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Table of Contents

-
-

1 Gravimeter - Simscape Model

+
+

1 Gravimeter - Simscape Model

-
-

1.1 Introduction

+
+

1.1 Introduction

-
+

gravimeter_model.png

Figure 1: Model of the gravimeter

@@ -93,8 +93,8 @@
-
-

1.2 Simscape Model - Parameters

+
+

1.2 Simscape Model - Parameters

open('gravimeter.slx')
@@ -125,8 +125,8 @@ g = 0; % Gravity [m/s2]
 
-
-

1.3 System Identification - Without Gravity

+
+

1.3 System Identification - Without Gravity

%% Name of the Simulink File
@@ -148,7 +148,7 @@ G.OutputName = {'Ax1', 
 
-
+
 pole(G)
 ans =
       -0.000473481142385795 +      21.7596190728632i
@@ -173,7 +173,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
 
 
 
-
+

open_loop_tf.png

Figure 2: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers

@@ -181,8 +181,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
-
-

1.4 System Identification - With Gravity

+
+

1.4 System Identification - With Gravity

g = 9.80665; % Gravity [m/s2]
@@ -199,7 +199,7 @@ Gg.OutputName = {'Ax1', 
 We can now see that the system is unstable due to gravity.
 

-
+
 pole(Gg)
 ans =
           -10.9848275341252 +                     0i
@@ -211,7 +211,7 @@ ans =
 
-
+

open_loop_tf_g.png

Figure 3: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity

@@ -219,12 +219,12 @@ ans =
-
-

1.5 Analytical Model

+
+

1.5 Analytical Model

-
-

1.5.1 Parameters

+
+

1.5.1 Parameters

Bode options. @@ -256,8 +256,8 @@ Frequency vector.

-
-

1.5.2 Generation of the State Space Model

+
+

1.5.2 Generation of the State Space Model

Mass matrix @@ -358,11 +358,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.

-
-

1.5.3 Comparison with the Simscape Model

+
+

1.5.3 Comparison with the Simscape Model

-
+

gravimeter_analytical_system_open_loop_models.png

Figure 4: Comparison of the analytical and the Simscape models

@@ -370,8 +370,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
-
-

1.5.4 Analysis

+
+

1.5.4 Analysis

% figure
@@ -439,8 +439,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
 
-
-

1.5.5 Control Section

+
+

1.5.5 Control Section

system_dec_10Hz = freqresp(system_dec,2*pi*10);
@@ -580,8 +580,8 @@ legend('Control OFF','D
 
-
-

1.5.6 Greshgorin radius

+
+

1.5.6 Greshgorin radius

system_dec_freq = freqresp(system_dec,w);
@@ -627,8 +627,8 @@ ylabel('Greshgorin radius [-]');
 
-
-

1.5.7 Injecting ground motion in the system to have the output

+
+

1.5.7 Injecting ground motion in the system to have the output

Fr = logspace(-2,3,1e3);
@@ -684,15 +684,15 @@ rot = PHI(:,11,11);
 
-
-

2 Gravimeter - Functions

+
+

2 Gravimeter - Functions

-
-

2.1 align

+
+

2.1 align

- +

@@ -721,11 +721,11 @@ This Matlab function is accessible here.

-
-

2.2 pzmap_testCL

+
+

2.2 pzmap_testCL

- +

@@ -774,11 +774,11 @@ This Matlab function is accessible here.

-
-

3 Stewart Platform - Simscape Model

+
+

3 Stewart Platform - Simscape Model

-In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 5. +In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 5.

@@ -791,7 +791,7 @@ Some notes about the system: -

+

SP_assembly.png

Figure 5: Stewart Platform CAD View

@@ -801,22 +801,22 @@ Some notes about the system: The analysis of the SVD control applied to the Stewart platform is performed in the following sections:

