Gonzalez et al 2011a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 447947970 to Gonzalez et al 2011a

2018-04-13T11:50:08Z

Cinmemj moved page Draft Samper 447947970 to Gonzalez et al 2011a

Cinmemj at 11:46, 13 April 2018

2018-04-13T11:46:39Z

Cinmemj at 11:43, 13 April 2018

2018-04-13T11:43:18Z

Cinmemj at 11:41, 13 April 2018

2018-04-13T11:41:46Z

Cinmemj at 11:04, 13 April 2018

2018-04-13T11:04:17Z

Cinmemj at 11:01, 13 April 2018

2018-04-13T11:01:53Z

Cinmemj at 11:00, 13 April 2018

2018-04-13T11:00:40Z

Cinmemj at 10:59, 13 April 2018

2018-04-13T10:59:54Z

Cinmemj at 10:40, 13 April 2018

2018-04-13T10:40:02Z

Cinmemj at 10:31, 13 April 2018

2018-04-13T10:31:45Z

@@ Line 507: / Line 507: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
- [[Image:draft_Samper_447947970-image85-c.png|270px]] </div>
+[[File:Draft_Samper_447947970_3847_Table4.png|270px]] </div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

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-<span style="text-align: center; font-size: 75%;">Figure 11: Deformed geometries obtained for no-brake deflection (left) and left brake deflection (right).</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 11''': Deformed geometries obtained for no-brake deflection (left) and left brake deflection (right).</span></div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 499: / Line 500: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Table 3: Main characteristics of the CIMSA parachute-payload system.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 3''': Main characteristics of the CIMSA parachute-payload system.</span></div>
 The analyses performed with the frozen equilibrium geometries are presented in Table 4. According to the linearized procedure assumed when running with the frozen geometry, the angle variations around the equilibrium positions are kept small.
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@@ Line 507: / Line 510: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Table 4: Runs computed for obtaining the aerodynamic derivatives.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 4''': Runs computed for obtaining the aerodynamic derivatives.</span></div>
 The aerodynamic derivatives are obtained from forces and moments data, computed at different flight conditions, by means of polynomial regression. Figure 12 shows a sample of the curves obtained for alpha and beta sweeps (pitch and yaw). The set of aerodynamic derivatives computed for the CIMSA parachute is presented in Table 5.
@@ Line 517: / Line 521: @@
 | [[Image:draft_Samper_447947970-image87.png|center|282px]]
 |}
 {|
 |-
@@ Line 528: / Line 529: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Figure 12: A sample of alpha and beta sweeps computed for the CIMSA parachute.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 12''': A sample of alpha and beta sweeps computed for the CIMSA parachute.</span></div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 534: / Line 536: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Table 5: Estimated derivatives for the CIMSA parachute-payload system.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 5''': Estimated derivatives for the CIMSA parachute-payload system.</span></div>
 <span id='_Toc290547657'></span>
@@ Line 541: / Line 543: @@
 The derivatives and inertial parameters listed in Tables 5 and 3 respectively are introduced in the dynamic simulator in order to perform trajectory and control analyses. In this example lateral and altitude control strategies are applied for the same payload drop conditions employed in the Application Example 1. The simulation results, though significantly different to that obtained for the Snowflake, are within normal expectations for a heavier parachute-payload system. It should be noticed that the performance for precision landing is heavily influenced by the control parameters; in this test case those listed in Table 6 are found to be acceptable although they could be further improved.
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@@ Line 546: / Line 549: @@
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-<span style="text-align: center; font-size: 75%;">Figure 13: Trajectory plot computed for the CIMSA parachute-payload system.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 13''': Trajectory plot computed for the CIMSA parachute-payload system.</span></div>
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@@ Line 552: / Line 556: @@
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-<span style="text-align: center; font-size: 75%;">Table 6: CIMSA parachute-payload system control parameters</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 6''': CIMSA parachute-payload system control parameters</span></div>
 <span id='_Toc290547658'></span>

← Older revision		Revision as of 11:41, 13 April 2018
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−	[[~~Image~~:~~draft_Samper_447947970-picture-Group 8~~.~~svg\|center~~\|600px]]	+	[[File:Draft_Samper_447947970_9658_Fig5.png\|600px]]

