot <math display="inline">{R}_{j}</math>) el vector de información que se transfiere al robot <math display="inline">{R}_{i}</math> mediante una transformación <math display="inline">{\varepsilon}_{ij}:\, {\mathit{\mathbb{R}}}^{{s}_{j}}\rightarrow {\mathit{\mathbb{R}}}^{{p}_{i}}</math>. La Figura 4 muestra las conexiones dirigidas de manera generalizada para un par de robots <math display="inline">{R}_{i}</math> y <math display="inline">{R}_{j}</math>. |
<div id='img-4'></div> | <div id='img-4'></div> | ||
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| [[Image:draft_Aparicio Nogué_723966060-image5.jpeg|336px]] | | [[Image:draft_Aparicio Nogué_723966060-image5.jpeg|336px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 4 | + | | colspan="1" | '''Figura 4.''' Grafo o esquema generalizado de conexiones dirigidas en los robots <math display="inline">{R}_{i}</math> y <math display="inline">{R}_{j}</math> de una red. |
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{\varepsilon}}}_{i}(\mathit{\boldsymbol{y}})=c\sum _{j=1}^{N}{\mathit{\boldsymbol{\varepsilon}}}_{ij}({\mathit{\boldsymbol{y}}}_{j})</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{\varepsilon}}}_{i}(\mathit{\boldsymbol{y}})=c\sum _{j=1}^{N}{\mathit{\boldsymbol{\varepsilon}}}_{ij}({\mathit{\boldsymbol{y}}}_{j})</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" |(4) | | style="width: 5px;text-align: right;white-space: nowrap;" |(4) | ||
|} | |} | ||
|} | |} | ||
| − | donde <math display="inline">c>0</math> es la denominada fuerza global de acoplamiento, <math display="inline">{\mathit{\boldsymbol{\varepsilon}}}_{ij}({\mathit{\boldsymbol{y}}}_{j})=</math><math>{g}_{ij}{\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{y}}}_{j}</math>, con <math display="inline">{g}_{ij}</math> elemento de la matriz de configuración externa de red <math display="inline">\mathit{\boldsymbol{\, G}}\in {\mathit{\mathbb{R}}}^{ | + | donde <math display="inline">c>0</math> es la denominada fuerza global de acoplamiento, <math display="inline">{\mathit{\boldsymbol{\varepsilon}}}_{ij}({\mathit{\boldsymbol{y}}}_{j})=</math><math>{g}_{ij}{\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{y}}}_{j}</math>, con <math display="inline">{g}_{ij}</math> elemento de la matriz de configuración externa de red <math display="inline">\mathit{\boldsymbol{\, G}}\in {\mathit{\mathbb{R}}}^{N\times N}</math> obtenida mediante |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\mathit{\boldsymbol{G}}=\mathit{\boldsymbol{\cal A}}-{\Delta }_{ent}</math> | + | | style="text-align: center;" | <math display="inline">\mathit{\boldsymbol{G}}=\mathit{\boldsymbol{\cal A}}-{\Delta }_{ent}</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" |(5) | | style="width: 5px;text-align: right;white-space: nowrap;" |(5) | ||
|} | |} | ||
|} | |} | ||
| − | donde <math display="inline">\mathit{\boldsymbol{\cal A}}{\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{ | + | donde <math display="inline">\mathit{\boldsymbol{\cal A}}{\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{N\times N}</math> es la matriz de adyacencia ponderada con sus elementos <math display="inline">{a}_{ij}>0</math> si existe una conexión entre el robot <math display="inline">{R}_{i}</math> y el robot <math display="inline">{R}_{j}</math> y <math display="inline">{a}_{ij}=</math><math>0</math> si no existe la conexión (<math display="inline">{a}_{ij}</math> es la magnitud de la conexión y en particular <math display="inline">{a}_{ii}=</math><math>0</math>); <math display="inline">{\Delta }_{ent}=</math><math>diag\lbrace {d}_{ii}\rbrace {\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{N\times N}</math> es la matriz de grado de entrada con <math display="inline">{d}_{ii}=</math><math>\sum _{j=1}^{N}{a}_{ij}</math> la suma de los pesos de las conexiones que ingresan a cada robot <math display="inline">{R}_{i}</math> (note que de esta forma <math display="inline">\mathit{\boldsymbol{G}}</math> es difundida puesto que la suma de sus elementos por renglón es nula); y <math display="inline">{\mathit{\boldsymbol{\Gamma }}}_{ij}\in {\mathit{\mathbb{R}}}^{{p}_{i}\times {s}_{j}}</math> es la matriz de configuración interna que expresa las proporciones de contribución de las variables de salida <math display="inline">{\mathit{\boldsymbol{y}}}_{j}</math> hacia <math display="inline">{\mathit{\boldsymbol{u}}}_{i}</math> y en consecuencia, define la relación interna entre las variables del robot <math display="inline">{R}_{j}</math> hacia las variables del robot <math display="inline">{R}_{i}</math>. |
De esta manera el acoplamiento difusivo estático para generar <math display="inline">{\mathit{\boldsymbol{u}}}_{i}</math> es | De esta manera el acoplamiento difusivo estático para generar <math display="inline">{\mathit{\boldsymbol{u}}}_{i}</math> es | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{y}}}_{j}+</math><math>{\mathit{\boldsymbol{\vartheta }}}_{i},\quad i=1,2,\ldots ,N</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{y}}}_{j}+</math><math>{\mathit{\boldsymbol{\vartheta }}}_{i},\quad i=1,2,\ldots ,N</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" |(6) | | style="width: 5px;text-align: right;white-space: nowrap;" |(6) | ||
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image6.jpeg|600px]] | | [[Image:draft_Aparicio Nogué_723966060-image6.jpeg|600px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 5 | + | | colspan="1" | '''Figura 5.''' Diagrama a bloques para una red de robots con acoplamiento difusivo estático. |
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image7.jpeg|426px]] | | [[Image:draft_Aparicio Nogué_723966060-image7.jpeg|426px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 6 | + | | colspan="1" | '''Figura 6.''' Esquema generalizado de conexiones dirigidas equivalente al diagrama a bloques de la Figura 5. |
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image8.jpeg|174px]] | | [[Image:draft_Aparicio Nogué_723966060-image8.jpeg|174px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 7 | + | | colspan="1" | '''Figura 7.''' Grafo para el ejemplo de formulación de red de robots. Observe que las autoconexiones en cada robot no son mostradas. |
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image9.jpeg|234px]] | | [[Image:draft_Aparicio Nogué_723966060-image9.jpeg|234px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 8 | + | | colspan="1" | '''Figura 8.''' Detalle de los <math display="inline">N=</math><math>4</math> robots distintos para el ejemplo de formulación de red. |
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\mathit{\boldsymbol{A=}}\left[ \begin{matrix}0\\0\\2\\0\end{matrix}\, \, \begin{matrix}3\\0\\0\\5\end{matrix}\, \, \begin{matrix}0\\4\\0\\1\end{matrix}\, \, \begin{matrix}0\\0\\0\\0\end{matrix}\right]</math>, y <math display="inline">{\Delta }_{ent}=</math><math>diag\lbrace 3,4,2,6\rbrace</math> | + | | style="text-align: center;" | <math display="inline">\mathit{\boldsymbol{A=}}\left[ \begin{matrix}0\\0\\2\\0\end{matrix}\, \, \begin{matrix}3\\0\\0\\5\end{matrix}\, \, \begin{matrix}0\\4\\0\\1\end{matrix}\, \, \begin{matrix}0\\0\\0\\0\end{matrix}\right]</math>, y <math display="inline">{\Delta }_{ent}=</math><math>diag\lbrace 3,4,2,6\rbrace</math>. |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
|} | |} | ||
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|} | |} | ||
| − | con <math display="inline">{\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{{r}_{i} | + | con <math display="inline">{\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{{r}_{i}\times {s}_{j}}</math> cada matriz de configuración interna y con <math display="inline">\mathit{\boldsymbol{0}}</math> una matriz cero de dimensión adecuada; es decir |
| Line 249: | Line 245: | ||
con cada <math display="inline">{\gamma }_{ij,kl}\geq 0</math> según la proporción de contribución de las variables <math display="inline">{\mathit{\boldsymbol{y}}}_{j}</math> hacia <math display="inline">{\mathit{\boldsymbol{u}}}_{i}</math>. | con cada <math display="inline">{\gamma }_{ij,kl}\geq 0</math> según la proporción de contribución de las variables <math display="inline">{\mathit{\boldsymbol{y}}}_{j}</math> hacia <math display="inline">{\mathit{\boldsymbol{u}}}_{i}</math>. | ||
| − | De esta manera, se completa (6) para la red de la | + | De esta manera, se completa (6) para la red de la Figura 7 y la Figura 9 muestra su diagrama a bloques equivalente |
<div id='img-9'></div> | <div id='img-9'></div> | ||
| Line 256: | Line 252: | ||
| [[Image:draft_Aparicio Nogué_723966060-image10.jpeg|600px]] | | [[Image:draft_Aparicio Nogué_723966060-image10.jpeg|600px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 9 | + | | colspan="1" | '''Figura 9.''' Diagrama a bloques equivalente para el ejemplo de red heterogénea con 4 robots. |
|} | |} | ||
| Line 268: | Line 264: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}\mathit{\boldsymbol{q}}=\left[ \begin{matrix}cos{(\theta }_{R})&- | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}\mathit{\boldsymbol{q}}=\left[ \begin{matrix}\text {cos}{(\theta }_{R})&-l\text {sen}{(\theta }_{R})\\ \text {sen}{(\theta }_{R})&l\text {cos}{(\theta }_{R})\end{matrix}\right] \left[ \begin{matrix}{v}_{R}\\{\omega }_{R}\end{matrix}\right]</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
|- | |- | ||
| Line 283: | Line 279: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}\mathit{\boldsymbol{q}}=\mathit{\boldsymbol{u}}</math> | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}\mathit{\boldsymbol{q}}=\mathit{\boldsymbol{u}}</math>. |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(11) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(11) | ||
|} | |} | ||
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
| − | | [[Image:draft_Aparicio Nogué_723966060-image11.jpeg| | + | | [[Image:draft_Aparicio Nogué_723966060-image11.jpeg|250px]] |
| + | | [[Image:draft_Aparicio Nogué_723966060-image12.jpeg|250px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan=" | + | | colspan="2" | '''Figura 10.''' (a) Robot tipo uniciclo de tracción diferencial, (b) Vista superior del robot. |
|} | |} | ||
| Line 305: | Line 302: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}\mathit{\boldsymbol{q}}_{\mathit{\boldsymbol{i}}}=</math><math>{\mathit{\boldsymbol{u}}}_{\mathit{\boldsymbol{i}}}</math> | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}\mathit{\boldsymbol{q}}_{\mathit{\boldsymbol{i}}}=</math><math>{\mathit{\boldsymbol{u}}}_{\mathit{\boldsymbol{i}}}</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(12) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(12) | ||
|} | |} | ||
| Line 316: | Line 313: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}{+\mathit{\boldsymbol{\vartheta }}}_{i},\quad i=</math><math>1,2,\ldots ,N</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}{+\mathit{\boldsymbol{\vartheta }}}_{i},\quad i=</math><math>1,2,\ldots ,N</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(13) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(13) | ||
|} | |} | ||
| Line 330: | Line 327: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\text{lim}_{t\rightarrow\infty}\,\|\mathit{\boldsymbol{e}}_{i}(t)\|=0,\quad i=</math><math>1,2,\ldots ,N</math> | + | | style="text-align: center;" | <math display="inline">\text{lim}_{t\rightarrow\infty}\,\|\mathit{\boldsymbol{e}}_{i}(t)\|=0,\quad i=</math><math>1,2,\ldots ,N</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(14) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(14) | ||
|} | |} | ||
|} | |} | ||
| − | donde <math display="inline">{\mathit{\boldsymbol{e}}}_{i}=</math><math>{[{e}_{ix}\, {e}_{iy}]}^{T}={\mathit{\boldsymbol{q}}}_{i}-\mathit{\boldsymbol{s}}-</math><math>{\mathit{\boldsymbol{\varphi }}}_{i}</math>es el error de formación y se ha insertado un líder virtual (sistema de referencia de red) con sistema dinámico <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{s}}}=</math><math>\mathit{\boldsymbol{f}}(\mathit{\boldsymbol{s}})</math> y un vector de posiciones finales deseadas constantes <math display="inline">{\mathit{\boldsymbol{\varphi }}}_{i}=</math><math>{[{\varphi }_{ix}\, \, {\varphi }_{iy}]}^{T}\, \in \, {\mathit{\mathbb{R}}}^{2}</math> para cada <math display="inline"> | + | donde <math display="inline">{\mathit{\boldsymbol{e}}}_{i}=</math><math>{[{e}_{ix}\, {e}_{iy}]}^{T}={\mathit{\boldsymbol{q}}}_{i}-\mathit{\boldsymbol{s}}-</math><math>{\mathit{\boldsymbol{\varphi }}}_{i}</math> es el error de formación y se ha insertado un líder virtual (sistema de referencia de red) con sistema dinámico <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{s}}}=</math><math>\mathit{\boldsymbol{f}}(\mathit{\boldsymbol{s}})</math> y un vector de posiciones finales deseadas constantes <math display="inline">{\mathit{\boldsymbol{\varphi }}}_{i}=</math><math>{[{\varphi }_{ix}\, \, {\varphi }_{iy}]}^{T}\, \in \, {\mathit{\mathbb{R}}}^{2}</math> para cada <math display="inline">{i}</math>-ésimo robot respecto al líder. El vector <math display="inline">\mathit{\boldsymbol{s}}=</math><math>{[{s}_{x}\, \, {s}_{y}]}^{T}</math> puede ser un punto fijo o una trayectoria [11]. La derivada de <math display="inline">{\mathit{\boldsymbol{e}}}_{i}</math> es <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}=</math> <math>{\overset{\cdot}{\mathit{\boldsymbol{q}}}}_{i}-\overset{\cdot}{\mathit{\boldsymbol{s}}}</math> o bien sustituyéndole (12) |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
| Line 342: | Line 339: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}={\mathit{\boldsymbol{u}}}_{i}-</math><math>\overset{\cdot}{\mathit{\boldsymbol{s}}}</math>. |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(15) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(15) | ||