    -
  • Section 3.1: The parameters of the Simscape model of the Stewart platform are defined
  • -
  • Section 3.2: The plant is identified from the Simscape model and the system coupling is shown
  • -
  • Section 3.3: The plant is first decoupled using the Jacobian
  • -
  • Section 3.4: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
  • -
  • Section 3.5: The decoupling is performed thanks to the SVD
  • -
  • Section 3.6: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
  • -
  • Section 3.7: The dynamics of the decoupled plants are shown
  • -
  • Section 3.8: A diagonal controller is defined to control the decoupled plant
  • -
  • Section 3.9: Finally, the closed loop system properties are studied
  • +
  • Section 3.1: The parameters of the Simscape model of the Stewart platform are defined
  • +
  • Section 3.2: The plant is identified from the Simscape model and the system coupling is shown
  • +
  • Section 3.3: The plant is first decoupled using the Jacobian
  • +
  • Section 3.4: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
  • +
  • Section 3.5: The decoupling is performed thanks to the SVD
  • +
  • Section 3.6: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
  • +
  • Section 3.7: The dynamics of the decoupled plants are shown
  • +
  • Section 3.8: A diagonal controller is defined to control the decoupled plant
  • +
  • Section 3.9: Finally, the closed loop system properties are studied
-
-

3.1 Simscape Model - Parameters

+
+

3.1 Simscape Model - Parameters

- +

open('drone_platform.slx');
@@ -864,14 +864,14 @@ Kc = tf(zeros(6));
 
-
+

stewart_simscape.png

Figure 6: General view of the Simscape Model

-
+

stewart_platform_details.png

Figure 7: Simscape model of the Stewart platform

@@ -879,15 +879,15 @@ Kc = tf(zeros(6));
-
-

3.2 Identification of the plant

+
+

3.2 Identification of the plant

- +

-The plant shown in Figure 8 is identified from the Simscape model. +The plant shown in Figure 8 is identified from the Simscape model.

@@ -903,7 +903,7 @@ The outputs are the 6 accelerations measured by the inertial unit.

-
+

stewart_platform_plant.png

Figure 8: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform

@@ -940,7 +940,7 @@ State-space model with 6 outputs, 12 inputs, and 24 states.

-The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 9. +The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 9.

@@ -948,7 +948,7 @@ One can easily see that the system is strongly coupled.

-
+

stewart_platform_coupled_plant.png

Figure 9: Magnitude of all 36 elements of the transfer function matrix \(\bm{G}\)

@@ -956,17 +956,17 @@ One can easily see that the system is strongly coupled.
-
-

3.3 Physical Decoupling using the Jacobian

+
+

3.3 Physical Decoupling using the Jacobian

- -Consider the control architecture shown in Figure 10. + +Consider the control architecture shown in Figure 10. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

-
+

plant_decouple_jacobian.png

Figure 10: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

@@ -990,11 +990,11 @@ Gx.InputName = {'Dwx',
-
-

3.4 Real Approximation of \(G\) at the decoupling frequency

+
+

3.4 Real Approximation of \(G\) at the decoupling frequency

- +

@@ -1174,11 +1174,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown

-
-

3.5 SVD Decoupling

+
+

3.5 SVD Decoupling

- +

@@ -1192,11 +1192,11 @@ First, the Singular Value Decomposition of \(H_1\) is performed:

-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 11. +The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 11.

-
+

plant_decouple_svd.png

Figure 11: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

@@ -1209,11 +1209,11 @@ The decoupled plant is then:
-
-

3.6 Verification of the decoupling using the “Gershgorin Radii”

+
+

3.6 Verification of the decoupling using the “Gershgorin Radii”

- +

@@ -1229,7 +1229,7 @@ This is computed over the following frequencies.

-
+

simscape_model_gershgorin_radii.png

Figure 12: Gershgorin Radii of the Coupled and Decoupled plants

@@ -1237,30 +1237,30 @@ This is computed over the following frequencies.
-
-

3.7 Obtained Decoupled Plants

+
+

3.7 Obtained Decoupled Plants

- +

-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 13. +The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 13.

-
+

simscape_model_decoupled_plant_svd.png

Figure 13: Decoupled Plant using SVD

-Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 14. +Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 14.

-
+

simscape_model_decoupled_plant_jacobian.png

Figure 14: Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

@@ -1268,27 +1268,15 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
-
-

3.8 Diagonal Controller

+
+

3.8 Diagonal Controller

- +

-The controller \(K_c\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\). -

- -
-
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
-C_g = 50; % DC Gain
-
-Kc = eye(6)*C_g/(s+wc);
-
-
- -

-The control diagram for the centralized control is shown in Figure 15. +The control diagram for the centralized control is shown in Figure 15.