	<span style="text-align: center; font-size: 75%;">'''Figure 5'''. Left: influence of the design parameter k on the guided trajectory for a South release heading (180º). Right: automatic control response to different release heading angles (k = 0.8).</span>		<span style="text-align: center; font-size: 75%;">'''Figure 5'''. Left: influence of the design parameter k on the guided trajectory for a South release heading (180º). Right: automatic control response to different release heading angles (k = 0.8).</span>

@@ Line 387: / Line 387: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Figure 3: Snowflake parafoil-payload system photo, geometry schematics and payload.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 3''': Snowflake parafoil-payload system photo, geometry schematics and payload.</span></div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 393: / Line 394: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Table 1: Main characteristics of the Snowflake parachute-payload system.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 1''': Main characteristics of the Snowflake parachute-payload system.</span></div>
 <span id="fn-1"></span>([[#fnc-1|<sup>1</sup>]])  <span style="text-align: center; font-size: 75%;">It must be noticed that the data was provided by Prof. Slegers from the University of Alabama in a personal communication to the authors</span>
@@ Line 414: / Line 415: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Figure 4: Trajectory plots, flat view (left) and 3D view (right).</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 4''': Trajectory plots, flat view (left) and 3D view (right).</span></div>
 Next, the performance of the control algorithm is assessed by guiding the parachute-payload system through a flight path defined by waypoints '''W''' ''<sub>p1</sub>'' and '''W''' ''<sub>p2</sub>''. Figure 5 shows three flight paths computed for different values of design parameter ''k'' and heading angles.
@@ Line 421: / Line 423: @@
 [[Image:draft_Samper_447947970-picture-Group 8.svg|center|600px]]
-<span style="text-align: center; font-size: 75%;">Figure 5. Left: influence of the design parameter k on the guided trajectory for a South release heading (180º). Right: automatic control response to different release heading angles (k = 0.8).</span>
+<span style="text-align: center; font-size: 75%;">'''Figure 5'''. Left: influence of the design parameter k on the guided trajectory for a South release heading (180º). Right: automatic control response to different release heading angles (k = 0.8).</span>
 Figure 5 (left) shows that the trajectory is noticeably affected by the design parameter ''k''; in the present example smaller values of ''k'' lead to a faster track interception but with a considerable overshoot. Therefore, this parameter should be adjusted depending on the mission requirements As regards initial heading, Figure 5 (right) demonstrates that it is successfully managed by the control system even under the most adverse release conditions.
 The second test applies full lateral and altitude control to a typical payload drop. The system is deployed at 2000 meters of altitude, 2200 meters away from the target, with wind coming from the south at 3 meters per second. Initially, the system navigates directly to the target regardless of the excess of altitude (Phase 1). When the horizontal distance to the target is reduced by 75% and spiral descent is performed until reaching ''h<sub>ideal</sub>'' (Phase 2). At this point, navigation to the target is re-engaged and glideslope control is activated (incidence angle variation, Phase 3). The computed trajectory is depicted in Figures 6 and 7. It should be noticed that although the parafoil touchdown took place at an acceptable distance from the target (74 meters), the precision could be further improved by fine tuning of the control gains. The gain parameters used in this simulation are presented in Table 2.
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@@ Line 437: / Line 441: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Figure 6: Trajectory plots, flat view (left) and lateral view (right).</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 6''': Trajectory plots, flat view (left) and lateral view (right).</span></div>
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@@ Line 443: / Line 448: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Table 2: Snowflake parachute-payload system control parameters</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 2''': Snowflake parachute-payload system control parameters</span></div>
 Finally, a similar test case is solved but this time considering wind effects. The simulation results are shown in Figure 8, where it can be observed that the control system has a very good performance in the no-wind condition but deteriorates with wing, a situation in which a strong dependance on its intensity and direction is identified. This behavior can be enhanced by taking into account wind conditions and relative position of the parafoil with respect to the target when the switch from flight phase 2 to phase 3 is computed. These strategies are currently a topic of investigation of many research groups, ''cf.'' Yakimenko<sup>29</sup> for an overview.
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@@ Line 451: / Line 458: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Figure 7: 3D trajectory plot.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 7''': 3D trajectory plot.</span></div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 457: / Line 465: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Figure 8: Trajectory plots computed for different wind conditions.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 8''': Trajectory plots computed for different wind conditions.</span></div>
 <span id='_Toc290547654'></span>

