Add physical interpretation of coupling

figs/detail_control_coupled_plant_bode.pdf

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nass-control.org

@@ -366,6 +366,150 @@ exportFig('figs/detail_control_hinf_filters_results_mixed_sensitivity.pdf', 'wid
 #+RESULTS:
 [[file:figs/detail_control_hinf_filters_results_mixed_sensitivity.png]]
 
+*** Analytical formulas for test model
+
+#+begin_src matlab
+%% Analytical Formula for the Jacobian Matrix
+% Create symbolic variables for all parameters
+syms l h la ha m I real
+syms c1 c2 c3 real
+syms k1 k2 k3 real
+
+% Unit vectors of the actuators (symbolic)
+s1 = [1; 0];  % Actuator 1 direction (horizontal)
+s2 = [0; 1];  % Actuator 2 direction (vertical)
+s3 = [0; 1];  % Actuator 3 direction (vertical)
+
+% Location of the joints with respect to the center of mass (symbolic)
+Mb1 = [-l/2; -ha];    % Joint 1 position vector
+Mb2 = [-la;  -h/2];   % Joint 2 position vector
+Mb3 = [ la;  -h/2];   % Joint 3 position vector
+
+% Jacobian matrix (Center of Mass)
+J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
+         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
+         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
+
+% Display the symbolic Jacobian matrix
+disp('Symbolic Jacobian Matrix (J_CoM):');
+disp(J_CoM);
+
+% Jacobian at the Center of stiffness {K}
+Mb1 = [-l/2; 0];
+Mb2 = [-la; -h/2+ha];
+Mb3 = [ la; -h/2+ha];
+
+J_CoK = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
+         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
+         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
+
+% Display the symbolic Jacobian matrix
+disp('Symbolic Jacobian Matrix (J_CoK):');
+disp(J_CoK);
+#+end_src
+
+#+begin_src matlab
+%% Analytical Formula for the Modal Decoupling
+syms l h la ha m I k c s real
+syms omega1 omega2 omega3 real % Natural frequencies
+
+% Unit vectors of the actuators
+s1 = [1; 0];  % Actuator 1 direction (horizontal)
+s2 = [0; 1];  % Actuator 2 direction (vertical)
+s3 = [0; 1];  % Actuator 3 direction (vertical)
+
+% Location of the joints with respect to the center of mass (symbolic)
+Mb1 = [-l/2; -ha];    % Joint 1 position vector
+Mb2 = [-la; -h/2];    % Joint 2 position vector
+Mb3 = [la; -h/2];     % Joint 3 position vector
+
+% Calculate the Jacobian matrix (Center of Mass) symbolically
+J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
+         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
+         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
+
+disp('Symbolic Jacobian Matrix (J_CoM):');
+disp(J_CoM);
+
+% Define system matrices
+M = diag([m, m, I]);
+K_struts = diag([k, k, k]);
+C_struts = diag([c, c, c]);
+
+% Transform stiffness and damping to Cartesian space
+K = J_CoM' * K_struts * J_CoM;
+C = J_CoM' * C_struts * J_CoM;
+
+disp('Mass Matrix (M):');
+disp(M);
+
+disp('Stiffness Matrix (K):');
+disp(K);
+
+disp('Damping Matrix (C):');
+disp(C);
+
+% Define the plant in the frame of the struts
+% G_L = J_CoM * inv(M*s^2 + C*s + K) * J_CoM'
+D_cart = M*s^2 + C*s + K; % Denominator in Cartesian space
+disp('Dynamic Matrix in Cartesian Space (M*s^2 + C*s + K):');
+disp(D_cart);
+
+% Modal Decomposition
+% Calculate the eigenvalues and eigenvectors of M\K
+% For a symbolic approach, we'll use the general form of eigenvectors
+% [V,D] = eig(M\K)
+
+% Instead of direct symbolic eigendecomposition (which is complex),
+% we'll use known properties of modal analysis for analytical expressions
+
+% First, calculate M\K (inverse mass matrix times stiffness matrix)
+MK = simplify(M\K);
+disp('M\K Matrix:');
+disp(MK);
+
+% For a mechanical system with 3 DOF, we expect 3 eigenmodes
+% Let's define symbolic eigenvectors in a general form
+% According to vibration theory, the eigenvectors should be orthogonal with respect to M
+
+% Define symbolic eigenvectors
+V = sym('v', [3, 3]);
+
+% Define the symbolic eigenvalues (squared natural frequencies)
+D = diag([omega1^2, omega2^2, omega3^2]);
+
+% The eigenvectors should satisfy the equation (M\K)*V = V*D
+% This is equivalent to K*V = M*V*D
+% We can derive this symbolically, but it's complex for 3D systems
+
+% For an analytical approach, we can use physics to guide us
+% For this system, we expect modes corresponding to:
+% 1. Horizontal translation
+% 2. Vertical translation
+% 3. Rotation
+
+% Calculate modal mass matrix (mu = V'*M*V)
+mu = simplify(V' * M * V);
+disp('Modal Mass Matrix (mu):');
+disp(mu);
+
+% Modal output matrix
+Cm = simplify(J_CoM * V);
+disp('Modal Output Matrix (Cm):');
+disp(Cm);
+
+% Modal input matrix
+Bm = simplify(inv(mu) * V' * J_CoM');
+disp('Modal Input Matrix (Bm):');
+disp(Bm);
+
+% Plant in the modal space
+% For a fully decoupled system, Gm should be diagonal
+Gm = simplify(inv(Cm) * J_CoM * inv(D_cart) * J_CoM' * inv(Bm'));
+disp('Plant in Modal Space (Gm):');
+disp(Gm);
+#+end_src
+
 ** DONE [#A] Fix the outline
 CLOSED: [2025-04-03 Thu 12:01]
 