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{\vartheta }}}_{i}=-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-</math><math>c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+\overset{\cdot}{\mathit{\boldsymbol{s}}},\quad i=</math><math>1,2,\ldots ,N</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{\vartheta }}}_{i}=-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-</math><math>c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+\overset{\cdot}{\mathit{\boldsymbol{s}}},\quad i=</math><math>1,2,\ldots ,N</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(16) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(16) | ||
|} | |} | ||
|} | |} | ||
| − | donde <math display="inline">{d}_{i}</math> es la ganancia del controlador para el <math display="inline"> | + | donde <math display="inline">{d}_{i}</math> es la ganancia del controlador para el <math display="inline">{i}</math>-ésimo robot. Así (13), luego de sustituirle (16) resulta |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
| Line 368: | Line 365: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}-</math><math>c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+</math><math>\overset{\cdot}{\mathit{\boldsymbol{s}}},\quad i=1,2,\ldots ,N</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}-</math><math>c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+</math><math>\overset{\cdot}{\mathit{\boldsymbol{s}}},\quad i=1,2,\ldots ,N</math>. |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(17) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(17) | ||
|} | |} | ||
| Line 380: | Line 377: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_ | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}=</math><math>c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{e}}}_{j}-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i},\quad i=</math><math>1,2,\ldots ,N</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(18) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(18) | ||
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{e}}}=-\mathit{\boldsymbol{Fe}}</math> | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{e}}}=-\mathit{\boldsymbol{Fe}}</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(19) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(19) | ||
|} | |} | ||
| Line 403: | Line 400: | ||
Para demostrar el cumplimiento del objetivo de formación (14) con la ley de control (16) en el sistema expresado por (13)-(12), considérense los siguientes lemas: | Para demostrar el cumplimiento del objetivo de formación (14) con la ley de control (16) en el sistema expresado por (13)-(12), considérense los siguientes lemas: | ||
| − | ''Lema 1'' [25]. Un sistema homogéneo <math display="inline">\mathit{\boldsymbol{ | + | ''Lema 1'' [25]. Un sistema homogéneo <math display="inline">\mathit{\boldsymbol{Ay}}=</math><math>\mathit{\boldsymbol{0}}</math>, donde <math display="inline">\mathit{\boldsymbol{A}}\in {\mathit{\mathbb{R}}}^{r\times r}</math>, posee una solución única (la solución trivial <math display="inline">\mathit{\boldsymbol{y}}=</math><math>\mathit{\boldsymbol{0}}</math>) si y sólo si el <math display="inline">rank\left( \mathit{\boldsymbol{A}}\right) =</math><math>r</math>. |
| − | ''Lema 2'' [26]. Sea <math display="inline">\mathit{\boldsymbol{A}}\in {\mathit{\mathbb{R}}}^{ | + | ''Lema 2'' [26]. Sea <math display="inline">\mathit{\boldsymbol{A}}\in {\mathit{\mathbb{R}}}^{m\times n}</math> con valores singulares <math display="inline">{\sigma }_{1}\geq \ldots \geq {\sigma }_{r}>0</math> y <math display="inline">\mathit{\boldsymbol{B}}\in {\mathit{\mathbb{R}}}^{p\times q}</math> con valores singulares <math display="inline">{\delta }_{1}\geq \ldots \geq {\delta }_{s}>0</math>. Entonces <math display="inline">\mathit{\boldsymbol{A}}\otimes \mathit{\boldsymbol{B}}</math> (ó <math display="inline">\mathit{\boldsymbol{B}}\otimes \mathit{\boldsymbol{A}}</math>) tiene <math display="inline">rs</math> valores singulares <math display="inline">{\sigma }_{1}{\delta }_{1}\geq \ldots \geq {\sigma }_{r}{\delta }_{s}>0</math> y <math display="inline">rank\left( \mathit{\boldsymbol{A}}\otimes \mathit{\boldsymbol{B}}\right) =</math><math>rank\left( \mathit{\boldsymbol{B}}\otimes \mathit{\boldsymbol{A}}\right) =</math><math>rank\left( \mathit{\boldsymbol{A}}\right) rank\mathit{\boldsymbol{(B)}}</math>. NOTA: Si las matrices son cuadradas puede intercambiarse "valores singulares" por "valores característicos". |
| − | ''Lema 3'' [27]. Sea <math display="inline">\mathit{\boldsymbol{P}}\in {\mathit{\mathbb{R}}}^{ | + | ''Lema 3'' [27]. Sea <math display="inline">\mathit{\boldsymbol{P}}\in {\mathit{\mathbb{R}}}^{N\times N}</math> una matriz cuadrada y haciendo <math display="inline">\mathit{\boldsymbol{A}}=</math><math>\frac{1}{2}[\mathit{\boldsymbol{P}}+{\mathit{\boldsymbol{P}}}^{T}]</math>, el Teorema de Silvester establece que <math display="inline">\mathit{\boldsymbol{P}}</math> es definida positiva (<math display="inline">\mathit{\boldsymbol{P}}>0</math>) si y sólo si: |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
| Line 414: | Line 411: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\mathrm{{\Delta }_{1}=det}\,\left[ {a}_{11}\right] >0,\, \, {\Delta }_{2}=</math><math>det\left[ \begin{matrix}{a}_{11}&{a}_{12}\\{a}_{21}&{a}_{22}\end{matrix}\right] >0,\ldots ,\mathrm{{\Delta }_{N}=det}\,\left[ \mathit{\boldsymbol{A}}\right] >0</math> | + | | style="text-align: center;" | <math display="inline">\mathrm{{\Delta }_{1}=det}\,\left[ {a}_{11}\right] >0,\, \, {\Delta }_{2}=</math><math>det\left[ \begin{matrix}{a}_{11}&{a}_{12}\\{a}_{21}&{a}_{22}\end{matrix}\right] >0,\ldots ,\mathrm{{\Delta }_{N}=det}\,\left[ \mathit{\boldsymbol{A}}\right] >0</math>. |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"| | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"| | ||
|} | |} | ||
|} | |} | ||
| − | NOTA: Si <math display="inline">\mathit{\boldsymbol{P}}</math> es definida positiva (<math display="inline">\mathit{\boldsymbol{P}}>0</math>) entonces <math display="inline" | + | NOTA: Si <math display="inline">\mathit{\boldsymbol{P}}</math> es definida positiva (<math display="inline">\mathit{\boldsymbol{P}}>0</math>) entonces <math display="inline">\mathit{\boldsymbol{-P}}</math> es definida negativa (<math display="inline">\mathit{\boldsymbol{-P}}<0</math>). Si <math display="inline">\mathit{\boldsymbol{P}}</math> es semidefinida positiva (<math display="inline">\mathit{\boldsymbol{P}}\geq 0</math>) entonces <math display="inline">\mathit{\boldsymbol{-P}}</math> es semidefinida negativa (<math display="inline">\mathit{\boldsymbol{-P}}\leq 0</math>). |
| − | ''Lema 4 ''[28]. Sean <math display="inline">\mathit{\boldsymbol{P}}\in {\mathit{\mathbb{R}}}^{ | + | ''Lema 4 ''[28]. Sean <math display="inline">\mathit{\boldsymbol{P}}\in {\mathit{\mathbb{R}}}^{N\times N}</math> y <math display="inline">\mathit{\boldsymbol{\Gamma }}\in {\mathit{\mathbb{R}}}^{n\times n}</math> matrices definidas (semidefinidas) positivas, entonces <math display="inline">\mathit{\boldsymbol{P}}\otimes \mathit{\boldsymbol{\Gamma }}\in {\mathit{\mathbb{R}}}^{Nn\times Nn}</math> es una matriz definida (semidefinida) positiva. |
| − | Ahora, | + | Ahora, considere el punto de equilibrio <math display="inline">\mathit{\boldsymbol{e}}=</math><math>\mathit{\boldsymbol{0}}</math> de (19) y, para el análisis de su estabilidad, considérese la siguiente función candidata de Lyapunov |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
| Line 430: | Line 427: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\mathit{\boldsymbol{V}}=\frac{1}{2}{\mathit{\boldsymbol{e}}}^{T}\mathit{\boldsymbol{e}}>0\, \forall \, \mathit{\boldsymbol{e}}\not =0</math> | + | | style="text-align: center;" | <math display="inline">\mathit{\boldsymbol{V}}=\frac{1}{2}{\mathit{\boldsymbol{e}}}^{T}\mathit{\boldsymbol{e}}>\mathit{\boldsymbol{0}}\, \forall \, \mathit{\boldsymbol{e}}\not =\mathit{\boldsymbol{0}}</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(20) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(20) | ||
|} | |} | ||
| Line 442: | Line 439: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}={\mathit{\boldsymbol{e}}}^{T}\overset{\cdot}{\mathit{\boldsymbol{e} | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}={\mathit{\boldsymbol{e}}}^{T}\overset{\cdot}{\mathit{\boldsymbol{e}}}</math>. |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(21) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(21) | ||
|} | |} | ||
| Line 454: | Line 451: | ||
{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}=-{\mathit{\boldsymbol{e}}}^{T}\mathit{\boldsymbol{Fe}}</math> | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}=-{\mathit{\boldsymbol{e}}}^{T}\mathit{\boldsymbol{Fe}}</math>. |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(22) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(22) | ||
|} | |} | ||
|} | |} | ||
| − | De manera que si <math display="inline">\mathit{\boldsymbol{F}}>0</math> entonces <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}\mathit{\boldsymbol{<}}0</math> <math display="inline">\forall \, e\not =0</math>, significando que <math display="inline">\mathit{\boldsymbol{e}}=</math><math>0</math> es un punto de equilibrio asintóticamente estable y con esto queda demostrado el cumplimiento del objetivo de formación (14). | + | De manera que si <math display="inline">\mathit{\boldsymbol{F}}>0</math> entonces <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}\mathit{\boldsymbol{<}}0</math> <math display="inline">\forall \, \mathit{\boldsymbol{e}}\not = \mathit{\boldsymbol{0}}</math>, significando que <math display="inline">\mathit{\boldsymbol{e}}= </math><math>\mathit{\boldsymbol{0}}</math> es un punto de equilibrio asintóticamente estable y con esto queda demostrado el cumplimiento del objetivo de formación (14). |
Como ejemplos considérense las siguientes topologías típicas para una red con <math display="inline">N</math> robots. | Como ejemplos considérense las siguientes topologías típicas para una red con <math display="inline">N</math> robots. | ||
| Line 492: | Line 489: | ||
| [[Image:draft_Aparicio Nogué_723966060-image13.jpeg|216px]] | | [[Image:draft_Aparicio Nogué_723966060-image13.jpeg|216px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 11 | + | | colspan="1" | '''Figura 11.''' Una red de <math display="inline">N</math> robots con Topología Estrella. |
|} | |} | ||
| Line 509: | Line 506: | ||
|} | |} | ||
| − | de forma que <math display="inline">{\Delta }_{1}=</math><math>{d}_{1}</math> ( | + | de forma que <math display="inline">{\Delta }_{1}=</math><math>{d}_{1}</math> (<math display="inline">{\Delta }_{1}>0</math> si <math display="inline">{d}_{1}>0</math>), <math display="inline">{\Delta }_{2}=</math><math>{d}_{1}-\frac{1}{4}</math> (<math display="inline">{\Delta }_{2}>0</math> si <math display="inline">{d}_{1}>\frac{1}{4)}</math>,…, <math display="inline">{\Delta }_{i}=</math><math>{d}_{1}-\frac{i-1}{4}</math> (<math display="inline">{\Delta }_{i}>0</math> si <math display="inline">{d}_{1}>\frac{i-1}{4}</math>),…, <math display="inline">{\Delta }_{N}=</math><math>{d}_{1}-\frac{N-1}{4}</math> (<math display="inline">{\Delta }_{N}>0</math> si <math display="inline">{d}_{1}>\frac{N-1}{4}</math>). |
De manera que apoyándonos en el ''Lema 4'', entonces si | De manera que apoyándonos en el ''Lema 4'', entonces si | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{d}_{1}>\frac{ | + | | style="text-align: center;" | <math display="inline">{d}_{1}>\frac{N-1}{4}</math>, |
| style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(26) | | style="text-align: right;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(26) | ||
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image14.jpeg|150px]] | | [[Image:draft_Aparicio Nogué_723966060-image14.jpeg|150px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 12 | + | | colspan="1" | '''Figura 12.''' Una red de <math display="inline">N</math> robots con Topología Anillo. |
|} | |} | ||
| − | Nótese que <math display="inline">rank(\mathit{\boldsymbol{G}})=N-</math><math>1</math> (lo cual puede comprobarse directamente al aplicar el procedimiento de eliminación de Gauss). Para que <math display="inline">rank\left( \mathit{\boldsymbol{D-G}}\right)</math> sea completo, por la simetría de la topología, resulta suficiente cualquier <math display="inline">{d}_{i}\not =0</math> | + | Nótese que <math display="inline">rank(\mathit{\boldsymbol{G}})=N-</math><math>1</math> (lo cual puede comprobarse directamente al aplicar el procedimiento de eliminación de Gauss). Para que <math display="inline">rank\left( \mathit{\boldsymbol{D-G}}\right)</math> sea completo, por la simetría de la topología, resulta suficiente cualquier <math display="inline">{d}_{i}\not =0.</math> Supóngase <math display="inline">\mathit{\boldsymbol{D=}}diag\mathit{\boldsymbol{\lbrace }}{d}_{1}\mathit{\boldsymbol{,\, }}0,\ldots ,0\rbrace</math> de manera que si <math display="inline">{d}_{1}\not =0</math> y apoyándose en los ''Lemas 1'' y ''2'', <math display="inline">\mathit{\boldsymbol{e}}=</math><math>0</math> es el único punto de equilibrio de (19). |
Ahora, sustituyendo la <math display="inline">\mathit{\boldsymbol{D}}</math> supuesta en (28) y aplicando el'' Lema 3 ''con <math display="inline">\mathit{\boldsymbol{A}}=</math><math>\left[ \left( \mathit{\boldsymbol{D-G}}\right) {+\left( \mathit{\boldsymbol{D-G}}\right) }^{T}\right] /2</math> se tiene | Ahora, sustituyendo la <math display="inline">\mathit{\boldsymbol{D}}</math> supuesta en (28) y aplicando el'' Lema 3 ''con <math display="inline">\mathit{\boldsymbol{A}}=</math><math>\left[ \left( \mathit{\boldsymbol{D-G}}\right) {+\left( \mathit{\boldsymbol{D-G}}\right) }^{T}\right] /2</math> se tiene | ||
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| [[Image:draft_Aparicio Nogué_723966060-image15.jpeg|180px]] | | [[Image:draft_Aparicio Nogué_723966060-image15.jpeg|180px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 13 | + | | colspan="1" | '''Figura 13.''' Una red de <math display="inline">N</math> robots con Topología Malla. |