@@ Line 306: / Line 306: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">Figure 2. Lateral control strategy.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 2'''. Lateral control strategy.</span></div>
 The control strategy proceeds as follows. For a given parachute track position <math display="inline">\left(X_{track},Y_{track}\right)</math> , measured from '''W''' ''<sub>p2</sub>'', the vehicle ground velocity vector  <math display="inline">V_{ground}</math> is pointed in the direction of a line intercepting the track at point ''C''. The interception point ''C'' is determined by considering that the distance on the track line from this point to ''W<sub>p2</sub>'' is, at any instant time, equal to <math display="inline">(1-k)\mbox{ }X_{track}</math> , being ''k'' a design parameter. From the geometry of the similar triangles ''OAB'' and ''OCD'', the parachute position and velocity is established according to
@@ Line 334: / Line 335: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(34)
 |}
 where the proportional gain <math display="inline">K_{r}</math> should be determined iteratively in order to achieve good tracking without overshoots. Moreover, the yaw-rate is limited to  <math display="inline">R_{\max}=\pm 0.2</math> rad/s with the aim of avoiding numerical misbehaviours. To command the parachute, brakes are actuated by comparison of the “desired yaw-rate” with the actual yaw-rate. If there is a difference, a brake input is generated left or right, depending on the sign resulting from the comparison. This process is summarized in the following expression,

← Older revision	Revision as of 11:50, 13 April 2018
(No difference)

@@ Line 351: / Line 351: @@
 Parafoil altitude control deals with potential energy management. The altitude control strategy adopted in this work divides the parafoil flight in three parts: an initial straight down gliding phase followed by a spiral descent phase and a final straight down approach towards the target. The first phase begins immediately after the parachute is deployed and extends until the target distance is reduced by a specified percentage. During this phase the parafoil uses only lateral control to navigate to the target and the travelled distance is recorded in order to evaluate an average parachute glide slope. At the beginning of the second phase, the control system computes an “ideal” flight altitude (''h<sub>ideal</sub>'') according to
 {| style="width: 100%; text-align: left;"
@@ Line 358: / Line 357: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(36)
 |}
 where D<sub>target</sub> is the remaining distance to target and GS<sub>Phase_1</sub> is the average glideslope measured at the initial phase. If the real parachute altitude'' h<sub>real</sub>'' is greater than ''h<sub>ideal</sub>'', a spiral descent is initiated by a specified constant brake input which holds until ''h<sub>real</sub>'' < ''h<sub>ideal</sub>''. Otherwise, the descent is omitted because the parachute altitude could be insufficient to reach the target. The third flight phase begins when ''h<sub>real</sub>'' &#x2264; ''h<sub>ideal</sub>'' and involves the final approach. During this stage both lateral and glideslope controls are applied. The latter is implemented through a variable incidence angle which is determined by the following proportional control
 {| style="width: 100%; text-align: left;"
@@ Line 368: / Line 365: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(37)
 |}
 where ''k<sub>p</sub>'' is the proportional gain. The incidence angle is increased (to the negative side) if the parafoil is higher than ''h<sub>ideal</sub>'' and decreased in the opposite case. This allows achieving a very fine adjustment of the glide slope as the parachute approaches to the target.

@@ Line 310: / Line 310: @@
 The control strategy proceeds as follows. For a given parachute track position <math display="inline">\left(X_{track},Y_{track}\right)</math> , measured from '''W''' ''<sub>p2</sub>'', the vehicle ground velocity vector  <math display="inline">V_{ground}</math> is pointed in the direction of a line intercepting the track at point ''C''. The interception point ''C'' is determined by considering that the distance on the track line from this point to ''W<sub>p2</sub>'' is, at any instant time, equal to <math display="inline">(1-k)\mbox{ }X_{track}</math> , being ''k'' a design parameter. From the geometry of the similar triangles ''OAB'' and ''OCD'', the parachute position and velocity is established according to
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+{| style="width: 100%; text-align: left;"
-  <math>\frac{{\dot{X}}_{track}}{k\cdot X_{track}}=\frac{{\dot{Y}}_{track}}{Y_{track}}</math> ()</div>
+|-
 Accordingly, an error ''E'' is defined by
-If Eq. (33) is not satisfied (meaning that there is a heading error), the magnitude and sign of error  [[Image:draft_Samper_447947970-image58.png|12px]] is used to define a parachute brake command. This is done in two steps: first a “desired yaw-rate” is defined and then parachute brakes are pulled in order to obtain it. The “desired yaw-rate” is defined as follows,
+{| style="width: 100%; text-align: left;"
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+If Eq. (33) is not satisfied (meaning that there is a heading error), the magnitude and sign of error  ''E'' is used to define a parachute brake command. This is done in two steps: first a “desired yaw-rate” is defined and then parachute brakes are pulled in order to obtain it. The “desired yaw-rate” is defined as follows,
-  [[Image:draft_Samper_447947970-image61.png|150px]] ()</div>
-where  [[Image:draft_Samper_447947970-image62.png|36px]] is a constant taking into account the kinematics of the parachute brake line.
+{| style="width: 100%; text-align: left;"
 <span id='_Toc280563188'></span><span id='_Toc290547651'></span>
@@ Line 336: / Line 352: @@
 Parafoil altitude control deals with potential energy management. The altitude control strategy adopted in this work divides the parafoil flight in three parts: an initial straight down gliding phase followed by a spiral descent phase and a final straight down approach towards the target. The first phase begins immediately after the parachute is deployed and extends until the target distance is reduced by a specified percentage. During this phase the parafoil uses only lateral control to navigate to the target and the travelled distance is recorded in order to evaluate an average parachute glide slope. At the beginning of the second phase, the control system computes an “ideal” flight altitude (''h<sub>ideal</sub>'') according to
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
- [[Image:draft_Samper_447947970-image63.png|102px]] ()</div>
+{| style="width: 100%; text-align: left;"
 where D<sub>target</sub> is the remaining distance to target and GS<sub>Phase_1</sub> is the average glideslope measured at the initial phase. If the real parachute altitude'' h<sub>real</sub>'' is greater than ''h<sub>ideal</sub>'', a spiral descent is initiated by a specified constant brake input which holds until ''h<sub>real</sub>'' < ''h<sub>ideal</sub>''. Otherwise, the descent is omitted because the parachute altitude could be insufficient to reach the target. The third flight phase begins when ''h<sub>real</sub>'' &#x2264; ''h<sub>ideal</sub>'' and involves the final approach. During this stage both lateral and glideslope controls are applied. The latter is implemented through a variable incidence angle which is determined by the following proportional control
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
- [[Image:draft_Samper_447947970-image64.png|150px]] ()</div>
+{| style="width: 100%; text-align: left;"
 where ''k<sub>p</sub>'' is the proportional gain. The incidence angle is increased (to the negative side) if the parafoil is higher than ''h<sub>ideal</sub>'' and decreased in the opposite case. This allows achieving a very fine adjustment of the glide slope as the parachute approaches to the target.