@@ -1828,33 +1972,32 @@ From [[cite:&thayer02_six_axis_vibrat_isolat_system]]:
 Experimental closed-loop control results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies.
 #+end_quote
 
-- [ ] Review of [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Decoupling Strategies][Decoupling Strategies]] for stewart platforms
+- [X] Review of [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Decoupling Strategies][Decoupling Strategies]] for stewart platforms
 - [ ] Add some citations about different methods
+- [ ] Maybe transform table into text
 
 #+name: tab:detail_control_decoupling_review
 #+caption: Litterature review about decoupling strategy for Stewart platform control
 #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccc
 #+attr_latex: :center t :booktabs t :font \scriptsize
 | *Actuators*   | *Sensors*                                          | *Control*                                                                             | *Reference*                                                                                                            |
-|---------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
+|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
 | APA           | Eddy current displacement                          | *Decentralized* (struts) PI + LPF control                                             | [[cite:&furutani04_nanom_cuttin_machin_using_stewar]]                                                                      |
 | PZT Piezo     | Strain Gauge                                       | Decentralized position feedback                                                       | [[cite:&du14_piezo_actuat_high_precis_flexib]]                                                                             |
-|---------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
+|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
 | Voice Coil    | Force                                              | *Cartesian frame* decoupling                                                          | [[cite:&obrien98_lesson]]                                                                                                  |
 | Voice Coil    | Force                                              | Cartesian Frame, Jacobians, IFF                                                       | [[cite:&mcinroy99_dynam;&mcinroy99_precis_fault_toler_point_using_stewar_platf;&mcinroy00_desig_contr_flexur_joint_hexap]] |
 | Hydraulic     | LVDT                                               | Decentralized (strut) vs Centralized (cartesian)                                      | [[cite:&kim00_robus_track_contr_desig_dof_paral_manip]]                                                                    |
 | Voice Coil    | Accelerometer (collocated), ext. Rx/Ry sensors     | Cartesian acceleration feedback (isolation) + 2DoF pointing control (external sensor) | [[cite:&li01_simul_vibrat_isolat_point_contr]]                                                                             |
 | Voice Coil    | Accelerometer in each leg                          | Centralized Vibration Control, PI, Skyhook                                            | [[cite:&abbas14_vibrat_stewar_platf]]                                                                                      |
-|---------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
+|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
 | Voice Coil    | Geophone + Eddy Current (Struts, collocated)       | Decentralized (Sky Hook) + Centralized (*modal*) Control                              | [[cite:&pu11_six_degree_of_freed_activ]]                                                                                   |
 | Piezoelectric | Force, Position                                    | Vibration isolation, Model-Based, *Modal control*: 6x PI controllers                  | [[cite:&yang19_dynam_model_decoup_contr_flexib]]                                                                           |
-|---------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
+|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
 | PZT           | Geophone (struts)                                  | *H-Infinity* and mu-synthesis                                                         | [[cite:&lei08_multi_objec_robus_activ_vibrat]]                                                                             |
 | Voice Coil    | Force sensors (struts) + accelerometer (cartesian) | Decentralized Force Feedback + Centralized H2 control based on accelerometers         | [[cite:&xie17_model_contr_hybrid_passiv_activ]]                                                                            |
 | Voice Coil    | Accelerometers                                     | MIMO H-Infinity, active damping                                                       | [[cite:&jiao18_dynam_model_exper_analy_stewar]]                                                                            |
 
-Assumptions:
-- parallel manipulators
 
 The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators.
 
@@ -1893,67 +2036,14 @@ freqs = logspace(0, 3, 1000);
 ** Test Model
 <<ssec:detail_control_decoupling_comp_model>>
 
-- a test model will be used to compare all the decoupling strategies
+- Instead of comparing the decoupling strategies using the Stewart platform, a similar yet much simpler parallel manipulator is used instead
 - to render the analysis simpler, the system of Figure ref:fig:detail_control_model_test_decoupling_detail is used
 - It has 3DoF, and has 3 parallels struts whose model is shown in Figure ref:fig:detail_control_strut_model
 - It is quite similar to the Stewart platform (parallel architecture, as many struts as DoF)
 
-- [ ] Write the equation of motion
+Two frames are defined:
-  - Write the equation of motion at the center of mass
+- $\{M\}$ with origin $O_M$ at the Center of mass of the solid body
-  - mass + Inertia, stiffness, damping, actuator forces (mapped using Jacobian)
+- $\{K\}$ with origin $O_K$ at the Center of mass of the parallel manipulator
-  - Write it in matrix form
-- [X] Show the used parameters (table ?)
-
-\begin{equation}
-  \bm{M}_{\{M\}} = \begin{bmatrix}
-    m & 0 & 0 \\
-    0 & m & 0 \\
-    0 & 0 & I
-  \end{bmatrix}
-\end{equation}
-
-\begin{equation}
-  \bm{\mathcal{K}} = \begin{bmatrix}
-    k & 0 & 0 \\
-    0 & k & 0 \\
-    0 & 0 & k
-  \end{bmatrix}, \quad \bm{\mathcal{C}} = \begin{bmatrix}
-    c & 0 & 0 \\
-    0 & c & 0 \\
-    0 & 0 & c
-  \end{bmatrix}
-\end{equation}
-
-\begin{equation}
-  \bm{J}_{\{M\}} = \begin{bmatrix}
-    1 & 0 &  h_a \\
-    0 & 1 & -l_a \\
-    0 & 1 &  l_a \\
-  \end{bmatrix}
-\end{equation}
-
-Recall the Jacobian relationships:
-Let's link forces and torques applied at the CoM
-
-\begin{equation}
-M_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}} + \bm{J}_{\{M\}}^t \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}} + \bm{J}_{\{M\}}^t \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}} = \bm{\mathcal{F}}_{\{M\}}
-\end{equation}
-
-with $\bm{\mathcal{X}}_{\{M\}}$ the two translation and one rotation expressed with respect to the center of mass.
-$\bm{\mathcal{F}}_{\{M\}}$ forces and torque applied at the center of mass.
-
-\begin{equation}
-  \bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix}
-    x \\
-    y \\
-    R_z
-  \end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix}
-    F_x \\
-    F_y \\
-    M_z
-  \end{bmatrix}
-\end{equation}
-
 
 #+name: fig:detail_control_model_test_decoupling_detail
 #+caption: 3DoF model used to study decoupling strategies
@@ -1973,21 +2063,71 @@ $\bm{\mathcal{F}}_{\{M\}}$ forces and torque applied at the center of mass.
 #+end_subfigure
 #+end_figure
 
-$\tau_i$  are the actuator forces
+First, the equation of motion are derived.
-$\mathcal{L}_i$ are the relative displacement of the struts
+Expressing the second law of Newton on the suspended mass, expressed at its center of mass gives
 