@@ -1297,55 +1285,71 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied

-
+

centralized_control.png

Figure 15: Control Diagram for the Centralized control

-The feedback system is computed as shown below. -

-
-
G_cen = feedback(G, inv(J')*Kc, [7:12], [1:6]);
-
-
- -

-The SVD control architecture is shown in Figure 16. +The SVD control architecture is shown in Figure 16. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

-
+

svd_control.png

Figure 16: Control Diagram for the SVD control

+

-The feedback system is computed as shown below. +We choose the controller to be a low pass filter: +\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]

+ +

+\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\) +

+
-
G_svd = feedback(G, pinv(V')*Kc*pinv(U), [7:12], [1:6]);
+
wc = 2*pi*80;
+w0 = 2*pi*0.1;
+
+
+ +
+
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_cen = K_cen*Gx;
+G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
+
+
+ +
+
K_svd = diag(1./diag(abs(evalfr(Gd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_svd = K_svd*Gd;
+G_svd = feedback(G, pinv(V')*K_svd*pinv(U), [7:12], [1:6]);
 

-Let’s look at the Loop Gains. +The obtained diagonal elements of the loop gains are shown in Figure 17.

-
-
L_svd =
-
+ +
+

stewart_comp_loop_gain_diagonal.png +

+

Figure 17: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

-
-

3.9 Closed-Loop system Performances

+
+

3.9 Closed-Loop system Performances

- +

@@ -1376,14 +1380,14 @@ ans =

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 18.

-
+

stewart_platform_simscape_cl_transmissibility.png

-

Figure 17: Obtained Transmissibility

+

Figure 18: Obtained Transmissibility

@@ -1391,7 +1395,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well

Author: Dehaeze Thomas

-

Created: 2020-11-06 ven. 17:02

+

Created: 2020-11-06 ven. 18:00

diff --git a/index.org b/index.org index dec1674..2d0d0a6 100644 --- a/index.org +++ b/index.org @@ -1229,9 +1229,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of ** Diagonal Controller <> -The controller $K_c$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$. - -#+begin_src matlab +#+begin_src matlab :exports none :tangle no wc = 2*pi*0.1; % Crossover Frequency [rad/s] C_g = 50; % DC Gain @@ -1268,11 +1266,6 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied #+RESULTS: [[file:figs/centralized_control.png]] -The feedback system is computed as shown below. -#+begin_src matlab - G_cen = feedback(G, inv(J')*Kc, [7:12], [1:6]); -#+end_src - The SVD control architecture is shown in Figure [[fig:svd_control]]. The matrices $U$ and $V$ are used to decoupled the plant $G$. #+begin_src latex :file svd_control.pdf :tangle no :exports results @@ -1301,17 +1294,89 @@ The matrices $U$ and $V$ are used to decoupled the plant $G$. #+RESULTS: [[file:figs/svd_control.png]] -The feedback system is computed as shown below. -#+begin_src matlab - G_svd = feedback(G, pinv(V')*Kc*pinv(U), [7:12], [1:6]); -#+end_src -Let's look at the Loop Gains. +We choose the controller to be a low pass filter: +\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \] + +$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$ #+begin_src matlab - L_svd = + wc = 2*pi*80; + w0 = 2*pi*0.1; #+end_src +#+begin_src matlab + K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); + L_cen = K_cen*Gx; + G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]); +#+end_src + +#+begin_src matlab + K_svd = diag(1./diag(abs(evalfr(Gd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); + L_svd = K_svd*Gd; + G_svd = feedback(G, pinv(V')*K_svd*pinv(U), [7:12], [1:6]); +#+end_src + +The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]]. + +#+begin_src matlab :exports none + freqs = logspace(-1, 2, 1000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile([2, 1]); + hold on; + plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$'); + for i_in_out = 2:6 + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off'); + end + + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ... + 'DisplayName', '$L_{J}(i,i)$'); + for i_in_out = 2:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + legend('location', 'northwest'); + ylim([5e-2, 2e3]) + + % Phase + ax2 = nexttile; + hold on; + for i_in_out = 1:6 + set(gca,'ColorOrderIndex',1) + plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz')))); + end + set(gca,'ColorOrderIndex',2) + for i_in_out = 1:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180:90:360]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/stewart_comp_loop_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:stewart_comp_loop_gain_diagonal +#+caption: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one +#+RESULTS: +[[file:figs/stewart_comp_loop_gain_diagonal.png]] + ** Closed-Loop system Performances <> @@ -1389,7 +1454,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well linkaxes([ax1,ax2,ax3,ax4],'xy'); xlim([freqs(1), freqs(end)]); - ylim([1e-5, 1e2]); + ylim([1e-3, 1e2]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace