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/Root 169 0 R - /Info 1 0 R - /ID [<4302AE240377954BF0D27763CD000DD1> <4302AE240377954BF0D27763CD000DD1>] - /Size 170 ->> -startxref -70249 -%%EOF diff --git a/figs/stewart_platform_translations.png b/figs/stewart_platform_translations.png deleted file mode 100644 index 9a547b2..0000000 Binary files a/figs/stewart_platform_translations.png and /dev/null differ diff --git a/figs/stewart_platform_transmissibility.pdf b/figs/stewart_platform_transmissibility.pdf deleted file mode 100644 index 7d8562d..0000000 Binary files a/figs/stewart_platform_transmissibility.pdf and /dev/null differ diff --git a/figs/stewart_platform_transmissibility.png b/figs/stewart_platform_transmissibility.png deleted file mode 100644 index 1de411d..0000000 Binary files a/figs/stewart_platform_transmissibility.png and /dev/null differ diff --git a/index.html b/index.html index 18916af..61cb403 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + SVD Control @@ -35,56 +35,57 @@

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

@@ -92,8 +93,8 @@
-
-

1.2 Simscape Model - Parameters

+
+

1.2 Simscape Model - Parameters

open('gravimeter.slx')
@@ -124,8 +125,8 @@ g = 0; % Gravity [m/s2]
-
-

1.3 System Identification - Without Gravity

+
+

1.3 System Identification - Without Gravity

%% Name of the Simulink File
@@ -147,7 +148,7 @@ G.OutputName = {'Ax1',
-
+
 pole(G)
 ans =
       -0.000473481142385795 +      21.7596190728632i
@@ -172,7 +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

@@ -180,8 +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]
@@ -198,7 +199,7 @@ Gg.OutputName = {'Ax1', 
 We can now see that the system is unstable due to gravity.
 

-
+
 pole(Gg)
 ans =
           -10.9848275341252 +                     0i
@@ -210,7 +211,7 @@ ans =
 

 
 
-

+

 
open_loop_tf_g.png
 

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

@@ -218,12 +219,12 @@ ans =
 

 

 
-

-
1.5 Analytical Model

+

+
1.5 Analytical Model

 

 

-

-
1.5.1 Parameters

+

+
1.5.1 Parameters

 

 

 Bode options.
@@ -255,8 +256,8 @@ Frequency vector.
 

 

 
-

-
1.5.2 Generation of the State Space Model

+

+
1.5.2 Generation of the State Space Model

 

 

 Mass matrix
@@ -357,11 +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

@@ -369,8 +370,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
 

 

 
-

-
1.5.4 Analysis

+

+
1.5.4 Analysis

 

 

 
% figure
@@ -438,8 +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);
@@ -579,8 +580,8 @@ legend('Control OFF','D
 

 

 
-

-
1.5.6 Greshgorin radius

+

+
1.5.6 Greshgorin radius

 

 

 
system_dec_freq = freqresp(system_dec,w);
@@ -626,8 +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);
@@ -683,15 +684,15 @@ rot = PHI(:,11,11);
 

 

 
-

-
2 Gravimeter - Functions

+

+
2 Gravimeter - Functions

 

 

-

-
2.1 align

+

+
2.1 align

 

 

-
+
 

 
 

@@ -720,11 +721,11 @@ This Matlab function is accessible here.
 

 
 
-

-
2.2 pzmap_testCL

+

+
2.2 pzmap_testCL

 

 

-
+
 

 
 

@@ -773,15 +774,24 @@ 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.
 

 
+

+Some notes about the system:
+

+
    
+
  • 6 voice coils actuators are used to control the motion of the top platform.
    • 
+
  • the motion of the top platform is measured with a 6-axis inertial unit (3 acceleration + 3 angular accelerations)
    • 
+
  • the control objective is to isolate the top platform from vibrations coming from the bottom platform
    • 
+

 
-

+
+

 
SP_assembly.png
 

 
Figure 5: Stewart Platform CAD View

@@ -791,21 +801,22 @@ In this analysis, we wish to applied SVD control to the Stewart Platform shown i
 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 centralized plant is computed thanks to the Jacobian
    • 
-
  • Section 3.3: The identified Dynamics is shown
    • 
-
  • 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. The effectiveness of the decoupling is verified using the Gershorin Radii
    • 
-
  • Section 3.6: The dynamics of the decoupled plant is shown
    • 
-
  • Section 3.7: A diagonal controller is defined to control the decoupled plant
    • 
-
  • Section 3.8: 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');
@@ -813,7 +824,7 @@ The analysis of the SVD control applied to the Stewart platform is performed in
 