-Two frames are defined:
+\begin{equation}
-- $\{M\}$ with origin $O_M$ at the Center of mass of the solid body
+  M_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t)
-- $\{K\}$ with origin $O_K$ at the Center of mass of the parallel manipulator
+\end{equation}
+
+with $\bm{\mathcal{X}}_{\{M\}}$ the two translation and one rotation expressed with respect to the center of mass and $\bm{\mathcal{F}}_{\{M\}}$ forces and torque applied at the center of mass.
+
+\begin{equation}
+  \bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix}
+    x \\
+    y \\
+    R_z
+  \end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix}
+    F_x \\
+    F_y \\
+    M_z
+  \end{bmatrix}
+\end{equation}
+
+In order to map the spring, damping and actuator forces to XY forces and Z torque expressed at the center of mass, the Jacobian matrix $\bm{J}_{\{M\}}$ is used.
+
+\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM}
+  \bm{J}_{\{M\}} = \begin{bmatrix}
+    1 & 0 &  h_a \\
+    0 & 1 & -l_a \\
+    0 & 1 &  l_a \\
+  \end{bmatrix}
+\end{equation}
+
+Then, the equation of motion linking the actuator forces $\tau$ to the motion of the mass $\bm{\mathcal{X}}_{\{M\}}$ is obtained.
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_cartesian}
+  M_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^t \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^t \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^t \bm{\tau}(t)
+\end{equation}
+
+Matrices representing the payload inertia as well as the actuator stiffness and damping are shown in
+
+\begin{equation}
+  \bm{M}_{\{M\}} = \begin{bmatrix}
+    m & 0 & 0 \\
+    0 & m & 0 \\
+    0 & 0 & I
+  \end{bmatrix}, \quad
+  \bm{\mathcal{K}} = \begin{bmatrix}
+    k & 0 & 0 \\
+    0 & k & 0 \\
+    0 & 0 & k
+  \end{bmatrix}, \quad
+  \bm{\mathcal{C}} = \begin{bmatrix}
+    c & 0 & 0 \\
+    0 & c & 0 \\
+    0 & 0 & c
+  \end{bmatrix}
+\end{equation}
+
+Parameters used for the following analysis are summarized in table ref:tab:detail_control_decoupling_test_model_params.
 
 #+name: tab:detail_control_decoupling_test_model_params
 #+caption: Model parameters
 #+attr_latex: :environment tabularx :width 0.9\linewidth :align cXc
 #+attr_latex: :center t :booktabs t :font \scriptsize
 | *Parameter* | *Description*              | *Value*           |
-|-------------+---------------------------------------------------+-------------------|
+|-------------+----------------------------+-------------------|
-| $b_i$       | Location of the joints on the top platform        | $b_1 = $          |
-| $\hat{s}_i$ | Unit vector corresponding to the struts direction |                   |
 | $l_a$       |                            | $0.5\,m$          |
 | $h_a$       |                            | $0.2\,m$          |
 | $k$         | Actuator stiffness         | $10\,N/\mu m$     |
@@ -1995,149 +2135,12 @@ Two frames are defined:
 | $m$         | Payload mass               | $40\,\text{kg}$   |
 | $I$         | Payload rotational inertia | $5\,\text{kg}m^2$ |
 
-#+begin_src matlab
+** Control in the frame of the struts
-%% Analytical Formula for the Jacobian Matrix
-% Create symbolic variables for all parameters
-syms l h la ha m I real
-syms c1 c2 c3 real
-syms k1 k2 k3 real
 
-% Unit vectors of the actuators (symbolic)
+Let's first study the obtained dynamics in the frame of the struts.
-s1 = [1; 0];  % Actuator 1 direction (horizontal)
+The equation of motion linking actuator forces $\bm{\mathcal{\tau}}$ to strut relative motion $\bm{\mathcal{L}}$ is obtained from eqref:eq:detail_control_decoupling_plant_cartesian by mapping the cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix $\bm{J}_{\{M\}}$ eqref:eq:detail_control_decoupling_jacobian_CoM .
-s2 = [0; 1];  % Actuator 2 direction (vertical)
-s3 = [0; 1];  % Actuator 3 direction (vertical)
 
-% Location of the joints with respect to the center of mass (symbolic)
+The transfer function from $\bm{\mathcal{\tau}}$ to $\bm{\mathcal{L}}$ is shown in equation eqref:eq:detail_control_decoupling_plant_decentralized.
-Mb1 = [-l/2; -ha];    % Joint 1 position vector
-Mb2 = [-la;  -h/2];   % Joint 2 position vector
-Mb3 = [ la;  -h/2];   % Joint 3 position vector
-
-% Jacobian matrix (Center of Mass)
-J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
-         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
-         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
-
-% Display the symbolic Jacobian matrix
-disp('Symbolic Jacobian Matrix (J_CoM):');
-disp(J_CoM);
-
-% Jacobian at the Center of stiffness {K}
-Mb1 = [-l/2; 0];
-Mb2 = [-la; -h/2+ha];
-Mb3 = [ la; -h/2+ha];
-
-J_CoK = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
-         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
-         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
-
-% Display the symbolic Jacobian matrix
-disp('Symbolic Jacobian Matrix (J_CoK):');
-disp(J_CoK);
-#+end_src
-
-#+begin_src matlab
-%% Analytical Formula for the Modal Decoupling
-syms l h la ha m I k c s real
-syms omega1 omega2 omega3 real % Natural frequencies
-
-% Unit vectors of the actuators
-s1 = [1; 0];  % Actuator 1 direction (horizontal)
-s2 = [0; 1];  % Actuator 2 direction (vertical)
-s3 = [0; 1];  % Actuator 3 direction (vertical)
-
-% Location of the joints with respect to the center of mass (symbolic)
-Mb1 = [-l/2; -ha];    % Joint 1 position vector
-Mb2 = [-la; -h/2];    % Joint 2 position vector
-Mb3 = [la; -h/2];     % Joint 3 position vector
-
-% Calculate the Jacobian matrix (Center of Mass) symbolically
-J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
-         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
-         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
-
-disp('Symbolic Jacobian Matrix (J_CoM):');
-disp(J_CoM);
-
-% Define system matrices
-M = diag([m, m, I]);
-K_struts = diag([k, k, k]);
-C_struts = diag([c, c, c]);
-
-% Transform stiffness and damping to Cartesian space
-K = J_CoM' * K_struts * J_CoM;
-C = J_CoM' * C_struts * J_CoM;
-
-disp('Mass Matrix (M):');
-disp(M);
-
-disp('Stiffness Matrix (K):');
-disp(K);
-
-disp('Damping Matrix (C):');
-disp(C);
-
-% Define the plant in the frame of the struts
-% G_L = J_CoM * inv(M*s^2 + C*s + K) * J_CoM'
-D_cart = M*s^2 + C*s + K; % Denominator in Cartesian space
-disp('Dynamic Matrix in Cartesian Space (M*s^2 + C*s + K):');
-disp(D_cart);
-
-% Modal Decomposition
-% Calculate the eigenvalues and eigenvectors of M\K
-% For a symbolic approach, we'll use the general form of eigenvectors
-% [V,D] = eig(M\K)
-
-% Instead of direct symbolic eigendecomposition (which is complex),
-% we'll use known properties of modal analysis for analytical expressions
-
-% First, calculate M\K (inverse mass matrix times stiffness matrix)
-MK = simplify(M\K);
-disp('M\K Matrix:');
-disp(MK);
-
-% For a mechanical system with 3 DOF, we expect 3 eigenmodes
-% Let's define symbolic eigenvectors in a general form
-% According to vibration theory, the eigenvectors should be orthogonal with respect to M
-
-% Define symbolic eigenvectors
-V = sym('v', [3, 3]);
-
-% Define the symbolic eigenvalues (squared natural frequencies)
-D = diag([omega1^2, omega2^2, omega3^2]);
-
-% The eigenvectors should satisfy the equation (M\K)*V = V*D
-% This is equivalent to K*V = M*V*D
-% We can derive this symbolically, but it's complex for 3D systems
-
-% For an analytical approach, we can use physics to guide us
-% For this system, we expect modes corresponding to:
-% 1. Horizontal translation
-% 2. Vertical translation
-% 3. Rotation
-
-% Calculate modal mass matrix (mu = V'*M*V)
-mu = simplify(V' * M * V);
-disp('Modal Mass Matrix (mu):');
-disp(mu);
-
-% Modal output matrix
-Cm = simplify(J_CoM * V);
-disp('Modal Output Matrix (Cm):');
-disp(Cm);
-
-% Modal input matrix
-Bm = simplify(inv(mu) * V' * J_CoM');
-disp('Modal Input Matrix (Bm):');
-disp(Bm);
-
-% Plant in the modal space
-% For a fully decoupled system, Gm should be diagonal
-Gm = simplify(inv(Cm) * J_CoM * inv(D_cart) * J_CoM' * inv(Bm'));
-disp('Plant in Modal Space (Gm):');
-disp(Gm);
-#+end_src
-
-** Decentralized Plant / Control in the frame of the struts
 
