Ortega et al 2010a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 456974532 to Ortega et al 2010a

2018-05-30T14:17:21Z

Cinmemj moved page Draft Samper 456974532 to Ortega et al 2010a

Cinmemj at 13:48, 30 May 2018

2018-05-30T13:48:58Z

Cinmemj at 13:47, 30 May 2018

2018-05-30T13:47:10Z

Cinmemj at 13:43, 30 May 2018

2018-05-30T13:43:06Z

Cinmemj at 13:39, 30 May 2018

2018-05-30T13:39:53Z

Cinmemj at 13:38, 30 May 2018

2018-05-30T13:38:23Z

Cinmemj at 12:58, 30 May 2018

2018-05-30T12:58:03Z

Cinmemj at 12:57, 30 May 2018

2018-05-30T12:57:11Z

Cinmemj at 12:39, 30 May 2018

2018-05-30T12:39:03Z

Cinmemj at 12:28, 30 May 2018

2018-05-30T12:28:25Z

← Older revision		Revision as of 13:48, 30 May 2018
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−	where ~~δ~~<~~sub~~>ij</~~sub~~> denotes the Kronecker’s delta function. In order to form the matrix and load vectors appearing in Eq. <span id='cite-ZEqnNum930710'></span>[[#ZEqnNum930710\|(8)]] an element-by-element approach is adopted. As the shape function of a node ''k'' is nonzero only inside elements containing said node, the integrals need only evaluated in the appropriate elements, e.g.	+	where <math display="inline">\delta_{ij}</math> denotes the Kronecker’s delta function. In order to form the matrix and load vectors appearing in Eq. <span id='cite-ZEqnNum930710'></span>[[#ZEqnNum930710\|(8)]] an element-by-element approach is adopted. As the shape function of a node ''k'' is nonzero only inside elements containing said node, the integrals need only evaluated in the appropriate elements, e.g.

	{\| class="formulaSCP" style="width: 100%; text-align: center;"		{\| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 140: / Line 140: @@
-being  <math display="inline">\tilde{u}</math> the approximate solution and ''N''<sup>k</sup> the interpolation (shape) function corresponding to the k<sup>th</sup> node of an element (from now on supra-indexes will indicate nodal values). Then, as the virtual strain field is a linear function of the virtual displacement field, it is also a linear function of  <math display="inline">\delta u_i^l</math> . Therefore, it is possible to write
+being  <math display="inline">\tilde{u}</math> the approximate solution and <math display="inline">N^k</math> the interpolation (shape) function corresponding to the <math display="inline">k^{th}</math> node of an element (from now on supra-indexes will indicate nodal values). Then, as the virtual strain field is a linear function of the virtual displacement field, it is also a linear function of  <math display="inline">\delta u_i^l</math>. Therefore, it is possible to write
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 170: / Line 170: @@
 |}
-where  <math display="inline">a=1,2,3</math> ,  <math>b=1,...,n_{nod}</math> and summation is assumed over the j and k indices. These equations can also be written in matrix form as
+where  <math display="inline">a=1,2,3</math>,  <math>b=1,...,n_{nod}</math> and summation is assumed over the j and k indices. These equations can also be written in matrix form as
 <span id='ZEqnNum930710'></span>

@@ Line 1,395: / Line 1,395: @@
 <span style="text-align: center; font-size: 75%;"><span id="fn-4"></span>([[#fnc-4|<sup>4</sup>]])  Data provided by Prof. Costello from Georgia Tech in personal communication to the authors.</span>
-==6. CONCLUSIONS==
+=6. CONCLUSIONS=
 The role parachutes have in many civil, humanitarian and military applications call for new and improved computational tools aimed at tackling the current lack of software applications in the field. New developments carried out at CIMNE to this end have been presented in this work. These developments involve a coupled fluid-structural solver intended for the unsteady simulation of ram-air type parachutes and a set of complementary tools aimed at studying trajectory and control systems effects. The coupled solution approach, based on an unsteady low-order panel method for aerodynamics, in conjunction with an explicit dynamic FE solver for the structure, has succeeded in solving ram-air parachute problems. This finding has been supported by the numerical examples presented and other many simulations performed to date. In addition, though at a preliminary stage, the complementary tools intended for trajectory and control effects analyses have shown their potential for the design, analysis and evaluation of guided parachute systems. The robustness and efficiency of the new coupled fluid-structural code have been highly improved with respect to the previous developments, at the same time that the scope of its potential applications has been widely extended. As it has been highlighted throughout this work, the challenges involved in the numerical simulation of parachutes are not minor; however, the numerical results obtained to date encourage us to go further in the development of numerical tools which are currently necessary as well as important contributions to the analysis and evaluation of parachute systems.