 
 

-Definition of spring parameters
+Definition of spring parameters:
 

 

 
kx = 0.5*1e3/3; % [N/m]
@@ -835,7 +846,7 @@ Gravity:
 

 
 

-We load the Jacobian (previously computed from the geometry).
+We load the Jacobian (previously computed from the geometry):
 

 

 
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
@@ -854,25 +865,45 @@ Kc = tf(zeros(6));
 

 

 
-

-
3.2 Identification of the plant

+

+
3.2 Identification of the plant

 

 

-
+
 

 
 

-The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
+The plant shown in Figure 6 is identified from the Simscape model.
 

+
+

+The inputs are:
+

+
    
+
  • \(D_w\) translation and rotation of the bottom platform (with respect to the center of mass of the top platform)
    • 
+
  • \(\tau\) the 6 forces applied by the voice coils
    • 
+

+
+

+The outputs are the 6 accelerations measured by the inertial unit.
+

+
+
+

+
stewart_platform_plant.png
+

+
Figure 6: 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

+

+
 

 
%% Name of the Simulink File
 mdl = 'drone_platform';
 
 %% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1; % Ground Motion
+io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1; % Actuator Forces
+io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
 
 G = linearize(mdl, io);
 G.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
@@ -895,19 +926,47 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
 
 
 

-The “centralized” plant \(\bm{G}_x\) is now computed (Figure 6).
+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 7.
+

+
+

+One can easily see that the system is strongly coupled.
 

 
 
-

-
centralized_control.png
+

+
stewart_platform_coupled_plant.png
 

-
Figure 6: Centralized control architecture

+
Figure 7: Magnitude of all 36 elements of the transfer function matrix \(\bm{G}\)

+

+

+

+
+

+
3.3 Physical Decoupling using the Jacobian

+

+

+
+Consider the control architecture shown in Figure 8.
+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 8: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

 

 
 

-Thanks to the Jacobian, we compute the transfer functions in the inertial frame (transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform).
+We define a new plant:
+\[ G_x(s) = G(s) J^{-T} \]
 

+
+

+\(G_x(s)\) correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
+

+
 

 
Gx = G*blkdiag(eye(6), inv(J'));
 Gx.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
@@ -917,38 +976,15 @@ Gx.InputName  = {'Dwx', 
 

 
-

-
3.3 Obtained Dynamics

-

-

-
-

-
-
-

-
stewart_platform_translations.png
-

-
Figure 7: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform

-

-
-
-

-
stewart_platform_rotations.png
-

-
Figure 8: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform

-

-

-

-
-

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

+

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

 

 

-
+
 

 
 

-Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_c(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
+Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
 

 

 
wc = 2*pi*30; % Decoupling frequency [rad/s]
@@ -1124,11 +1160,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown
 

 

 
-

-
3.5 Verification of the decoupling using the “Gershgorin Radii”

+

+
3.5 SVD Decoupling

 

 

-
+
 

 
 

@@ -1142,8 +1178,32 @@ First, the Singular Value Decomposition of \(H_1\) is performed:
 

 
 

-Then, the “Gershgorin Radii” is computed for the plant \(G_c(s)\) and the “SVD Decoupled Plant” \(G_d(s)\):
-\[ G_d(s) = U^T G_c(s) V \]
+The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 9.
+

+
+
+

+
plant_decouple_svd.png
+

+
Figure 9: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

+

+
+

+The decoupled plant is then:
+\[ G_{SVD}(s) = U^T G(s) V \]
+

+

+

+
+

+
3.6 Verification of the decoupling using the “Gershgorin Radii”

+

+

+
+

+
+

+The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\):
 

 
 

@@ -1154,94 +1214,55 @@ This is computed over the following frequencies.
 