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image16.jpeg|162px]] | | [[Image:draft_Aparicio Nogué_723966060-image16.jpeg|162px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 14 | + | | colspan="1" | '''Figura 14.''' Topología externa de una RR´s no típica con 5 robots tipo uniciclo de tracción diferencial. |
|} | |} | ||
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Para la simulación de la red formada por cinco robots se utilizó MATLAB R2011b. El experimento se realizó con cinco robots tipo uniciclo modelo YSR-A de la empresa Yujin que reciben las consignas de velocidad mediante módulos de comunicación inalámbrica operando a 418 MHz y a una tasa de 19.2 Kbps. Los módulos de comunicación se conectan a un procesador central vía puerto serie. Este procesador central ejecuta una aplicación de tiempo real en ambiente RTLinux, auxiliado de una tarjeta de video Leonardo de la empresa Arvoo y con programación basada en [33]. Para efectos de identificar la postura de los robots se colocaron dos puntos marca sobre cada robot, uno para la posición objetivo y el otro como auxiliar para el cálculo de su orientación. Las imágenes se registran mediante una cámara UF-1000CL de la empresa UNIQ cuyo eje óptico apunta perpendicularmente al plano de movimiento de los robots (ver Figura 10). La cámara es de alta velocidad y se configuró a razón de 200 cuadros por segundo, obteniéndose un periodo estricto de muestreo en los experimentos de 0.005 s. | Para la simulación de la red formada por cinco robots se utilizó MATLAB R2011b. El experimento se realizó con cinco robots tipo uniciclo modelo YSR-A de la empresa Yujin que reciben las consignas de velocidad mediante módulos de comunicación inalámbrica operando a 418 MHz y a una tasa de 19.2 Kbps. Los módulos de comunicación se conectan a un procesador central vía puerto serie. Este procesador central ejecuta una aplicación de tiempo real en ambiente RTLinux, auxiliado de una tarjeta de video Leonardo de la empresa Arvoo y con programación basada en [33]. Para efectos de identificar la postura de los robots se colocaron dos puntos marca sobre cada robot, uno para la posición objetivo y el otro como auxiliar para el cálculo de su orientación. Las imágenes se registran mediante una cámara UF-1000CL de la empresa UNIQ cuyo eje óptico apunta perpendicularmente al plano de movimiento de los robots (ver Figura 10). La cámara es de alta velocidad y se configuró a razón de 200 cuadros por segundo, obteniéndose un periodo estricto de muestreo en los experimentos de 0.005 s. | ||
| − | En la Tabla 1 se muestran las condiciones iniciales y posiciones finales deseadas para el control de formación de la red de la Figura 14; utilizando un líder virtual con dinámica <math display="inline">\mathit{\boldsymbol{s}} | + | En la Tabla 1 se muestran las condiciones iniciales y posiciones finales deseadas para el control de formación de la red de la Figura 14; utilizando un líder virtual con dinámica <math display="inline">\overset{\cdot}\mathit{\boldsymbol{s}}=</math><math>0</math> y con condición inicial <math display="inline">\mathit{\boldsymbol{s}}(0)=</math><math>{[0\, \, 0]}^{T}</math> m, una fuerza global de acoplamiento <math display="inline">c=</math><math>1.8</math>, ganancias <math display="inline">\mathit{\boldsymbol{D}}=</math><math>diag\lbrace 2\, \, 0\, \, 0\, \, 0\, \, 0\rbrace</math> y una distancia <math display="inline">l=</math><math>0.036</math> m entre marcas sobre cada robot. Obsérvese que se desea una formación en línea recta con un ángulo de |
<math display="inline">\pi /4</math> rad respecto al marco global <math display="inline">{\Sigma }_{W}</math>. | <math display="inline">\pi /4</math> rad respecto al marco global <math display="inline">{\Sigma }_{W}</math>. | ||
| − | < | + | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;"> |
| + | '''Tabla 1'''. Condiciones de simulación y experimentación para la RR´s de la Figura 14.</div><br> | ||
| − | {| style="border-collapse: collapse; width: 80%; margin: 0 auto; text-align: center;" | + | {| style="border-collapse: collapse; width: 80%; margin: 0 auto; text-align: center;font-size:85%;" |
|- | |- | ||
| style="border: 1pt solid black;vertical-align: top; width: 25px"| <math>i</math> | | style="border: 1pt solid black;vertical-align: top; width: 25px"| <math>i</math> | ||
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| style="border: 1pt solid black;text-align: center;vertical-align: top;"|-0.30 | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|-0.30 | ||
| style="border: 1pt solid black;text-align: center;vertical-align: top;"|-0.30 | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|-0.30 | ||
| − | |} | + | |}<br /> |
| + | |||
| − | La Figura 15(a) muestra la evolución de los errores en simulación, los cuales tienden a cero a medida que el tiempo transcurre. Una apreciación visual simple destaca que es suficiente <math display="inline">t\approx 5</math> s para que el error pueda considerarse prácticamente cero. En la Figura 15(b) se presentan las gráficas de los errores en el experimento. Cabe destacar que el umbral de reacción de los robots se encuentra por encima de 0.0156 m/s de manera que debajo de dicho valor no hay movimiento del robot. Lo anterior es lo que justifica que en el experimento exista un error en estado estacionario, es decir, cuando la ley de control demanda velocidades pequeñas (pues el robot está cerca del objetivo) se cae dentro del umbral de no reacción del robot deteniendo su movimiento. La norma de error más grande lo presenta <math display="inline">{R}_{5}</math> con <math display="inline">\left\| {e}_{5}\right\| =</math><math>0.0105</math> m. | + | La Figura 15(a) muestra la evolución de los errores en simulación, los cuales tienden a cero a medida que el tiempo transcurre. Una apreciación visual simple destaca que es suficiente <math display="inline">t\approx 5</math> s para que el error pueda considerarse prácticamente cero. En la Figura 15(b) se presentan las gráficas de los errores en el experimento. Cabe destacar que el umbral de reacción de los robots se encuentra por encima de <math display="inline">0.0156</math> m/s de manera que debajo de dicho valor no hay movimiento del robot. Lo anterior es lo que justifica que en el experimento exista un error en estado estacionario, es decir, cuando la ley de control demanda velocidades pequeñas (pues el robot está cerca del objetivo) se cae dentro del umbral de no reacción del robot deteniendo su movimiento. La norma de error más grande lo presenta <math display="inline">{R}_{5}</math> con <math display="inline">\left\| {e}_{5}\right\| =</math><math>0.0105</math> m. |
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| [[Image:draft_Aparicio Nogué_723966060-image17.png|312px]] [[Image:draft_Aparicio Nogué_723966060-image18.png|318px]] | | [[Image:draft_Aparicio Nogué_723966060-image17.png|312px]] [[Image:draft_Aparicio Nogué_723966060-image18.png|318px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 15 | + | | colspan="1" | '''Figura 15.''' Evolución de los errores: (a) En simulación, (b) En experimento. |
|} | |} | ||
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| [[Image:draft_Aparicio Nogué_723966060-image19.png|372px]] | | [[Image:draft_Aparicio Nogué_723966060-image19.png|372px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 16 | + | | colspan="1" | '''Figura 16.''' Trayectorias para la formación en línea recta en el marco global <math display="inline">{\Sigma }_{W}</math>. |
|} | |} | ||
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
| − | | [[Image:draft_Aparicio Nogué_723966060-image20.png|312px]] [[Image:draft_Aparicio Nogué_723966060-image21.png|312px]] | + | | [[Image:draft_Aparicio Nogué_723966060-image20.png|312px]] |
| + | | [[Image:draft_Aparicio Nogué_723966060-image21.png|312px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan=" | + | | colspan="2" | '''Figura 17.''' Gráficas de velocidades lineales de los robots: (a) Simulación, (b) Experimento. |
|} | |} | ||
| + | |||
<div id='img-18'></div> | <div id='img-18'></div> | ||
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
| − | | [[Image:draft_Aparicio Nogué_723966060-image22.png|312px]] [[Image:draft_Aparicio Nogué_723966060-image23.png|312px]] | + | | [[Image:draft_Aparicio Nogué_723966060-image22.png|312px]] |
| + | | [[Image:draft_Aparicio Nogué_723966060-image23.png|312px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan=" | + | | colspan="2" | '''Figura 18.''' Gráficas de velocidades angulares de los robots: (a) Simulación, (b) Experimento. |
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{\vartheta }}}_{i}=-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-</math><math>c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+\overset{\cdot}{\mathit{\boldsymbol{s}}}+</math><math>{\mathit{\boldsymbol{u}}}_{ri}</math>, <math display="inline">i=</math><math>1,2,\ldots N</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{\vartheta }}}_{i}=-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-</math><math>c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+\overset{\cdot}{\mathit{\boldsymbol{s}}}+</math><math>{\mathit{\boldsymbol{u}}}_{ri}</math>, <math display="inline">i=</math><math>1,2,\ldots N</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" |(35) | | style="width: 5px;text-align: right;white-space: nowrap;" |(35) | ||
|} | |} | ||
|} | |} | ||
| − | donde <math display="inline">{\mathit{\boldsymbol{u}}}_{ri}=</math><math>{h}_{i}{[-{e}_{iy}\, \, {e}_{ix}]}^{T}</math>con <math display="inline">{h}_{i}\, \in \, \mathit{\mathbb{R}}</math>un factor de repulsión. De esta manera (13) resulta | + | donde <math display="inline">{\mathit{\boldsymbol{u}}}_{ri}=</math><math>{h}_{i}{[-{e}_{iy}\, \, {e}_{ix}]}^{T}</math>con <math display="inline">{h}_{i}\, \in \, \mathit{\mathbb{R}}</math> un factor de repulsión. De esta manera (13) resulta |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}-</math><math>c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+</math><math>\overset{\cdot}{\mathit{\boldsymbol{s}}}+{h}_{i}{\left[ -{e}_{iy}\, \, {e}_{ix}\right] }^{T},\quad i=</math><math>1,2,\ldots N</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{u}}}_{i}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}-</math><math>c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+</math><math>\overset{\cdot}{\mathit{\boldsymbol{s}}}+{h}_{i}{\left[ -{e}_{iy}\, \, {e}_{ix}\right] }^{T},\quad i=</math><math>1,2,\ldots N</math>. |
| style="width: 5px;text-align: right;white-space: nowrap;" |(36) | | style="width: 5px;text-align: right;white-space: nowrap;" |(36) | ||
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}=</math><math>c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{e}}}_{j}-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}+</math><math>{h}_{i}{\left[ -{e}_{iy}\, \, {e}_{ix}\right] }^{T},\quad i=1,2,\ldots N</math>. |
| style="width: 5px;text-align: right;white-space: nowrap;" |(37) | | style="width: 5px;text-align: right;white-space: nowrap;" |(37) | ||
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}=-c{\mathit{\boldsymbol{e}}}^{T}\lbrace [\mathit{\boldsymbol{D-G}}]\otimes {\mathit{\boldsymbol{I}}}_{2}\rbrace \mathit{\boldsymbol{e}}+</math><math>c{\mathit{\boldsymbol{e}}}^{T}[\mathit{\boldsymbol{H}}\otimes {\mathit{\boldsymbol{I}}}_{2}]{\mathit{\boldsymbol{e}}}^{\bot}</math> | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{V}}}=-c{\mathit{\boldsymbol{e}}}^{T}\lbrace [\mathit{\boldsymbol{D-G}}]\otimes {\mathit{\boldsymbol{I}}}_{2}\rbrace \mathit{\boldsymbol{e}}+</math><math>c{\mathit{\boldsymbol{e}}}^{T}[\mathit{\boldsymbol{H}}\otimes {\mathit{\boldsymbol{I}}}_{2}]{\mathit{\boldsymbol{e}}}^{\bot}</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" |(38) | | style="width: 5px;text-align: right;white-space: nowrap;" |(38) | ||
|} | |} | ||
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donde <math display="inline">\mathit{\boldsymbol{\, H}}=</math><math>diag\lbrace {h}_{1},{h}_{2},\ldots {h}_{N}\rbrace</math> y <math display="inline">{\mathit{\boldsymbol{e}}}^{\bot}=</math><math>{[-{e}_{1y}\, \, {e}_{1x}\, \, -{e}_{2y}\, \, {e}_{2x}\ldots \, -{e}_{Ny}\, \, {e}_{Nx}]}^{T}</math>. | donde <math display="inline">\mathit{\boldsymbol{\, H}}=</math><math>diag\lbrace {h}_{1},{h}_{2},\ldots {h}_{N}\rbrace</math> y <math display="inline">{\mathit{\boldsymbol{e}}}^{\bot}=</math><math>{[-{e}_{1y}\, \, {e}_{1x}\, \, -{e}_{2y}\, \, {e}_{2x}\ldots \, -{e}_{Ny}\, \, {e}_{Nx}]}^{T}</math>. | ||
| − | La contribución<math display="inline">c\mathit{\boldsymbol{e}}^{T}[\mathit{\boldsymbol{H}}\otimes \mathit{\boldsymbol{I}}_{2}]\mathit{\boldsymbol{e}}^{\bot}</math> en (38) es | + | La contribución <math display="inline">c\mathit{\boldsymbol{e}}^{T}[\mathit{\boldsymbol{H}}\otimes \mathit{\boldsymbol{I}}_{2}]\mathit{\boldsymbol{e}}^{\bot}</math> en (38) es |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math>c{\mathit{\boldsymbol{e}}}^{T}[\mathit{\boldsymbol{H}}\otimes {\mathit{\boldsymbol{I}}}_{2}]{\mathit{\boldsymbol{e}}}^{\bot}</math> <math display="inline">=c{\left[ {e}_{1x}\ | + | | style="text-align: center;" | <math>c{\mathit{\boldsymbol{e}}}^{T}[\mathit{\boldsymbol{H}}\otimes {\mathit{\boldsymbol{I}}}_{2}]{\mathit{\boldsymbol{e}}}^{\bot}</math> <math display="inline">=c{\left[ \begin{matrix}{e}_{1x}\\{e}_{1y}\\{e}_{2x}\\{e}_{2y}\\\vdots \\{e}_{Nx}\\{e}_{Ny}\end{matrix}\right]}^{T}\,\left[ \begin{matrix}{h}_{1}\\0\\0\\0\\\vdots \\0\\0\end{matrix}\, \, \begin{matrix}0\\{h}_{1}\\0\\0\\\vdots \\0\\0\end{matrix}\, \, \begin{matrix}0\\0\\{h}_{2}\\0\\\vdots \\0\\0\end{matrix}\, \, \begin{matrix}0\\0\\0\\{h}_{2}\\\vdots \\0\\0\end{matrix}\, \, \begin{matrix}\ldots \\\ldots \\\ldots \\\ldots \\\ddots \\0\\0\end{matrix}\, \, \begin{matrix}0\\0\\0\\0\\\vdots \\{h}_{N}\\0\end{matrix}\, \, \begin{matrix}0\\0\\0\\0\\\vdots \\0\\{h}_{N}\end{matrix}\right] \,\left[ \begin{matrix}-{e}_{1y}\\{e}_{1x}\\-{e}_{2y}\\{e}_{2x}\\\vdots \\-{e}_{Ny}\\{e}_{Nx}\end{matrix}\right]</math> |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