 #+begin_src latex :file detail_control_decoupling_control_struts.pdf
 \begin{tikzpicture}
@@ -2150,18 +2153,20 @@ disp(Gm);
 #+end_src
 
 #+RESULTS:
-[[file:figs/detail_control_jacobian_decoupling_arch.png]]
+[[file:figs/detail_control_decoupling_control_struts.png]]
 
-\begin{equation}
+\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized}
   \frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-t} M_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1}
 \end{equation}
 
-At low frequency the plant converges to a diagonal constant matrix whose diagonal elements are linked to the actuator stiffnesses.
+At low frequency the plant converges to a diagonal constant matrix whose diagonal elements are linked to the actuator stiffnesses eqref:eq:detail_control_decoupling_plant_decentralized_low_freq.
 
-\begin{equation}
+\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq}
   \bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}}
 \end{equation}
 
+At high frequency, the plant converges to the mass matrix mapped in the frame of the struts, which is in general highly non-diagonal.
+
 #+begin_src matlab
 %% Compute Equation of motion
 l = 1; h=2;
@@ -2214,6 +2219,8 @@ G_L = J_CoM*inv(M*s^2 + C*s + K)*J_CoM';
 #+end_src
 
 The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is shown in Figure ref:fig:detail_control_coupled_plant_bode.
+This confirms that at low frequency (below the first suspension mode), the plant is well decoupled.
+Depending on the symmetry in the system, some diagonal elements may be equal (such as for struts 2 and 3 in this example).
 
 #+begin_src matlab :exports none
 figure;
@@ -2253,35 +2260,35 @@ exportFig('figs/detail_control_coupled_plant_bode.pdf', 'width', 'full', 'height
 #+RESULTS:
 [[file:figs/detail_control_coupled_plant_bode.png]]
 
+
 ** Jacobian Decoupling
 <<ssec:detail_control_comp_jacobian>>
 **** Jacobian Matrix
 
-The Jacobian matrix can be used to:
+As already explained, the Jacobian matrix can be used to both convert strut velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$ and Convert actuators forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$ eqref:eq:detail_control_decoupling_jacobian.
-- Convert joints velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$:
-  \[ \dot{\bm{\mathcal{X}}}_{\{O\}} = J_{\{O\}} \dot{\bm{\mathcal{L}}} \]
-- Convert actuators forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$:
-  \[ \bm{\mathcal{F}}_{\{O\}} = J_{\{O\}}^T \bm{\tau} \]
-with $\{O\}$ any chosen frame.
 
-By wisely choosing frame $\{O\}$, we can obtain nice decoupling for plant:
+\begin{subequations}\label{eq:detail_control_decoupling_jacobian}
-\begin{equation}
+  \begin{align}
-  \bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}
+    \dot{\bm{\mathcal{X}}}_{\{O\}} &= J_{\{O\}} \dot{\bm{\mathcal{L}}}, \quad \dot{\bm{\mathcal{L}}} = J_{\{O\}}^{-1} \dot{\bm{\mathcal{X}}}_{\{O\}} \\
-\end{equation}
+    \bm{\mathcal{F}}_{\{O\}}       &= J_{\{O\}}^t \bm{\tau},            \quad \bm{\tau} = J_{\{O\}}^{-t} \bm{\mathcal{F}}_{\{O\}}
+  \end{align}
+\end{subequations}
 
-The obtained plan corresponds to forces/torques applied on origin of frame $\{O\}$ to the translation/rotation of the payload expressed in frame $\{O\}$.
+The obtained plan (Figure ref:fig:detail_control_jacobian_decoupling_arch) has inputs and outputs that have physical meaning:
+- $\bm{\mathcal{F}}_{\{O\}}$ are forces/torques applied on the payload at the origin of frame $\{O\}$
+- $\bm{\mathcal{X}}_{\{O\}}$ are translations/rotation of the payload expressed in frame $\{O\}$
 
 #+begin_src latex :file detail_control_decoupling_control_jacobian.pdf
 \begin{tikzpicture}
-  \node[block] (G) {$\bm{G}$};
+  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
   \node[block, left=0.6 of G] (Jt) {$J_{\{O\}}^{-T}$};
   \node[block, right=0.6 of G] (Ja) {$J_{\{O\}}^{-1}$};
 