 

 
-

-Gershgorin Radii for the coupled plant:
-

-

-
Gr_coupled = zeros(length(freqs), size(Gc,2));
 
-H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
-for out_i = 1:size(Gc,2)
-    Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
-end
-

-

-
-

-Gershgorin Radii for the decoupled plant using SVD:
-

-

-
Gd = U'*Gc*V;
-Gr_decoupled = zeros(length(freqs), size(Gd,2));
-
-H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
-for out_i = 1:size(Gd,2)
-    Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
-end
-

-

-
-

-Gershgorin Radii for the decoupled plant using the Jacobian:
-

-

-
Gj = Gc*inv(J');
-Gr_jacobian = zeros(length(freqs), size(Gj,2));
-
-H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
-
-for out_i = 1:size(Gj,2)
-    Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
-end
-

-

-
-
-

+

 
simscape_model_gershgorin_radii.png
 

-
Figure 9: Gershgorin Radii of the Coupled and Decoupled plants

+
Figure 10: Gershgorin Radii of the Coupled and Decoupled plants

 

 

 

 
-

-
3.6 Decoupled Plant

-

-

-
-

-
-

-Let’s see the bode plot of the decoupled plant \(G_d(s)\).
-\[ G_d(s) = U^T G_c(s) V \]
-

-
-
-

-
simscape_model_decoupled_plant_svd.png
-

-
Figure 10: Decoupled Plant using SVD

-

-
-
-

-
simscape_model_decoupled_plant_jacobian.png
-

-
Figure 11: Decoupled Plant using the Jacobian

-

-

-

-
-

-
3.7 Diagonal Controller

+

+
3.7 Obtained Decoupled Plants

 

 

-
+
 

 
 

-The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
+The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 11.
+

+
+
+

+
simscape_model_decoupled_plant_svd.png
+

+
Figure 11: 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 12.
+

+
+
+

+
simscape_model_decoupled_plant_jacobian.png
+

+
Figure 12: 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)

+

+

+

+
+

+
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 \(C_g\).
 

 
 

@@ -1253,7 +1274,7 @@ Kc = eye(6)*C_g/(s
 
 

-The control diagram for the centralized control is shown in Figure 6.
+The control diagram for the centralized control is shown in Figure 13.
 

 
 

@@ -1262,10 +1283,10 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
 

 
 
-

+

 
centralized_control.png
 

-
Figure 12: Control Diagram for the Centralized control

+
Figure 13: Control Diagram for the Centralized control

 

 
 

@@ -1277,14 +1298,14 @@ The feedback system is computed as shown below.
 

 
 

-The SVD control architecture is shown in Figure 13.
+The SVD control architecture is shown in Figure 14.
 The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
 

 
-

+

 
svd_control.png
 

-
Figure 13: Control Diagram for the SVD control

+
Figure 14: Control Diagram for the SVD control

 

 
 

@@ -1297,11 +1318,11 @@ The feedback system is computed as shown below.
 

 

 
-

-
3.8 Closed-Loop system Performances

-

+

+
3.9 Closed-Loop system Performances

+

 

-
+
 

 
 

@@ -1327,19 +1348,19 @@ ans =
 
 ans =
   logical
-   0
+   1
 

 
 
 

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 14.
+The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 15.
 

 
 
-

+

 
stewart_platform_simscape_cl_transmissibility.png
 

-
Figure 14: Obtained Transmissibility

+
Figure 15: Obtained Transmissibility

 

 

 

@@ -1347,7 +1368,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
 

 

 
Author: Dehaeze Thomas

-
Created: 2020-11-06 ven. 15:06

+
Created: 2020-11-06 ven. 16:58

 

 
 
diff --git a/index.org b/index.org
index 41627d7..3d19c48 100644
--- a/index.org
+++ b/index.org
@@ -693,17 +693,23 @@ This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]].
 
 In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure [[fig:SP_assembly]].
 