|- | |- | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">=c\lbrace -{h}_{1}{e}_{1x}{e}_{1y}+{h}_{1}{e}_{1x}{e}_{1y}-</math><math>{h}_{2}{e}_{2x}{e}_{2y}+{h}_{2}{e}_{2x}{e}_{2y}\ldots -{h}_{N}{e}_{Nx}{e}_{Ny}+{h}_{N}{e}_{Nx}{e}_{Ny}\rbrace | + | | style="text-align: center;" | <math display="inline">=c\lbrace -{h}_{1}{e}_{1x}{e}_{1y}+{h}_{1}{e}_{1x}{e}_{1y}-</math><math>{h}_{2}{e}_{2x}{e}_{2y}+{h}_{2}{e}_{2x}{e}_{2y}\ldots -{h}_{N}{e}_{Nx}{e}_{Ny}+{h}_{N}{e}_{Nx}{e}_{Ny}\rbrace =0</math>, |
| − | + | | style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(39) | |
| − | + | ||
| − | + | ||
| − | | style="text-align: right;vertical-align: | + | |
|} | |} | ||
|} | |} | ||
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y (38) resulta equivalente a (22). Con esto se demuestra que el objetivo de formación (14) se sigue satisfaciendo. | y (38) resulta equivalente a (22). Con esto se demuestra que el objetivo de formación (14) se sigue satisfaciendo. | ||
| − | ===5.2. Cálculo de la matriz de repulsión | + | ===5.2. Cálculo de la matriz de repulsión '''H'''=== |
Considérese la Figura 19 en la que se representa el plano <math display="inline">{\Sigma }_{W}</math> de movimiento de los robots. Obsérvese la posición final deseada <math display="inline">{\mathit{\boldsymbol{\varphi }}}_{i}</math> para el punto objetivo <math display="inline">{\mathit{\boldsymbol{q}}}_{i}</math> del robot <math display="inline">{R}_{i}</math>, y obsérvese el punto objetivo <math display="inline">{\mathit{\boldsymbol{q}}}_{j}</math> del robot <math display="inline">{R}_{j}</math>. El espacio físico que ocupa cada robot se delimita por un círculo de radio <math display="inline">r</math>. Ahora note el punto <math display="inline">{\mathit{\boldsymbol{O}}}_{ij}\in \, {\mathit{\mathbb{R}}}^{2}</math> a una distancia <math display="inline">2r</math> de la posición del robot <math display="inline">{R}_{j}</math> sobre la línea que une los puntos <math display="inline">{\mathit{\boldsymbol{q}}}_{i}</math> y <math display="inline">{\mathit{\boldsymbol{q}}}_{j}</math>, el cual expresa la mínima distancia permitida al robot <math display="inline">{R}_{i}</math>para no colisionar con el robot <math display="inline">{R}_{j}</math>; es decir | Considérese la Figura 19 en la que se representa el plano <math display="inline">{\Sigma }_{W}</math> de movimiento de los robots. Obsérvese la posición final deseada <math display="inline">{\mathit{\boldsymbol{\varphi }}}_{i}</math> para el punto objetivo <math display="inline">{\mathit{\boldsymbol{q}}}_{i}</math> del robot <math display="inline">{R}_{i}</math>, y obsérvese el punto objetivo <math display="inline">{\mathit{\boldsymbol{q}}}_{j}</math> del robot <math display="inline">{R}_{j}</math>. El espacio físico que ocupa cada robot se delimita por un círculo de radio <math display="inline">r</math>. Ahora note el punto <math display="inline">{\mathit{\boldsymbol{O}}}_{ij}\in \, {\mathit{\mathbb{R}}}^{2}</math> a una distancia <math display="inline">2r</math> de la posición del robot <math display="inline">{R}_{j}</math> sobre la línea que une los puntos <math display="inline">{\mathit{\boldsymbol{q}}}_{i}</math> y <math display="inline">{\mathit{\boldsymbol{q}}}_{j}</math>, el cual expresa la mínima distancia permitida al robot <math display="inline">{R}_{i}</math>para no colisionar con el robot <math display="inline">{R}_{j}</math>; es decir | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{O}}}_{ij}={\mathit{\boldsymbol{q}}}_{j}+</math><math>2r\frac{{\mathit{\boldsymbol{q}}}_{i}-{\mathit{\boldsymbol{q}}}_{j}}{\left\| {\mathit{\boldsymbol{q}}}_{i}-{\mathit{\boldsymbol{q}}}_{j}\right\| }</math> | + | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{O}}}_{ij}={\mathit{\boldsymbol{q}}}_{j}+</math><math>2r\frac{{\mathit{\boldsymbol{q}}}_{i}-{\mathit{\boldsymbol{q}}}_{j}}{\left\| {\mathit{\boldsymbol{q}}}_{i}-{\mathit{\boldsymbol{q}}}_{j}\right\| }</math>. |
| style="width: 5px;text-align: right;white-space: nowrap;" |(40) | | style="width: 5px;text-align: right;white-space: nowrap;" |(40) | ||
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| [[Image:draft_Aparicio Nogué_723966060-image24.png|246px]] | | [[Image:draft_Aparicio Nogué_723966060-image24.png|246px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan="1" | '''Figura 19 | + | | colspan="1" | '''Figura 19.''' Esquema para la evasión de obstáculos. |
|} | |} | ||
| − | Considérese además una matriz <math display="inline">{\mathit{\boldsymbol{H}}}_{r}\in \, {\mathit{\mathbb{R}}}^{ | + | Considérese además una matriz <math display="inline">{\mathit{\boldsymbol{H}}}_{r}\in \, {\mathit{\mathbb{R}}}^{N\times N}</math> con componentes del factor de repulsión <math display="inline">{h}_{ij}>0 </math> (para <math display="inline">i,j=</math><math>1,2,\ldots ,N</math> con <math display="inline">i\not =j</math>) si existe una trayectoria direccionada desde <math display="inline">{R}_{j}</math> hasta <math display="inline">{R}_{i}</math>, tal que el robot <math display="inline">{R}_{i}</math> disponga de la información de las variables de salida del robot <math display="inline">{R}_{j}</math> de forma directa o a través de otro robot en dicha trayectoria; <math display="inline">{h}_{ij}=</math><math>0</math> en caso contrario. Los elementos <math display="inline">{h}_{ii}=</math><math>0</math> (los componentes del factor de repulsión en la diagonal de <math display="inline">{\mathit{\boldsymbol{H}}}_{r}</math> son nulos). De esta manera, los componentes no nulos del factor de repulsión estarían dados por |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" | <math display="inline">{h}_{ij}=\frac{\alpha \mathrm{tanh}\,(\lambda \lbrace {q}_{iy}-L({q}_{ix})\rbrace \lbrace {\varphi }_{ix}-{q}_{jx}\rbrace )}{\left\| {q}_{i}-{O}_{ij}\right\| +\sigma }</math> | + | | style="text-align: center;" | <math display="inline">{h}_{ij}=\frac{\alpha \mathrm{tanh}\,(\lambda \lbrace {q}_{iy}-L({q}_{ix})\rbrace \lbrace {\varphi }_{ix}-{q}_{jx}\rbrace )}{\left\| {q}_{i}-{O}_{ij}\right\| +\sigma }</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" |(41) | | style="width: 5px;text-align: right;white-space: nowrap;" |(41) | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" | <math display="inline">L\left( {q}_{ix}\right) =\frac{{\varphi }_{iy}-{q}_{jy}}{{\varphi }_{ix}-{q}_{jx}}\left( {q}_{ix}-\right. </math><math>\left. {q}_{jx}\right) +{q}_{jy}</math> | + | | style="text-align: center;" | <math display="inline">L\left( {q}_{ix}\right) =\frac{{\varphi }_{iy}-{q}_{jy}}{{\varphi }_{ix}-{q}_{jx}}\left( {q}_{ix}-\right. </math><math>\left. {q}_{jx}\right) +{q}_{jy}</math>. |
| style="width: 5px;text-align: right;white-space: nowrap;" |(42) | | style="width: 5px;text-align: right;white-space: nowrap;" |(42) | ||
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En la Tabla 2 se muestran las condiciones iniciales y posiciones finales deseadas. Para el control de esta red se utilizó un líder virtual fijo en el origen del marco global, una fuerza global de acoplamiento <math display="inline">c=</math><math>1.25</math>, ganancia <math display="inline">\mathit{\boldsymbol{D}}=</math><math>diag\lbrace 1\, \, 0\, \, 0\, \, 0\, \, 0\rbrace</math>, <math display="inline">\alpha =</math><math>0.25,\, \, \lambda =12</math> y <math display="inline">\sigma =</math><math>0.001</math>. El radio de delimitación <math display="inline">r</math> fue igual a <math display="inline">0.1</math> m, el cual incluye una tolerancia de <math display="inline">0.02</math> m. | En la Tabla 2 se muestran las condiciones iniciales y posiciones finales deseadas. Para el control de esta red se utilizó un líder virtual fijo en el origen del marco global, una fuerza global de acoplamiento <math display="inline">c=</math><math>1.25</math>, ganancia <math display="inline">\mathit{\boldsymbol{D}}=</math><math>diag\lbrace 1\, \, 0\, \, 0\, \, 0\, \, 0\rbrace</math>, <math display="inline">\alpha =</math><math>0.25,\, \, \lambda =12</math> y <math display="inline">\sigma =</math><math>0.001</math>. El radio de delimitación <math display="inline">r</math> fue igual a <math display="inline">0.1</math> m, el cual incluye una tolerancia de <math display="inline">0.02</math> m. | ||
| − | |||
| − | {| style="border-collapse: collapse;" | + | <div class="auto" style="width: auto; text-align: center; margin-left: auto; margin-right: auto;font-size: 75%;"> |
| + | '''Tabla 2'''. Condiciones de simulación y experimentación para la RR´s de la Figura 14 que incluye la ley con procedimiento para la no colisión.</div><br /> | ||
| + | |||
| + | {| style="border-collapse: collapse; width: 80%; margin: 0 auto; text-align: center;font-size:85%;" | ||
|- | |- | ||
| style="border: 1pt solid black;vertical-align: top;"| <math>i</math> | | style="border: 1pt solid black;vertical-align: top;"| <math>i</math> | ||
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| style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.3 | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.3 | ||
| style="border: 1pt solid black;text-align: center;vertical-align: top;"|-0.3 | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|-0.3 | ||
| − | |} | + | |}<br /> |
En el proceso de simulación y experimentación se han calculado las normas de separación <math display="inline">{N}_{ij}=</math><math>\left\| {q}_{i}-{q}_{j}\right\|</math> entre todos los robots con la finalidad de apreciar el desempeño del procedimiento para evitar colisiones en conjunto con la ley para la formación. La Figura 20 presenta las magnitudes de las normas <math display="inline">{N}_{ij}</math> de separación. Debe esperarse que ninguna norma resulte menor a <math display="inline">2\left( r-\right. </math><math>\left. 0.02\right) =0.16</math> m. En la Figura 20(a), obtenida mediante simulación sin incluir el procedimiento para evitar colisiones, se muestra que existe un conflicto de colisión entre los robots <math display="inline">{R}_{1}</math> y <math display="inline">{R}_{2}</math> (en algún momento <math display="inline">{N}_{21}<0.16</math> m). La Figura 20(b), también mediante simulación, muestra que este conflicto se elimina al incluir la acción para la no colisión. Nótese además que entre <math display="inline">{R}_{1}</math> y <math display="inline">{R}_{3}</math> existe una condición inicial cercana a <math display="inline">2(r-</math><math>0.02)</math> sin causar colisión en la evolución de las trayectorias de los robots. La Figura 20(c), obtenida mediante el experimento, muestra que con las ganancias utilizadas se resuelve la colisión entre <math display="inline">{R}_{1}</math> y <math display="inline">{R}_{2}</math> y no se genera ninguna otra colisión. | En el proceso de simulación y experimentación se han calculado las normas de separación <math display="inline">{N}_{ij}=</math><math>\left\| {q}_{i}-{q}_{j}\right\|</math> entre todos los robots con la finalidad de apreciar el desempeño del procedimiento para evitar colisiones en conjunto con la ley para la formación. La Figura 20 presenta las magnitudes de las normas <math display="inline">{N}_{ij}</math> de separación. Debe esperarse que ninguna norma resulte menor a <math display="inline">2\left( r-\right. </math><math>\left. 0.02\right) =0.16</math> m. En la Figura 20(a), obtenida mediante simulación sin incluir el procedimiento para evitar colisiones, se muestra que existe un conflicto de colisión entre los robots <math display="inline">{R}_{1}</math> y <math display="inline">{R}_{2}</math> (en algún momento <math display="inline">{N}_{21}<0.16</math> m). La Figura 20(b), también mediante simulación, muestra que este conflicto se elimina al incluir la acción para la no colisión. Nótese además que entre <math display="inline">{R}_{1}</math> y <math display="inline">{R}_{3}</math> existe una condición inicial cercana a <math display="inline">2(r-</math><math>0.02)</math> sin causar colisión en la evolución de las trayectorias de los robots. La Figura 20(c), obtenida mediante el experimento, muestra que con las ganancias utilizadas se resuelve la colisión entre <math display="inline">{R}_{1}</math> y <math display="inline">{R}_{2}</math> y no se genera ninguna otra colisión. | ||
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
| − | | [[Image:draft_Aparicio Nogué_723966060-image25.png| | + | | [[Image:draft_Aparicio Nogué_723966060-image25.png|300px]] |
| + | | [[Image:draft_Aparicio Nogué_723966060-image26.png|300px]] | ||
| + | |- | ||
| + | | colspan="2" | [[Image:draft_Aparicio Nogué_723966060-image27.png|300px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan=" | + | | colspan="2" | '''Figura 20.''' Normas <math display="inline">\left\| {q}_{i}-\right. </math><math>\left. {q}_{j}\right\|</math> : (a) Simulada sin incluir el procedimiento para evitar colisiones, (b) Simulada incluyendo el procedimiento para evitar colisiones y (c) Experimento incluyendo el procedimiento para evitar colisiones. |
|} | |} | ||
| − | La Figura 21 grafica la evolución de los errores de formación. La Figura 21(a) muestra el comportamiento de la evolución de los errores en simulación y sugiere un tiempo de <math display="inline">t\approx</math> | + | La Figura 21 grafica la evolución de los errores de formación. La Figura 21(a) muestra el comportamiento de la evolución de los errores en simulación y sugiere un tiempo de <math display="inline">t\approx 8</math> s para alcanzar prácticamente el objetivo de formación fijado. La Figura 21(b) presenta la gráfica obtenida mediante el experimento. |
<div id='img-21'></div> | <div id='img-21'></div> | ||
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
| − | | [[Image:draft_Aparicio Nogué_723966060-image28.png|312px]] [[Image:draft_Aparicio Nogué_723966060-image29.png|312px]] | + | | [[Image:draft_Aparicio Nogué_723966060-image28.png|312px]] |
| + | | [[Image:draft_Aparicio Nogué_723966060-image29.png|312px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan=" | + | | colspan="2" | '''Figura 21.''' Evolución de los errores: (a) En simulación, (b) En el experimento. |
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
| − | | [[Image:draft_Aparicio Nogué_723966060-image30.png|318px]] [[Image:draft_Aparicio Nogué_723966060-image31.png|318px]] | + | | [[Image:draft_Aparicio Nogué_723966060-image30.png|318px]] |
| + | | [[Image:draft_Aparicio Nogué_723966060-image31.png|318px]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| − | | colspan=" | + | | colspan="2" | '''Figura 22.''' (a) Trayectorias para la formación simuladas sin incluir el procedimiento para evitar colisiones, (b) Trayectorias para la formación en simulación y experimentación incluyendo el procedimiento para evitar colisiones. |
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}={\mathit{\boldsymbol{u}}}_{i}-</math><math>\overset{\cdot}{\mathit{\boldsymbol{s}}}</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}=c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}+</math><math>{\mathit{\boldsymbol{\vartheta }}}_{i}-\overset{\cdot}{\mathit{\boldsymbol{s}}}</math>. |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" |<math>\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i | + | | style="text-align: center;" |<math>\begin{array}{l}\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}} & \displaystyle =c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{q}}}_{j}-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}+\overset{\cdot}{\mathit{\boldsymbol{s}}}-\overset{\cdot}{\mathit{\boldsymbol{s}}},\\ |
| − | + | & \displaystyle =c\sum _{j=1}^{N}{g}_{ij}({\mathit{\boldsymbol{e}}}_{j}+\mathit{\boldsymbol{s}}+{\mathit{\boldsymbol{\varphi }}}_{j})-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j},\\ | |
| − | + | \displaystyle & =c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{e}}}_{j}+c\sum _{j=1}^{N}{g}_{ij}\mathit{\boldsymbol{s}}+c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j}-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}-c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{\varphi }}}_{j},\\ | |
| − | + | \displaystyle & =c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{e}}}_{j}+c\sum _{j=1}^{N}{g}_{ij}\mathit{\boldsymbol{s}}-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}.\end{array}</math> | |
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_ | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}=</math><math>c\sum _{j=1}^{N}{g}_{ij}{\mathit{\boldsymbol{e}}}_{j}-c{d}_{i}{\mathit{\boldsymbol{e}}}_{i}</math>. |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
|} | |} | ||
|} | |} | ||
| − | El desarrollo de <math display="inline">{\mathit{\boldsymbol{e | + | El desarrollo de <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{e}}}_{i}</math> para <math display="inline">i=</math><math>1,2,\ldots ,N</math> es |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" |<math display="inline">{\mathit{\boldsymbol{e | + | | style="text-align: center;" |<math display="inline">\overset{\cdot}{\mathit{\boldsymbol{e}}}_{1}=c\left\{ {g}_{11}{e}_{1}+\right. </math><math>\left. {g}_{12}{e}_{2}+\ldots +{g}_{1N}{e}_{N}\right\} -c{d}_{1}{e}_{1}</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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| − | | style="text-align: center;" |<math display="inline">{\mathit{\boldsymbol{e | + | | style="text-align: center;" |<math display="inline">\overset{\cdot}{\mathit{\boldsymbol{e}}}_{2}=c\left\{ {g}_{21}{e}_{1}+\right. </math><math>\left. {g}_{22}{e}_{2}+\ldots +{g}_{2N}{e}_{N}\right\} -c{d}_{2}{e}_{2}</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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| − | | style="text-align: center;" | <math display="inline">{\mathit{\boldsymbol{e | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}{\mathit{\boldsymbol{e}}}_{N}=c\left\{ {g}_{N1}{e}_{1}+\right. </math><math>\left. {g}_{N2}{e}_{2}+\ldots +{g}_{NN}{e}_{N}\right\} -c{d}_{N}{e}_{N}</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
|} | |} | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
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| − | | style="text-align: center;" | <math display="inline">\left[ \begin{matrix}{\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{1}\\{\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{2}\\\vdots \\{\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{N}\end{matrix}\right] =</math><math>c\left[ \begin{matrix}{g}_{11}\\{g}_{21}\\\vdots \\{g}_{N1}\end{matrix}\quad \begin{matrix}{g}_{12}\\{g}_{22}\\\vdots \\{g}_{N2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}\\{g}_{2N}\\\vdots \\{g}_{NN}\end{matrix}\right] \left[ \begin{matrix}{\mathit{\boldsymbol{e}}}_{1}\\{\mathit{\boldsymbol{e}}}_{2}\\\vdots \\{\mathit{\boldsymbol{e}}}_{N}\end{matrix}\right] -</math><math>c\left[ \begin{matrix}{d}_{1}\\0\\\vdots \\0\end{matrix}\quad \begin{matrix}0\\{d}_{2}\\\vdots \\0\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0\\0\\\vdots \\{d}_{N}\end{matrix}\right] \left[ \begin{matrix}{\mathit{\boldsymbol{e}}}_{1}\\{\mathit{\boldsymbol{e}}}_{2}\\\vdots \\{\mathit{\boldsymbol{e}}}_{N}\end{matrix}\right]</math> | + | | style="text-align: center;" | <math display="inline">\left[ \begin{matrix}{\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{1}\\{\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{2}\\\vdots \\{\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{N}\end{matrix}\right] =</math><math>c\left[ \begin{matrix}{g}_{11}\\{g}_{21}\\\vdots \\{g}_{N1}\end{matrix}\quad \begin{matrix}{g}_{12}\\{g}_{22}\\\vdots \\{g}_{N2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}\\{g}_{2N}\\\vdots \\{g}_{NN}\end{matrix}\right] \left[ \begin{matrix}{\mathit{\boldsymbol{e}}}_{1}\\{\mathit{\boldsymbol{e}}}_{2}\\\vdots \\{\mathit{\boldsymbol{e}}}_{N}\end{matrix}\right] -</math><math>c\left[ \begin{matrix}{d}_{1}\\0\\\vdots \\0\end{matrix}\quad \begin{matrix}0\\{d}_{2}\\\vdots \\0\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0\\0\\\vdots \\{d}_{N}\end{matrix}\right] \left[ \begin{matrix}{\mathit{\boldsymbol{e}}}_{1}\\{\mathit{\boldsymbol{e}}}_{2}\\\vdots \\{\mathit{\boldsymbol{e}}}_{N}\end{matrix}\right]</math> |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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| − | | style="text-align: center;" | <math display="inline">=c\left( \left[ \begin{matrix}{g}_{11}\\{g}_{21}\\\vdots \\{g}_{N1}\end{matrix}\quad \begin{matrix}{g}_{12}\\{g}_{22}\\\vdots \\{g}_{N2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}\\{g}_{2N}\\\vdots \\{g}_{NN}\end{matrix}\right] -\right. </math><math>\left. \left[ \begin{matrix}{d}_{1}\\0\\\vdots \\0\end{matrix}\quad \begin{matrix}0\\{d}_{2}\\\vdots \\0\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0\\0\\\vdots \\{d}_{N}\end{matrix}\right] \right) \left[ \begin{matrix}{\mathit{\boldsymbol{e}}}_{1}\\{\mathit{\boldsymbol{e}}}_{2}\\\vdots \\{\mathit{\boldsymbol{e}}}_{N}\end{matrix}\right]</math> | + | | style="text-align: center;" | <math display="inline">=c\left( \left[ \begin{matrix}{g}_{11}\\{g}_{21}\\\vdots \\{g}_{N1}\end{matrix}\quad \begin{matrix}{g}_{12}\\{g}_{22}\\\vdots \\{g}_{N2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}\\{g}_{2N}\\\vdots \\{g}_{NN}\end{matrix}\right] -\right. </math><math>\left. \left[ \begin{matrix}{d}_{1}\\0\\\vdots \\0\end{matrix}\quad \begin{matrix}0\\{d}_{2}\\\vdots \\0\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0\\0\\\vdots \\{d}_{N}\end{matrix}\right] \right) \left[ \begin{matrix}{\mathit{\boldsymbol{e}}}_{1}\\{\mathit{\boldsymbol{e}}}_{2}\\\vdots \\{\mathit{\boldsymbol{e}}}_{N}\end{matrix}\right]</math> |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\left[ \begin{matrix}{\overset{\cdot}{e}}_{11}\\{\overset{\cdot}{e}}_{12}\\{\overset{\cdot}{e}}_{21}\\\vdots \\{\overset{\cdot}{e}}_{N2}\end{matrix}\right] =</math><math>c\left( \left[ \begin{matrix}{g}_{11}{I}_{2}\\{g}_{21}{I}_{2}\\\vdots \\{g}_{N1}{I}_{2}\end{matrix}\quad \begin{matrix}{g}_{12}{I}_{2}\\{g}_{22}{I}_{2}\\\vdots \\{g}_{N2}{I}_{2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}{I}_{2}\\{g}_{2N}{I}_{2}\\\vdots \\{g}_{NN}{I}_{2}\end{matrix}\right] -\right. </math><math>\left. \left[ \begin{matrix}{d}_{1}{I}_{2}\\0{I}_{2}\\\vdots \\0{I}_{2}\end{matrix}\quad \begin{matrix}0{I}_{2}\\{d}_{2}{I}_{2}\\\vdots \\0{I}_{2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0{I}_{2}\\0{I}_{2}\\\vdots \\{d}_{N}{I}_{2}\end{matrix}\right] \right) \left[ \begin{matrix}{e}_{11}\\{e}_{12}\\{e}_{21}\\\vdots \\{e}_{N2}\end{matrix}\right]</math> | + | | style="text-align: center;" | <math display="inline">\left[ \begin{matrix}{\overset{\cdot}{e}}_{11}\\{\overset{\cdot}{e}}_{12}\\{\overset{\cdot}{e}}_{21}\\\vdots \\{\overset{\cdot}{e}}_{N2}\end{matrix}\right] =</math><math>c\left( \left[ \begin{matrix}{g}_{11}{I}_{2}\\{g}_{21}{I}_{2}\\\vdots \\{g}_{N1}{I}_{2}\end{matrix}\quad \begin{matrix}{g}_{12}{I}_{2}\\{g}_{22}{I}_{2}\\\vdots \\{g}_{N2}{I}_{2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}{I}_{2}\\{g}_{2N}{I}_{2}\\\vdots \\{g}_{NN}{I}_{2}\end{matrix}\right] -\right. </math><math>\left. \left[ \begin{matrix}{d}_{1}{I}_{2}\\0{I}_{2}\\\vdots \\0{I}_{2}\end{matrix}\quad \begin{matrix}0{I}_{2}\\{d}_{2}{I}_{2}\\\vdots \\0{I}_{2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0{I}_{2}\\0{I}_{2}\\\vdots \\{d}_{N}{I}_{2}\end{matrix}\right] \right) \left[ \begin{matrix}{e}_{11}\\{e}_{12}\\{e}_{21}\\\vdots \\{e}_{N2}\end{matrix}\right]</math> |
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{| style="text-align: center; margin:auto;width: 100%;" | {| style="text-align: center; margin:auto;width: 100%;" | ||
|- | |- | ||
| − | | style="text-align: center;" | <math display="inline">\left[ \begin{matrix}{\overset{\cdot}{e}}_{11}\\{\overset{\cdot}{e}}_{12}\\{\overset{\cdot}{e}}_{21}\\\vdots \\{\overset{\cdot}{e}}_{N2}\end{matrix}\right] =</math><math>c\left( \left( \left[ \begin{matrix}{g}_{11}\\{g}_{21}\\\vdots \\{g}_{N1}\end{matrix}\quad \begin{matrix}{g}_{12}\\{g}_{22}\\\vdots \\{g}_{N2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}\\{g}_{2N}\\\vdots \\{g}_{NN}\end{matrix}\right] -\right. \right. </math><math>\left. \left. \left[ \begin{matrix}{d}_{1}\\0\\\vdots \\0\end{matrix}\quad \begin{matrix}0\\{d}_{2}\\\vdots \\0\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0\\0\\\vdots \\{d}_{N}\end{matrix}\right] \right) \otimes {I}_{2}\right) \left[ \begin{matrix}{e}_{11}\\{e}_{12}\\{e}_{21}\\\vdots \\{e}_{N2}\end{matrix}\right]</math> | + | | style="text-align: center;" | <math display="inline">\left[ \begin{matrix}{\overset{\cdot}{e}}_{11}\\{\overset{\cdot}{e}}_{12}\\{\overset{\cdot}{e}}_{21}\\\vdots \\{\overset{\cdot}{e}}_{N2}\end{matrix}\right] =</math><math>c\left( \left( \left[ \begin{matrix}{g}_{11}\\{g}_{21}\\\vdots \\{g}_{N1}\end{matrix}\quad \begin{matrix}{g}_{12}\\{g}_{22}\\\vdots \\{g}_{N2}\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}{g}_{1N}\\{g}_{2N}\\\vdots \\{g}_{NN}\end{matrix}\right] -\right. \right. </math><math>\left. \left. \left[ \begin{matrix}{d}_{1}\\0\\\vdots \\0\end{matrix}\quad \begin{matrix}0\\{d}_{2}\\\vdots \\0\end{matrix}\quad \begin{matrix}\ldots \\\ldots \\\ddots \\\ldots \end{matrix}\quad \begin{matrix}0\\0\\\vdots \\{d}_{N}\end{matrix}\right] \right) \otimes {I}_{2}\right) \left[ \begin{matrix}{e}_{11}\\{e}_{12}\\{e}_{21}\\\vdots \\{e}_{N2}\end{matrix}\right]</math> |
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| − | | style="text-align: center;" | <math display="inline">\ | + | | style="text-align: center;" | <math display="inline">\overset{\cdot}\mathit{\boldsymbol{e}}= c\left( \left( \mathit{\boldsymbol{G}}-\right. \right. </math><math>\left. \left. \mathit{\boldsymbol{D}}\right) \otimes {\mathit{\boldsymbol{I}}}_{2}\right) \mathit{\boldsymbol{e}}</math>, |
| style="width: 5px;text-align: right;white-space: nowrap;" | | | style="width: 5px;text-align: right;white-space: nowrap;" | | ||
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|- | |- | ||
| − | | style="text-align: center;" | <math>\overset{\cdot}{\mathit{\boldsymbol{e}}} | + | | style="text-align: center;" | <math>\begin{array}{ll}\overset{\cdot}{\mathit{\boldsymbol{e}}} & =-c\left( \left( \mathit{\boldsymbol{D}}-\mathit{\boldsymbol{G}}\right) \otimes {\mathit{\boldsymbol{I}}}_{2}\right) \mathit{\boldsymbol{e}},\\ |
| − | + | & =-\mathit{\boldsymbol{F}}\mathit{\boldsymbol{e}}.\blacksquare \end{array}</math> | |
| − | + | ||
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==Referencias== | ==Referencias== | ||
| + | <div class="auto" style="width: auto; margin-left: auto; margin-right: auto;font-size: 85%;"> | ||
[1] T. M. Cheng, A. V. Savkin, Decentralized control of multi-agent systems for swarming with a given geometric pattern, Computers and Mathematics with Applications, 61(4) (2011) 731–744. | [1] T. M. Cheng, A. V. Savkin, Decentralized control of multi-agent systems for swarming with a given geometric pattern, Computers and Mathematics with Applications, 61(4) (2011) 731–744. | ||
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[33] E. Bugarín, R. Kelly, RTSVC: Real-Time System for Visual Control of Robots, Wiley Periodicals, Inc. Int J Imaging Syst Technol, Vol. 18, 251–256 (2008). | [33] E. Bugarín, R. Kelly, RTSVC: Real-Time System for Visual Control of Robots, Wiley Periodicals, Inc. Int J Imaging Syst Technol, Vol. 18, 251–256 (2008). | ||
| + | </div> | ||
Latest revision as of 15:23, 16 March 2018
Resumen
En este artículo se describe una propuesta de control para un objetivo de formación de una red de robots en trabajo colaborativo con acoplamiento difusivo estático utilizando herramientas derivadas de la teoría de grafos. Dicha propuesta se garantiza mediante la teoría de estabilidad de Lyapunov y se valida con resultados en simulación y en experimentación. Como parte importante de este trabajo, se detalla a manera de tutorial el procedimiento para la formulación de la red de robots abordada y también se propone una ley de control de formación con procedimiento para la no colisión entre los robots que forman la red utilizando la misma topología de red.
Palabras clave: Formación de redes de robots, acoplamiento difusivo estático, teoría de estabilidad de Lyapunov, validación experimental, no colisión.
Abstract
This article describes a control proposal for a formation objective of a robot network in collaborative work with static diffusive coupling using tools derived from graph theory. This proposal is guaranteed by the stability theory of Lyapunov and it is validated with results in simulation and experimentation. As an important part of this work, it is detailed like a tutorial the procedure for the formulation of the robot network addressed and also it is proposed a formation control law with procedure for non-collision between the robots that form the network using the same network topology.
1. Introducción
Desde principios del presente siglo se han incrementado notablemente los esfuerzos por coordinar y sincronizar grupos de robots que forman redes para el trabajo colaborativo. Estas Redes de Robots (RR´s) se justifican en misiones que no son posibles llevarlas a cabo con robots actuando de forma individual por razones de su propia capacidad. Entre los aspectos de estudio de mayor interés destaca la propuesta de leyes de control que gobiernen el comportamiento colectivo. Para muchas misiones de estas RR´s, la etapa primaria es su formación con un patrón geométrico o una distribución determinada [1]. La formación consiste en un posicionamiento relativo espacial con respecto a una o más referencias, fijas o móviles, para cada uno de los robots que forman parte de la red [2]. Para este problema de formación se han utilizado distintas metodologías [2-8]. La mayor parte de estas metodologías se fundamenta en la teoría de grafos cuyos inicios están documentados desde el siglo XVIII y que en la década de 1960 se extiende para aplicaciones en redes complejas [9-15]. Mediante la teoría de grafos es posible expresar interacciones entre sistemas dinámicos de manera general y organizada, facilitando la formulación de leyes de control para gobernar el comportamiento entero de una red.
Dentro del campo de la teoría de grafos y redes complejas, el acoplamiento difusivo estático es una estrategia ampliamente utilizada para realizar la interconexión entre los nodos (robots) de la red [16,17]. El principio de acoplamiento difusivo estático es un esquema de retroalimentación constante [17] entre los robots que forman parte de la red, pudiendo ser esta conectividad global o parcial (respectivamente: todos los robots se conectan con el resto de robots o sólo parte de ellos). La retroalimentación consiste en insertar como parte de la entrada del sistema dinámico del robot una señal que es la diferencia entre las variables de salida del robot de interés y las variables de salida del resto de robots en la red [18]. De este tipo de acoplamiento se han derivado distintas metodologías con el propósito de forzar una respuesta deseada en los robots, denominada estado de sincronía[14]; ó bien, una respuesta de sincronía natural, denominada consenso [19-21]. Una modalidad de estrategia de control que destaca para redes de gran dimensión es el control pinning (líderes referentes), que propone una acción de control en sólo un reducido número de sistemas dinámicos de la red y que inducen a la red entera hacia un estado de sincronía [11,14-15,22]. Estas estrategias se pueden llevar a cabo tanto mediante el control centralizado como con el control distribuido [23]. Con el control centralizado el procesamiento reside en un procesador que se encarga de monitorear las variables de estado de cada robot y de comunicar las acciones de control necesarias para alcanzar el objetivo de movimiento. Con el control distribuido el procesamiento reside en dos o más procesadores con acceso total o parcial a la información de variables de estado de la red; típicamente los robots llevan sus procesadores a bordo.
La formación de robots es aún un problema abierto [3-5], especialmente para RR´s de gran dimensión y RR´s heterogéneas. Las redes heterogéneas de robots están integradas por robots con distintos modelos dinámicos, con distintas dimensiones de vectores de variables de estado, con distinta dimensión de variables de salida o una combinación de las condiciones anteriores [16,17]. Las soluciones propuestas aún son parciales y requieren una integración para resolver problemas íntimamente relacionados. Tales problemas, además de los ya mencionados, son la evasión de obstáculos, la no colisión entre robots, las saturaciones en los accionamientos para el control, la diversidad de escenarios y las limitaciones en los enlaces de comunicación para el control [14-15].
Las contribuciones del presente trabajo se aplican en la formación de una RR´s utilizando herramientas derivadas de la teoría de grafos. Estas contribuciones son: (i) un tutorial para la formulación de RR´s mediante el acoplamiento difusivo estático en el que se describen de manera detallada los elementos y variables que intervienen en la red, (ii) una ley de control para una formación deseada de la RR´s bajo esta formulación y (iii) un procedimiento para la no colisión entre los robots que forman la red utilizando la misma topología de la RR´s. La metodología desarrollada se ejemplifica con simulaciones y experimentos mediante la construcción de una red de Robots Móviles Terrestres a base de Ruedas (RMTR’s).
El resto de este documento está organizado como sigue. En la sección 2 se expone un tutorial que describe paso a paso la formulación de una RR´s basado en el modelo para cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} -ésimo robot que forma parte de la red y la estrategia de acoplamiento difusivo estático. En la sección 3 se propone la ley de control para la formación de una red de robots construida con robots tipo uniciclo. La efectividad de la ley de control se demuestra utilizando la teoría de estabilidad de Lyapunov. En la sección 4 se presentan los resultados de simulación y experimentación. En la sección 5 se describe un procedimiento para evitar las colisiones entre robots de la red ejemplificada en la sección 4 y se presentan resultados de simulación y experimentación. Finalmente, en la sección 6 se presentan las conclusiones generales.
2. Formulación de una red de robots
En esta sección se describe a manera de tutorial una metodología para la formulación de una RR´s. En la sección 2.1 se presenta el modelo de un robot individual. En la sección 2.2 se describe el modelo del Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} -ésimo robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
en una red de robots considerando la contribución de todos los robots que pertenecen a la red sobre las entradas de dicho robot. En la sección 2.3 se muestra un esquema de red general de robots. En la sección 2.4 se explican los elementos necesarios para la formulación de una red de robots mediante el acoplamiento difusivo estático y la sección 2.5 expone un ejemplo detallado para la formulación de una red de robots con la estrategia descrita.
2.1 Modelo de un robot individual
El modelo de un robot individual resulta por lo general en una o más ecuaciones diferenciales no lineales que pueden expresarse de la forma
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{x}}\in \, {\mathit{\mathbb{R}}}^{r}}
es el vector de variables de estado, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y\, \in \, }}{\mathit{\mathbb{R}}}^{s}}
es el vector de variables de salida, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
denota la variable tiempo continuo, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}}\in \, {\mathit{\mathbb{R}}}^{p}}
es el vector de variables de entrada. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{f}}:\mathit{\mathbb{R}}\times {\mathit{\mathbb{R}}}^{r}\times {\mathit{\mathbb{R}}}^{p}\rightarrow {\mathit{\mathbb{R}}}^{r}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{h}}:\mathit{\boldsymbol{\, }}\mathit{\mathbb{R}}\times {\mathit{\mathbb{R}}}^{r}\times {\mathit{\mathbb{R}}}^{p}\rightarrow {\mathit{\mathbb{R}}}^{s}}
son funciones continuas. En la Figura 1 se muestra el diagrama a bloques de un robot individual en correspondencia con (1).
 
|
| Figura 1. (a) Diagrama a bloques de un robot individual, (b) Representación simplificada |
2.2 Modelo del i-ésimo robot en una red de robots
Para un robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
que pertenece a una red se puede proponer un nuevo vector de entrada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\vartheta }}}_{i}\mathit{\boldsymbol{\in \, }}{\mathit{\mathbb{R}}}^{{p}_{{\vartheta }_{i}}}}
, de manera que su modelo quedaría como
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\Upsilon }}}_{i}:\mathit{\boldsymbol{\, }}\mathit{\mathbb{R}}\times {\mathit{\mathbb{R}}}^{s}\times {\mathit{\mathbb{R}}}^{{p}_{{\vartheta }_{i}}}\rightarrow {\mathit{\mathbb{R}}}^{{p}_{i}}}
es un mapeo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\vartheta }}}_{i}}
hacia el vector de excitación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{u}}}_{i}}
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}}
la concatenación de vectores de variables de salida de cada robot de la red, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sum _{i=1}^{N}{s}_{i}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
la cantidad de robots que integran la red. La Figura 2 muestra el modelo de (2) de manera simplificada.
|
| Figura 2. Modelo de un Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
-ésimo robot que pertenece a una red. |
2.3 Modelo de una red de robots
En consecuencia, de acuerdo a (2), la Figura 3 muestra el diagrama a bloques de una red de robots.
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| Figura 3. Diagrama a bloques de una red de robots. |
2.4 Estrategia de acoplamiento difusivo estático
Una red de robots también puede ser expresada mediante el conjunto de robots y de conexiones que la conforman mediante
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ {R}_{i}:i=1,2,\ldots ,N\right\}
es el conjunto de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
robots y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varepsilon=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ {\varepsilon}_{ij}({\mathit{\boldsymbol{y}}}_{j}):i,j=1,2,\ldots ,N\right\}
es el conjunto de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N\times N}
conexiones dirigidas de la red. Una conexión dirigida Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \, {\varepsilon}_{ij}({\mathit{\boldsymbol{y}}}_{j})\in {\mathit{\mathbb{R}}}^{{p}_{i}}}
es una función que corresponde a un par de robots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ({R}_{i},{R}_{j})}
siendo la salida Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{y}}}_{j}\in {\mathit{\mathbb{R}}}^{{s}_{j}}}
(salida del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
) el vector de información que se transfiere al robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
mediante una transformación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varepsilon}_{ij}:\, {\mathit{\mathbb{R}}}^{{s}_{j}}\rightarrow {\mathit{\mathbb{R}}}^{{p}_{i}}}
. La Figura 4 muestra las conexiones dirigidas de manera generalizada para un par de robots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
.
|
Figura 4. Grafo o esquema generalizado de conexiones dirigidas en los robots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
de una red.
|
De esta forma, una estrategia común para la construcción de la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\Upsilon }}}_{i}}
relacionada con (2) [11,15-17], corresponde a un sistema lineal de acoplamiento difusivo estático donde el conjunto de conexiones de todos los robots respecto al robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
se expresa como
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c>0}
es la denominada fuerza global de acoplamiento, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\varepsilon}}}_{ij}({\mathit{\boldsymbol{y}}}_{j})=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {g}_{ij}{\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{y}}}_{j} , con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {g}_{ij}}
elemento de la matriz de configuración externa de red Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\, G}}\in {\mathit{\mathbb{R}}}^{N\times N}}
obtenida mediante
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\cal A}}{\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{N\times N}}
es la matriz de adyacencia ponderada con sus elementos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{ij}>0}
si existe una conexión entre el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
y el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{ij}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
si no existe la conexión (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{ij}}
es la magnitud de la conexión y en particular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{ii}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0 ); Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{ent}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): diag\lbrace {d}_{ii}\rbrace {\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{N\times N}
es la matriz de grado de entrada con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{ii}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sum _{j=1}^{N}{a}_{ij}
la suma de los pesos de las conexiones que ingresan a cada robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
(note que de esta forma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{G}}}
es difundida puesto que la suma de sus elementos por renglón es nula); y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\Gamma }}}_{ij}\in {\mathit{\mathbb{R}}}^{{p}_{i}\times {s}_{j}}}
es la matriz de configuración interna que expresa las proporciones de contribución de las variables de salida Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{y}}}_{j}}
hacia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{u}}}_{i}}
y en consecuencia, define la relación interna entre las variables del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
hacia las variables del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
.
De esta manera el acoplamiento difusivo estático para generar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{u}}}_{i}}
es
|
donde se ha considerado en forma aditiva la nueva entrada de control Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\vartheta }}}_{i}} . La Figura 5 muestra el diagrama a bloques para una red formulada utilizando la estrategia (6), mientras que la Figura 6 presenta el esquema generalizado de conexiones dirigidas equivalente.
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| Figura 5. Diagrama a bloques para una red de robots con acoplamiento difusivo estático. |
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| Figura 6. Esquema generalizado de conexiones dirigidas equivalente al diagrama a bloques de la Figura 5. |
2.5 Ejemplo de formulación de una RR´s con acoplamiento difusivo estático
Ejemplo 2.1: Considérese la topología descrita en el grafo de la Figura 7 con los 4 robots distintos (red heterogénea) mostrados en la Figura 8.
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| Figura 7. Grafo para el ejemplo de formulación de red de robots. Observe que las autoconexiones en cada robot no son mostradas. |
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| Figura 8. Detalle de los Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4 robots distintos para el ejemplo de formulación de red. |
Considere las siguientes magnitudes de conexión: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{12}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{23}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{31}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{42}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 5
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {a}_{43}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
- tal que
|
De este modo, según (5), la matriz de conexión externa resulta
|
Ahora bien, la matriz de matrices de configuración interna debe ser
|
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\Gamma }}}_{ij}{\mathit{\boldsymbol{\, \, }}\in \mathit{\mathbb{R}}}^{{r}_{i}\times {s}_{j}}}
cada matriz de configuración interna y con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{0}}}
una matriz cero de dimensión adecuada; es decir
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con cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\gamma }_{ij,kl}\geq 0}
según la proporción de contribución de las variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{y}}}_{j}}
hacia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{u}}}_{i}}
.