← Older revision		Revision as of 13:39, 30 May 2018
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−	'''Test case 2''': in this example the parachute-payload system is guided when following a flight path between two specified waypoints <math>{\bf W}_{p1}</math> and <math>{\bf W}_{p2}</math>. The starting position and heading of the vehicle is shown with the parachute top-view in Figure 19. There, several flight paths computed for different values of parameter ''k'' and heading angles are shown. This test case reveals the potential these set of tools have for studying GNC strategies.	+	'''Test case 2''': in this example the parachute-payload system is guided when following a flight path between two specified waypoints <math>{\bf W}_{p1}</math> and <math>{\bf W}_{p2}</math>. The starting position and heading of the vehicle is shown with the parachute top-view in Figure 19. There, several flight paths computed for different values of parameter <math>k</math> and heading angles are shown. This test case reveals the potential these set of tools have for studying GNC strategies.

@@ Line 1,358: / Line 1,358: @@
 Two simulation cases involving a small ram-air parachute, subject to different flight conditions, are intended to show the performance of the dynamic model and the lateral control strategy. The geometrical, mass, inertial and aerodynamic characteristics<span id="fnc-4"></span><span style="text-align: center; font-size: 75%;">[[#fn-4|<sup>4</sup>]]</span> of the test parachute-payload system are presented in Table 1.
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size: 75%;">
-<span style="text-align: center; font-size: 75%;">Table 1. Characteristics of the parachute-payload system</span></div>
+[[Image:draft_Samper_456974532-image147.png|286px]]
-'''Test case 1:''' glide slope (GS) control systems are studied in this example along with standard brake inputs for different wind settings. The parachute-payload system is deployed from a 400 m altitude with heading north, fixed incidence angle &#x0393; = -12º and no wind. The computed trajectory is presented in Figure 17, where left and right brake inputs are applied.
+'''Table 1'''. Characteristics of the parachute-payload system</div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+'''Test case 1:''' glide slope (GS) control systems are studied in this example along with standard brake inputs for different wind settings. The parachute-payload system is deployed from a 400 m altitude with heading north, fixed incidence angle <math>\Gamma = -12^\circ</math> and no wind. The computed trajectory is presented in Figure 17, where left and right brake inputs are applied.
-<span style="text-align: center; font-size: 75%;">Figure 17. Computed trajectory (400 m altitude, &#x0393; = -12º and no wind).</span></div>
 Next, the same simulation is performed, but this time applying a 3 m/s wind coming from East. The computed results are displayed in Figure 18.
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size: 75%;">
-<span style="text-align: center; font-size: 75%;">Figure 18. Computed trajectory (400 m altitude, &#x0393; = -12º and 3 m/s wind from East).</span></div>
+[[Image:draft_Samper_456974532-image149.png|600px]]
-'''Test case 2''': in this example the parachute-payload system is guided when following a flight path between two specified waypoints '''W''' ''<sub>p1</sub>'' and '''W''' ''<sub>p2</sub>''. The starting position and heading of the vehicle is shown with the parachute top-view in Figure 19. There, several flight paths computed for different values of parameter ''k'' and heading angles are shown. This test case reveals the potential these set of tools have for studying GNC strategies.
+'''Test case 2''': in this example the parachute-payload system is guided when following a flight path between two specified waypoints <math>{\bf W}_{p1}</math>  and <math>{\bf W}_{p2}</math>. The starting position and heading of the vehicle is shown with the parachute top-view in Figure 19. There, several flight paths computed for different values of parameter ''k'' and heading angles are shown. This test case reveals the potential these set of tools have for studying GNC strategies.
 [[Image:draft_Samper_456974532-picture-Group 505.svg|center|600px]]
-<span style="text-align: center; font-size: 75%;">Figure 19. Left: influence of 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 19'''. Left: influence of parameter <math>k</math> on the guided trajectory for a South release heading (180º). Right: automatic control response to different release heading angles (<math>k = 0.8</math>).</div>
 ----

← Older revision	Revision as of 14:17, 30 May 2018
(No difference)