+Some notes about the system:
+- 6 voice coils actuators are used to control the motion of the top platform.
+- the motion of the top platform is measured with a 6-axis inertial unit (3 acceleration + 3 angular accelerations)
+- the control objective is to isolate the top platform from vibrations coming from the bottom platform
+
 #+name: fig:SP_assembly
 #+caption: Stewart Platform CAD View
 [[file:figs/SP_assembly.png]]
 
 The analysis of the SVD control applied to the Stewart platform is performed in the following sections:
 - Section [[sec:stewart_simscape]]: The parameters of the Simscape model of the Stewart platform are defined
-- Section [[sec:stewart_identification]]: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian
-- Section [[sec:stewart_dynamics]]: The identified Dynamics is shown
+- Section [[sec:stewart_identification]]: The plant is identified from the Simscape model and the system coupling is shown
+- Section [[sec:stewart_jacobian_decoupling]]: The plant is first decoupled using the Jacobian
 - Section [[sec:stewart_real_approx]]: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
-- Section [[sec:stewart_svd_decoupling]]: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii
-- Section [[sec:stewart_decoupled_plant]]: The dynamics of the decoupled plant is shown
+- Section [[sec:stewart_svd_decoupling]]: The decoupling is performed thanks to the SVD
+- Section [[sec:comp_decoupling]]: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
+- Section [[sec:stewart_decoupled_plant]]: The dynamics of the decoupled plants are shown
 - Section [[sec:stewart_diagonal_control]]: A diagonal controller is defined to control the decoupled plant
 - Section [[sec:stewart_closed_loop_results]]: Finally, the closed loop system properties are studied
 
@@ -765,7 +771,7 @@ First, the position of the "joints" (points of force application) are estimated
   open('drone_platform.slx');
 #+end_src
 
-Definition of spring parameters
+Definition of spring parameters:
 #+begin_src matlab
   kx = 0.5*1e3/3; % [N/m]
   ky = 0.5*1e3/3;
@@ -781,7 +787,7 @@ Gravity:
   g = 0;
 #+end_src
 
-We load the Jacobian (previously computed from the geometry).
+We load the Jacobian (previously computed from the geometry):
 #+begin_src matlab
   load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
 #+end_src
@@ -796,16 +802,44 @@ We initialize other parameters:
 ** Identification of the plant
 <>
 
-The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
+The plant shown in Figure [[fig:stewart_platform_plant]] is identified from the Simscape model.
+
+The inputs are:
+- $D_w$ translation and rotation of the bottom platform (with respect to the center of mass of the top platform)
+- $\tau$ the 6 forces applied by the voice coils
+
+The outputs are the 6 accelerations measured by the inertial unit.
+
+#+begin_src latex :file stewart_platform_plant.pdf :tangle no :exports results
+  \begin{tikzpicture}
+    \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
+    \node[above] at (G.north) {$\bm{G}$};
+
+    % Inputs of the controllers
+    \coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
+    \coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
+    % Connections and labels
+
+    \draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
+    \draw[<-] (inputu) -- ++(-0.8, 0) node[above right]{$\tau$};
+    \draw[->] (G.east) -- ++(0.8, 0)  node[above left]{$a$};
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:stewart_platform_plant
+#+caption: 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
+#+RESULTS:
+[[file:figs/stewart_platform_plant.png]]
+
 #+begin_src matlab
   %% Name of the Simulink File
   mdl = 'drone_platform';
 
   %% Input/Output definition
   clear io; io_i = 1;
-  io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1; % Ground Motion
+  io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1; % Actuator Forces
+  io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
 
   G = linearize(mdl, io);
   G.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
@@ -821,108 +855,93 @@ There are 24 states (6dof for the bottom platform + 6dof for the top platform).
 #+RESULTS:
 : State-space model with 6 outputs, 12 inputs, and 24 states.
 
-The "centralized" plant $\bm{G}_x$ is now computed (Figure [[fig:centralized_control]]).
+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 [[fig:stewart_platform_coupled_plant]].
 