@@ Line 50: / Line 50: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(2)
 |}
 where T<sub>IB</sub> is the transformation matrix from an inertial reference frame to the body reference frame
@@ Line 59: / Line 60: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(3)
 |}
 The equations of motion of the parachute-payload system can be obtained by summing up forces and moment contributions about the system centre of gravity (CG). Thus,
@@ Line 76: / Line 78: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(5)
 |}
 The following convention is used for the vector cross product of two vectors''' r''' = {r<sub>x</sub>  r<sub>y</sub>  r<sub>z</sub>}<sup>T</sup> and '''F''' = {F<sub>x</sub>  F<sub>y</sub>  F<sub>z</sub>}<sup>T</sup> both expressed in the  aerodynamic reference frame
@@ Line 84: / Line 87: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(6)
 |}
 The forces appearing in Eq. (4) are the weight ('''F'''<sub>W</sub>), the canopy aerodynamic load ('''F'''<sub>A</sub>), the payload aerodynamic load ('''F'''<sub>S</sub>) and the apparent mass forces ('''F'''<sub>AM</sub>). These contributions are developed in the following equations.
@@ Line 94: / Line 98: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(7)
 |}
 Then, before defining aerodynamic forces, the aerodynamic velocities and angles are written in terms of the wind components and the canopy relative angle of incidence. Thus,
@@ Line 131: / Line 136: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(12)
 |}
 The aerodynamic force is defined in terms of aerodynamic derivatives (these are considered known data). Thus,
@@ Line 139: / Line 145: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(13)
 |}
 with '''T'''<sub>AB</sub> being the transformation from the aerodynamic to canopy reference frame
@@ Line 147: / Line 154: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(14)
 |}
 The only force taken into account for the payload is the drag. This is expressed in terms of its own local velocity (in the body reference frame) by
@@ Line 155: / Line 163: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(15)
 |}
 Besides the forces contribution to the total system moment, there are two additional terms in Eq. (5) that correspond to aerodynamic and apparent mass moments. Similar to the aerodynamic forces, the aerodynamic moment is defined in terms of derivatives
@@ Line 163: / Line 172: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(16)
 |}
 As the parafoil moves through the air, it displaces and accelerates the fluid surrounding it. The apparent mass effect is defined as the reaction of the fluid surrounding the parafoil. In high mass-to-volume ratio bodies like airplanes this reaction is negligible, but in very light vehicles like parafoils it plays an important role and can significantly affect their behaviour. A common practice is to add this effect as additional mass and inertia terms named “apparent mass” and “apparent inertia” which can be derived from the total kinetic energy of the fluid set in motion along with the parafoil. According to Lamb<sup>27</sup>, the kinetic energy can be expressed for a typical parachute canopy having two planes of symmetry, x-z and y-z, by
@@ Line 171: / Line 181: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(17)
 |}
 where the seven constants are defined in terms of parachute geometric parameters. In addition, if we neglect the spanwise camber effect, the constant H can be considered to be zero (this is the approach adopted here). Therefore, using the formulae given by Lissaman and Brown<sup>24</sup>, the apparent mass terms A, B, C, P, Q and R can be computed as follows
@@ Line 214: / Line 225: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(23)
 |}
 Then, forces and moments due to apparent mass and inertia can be obtained by relating the momentum of the fluid surrounding the parafoil to its kinetic energy. Following Lissman and Brown,
@@ Line 265: / Line 277: @@
 |  style="text-align: right;vertical-align: center;width: 5px;white-space: nowrap;"|(30)
 |}
 where the following notation applies
@@ Line 273: / Line 286: @@
 |  style="text-align: right;vertical-align: center;width: 5px;white-space: nowrap;"|(31)
 |}
 In the matrix in Eq. (29) it is possible to observe that apparent mass effects couple linear and rotational dynamics, which would be decoupled otherwise (in models where apparent mass coefficients are constant).

@@ Line 43: / Line 43: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(1)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 51: / Line 52: @@
 where T<sub>IB</sub> is the transformation matrix from an inertial reference frame to the body reference frame
 {| style="width: 100%; text-align: left;"
@@ Line 66: / Line 68: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(4)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 99: / Line 102: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(8)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 105: / Line 109: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(9)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 111: / Line 116: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(10)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 119: / Line 125: @@
 where the transformation from body to canopy reference frames can be expressed by
 {| style="width: 100%; text-align: left;"
 |-
@@ Line 172: / Line 179: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(18)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 178: / Line 186: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(19)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 184: / Line 193: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(20)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 190: / Line 200: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(21)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 196: / Line 207: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(22)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 210: / Line 222: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(24)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 216: / Line 229: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(25)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 222: / Line 236: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(26)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 236: / Line 251: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(28)
 |}
 {| style="width: 100%; text-align: left;"
@@ Line 242: / Line 258: @@
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(29)
 |}
 {| style="width: 100%; text-align: left;"