@@ Line 1,346: / Line 1,346: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\lambda_{brake}=K_{brake} \cdot (r_{CMD}-\dot\Psi</math>
+| <math>\delta_{brake}=K_{brake} \cdot (r_{CMD}-\dot\psi)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (71)

@@ Line 1,291: / Line 1,291: @@
 ==5.2 Lateral Track Control Strategy==
-A heading tracking control strategy is implemented following the PID navigation system presented in [17] with the aim of guiding the parachute from a waypoint '''W''' ''<sub>p1</sub>'' (parachute deployment) to a specified target waypoint '''W''' ''<sub>p2</sub>'' through a yaw-rate command (see Figure 16).
+A heading tracking control strategy is implemented following the PID navigation system presented in [17] with the aim of guiding the parachute from a waypoint <math>{\bf W}_{p1} </math> (parachute deployment) to a specified target waypoint <math>{\bf W}_{p2} </math> through a yaw-rate command (see Figure 16).
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size: 75%;">
-<span style="text-align: center; font-size: 75%;">Figure 16. Lateral control strategy</span></div>
+[[Image:draft_Samper_456974532-image136.png|456px]]
-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
+'''Figure 16'''. Lateral control strategy</div>
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 1,319: / Line 1,320: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum155972'></span>  <math>E=k\cdot X_{track}{\dot{Y}}_{track}-Y_{track}{\dot{X}}_{track}=</math><math>0</math>
+| <span id='ZEqnNum155972'></span>  <math>E=k\cdot X_{track}{\dot{Y}}_{track}-Y_{track}{\dot{X}}_{track}=0</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (69)
@@ Line 1,338: / Line 1,339: @@
-where the proportional gain ''K<sub>R</sub>'' is determined iteratively in order to achieve good tracking without overshoots and the yaw-rate command is limited to  <math display="inline">R_{max}\mbox{ }=\mbox{ }\pm \mbox{ }0.2\mbox{ }\mbox{rad/s}</math> with the aim of avoiding numerical misbehaviours. Then, the desired yaw-rate is compared to the actual yaw-rate and left or right brake are applied according to
+where the proportional gain <math>K_{R} </math> is determined iteratively in order to achieve good tracking without overshoots and the yaw-rate command is limited to  <math display="inline">R_{max}\mbox{ }=\mbox{ }\pm \mbox{ }0.2\mbox{ }\mbox{rad/s}</math> with the aim of avoiding numerical misbehaviours. Then, the desired yaw-rate is compared to the actual yaw-rate and left or right brake are applied according to
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 1,345: / Line 1,346: @@
 {| style="text-align: center; margin:auto;"
 |-
-| [[Image:draft_Samper_456974532-image145.png|162px]]
+| <math>\lambda_{brake}=K_{brake} \cdot (r_{CMD}-\dot\Psi</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (71)
@@ Line 1,351: / Line 1,352: @@
-being  [[Image:draft_Samper_456974532-image146.png|36px]] a gain parameter accounting for the kinematics of the parachute brake.
+being   <math>K_{brake}</math> a gain parameter accounting for the kinematics of the parachute brake.
 ==5.3 Application examples==