De esta manera, se completa (6) para la red de la Figura 7 y la Figura 9 muestra su diagrama a bloques equivalente
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| Figura 9. Diagrama a bloques equivalente para el ejemplo de red heterogénea con 4 robots. |
3. Formación de una red de robots
En esta sección se presenta la propuesta para la formación de una red de robots construida con RMTR’s tipo uniciclo de tracción diferencial como el que se muestra en la Figura 10. Los robots se han modelado para expresar la cinemática de un punto objetivo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{q}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[{q}_{x}\, {q}_{y}]}^{T}\, \in \, \mathit{\mathbb{R}}^{2}
colocado en la parte superior del robot, centrado en referencia al eje que une sus ruedas en el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{O}}
pero desplazado una distancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l}
en dirección perpendicular a dicho eje (ver Figura 10). La cinemática de tal punto objetivo puede expresarse como [24]
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\theta }_{R}}
denota la orientación del robot en referencia al marco global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Sigma }_{W}}
,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {v}_{R}}
es la variable de entrada correspondiente a la velocidad lineal del robot y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\omega }_{R}}
su otra variable de entrada correspondiente a su velocidad angular. Note que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\Lambda }}{(\theta }_{R})}
es invertible si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l\not =0}
, de manera que si esto se cumple (10) en forma simplificada quedaría
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donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[{u}_{x}\, \, {u}_{y}]}^{T}=\mathit{\boldsymbol{\Lambda }}{(\theta }_{R}){[{v}_{R}\, \, {\omega }_{R}]}^{T}
representa el vector de entrada correspondiente a las velocidades lineales en direcciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {W}_{x}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {W}_{y}}
como se muestra en la Figura 10(b). La ecuación (11) es el modelo de nuestro robot individual, significando que en relación a (1): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{x}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{q}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{f}}\left( t,\mathit{\boldsymbol{x}},\mathit{\boldsymbol{u}}\right) =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{u}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{q}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{h}}\left( t,\mathit{\boldsymbol{x}},\mathit{\boldsymbol{u}}\right) =}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{x}}=\mathit{\boldsymbol{q}} .
|
|
| Figura 10. (a) Robot tipo uniciclo de tracción diferencial, (b) Vista superior del robot. | |
Ahora, considérense Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
robots en red con acoplamiento difusivo estático con dinámica (11), de tal modo que para el Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
-ésimo robot sería
|
y de acuerdo a (6)
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donde se ha sustituido Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{q}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\Gamma }}}_{ij}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{I}}}_{2}
por ser una red homogénea.
El objetivo de formación se propone como
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{e}}}_{i}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[{e}_{ix}\, {e}_{iy}]}^{T}={\mathit{\boldsymbol{q}}}_{i}-\mathit{\boldsymbol{s}}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{\varphi }}}_{i}
es el error de formación y se ha insertado un líder virtual (sistema de referencia de red) con sistema dinámico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{\mathit{\boldsymbol{s}}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{s}})
y un vector de posiciones finales deseadas constantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\varphi }}}_{i}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[{\varphi }_{ix}\, \, {\varphi }_{iy}]}^{T}\, \in \, {\mathit{\mathbb{R}}}^{2}
para cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {i}}
-ésimo robot respecto al líder. El vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{s}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[{s}_{x}\, \, {s}_{y}]}^{T}
puede ser un punto fijo o una trayectoria [11]. La derivada de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{e}}}_{i}}
es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overset{\cdot}{\mathit{\boldsymbol{q}}}}_{i}-\overset{\cdot}{\mathit{\boldsymbol{s}}}
o bien sustituyéndole (12)
|
3.1 Ley de control propuesta
La ley de control propuesta para el objetivo de formación es
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donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{i}}
es la ganancia del controlador para el Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {i}}
-ésimo robot. Así (13), luego de sustituirle (16) resulta
|
Ya que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{q}}}_{j}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{e}}}_{j}+\mathit{\boldsymbol{s}}+{\mathit{\boldsymbol{\varphi }}}_{j}
y que la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{G}}}
es difundida, la dinámica (15) del error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{{\mathit{\boldsymbol{e}}}_{i}}}
resulta
|
y en forma matricial (ver Apéndice A)
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{\mathit{\boldsymbol{e}}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[{\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{1}^{T}\, \, {\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{2}^{T}\, \ldots \, {\overset{\cdot}{\mathit{\boldsymbol{e}}}}_{N}^{T}]}^{T} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{F}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c\left( \mathit{\boldsymbol{D}}-\mathit{\boldsymbol{G}}\right) \otimes {\mathit{\boldsymbol{I}}}_{2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): diag\lbrace {d}_{1},\, {d}_{2},\, \ldots ,\, {d}_{N}\rbrace
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \otimes}
expresa el producto Kronecker [26].
3.2 Punto de equilibrio y prueba de estabilidad
Para demostrar el cumplimiento del objetivo de formación (14) con la ley de control (16) en el sistema expresado por (13)-(12), considérense los siguientes lemas:
Lema 1 [25]. Un sistema homogéneo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{Ay}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{0}} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}\in {\mathit{\mathbb{R}}}^{r\times r}} , posee una solución única (la solución trivial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{0}} ) si y sólo si el Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank\left( \mathit{\boldsymbol{A}}\right) =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r .
Lema 2 [26]. Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}\in {\mathit{\mathbb{R}}}^{m\times n}}
con valores singulares Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }_{1}\geq \ldots \geq {\sigma }_{r}>0}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{B}}\in {\mathit{\mathbb{R}}}^{p\times q}}
con valores singulares Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\delta }_{1}\geq \ldots \geq {\delta }_{s}>0}
. Entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}\otimes \mathit{\boldsymbol{B}}}
(ó Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{B}}\otimes \mathit{\boldsymbol{A}}}
) tiene Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rs}
valores singulares Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }_{1}{\delta }_{1}\geq \ldots \geq {\sigma }_{r}{\delta }_{s}>0}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank\left( \mathit{\boldsymbol{A}}\otimes \mathit{\boldsymbol{B}}\right) =}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left( \mathit{\boldsymbol{B}}\otimes \mathit{\boldsymbol{A}}\right) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left( \mathit{\boldsymbol{A}}\right) rank\mathit{\boldsymbol{(B)}} . NOTA: Si las matrices son cuadradas puede intercambiarse "valores singulares" por "valores característicos".
Lema 3 [27]. Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}\in {\mathit{\mathbb{R}}}^{N\times N}}
una matriz cuadrada y haciendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{2}[\mathit{\boldsymbol{P}}+{\mathit{\boldsymbol{P}}}^{T}] , el Teorema de Silvester establece que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}}
es definida positiva (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}>0}
) si y sólo si:
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NOTA: Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}}
es definida positiva (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}>0}
) entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{-P}}}
es definida negativa (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{-P}}<0}
). Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}}
es semidefinida positiva (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}\geq 0}
) entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{-P}}}
es semidefinida negativa (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{-P}}\leq 0}
).
Lema 4 [28]. Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}\in {\mathit{\mathbb{R}}}^{N\times N}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\Gamma }}\in {\mathit{\mathbb{R}}}^{n\times n}}
matrices definidas (semidefinidas) positivas, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}\otimes \mathit{\boldsymbol{\Gamma }}\in {\mathit{\mathbb{R}}}^{Nn\times Nn}}
es una matriz definida (semidefinida) positiva.
Ahora, considere el punto de equilibrio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{e}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{0}}
de (19) y, para el análisis de su estabilidad, considérese la siguiente función candidata de Lyapunov
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por lo que la derivada temporal de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{V}}}
es
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Tomando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{\mathit{\boldsymbol{e}}}}
de (19) y sustituyendo en (21) se tiene
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De manera que si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{F}}>0}
entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{\mathit{\boldsymbol{V}}}\mathit{\boldsymbol{<}}0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \forall \, \mathit{\boldsymbol{e}}\not = \mathit{\boldsymbol{0}}}
, significando que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{e}}= } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{0}}
es un punto de equilibrio asintóticamente estable y con esto queda demostrado el cumplimiento del objetivo de formación (14).
Como ejemplos considérense las siguientes topologías típicas para una red con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
robots.
Ejemplo 3.1: La red direccionada y no ponderada (magnitud de conexión unitaria) de la Figura 11 en configuración estrella con el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
en la raíz. Su matriz de configuración externa resulta
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Propóngase
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Figura 11. Una red de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
robots con Topología Estrella. |
Se ha considerado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): diag\lbrace {d}_{1},0,\ldots ,0\rbrace
ya que el robot en la raíz (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
) de la red es el único viable para ser líder, pues es el único que mantiene conexiones direccionadas con el resto de los robots. Obsérvese que (24) es una matriz triangular, por lo que si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}\not =0}
su rango es completo (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank(\mathit{\boldsymbol{D}}-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{G}})=N ). De esta manera, si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}\not =0}
y apoyándonos en los Lemas 1 y 2 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{e}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
es el único punto de equilibrio de (19).
Ahora, partiendo de (24) y aplicando el Lema 3 con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \left( \mathit{\boldsymbol{D-G}}\right) {+\left( \mathit{\boldsymbol{D-G}}\right) }^{T}\right] /2
se tiene
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de forma que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{1}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {d}_{1}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{1}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>0}
), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{2}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {d}_{1}-\frac{1}{4}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{2}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{1}{4)}}
,…, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{i}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {d}_{1}-\frac{i-1}{4}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{i}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{i-1}{4}}
),…, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{N}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {d}_{1}-\frac{N-1}{4}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{N}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{N-1}{4}}
).
De manera que apoyándonos en el Lema 4, entonces si
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entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{F}}>0}
- cumpliéndose el objetivo de formación (14) de forma global.
Ejemplo 3.2: La red direccionada y no ponderada de la Figura 12 en configuración anillo. Su matriz de configuración externa resulta
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Propóngase
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Figura 12. Una red de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
robots con Topología Anillo. |
Nótese que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank(\mathit{\boldsymbol{G}})=N-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
(lo cual puede comprobarse directamente al aplicar el procedimiento de eliminación de Gauss). Para que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank\left( \mathit{\boldsymbol{D-G}}\right)}
sea completo, por la simetría de la topología, resulta suficiente cualquier Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{i}\not =0.}
Supóngase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D=}}diag\mathit{\boldsymbol{\lbrace }}{d}_{1}\mathit{\boldsymbol{,\, }}0,\ldots ,0\rbrace}
de manera que si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}\not =0}
y apoyándose en los Lemas 1 y 2, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{e}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
es el único punto de equilibrio de (19).
Ahora, sustituyendo la Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D}}}
supuesta en (28) y aplicando el Lema 3 con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \left( \mathit{\boldsymbol{D-G}}\right) {+\left( \mathit{\boldsymbol{D-G}}\right) }^{T}\right] /2
se tiene
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de forma que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{i}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a{d}_{1}+b , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{i}{{2}^{i-1}}
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,2,\ldots ,N , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{(i+1)}{{2}^{i}}
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1,2,\ldots ,N-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N , siendo cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{i}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>0}
. Por lo tanto, apoyándonos en el Lema 4, entonces si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>0}
se cumple que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{F}}>0}
y el objetivo de formación (14) es satisfecho de forma global.
Ejemplo 3.3: La red direccionada y no ponderada de la Figura 13 en configuración malla. Su matriz de configuración externa resulta
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Propóngase
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Figura 13. Una red de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
robots con Topología Malla. |
Nótese que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank(\mathit{\boldsymbol{G}})=N-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
(lo cual puede comprobarse al aplicar el procedimiento de eliminación de Gauss). Para que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank\left( \mathit{\boldsymbol{D-G}}\right)}
sea completo resulta suficiente cualquier Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{i}\not =0}
. Supóngase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D=}}diag\mathit{\boldsymbol{\lbrace }}{d}_{1}\mathit{\boldsymbol{,\, }}0,\ldots ,0\rbrace}
de manera que si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}\not =0}
y apoyándose en los Lemas 1 y 2, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{e}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
es el único punto de equilibrio de (19).
Ahora, sustituyendo la Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D}}}
supuesta en (31) y aplicando el Lema 3 con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \left( \mathit{\boldsymbol{D-G}}\right) {+\left( \mathit{\boldsymbol{D-G}}\right) }^{T}\right] /2
se tiene
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\Delta }_{i}=(N}^{i-1}-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( i-1\right) {N}^{i-2}){d}_{1}+{N}^{i}-i{N}^{i-1},\quad i=1,2,\ldots ,N . De manera que para que cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{i}>0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{(i{N}^{i-1}-{N}^{i})}{\left( {N}^{i-1}-\left( i-1\right) {N}^{i-2}\right) } =N(i-N)/(N-i+1) }
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,2,\ldots ,N , es decir, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>0} . Por lo tanto, apoyándonos en el Lema 4, entonces si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>0}
se cumple que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{F}}>0}
y el objetivo de formación (14) es satisfecho de forma global.
4. Simulación y experimento
4.1. Caso particular de una red con 5 robots tipo uniciclo
La Figura 14 muestra una RR’s direccionada, no típica y no ponderada, compuesta por cinco robots tipo uniciclo con dinámica de sus puntos objetivo (12) y para la cual resulta su matriz de configuración externa
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| Figura 14. Topología externa de una RR´s no típica con 5 robots tipo uniciclo de tracción diferencial. |
Nótese que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank(\mathit{\boldsymbol{G}})=N-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
ya que su renglón 1 es nulo. Se propone la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D=}}diag\mathit{\boldsymbol{\lbrace }}{d}_{1}\mathit{\boldsymbol{,\, }}0,\ldots ,0\rbrace}
debido a que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
tiene trayectorias direccionadas hacia el resto de los robots. Para que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle rank\left( \mathit{\boldsymbol{D-G}}\right)}
sea completo resulta suficiente Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}\not =0}
(lo que se comprueba con el procedimiento de eliminación de Gauss) y apoyándose en los Lemas 1 y 2, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{e}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
es el único punto de equilibrio de (19).
Ahora, aplicando el Lema 3 en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D-G}}}
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{A}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \left( \mathit{\boldsymbol{D-G}}\right) {+\left( \mathit{\boldsymbol{D-G}}\right) }^{T}\right] /2
se tiene
|
donde resulta
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{1}={d}_{1}}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{1}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>0}
), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{2}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {d}_{1}-\frac{1}{4}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{2}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{1}{4}}
), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{3}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2{d}_{1}-\frac{3}{4} , (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{3}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{3}{8}}
), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{4}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{5}{2}{d}_{1}-\frac{23}{16}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{4}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{23}{40}}
) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{5}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{33}{16}{d}_{1}-\frac{21}{16}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{5}>0}
si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{21}{33}}
).
Por lo tanto, apoyándonos en el Lema 4, entonces si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {d}_{1}>\frac{21}{33}}
se cumple que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{F}}>0}
y el objetivo de formación (14) es satisfecho de forma global.
4.2. Simulación y experimento
Para la simulación de la red formada por cinco robots se utilizó MATLAB R2011b. El experimento se realizó con cinco robots tipo uniciclo modelo YSR-A de la empresa Yujin que reciben las consignas de velocidad mediante módulos de comunicación inalámbrica operando a 418 MHz y a una tasa de 19.2 Kbps. Los módulos de comunicación se conectan a un procesador central vía puerto serie. Este procesador central ejecuta una aplicación de tiempo real en ambiente RTLinux, auxiliado de una tarjeta de video Leonardo de la empresa Arvoo y con programación basada en [33]. Para efectos de identificar la postura de los robots se colocaron dos puntos marca sobre cada robot, uno para la posición objetivo y el otro como auxiliar para el cálculo de su orientación. Las imágenes se registran mediante una cámara UF-1000CL de la empresa UNIQ cuyo eje óptico apunta perpendicularmente al plano de movimiento de los robots (ver Figura 10). La cámara es de alta velocidad y se configuró a razón de 200 cuadros por segundo, obteniéndose un periodo estricto de muestreo en los experimentos de 0.005 s.