-#+name: fig:centralized_control
-#+caption: Centralized control architecture
-[[file:figs/centralized_control.png]]
+One can easily see that the system is strongly coupled.
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 2, 1000);
+
+  figure;
+
+  % Magnitude
+  hold on;
+  for i_in = 1:6
+      for i_out = [1:i_in-1, i_in+1:6]
+          plot(freqs, abs(squeeze(freqresp(G(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+               'HandleVisibility', 'off');
+      end
+  end
+  plot(freqs, abs(squeeze(freqresp(G(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+       'DisplayName', '$G(i,j)\ i \neq j$');
+  set(gca,'ColorOrderIndex',1)
+  for i_in_out = 1:6
+    plot(freqs, abs(squeeze(freqresp(G(i_in_out, 6+i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G(%d,%d)$', i_in_out, i_in_out));
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  ylim([1e-2, 1e5]);
+  legend('location', 'northwest');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/stewart_platform_coupled_plant.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:stewart_platform_coupled_plant
+#+caption: Magnitude of all 36 elements of the transfer function matrix $\bm{G}$
+#+RESULTS:
+[[file:figs/stewart_platform_coupled_plant.png]]
+
+** Physical Decoupling using the Jacobian
+<>
+Consider the control architecture shown in Figure [[fig:plant_decouple_jacobian]].
+The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.
+
+#+begin_src latex :file plant_decouple_jacobian.pdf :tangle no :exports results
+  \begin{tikzpicture}
+    \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
+
+    % Inputs of the controllers
+    \coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
+    \coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
+
+    \node[block, left=0.6 of inputu] (J) {$J^{-T}$};
+
+    % Connections and labels
+    \draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
+    \draw[->] (G.east) -- ++( 0.8, 0)  node[above left]{$a$};
+    \draw[->] (J.east) -- (inputu)  node[above left]{$\tau$};
+    \draw[<-] (J.west) -- ++(-0.8, 0) node[above right]{$\mathcal{F}$};
+
+    \begin{scope}[on background layer]
+      \node[fit={(J.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=8pt] (Gx) {};
+      \node[below right] at (Gx.north west) {$\bm{G}_x$};
+    \end{scope}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:plant_decouple_jacobian
+#+caption: Decoupled plant $\bm{G}_x$ using the Jacobian matrix $J$
+#+RESULTS:
+[[file:figs/plant_decouple_jacobian.png]]
+
+We define a new plant:
+\[ G_x(s) = G(s) J^{-T} \]
+
+$G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
 
-Thanks to the Jacobian, we compute the transfer functions in the inertial frame (transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform).
 #+begin_src matlab
   Gx = G*blkdiag(eye(6), inv(J'));
   Gx.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
                    'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
 #+end_src
 
-** Obtained Dynamics
-<>
-
-#+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(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
-  plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
-  plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
-  hold off;
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
-  legend('location', 'southeast');
-
-  % Phase
-  ax2 = nexttile;
-  hold on;
-  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
-  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
-  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
-  hold off;
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
-  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
-  ylim([-180, 180]);
-  yticks([-360:90:360]);
-
-  linkaxes([ax1,ax2],'x');
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/stewart_platform_translations.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
-#+end_src
-
-#+name: fig:stewart_platform_translations
-#+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
-#+RESULTS:
-[[file:figs/stewart_platform_translations.png]]
-
-#+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(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
-  plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
-  plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
-  hold off;
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Magnitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
-  legend('location', 'southeast');
-
-  % Phase
-  ax2 = nexttile;
-  hold on;
-  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
-  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
-  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
-  hold off;
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
-  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
-  ylim([-180, 180]);
-  yticks([-360:90:360]);
-
-  linkaxes([ax1,ax2],'x');
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/stewart_platform_rotations.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
-#+end_src
-
-#+name: fig:stewart_platform_rotations
-#+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
-#+RESULTS:
-[[file:figs/stewart_platform_rotations.png]]
-
 ** Real Approximation of $G$ at the decoupling frequency
 <>
 
-Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
+Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
 #+begin_src matlab
   wc = 2*pi*30; % Decoupling frequency [rad/s]
 
@@ -968,7 +987,7 @@ This can be verified below where only the real value of $G(\omega_c)$ is shown
 | -162.0 | -237.0 | -237.0 | -162.0 | 398.9 |  398.9 |
 |  220.6 | -220.6 |  220.6 | -220.6 | 220.6 | -220.6 |
 
-** Verification of the decoupling using the "Gershgorin Radii"
+** SVD Decoupling
 <>
 
 First, the Singular Value Decomposition of $H_1$ is performed:
@@ -978,26 +997,61 @@ First, the Singular Value Decomposition of $H_1$ is performed:
   [U,S,V] = svd(H1);
 #+end_src
 
-Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
-\[ G_d(s) = U^T G_c(s) V \]
+The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure [[fig:plant_decouple_svd]].
+
+#+begin_src latex :file plant_decouple_svd.pdf :tangle no :exports results
+  \begin{tikzpicture}
+    \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
+
+    % Inputs of the controllers
+    \coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
+    \coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
+
+    \node[block, left=0.6 of inputu] (V) {$V$};
+    \node[block, right=0.6 of G.east] (U) {$U^T$};
+
+    % Connections and labels
+    \draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
+    \draw[->] (G.east) -- (U.west) node[above left]{$a$};
+    \draw[->] (U.east) -- ++( 0.8, 0) node[above left]{$y$};
+    \draw[->] (V.east) -- (inputu) node[above left]{$\tau$};
+    \draw[<-] (V.west) -- ++(-0.8, 0) node[above right]{$u$};
+
+    \begin{scope}[on background layer]
+      \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=8pt] (Gsvd) {};
+      \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
+    \end{scope}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:plant_decouple_svd
+#+caption: Decoupled plant $\bm{G}_{SVD}$ using the Singular Value Decomposition
+#+RESULTS:
+[[file:figs/plant_decouple_svd.png]]
+
+The decoupled plant is then:
+\[ G_{SVD}(s) = U^T G(s) V \]
+
+** Verification of the decoupling using the "Gershgorin Radii"
+<>
+
+The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
 
 This is computed over the following frequencies.
 #+begin_src matlab
   freqs = logspace(-2, 2, 1000); % [Hz]
 #+end_src
 
-Gershgorin Radii for the coupled plant:
-#+begin_src matlab
+#+begin_src matlab :exports none
+  % Gershgorin Radii for the coupled plant:
   Gr_coupled = zeros(length(freqs), size(Gc,2));
 
   H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
   for out_i = 1:size(Gc,2)
       Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
   end
-#+end_src
 
-Gershgorin Radii for the decoupled plant using SVD:
-#+begin_src matlab
+  % Gershgorin Radii for the decoupled plant using SVD:
   Gd = U'*Gc*V;
   Gr_decoupled = zeros(length(freqs), size(Gd,2));
 
@@ -1005,10 +1059,8 @@ Gershgorin Radii for the decoupled plant using SVD:
   for out_i = 1:size(Gd,2)
       Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
   end
-#+end_src
 
-Gershgorin Radii for the decoupled plant using the Jacobian:
-#+begin_src matlab
+  % Gershgorin Radii for the decoupled plant using the Jacobian:
   Gj = Gc*inv(J');
   Gr_jacobian = zeros(length(freqs), size(Gj,2));
 
@@ -1037,12 +1089,12 @@ Gershgorin Radii for the decoupled plant using the Jacobian:
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   hold off;
   xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
-  legend('location', 'northeast');
+  legend('location', 'northwest');
   ylim([1e-3, 1e3]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/simscape_model_gershgorin_radii.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
+  exportFig('figs/simscape_model_gershgorin_radii.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
 #+end_src
 
 #+name: fig:simscape_model_gershgorin_radii
@@ -1050,36 +1102,56 @@ Gershgorin Radii for the decoupled plant using the Jacobian:
 #+RESULTS:
 [[file:figs/simscape_model_gershgorin_radii.png]]
 
-** Decoupled Plant
+** Obtained Decoupled Plants
 <>
 
-Let's see the bode plot of the decoupled plant $G_d(s)$.
-\[ G_d(s) = U^T G_c(s) V \]
+The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
 
-#+begin_src matlab :exports results
+#+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;
+  for i_in = 1:6
+      for i_out = [1:i_in-1, i_in+1:6]
+          plot(freqs, abs(squeeze(freqresp(Gd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+               'HandleVisibility', 'off');
+      end
+  end
+  plot(freqs, abs(squeeze(freqresp(Gd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
+       'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
+  set(gca,'ColorOrderIndex',1)
   for ch_i = 1:6
     plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
-         'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
-  end
-  for in_i = 1:5
-    for out_i = in_i+1:6
-      plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
-           'HandleVisibility', 'off');
-    end
+         'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
   end
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Magnitude'); xlabel('Frequency [Hz]');
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
   legend('location', 'northwest');
-  ylim([1e-3, 1e4]);
+  ylim([1e-1, 1e5])
+
+  % Phase
+  ax2 = nexttile;
+  hold on;
+  for ch_i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(ch_i, ch_i), 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/simscape_model_decoupled_plant_svd.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
+  exportFig('figs/simscape_model_decoupled_plant_svd.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
 #+end_src
 