@@ Line 1,202: / Line 1,202: @@
 |-
 |}
 '''Figure 15'''. Parachute-payload schematic</div>
@@ Line 1,216: / Line 1,217: @@
 \dot{y}\\
 \dot{z}
-\end{array}\right\}={\left[T_{IB}\right]}^T\left\{\begin{array}{c}
+\end{array}\right\}={\left[{\bf T}_{IB}\right]}^T\left\{\begin{array}{c}
 u\\
 v\\
@@ Line 1,238: / Line 1,239: @@
-where  <math display="inline">sin(\alpha )=s_{\alpha }</math> ,  <math display="inline">cos(\alpha )=c_{\alpha }</math> ,  <math display="inline">tan(\alpha )=t_{\alpha }</math> , T<sub>IB</sub> is the transformation matrix from the inertial reference frame to the body reference frame,  [[Image:draft_Samper_456974532-image126.png|48px]] are the Euler roll, pitch and yaw angles and  [[Image:draft_Samper_456974532-image127.png|42px]] are the angular velocities in a body reference frame.
+where  <math display="inline">sin(\alpha )=s_{\alpha }</math>,  <math display="inline">cos(\alpha )=c_{\alpha }</math>,  <math display="inline">tan(\alpha )=t_{\alpha }</math>, <math>T_{IB}</math> is the transformation matrix from the inertial reference frame to the body reference frame, <math>\phi ,\theta , \psi</math> are the Euler roll, pitch and yaw angles and  <math>p,q,r </math> are the angular velocities in a body reference frame.
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 1,245: / Line 1,246: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>T_{IB}=\left[\begin{array}{ccc}
+| <math>{\bf T}_{IB}=\left[\begin{array}{ccc}
 c_{\theta }c_{\psi } & c_{\theta }s_{\psi } & -s_{\theta }\\
 s_{\phi }s_{\theta }c_{\psi }-c_{\phi }s_{\psi } & s_{\phi }s_{\theta }s_{\psi }+c_{\phi }c_{\psi } & s_{\phi }c_{\theta }\\
@@ Line 1,267: / Line 1,268: @@
 \dot{v}\\
 \dot{w}
-\end{array}\right\}=\frac{1}{m}\left(F_W+F_A+F_S-F_{AM}\right)-S_{\omega }^B\left\{\begin{array}{c}
+\end{array}\right\}=\frac{1}{m}\left({\bf F}_W+{\bf F}_A+{\bf F}_S-{\bf F}_{AM}\right)-{\bf S}_{\omega }^B\left\{\begin{array}{c}
 u\\
 v\\
@@ Line 1,276: / Line 1,277: @@
 \dot{q}\\
 \dot{r}
-\end{array}\right\}={\left[I_T\right]}^{-1}\left(M_A+M_{AM}+S_{CG.P}^BF_A+S_{CG.S}^BF_S+S_{CG.M}^BF_{AM}-S_{\omega }^B\left[I_T\right]\left\{\begin{array}{c}
+\end{array}\right\}={\left[{\bf I}_T\right]}^{-1}\left({\bf M}_A+{\bf M}_{AM}+{\bf S}_{CG.P}^B {\bf F}_A+{\bf S}_{CG.S}^B {\bf F}_S+{\bf S}_{CG.M}^B {\bf F}_{AM}-S_{\omega }^B\left[{\bf I}_T\right]\left\{\begin{array}{c}
 u\\
 v\\
@@ Line 1,286: / Line 1,287: @@
 |}
-where the indices W, A, S and AM denote weight, aerodynamic, payload and apparent mass respectively.  [[Image:draft_Samper_456974532-image130.png|18px]] are cross-product matrices of vector  [[Image:draft_Samper_456974532-image131.png|12px]] expressed in the  [[Image:draft_Samper_456974532-image132.png|12px]] reference frame,  [[Image:draft_Samper_456974532-image133.png|18px]] is the total system inertia matrix and  [[Image:draft_Samper_456974532-image134.png|12px]] and  [[Image:draft_Samper_456974532-image135.png|18px]] stand for aerodynamic forces and moments. These terms are calculated by adding contributions of the different aerodynamic derivatives of the system [15]. The apparent mass forces and moments are computed according to the model developed by Barrows [16].
+where the indices <math>W,A,S </math> and <math>AM </math> denote weight, aerodynamic, payload and apparent mass respectively.  <math>S_r^A </math>are cross-product matrices of vector <math>r </math> expressed in the  <math>A </math> reference frame, <math>I_t</math> is the total system inertia matrix and  <math>F </math> and  <math>M </math> stand for aerodynamic forces and moments. These terms are calculated by adding contributions of the different aerodynamic derivatives of the system [15]. The apparent mass forces and moments are computed according to the model developed by Barrows [16].
 ==5.2 Lateral Track Control Strategy==

@@ Line 1,194: / Line 1,194: @@
 Figure 15 sketches the parachute-payload configuration analyzed. With the exception of movable brakes, the canopy is considered to have a fixed shape after being completely inflated. The combined system of the parachute canopy and the payload are modeled with 6 DoF, including three inertial position components of the total system mass center as well as the three Euler orientation angles. Orientation of the canopy with respect to the payload is defined by the incidence angle Γ and is considered a control variable. Rotation of the canopy about point C allows the tilting of the canopy lift and drag vectors, causing changes in the equilibrium glide slope.
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size: 75%;">
 {|
 |-
 | [[Image:draft_Samper_456974532-image120.png|264px]]
 | [[Image:draft_Samper_456974532-image121.png|center|192px]]
 |}
-</div>
+'''Figure 15'''. Parachute-payload schematic</div>
 The kinematic equations for the parachute-payload system are given by