En la Tabla 1 se muestran las condiciones iniciales y posiciones finales deseadas para el control de formación de la red de la Figura 14; utilizando un líder virtual con dinámica Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}\mathit{\boldsymbol{s}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
y con condición inicial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{s}}(0)=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[0\, \, 0]}^{T}
m, una fuerza global de acoplamiento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.8 , ganancias Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): diag\lbrace 2\, \, 0\, \, 0\, \, 0\, \, 0\rbrace
y una distancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.036
m entre marcas sobre cada robot. Obsérvese que se desea una formación en línea recta con un ángulo de
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \pi /4}
rad respecto al marco global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Sigma }_{W}}
.
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {q}_{ix}(0)}
[m] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {q}_{iy}(0)}
[m] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\theta }_{i}(0)}
[rad] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi }_{ix}}
[m] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi }_{iy}}
[m] |
| 1 | 0.1511 | 0.2286 | -1.1802 | 0.30 | 0.30 |
| 2 | 0.4243 | 0.0891 | -1.1310 | 0.15 | 0.15 |
| 3 | -0.2135 | -0.0247 | -0.7598 | 0 | 0 |
| 4 | 0.0496 | -0.0717 | -0.3286 | -0.15 | -0.15 |
| 5 | 0.1259 | -0.2499 | 2.8198 | -0.30 | -0.30 |
La Figura 15(a) muestra la evolución de los errores en simulación, los cuales tienden a cero a medida que el tiempo transcurre. Una apreciación visual simple destaca que es suficiente Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t\approx 5}
s para que el error pueda considerarse prácticamente cero. En la Figura 15(b) se presentan las gráficas de los errores en el experimento. Cabe destacar que el umbral de reacción de los robots se encuentra por encima de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0.0156}
m/s de manera que debajo de dicho valor no hay movimiento del robot. Lo anterior es lo que justifica que en el experimento exista un error en estado estacionario, es decir, cuando la ley de control demanda velocidades pequeñas (pues el robot está cerca del objetivo) se cae dentro del umbral de no reacción del robot deteniendo su movimiento. La norma de error más grande lo presenta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{5}}
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\| {e}_{5}\right\| =}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.0105
m.
 
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| Figura 15. Evolución de los errores: (a) En simulación, (b) En experimento. |
La Figura 16 muestra las trayectorias seguidas por los robots. Las trayectorias experimentales descritas difieren un tanto de las trayectorias simuladas, esto debido a factores no modelados, como deslizamientos, derrapes y umbrales de reacción. El objetivo final, sin embargo, se alcanza satisfactoriamente.
|
| Figura 16. Trayectorias para la formación en línea recta en el marco global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Sigma }_{W}}
. |
Las gráficas en las Figuras 17 y 18 muestran respectivamente la evolución de las velocidades lineales (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {v}_{{R}_{i}}} ) y angulares (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\omega }_{{R}_{i}}} ) de los robots en simulación (17(a) y 18(a)) y en experimentación (17(b) y 18(b)). Las trayectorias experimentales muestran que luego de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t\approx 4}
s permanece un pequeño valor constante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {v}_{{R}_{i}}}
y un pequeño valor oscilatorio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\omega }_{{R}_{i}}}
incapaces de causar movimiento práctico en los robots.
|
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| Figura 17. Gráficas de velocidades lineales de los robots: (a) Simulación, (b) Experimento. | |
|
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| Figura 18. Gráficas de velocidades angulares de los robots: (a) Simulación, (b) Experimento. | |
5. Ley de control con procedimiento para evitar colisiones
La evasión de obstáculos y la no colisión entre los robots de una red aún es un problema abierto, especialmente cuando se tiene un número elevado de robots en áreas reducidas. En la literatura pueden encontrarse estudios que abordan esta problemática [24,29-32].
5.1. Ley de control
Se propone que la ley de control (16) contenga además, en forma aditiva, una acción de repulsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{u}}}_{ri}\, \in \, {\mathit{\mathbb{R}}}^{2}} como procedimiento para evitar colisiones entre los robots. Para el planteamiento formulado en la Sección 3, la propuesta, inspirada en [31], es la siguiente (la cual incluye el vector adicional de acciones de control Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{U}}}_{r}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[{\mathit{\boldsymbol{u}}}_{r1}^{T}\, {\mathit{\boldsymbol{u}}}_{r2}^{T}\ldots {\mathit{\boldsymbol{u}}}_{rN}^{T}]}^{T}\in \, {\mathit{\mathbb{R}}}^{2N} ):
|
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{u}}}_{ri}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {h}_{i}{[-{e}_{iy}\, \, {e}_{ix}]}^{T} con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{i}\, \in \, \mathit{\mathbb{R}}}
un factor de repulsión. De esta manera (13) resulta
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Manteniendo como objetivo el control de formación expresado por (14), la derivada del error es
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Obsérvese que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{e}}}_{i}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
sigue siendo un punto de equilibrio. Ahora, considérese la función candidata de Lyapunov (20) y su derivada (21), entonces para este caso Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{\mathit{\boldsymbol{V}}}}
puede escribirse como
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donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\, H}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): diag\lbrace {h}_{1},{h}_{2},\ldots {h}_{N}\rbrace
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{e}}}^{\bot}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {[-{e}_{1y}\, \, {e}_{1x}\, \, -{e}_{2y}\, \, {e}_{2x}\ldots \, -{e}_{Ny}\, \, {e}_{Nx}]}^{T} .
La contribución Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c\mathit{\boldsymbol{e}}^{T}[\mathit{\boldsymbol{H}}\otimes \mathit{\boldsymbol{I}}_{2}]\mathit{\boldsymbol{e}}^{\bot}}
en (38) es
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y (38) resulta equivalente a (22). Con esto se demuestra que el objetivo de formación (14) se sigue satisfaciendo.
5.2. Cálculo de la matriz de repulsión H
Considérese la Figura 19 en la que se representa el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Sigma }_{W}}
de movimiento de los robots. Obsérvese la posición final deseada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\varphi }}}_{i}}
para el punto objetivo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{q}}}_{i}}
del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
, y obsérvese el punto objetivo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{q}}}_{j}}
del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
. El espacio físico que ocupa cada robot se delimita por un círculo de radio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r} . Ahora note el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{O}}}_{ij}\in \, {\mathit{\mathbb{R}}}^{2}}
a una distancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2r}
de la posición del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
sobre la línea que une los puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{q}}}_{i}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{q}}}_{j}}
, el cual expresa la mínima distancia permitida al robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}} para no colisionar con el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
- es decir
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| Figura 19. Esquema para la evasión de obstáculos. |
Considérese además una matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{H}}}_{r}\in \, {\mathit{\mathbb{R}}}^{N\times N}}
con componentes del factor de repulsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{ij}>0 }
(para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i,j=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,2,\ldots ,N
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i\not =j}
) si existe una trayectoria direccionada desde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
hasta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
, tal que el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{i}}
disponga de la información de las variables de salida del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{j}}
de forma directa o a través de otro robot en dicha trayectoria; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{ij}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
en caso contrario. Los elementos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{ii}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
(los componentes del factor de repulsión en la diagonal de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{H}}}_{r}}
son nulos). De esta manera, los componentes no nulos del factor de repulsión estarían dados por
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donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha ,\lambda >0\,}
son ganancias para ponderar la magnitud de la repulsión, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma}
es un escalar positivo (muy pequeño sólo para evitar singularidades) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L({q}_{ix})}
es la evaluación en el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {q}_{ix}}
de la ecuación de la línea recta que une los puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\varphi }}}_{i}} y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{q}}}_{j}}
- es decir
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Finalmente, los elementos de la diagonal en la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{H}}}
son Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{i}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sum _{j=1}^{N}{h}_{ij}\,
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,2,\ldots ,N .
5.3. Simulación y experimento considerando el procedimiento para evitar colisiones
Considérese la red mostrada en la Figura 14 y descrita en la Sección 4.2, de tal forma que su matriz de componentes del factor de repulsión es
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Por lo que la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{H=}}diag\lbrace 0,{h}_{2},{h}_{3},{h}_{4},{h}_{5}\rbrace}
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{2}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {h}_{21} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{3}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {h}_{31}+{h}_{32}+{h}_{34} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{4}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {h}_{41}+{h}_{42}+{h}_{43}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{5}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {h}_{51}+{h}_{52}+{h}_{53}+{h}_{54} .
En la Tabla 2 se muestran las condiciones iniciales y posiciones finales deseadas. Para el control de esta red se utilizó un líder virtual fijo en el origen del marco global, una fuerza global de acoplamiento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.25 , ganancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{D}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): diag\lbrace 1\, \, 0\, \, 0\, \, 0\, \, 0\rbrace , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.25,\, \, \lambda =12
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma =}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.001 . El radio de delimitación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
fue igual a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0.1}
m, el cual incluye una tolerancia de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0.02}
m.
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {q}_{ix}(0)}
[m] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {q}_{iy}(0)}
[m] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\theta }_{i}(0)}
[rad] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi }_{ix}}
[m] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi }_{iy}}
[m] |
| 1 | -0.1955 | 0.1706 | 0.0568 | 0 | 0 |
| 2 | -0.0390 | -0.0020 | -0.0267 | 0.3 | 0.3 |
| 3 | -0.1620 | 0.3492 | -3.0882 | -0.3 | 0.3 |
| 4 | -0.1135 | -0.3512 | 3.0847 | -0.3 | -0.3 |
| 5 | 0.0261 | -0.1846 | -0.0542 | 0.3 | -0.3 |
En el proceso de simulación y experimentación se han calculado las normas de separación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {N}_{ij}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\| {q}_{i}-{q}_{j}\right\|
entre todos los robots con la finalidad de apreciar el desempeño del procedimiento para evitar colisiones en conjunto con la ley para la formación. La Figura 20 presenta las magnitudes de las normas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {N}_{ij}}
de separación. Debe esperarse que ninguna norma resulte menor a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2\left( r-\right. }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 0.02\right) =0.16
m. En la Figura 20(a), obtenida mediante simulación sin incluir el procedimiento para evitar colisiones, se muestra que existe un conflicto de colisión entre los robots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{2}}
(en algún momento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {N}_{21}<0.16}
m). La Figura 20(b), también mediante simulación, muestra que este conflicto se elimina al incluir la acción para la no colisión. Nótese además que entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{3}}
existe una condición inicial cercana a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2(r-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.02)
sin causar colisión en la evolución de las trayectorias de los robots. La Figura 20(c), obtenida mediante el experimento, muestra que con las ganancias utilizadas se resuelve la colisión entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{2}}
y no se genera ninguna otra colisión.
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| Figura 20. Normas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\| {q}_{i}-\right. }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {q}_{j}\right\| : (a) Simulada sin incluir el procedimiento para evitar colisiones, (b) Simulada incluyendo el procedimiento para evitar colisiones y (c) Experimento incluyendo el procedimiento para evitar colisiones. | |
La Figura 21 grafica la evolución de los errores de formación. La Figura 21(a) muestra el comportamiento de la evolución de los errores en simulación y sugiere un tiempo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t\approx 8}
s para alcanzar prácticamente el objetivo de formación fijado. La Figura 21(b) presenta la gráfica obtenida mediante el experimento.
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| Figura 21. Evolución de los errores: (a) En simulación, (b) En el experimento. | |
En la Figura 22 se observan las posiciones iniciales y finales para cada robot; así como las trayectorias seguidas por cada uno de ellos. En la Figura 22(a) se muestran las trayectorias en simulación sin incluir la ley para la no colisión. En la Figura 22(b) se presentan las trayectorias de los robots cuando se incluye la ley para la no colisión, obsérvese que en el experimento el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
no sigue una línea recta (como en la simulación) debido a que se hace presente la acción para no colisionar con el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{3}}
, mismo que describe una trayectoria en arco para evitar colisionar tanto con el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
como con el robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{2}}
. El movimiento inicial del robot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{2}}
en dirección opuesta a su posición final deseada se debe a la dependencia con los estados de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{1}}
, que al acercarse a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {R}_{2}}
incrementa la acción de no colisión en este último. En general, las trayectorias de los cinco robots han sufrido cambios con la ley de no colisión en virtud de la naturaleza de la repulsión entre todos ellos. Para las trayectorias de este experimento y de igual manera que para las de la Figura 16, es notable el logro en una medida satisfactoria del objetivo propuesto.
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| Figura 22. (a) Trayectorias para la formación simuladas sin incluir el procedimiento para evitar colisiones, (b) Trayectorias para la formación en simulación y experimentación incluyendo el procedimiento para evitar colisiones. | |
6. Conclusiones
Se ha presentado un tutorial para formular una red de robots mediante la estrategia de acoplamiento difusivo estático, típica en teoría de grafos. Se ha propuesto una ley de control para lograr el objetivo de formación de una red homogénea de robots utilizando la metodología descrita. Se ha realizado la experimentación para demostrar la validez de la ley de control propuesta y se ha demostrado el cumplimiento del objetivo de formación mediante la teoría de estabilidad de Lyapunov. Se ha particularizado en la inserción de la ganancia de control en tan solo un robot que recibe la información de estados de un líder virtual con la condición de que exista una trayectoria direccionada desde tal robot hacia todos los robots restantes de la red. Finalmente, utilizando la misma topología externa de la red se ha propuesto y validado experimentalmente el objetivo de formación con procedimiento para evitar colisiones entre los robots.
Apéndice A
Detalle de la forma matricial de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{\mathit{\boldsymbol{e}}}}
Para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,2,\ldots ,N
y de acuerdo a (15) se tiene
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y considerando (13)
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Ahora, sustituyéndole la ley de control propuesta (16) tenemos (note que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{q}}}_{j}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{e}}}_{j}+\mathit{\boldsymbol{s}}+{\mathit{\boldsymbol{\varphi }}}_{j} )
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Dado que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{G}}}
es difundida
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El desarrollo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot}{\mathit{\boldsymbol{e}}}_{i}}
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,2,\ldots ,N
es
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ó bien en forma matricial
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y dado que el acoplamiento corresponde a una red homogénea con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\Gamma }}}_{ij}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {I}_{2}
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i,j=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,2,\ldots ,N
|
o bien
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que resulta en
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es decir
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Agradecimientos
Se agradece el apoyo financiero del Tecnológico Nacional de México, del Instituto Tecnológico de Ensenada, del PRODEP y del CONACYT (Proyecto 166654).
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Document information
Published on 03/01/18
Accepted on 07/05/17
Submitted on 12/12/16
Volume 34, Issue 1, 2018
DOI: 10.23967/j.rimni.2017.7.002
Licence: CC BY-NC-SA license
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