 #+name: fig:simscape_model_decoupled_plant_svd
@@ -1087,42 +1159,69 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
 #+RESULTS:
 [[file:figs/simscape_model_decoupled_plant_svd.png]]
 
-#+begin_src matlab :exports results
+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 [[fig:simscape_model_decoupled_plant_jacobian]].
+
+#+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;
-  for ch_i = 1:6
-    plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
-         'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
-  end
-  for in_i = 1:5
-    for out_i = in_i+1:6
-      plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
-           'HandleVisibility', 'off');
-    end
+  for i_in = 1:6
+      for i_out = [1:i_in-1, i_in+1:6]
+          plot(freqs, abs(squeeze(freqresp(Gx(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+               'HandleVisibility', 'off');
+      end
   end
+  plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
+       'DisplayName', '$G_x(i,j),\ i \neq j$');
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
+  plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
+  plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
+  plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
+  plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
+  plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Magnitude'); xlabel('Frequency [Hz]');
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
   legend('location', 'northwest');
-  ylim([1e-1, 1e6]);
-  set(gca, 'YMinorTick', 'on');
+  ylim([1e-2, 2e6])
+
+  % Phase
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+  ylim([0, 180]);
+  yticks([0:45:360]);
+
+  linkaxes([ax1,ax2],'x');
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
+  exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
 #+end_src
 
 #+name: fig:simscape_model_decoupled_plant_jacobian
-#+caption: Decoupled Plant using the Jacobian
+#+caption: 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)
 #+RESULTS:
 [[file:figs/simscape_model_decoupled_plant_jacobian.png]]
 
 ** Diagonal Controller
 <>
 
-The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
+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
   wc = 2*pi*0.1; % Crossover Frequency [rad/s]
@@ -1138,7 +1237,8 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
 
 #+begin_src latex :file centralized_control.pdf :tangle no :exports results
   \begin{tikzpicture}
-    \node[block={2cm}{1.5cm}] (G) {$G$};
+    \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
+    \node[above] at (G.north) {$\bm{G}$};
     \node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
     \node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$};
 
@@ -1218,7 +1318,7 @@ Let's first verify the stability of the closed-loop systems:
 #+RESULTS:
 : ans =
 :   logical
-:    0
+:    1
 
 The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
 
@@ -1233,21 +1333,16 @@ The obtained transmissibility in Open-loop, for the centralized control as well
   plot(freqs, abs(squeeze(freqresp(G(    'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
   plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
   plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(G(    'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
+  plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
+  plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
   legend('location', 'southwest');
 
-  % ax2 = nexttile;
-  % hold on;
-  % plot(freqs, abs(squeeze(freqresp(G(    'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
-  % plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
-  % plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
-  % hold off;
-  % set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  % ylabel('Transmissibility - $D_y/D_{w,y}$');  xlabel('Frequency [Hz]');
-
-  ax3 = nexttile;
+  ax2 = nexttile;
   hold on;
   plot(freqs, abs(squeeze(freqresp(G(    'Az', 'Dwz')/s^2, freqs, 'Hz'))));
   plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
@@ -1256,25 +1351,20 @@ The obtained transmissibility in Open-loop, for the centralized control as well
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
 
-  ax4 = nexttile;
+  ax3 = nexttile;
   hold on;
   plot(freqs, abs(squeeze(freqresp(G(    'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
   plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
   plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(G(    'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$');  xlabel('Frequency [Hz]');
 
-  % ax5 = nexttile;
-  % hold on;
-  % plot(freqs, abs(squeeze(freqresp(G(    'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
-  % plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
-  % plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
-  % hold off;
-  % set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  % ylabel('Transmissibility - $R_y/R_{w,y}$');  xlabel('Frequency [Hz]');
-
-  ax6 = nexttile;
+  ax4 = nexttile;
   hold on;
   plot(freqs, abs(squeeze(freqresp(G(    'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
   plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
@@ -1283,7 +1373,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('$R_z/R_{w,z}$');  xlabel('Frequency [Hz]');
 
-  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
+  linkaxes([ax1,ax2,ax3,ax4],'xy');
   xlim([freqs(1), freqs(end)]);
   ylim([1e-5, 1e2]);
 #+end_src