   % Connections and labels
-  \draw[<-] (Jt.west) -- ++(-1.2, 0) node[above right]{$\bm{\mathcal{F}}_{\{O\}}$};
+  \draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{O\}}$};
   \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
   \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
-  \draw[->] (Ja.east) -- ++( 1.2, 0)  node[above left]{$\bm{\mathcal{X}}_{\{O\}}$};
+  \draw[->] (Ja.east) -- ++( 1.4, 0)  node[above left]{$\bm{\mathcal{X}}_{\{O\}}$};
 
   \begin{scope}[on background layer]
     \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
@@ -2293,40 +2300,22 @@ The obtained plan corresponds to forces/torques applied on origin of frame $\{O\
 #+name: fig:detail_control_jacobian_decoupling_arch
 #+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{O\}}$ to $\bm{\mathcal{X}}_{\{O\}}$
 #+RESULTS:
-[[file:figs/detail_control_jacobian_decoupling_arch.png]]
+[[file:figs/detail_control_decoupling_control_jacobian.png]]
 
-The Jacobian matrix is only based on the geometry of the system and does not depend on the physical properties such as mass and stiffness.
+\begin{equation}\label{eq:detail_control_decoupling_plant_jacobian}
+  \frac{\bm{\mathcal{X}}_{\{O\}}}{\bm{\mathcal{F}}_{\{O\}}}(s) = \bm{G}_{\{O\}}(s) = \left( \bm{J}_{\{O\}}^t \bm{J}_{\{M\}}^{-T} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{O\}} s^2 + \bm{J}_{\{O\}}^t \bm{\mathcal{C}} \bm{J}_{\{O\}} s + \bm{J}_{\{O\}}^t \bm{\mathcal{K}} \bm{J}_{\{O\}} \right)^{-1}
+\end{equation}
 
-The inputs and outputs of the decoupled plant $\bm{G}_{\{O\}}$ have physical meaning:
+The frame $\{O\}$ can be any chosen frame, but the decoupling properties depends on the chosen frame $\{O\}$.
-- $\bm{\mathcal{F}}_{\{O\}}$ are forces/torques applied on the payload at the origin of frame $\{O\}$
+There are two natural choices: the center of mass $\{M\}$ and the center of stiffness $\{K\}$.
-- $\bm{\mathcal{X}}_{\{O\}}$ are translations/rotation of the payload expressed in frame $\{O\}$
+Note that the Jacobian matrix is only based on the geometry of the system and does not depend on the physical properties such as mass and stiffness.
-
-It is then easy to include a reference tracking input that specify the wanted motion of the payload in the frame $\{O\}$.
-
-Decoupling properties depends on the chosen frame $\{O\}$.
 
 **** Center Of Mass
 
-#+begin_src latex :file detail_control_decoupling_control_jacobian_CoM.pdf
+If the center of mass is chosen as the decoupling frame.
-\begin{tikzpicture}
+The Jacobian matrix and its inverse are expressed in eqref:eq:detail_control_decoupling_jacobian_CoM_inverse.
-  \node[block] (G) {$\bm{G}$};
-  \node[block, left=0.6 of G] (Jt) {$J_{\{M\}}^{-T}$};
-  \node[block, right=0.6 of G] (Ja) {$J_{\{M\}}^{-1}$};
 
-  % Connections and labels
+\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse}
-  \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$};
-  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
-  \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
-  \draw[->] (Ja.east) -- ++( 1.8, 0)  node[above left]{$\bm{\mathcal{X}}_{\{M\}}$};
-
-  \begin{scope}[on background layer]
-    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {};
-    \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$};
-  \end{scope}
-\end{tikzpicture}
-#+end_src
-
-\begin{equation}
   J_{\{M\}} = \begin{bmatrix}
     1 & 0 &  h_a \\
     0 & 1 & -l_a \\
@@ -2338,15 +2327,37 @@ Decoupling properties depends on the chosen frame $\{O\}$.
   \end{bmatrix}
 \end{equation}
 
-Analytical formula of the plant:
+#+begin_src latex :file detail_control_decoupling_control_jacobian_CoM.pdf
+\begin{tikzpicture}
+  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
+  \node[block, left=0.6 of G] (Jt) {$J_{\{M\}}^{-T}$};
+  \node[block, right=0.6 of G] (Ja) {$J_{\{M\}}^{-1}$};
 
-\begin{equation}
+  % Connections and labels
+  \draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$};
+  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
+  \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
+  \draw[->] (Ja.east) -- ++( 1.4, 0)  node[above left]{$\bm{\mathcal{X}}_{\{M\}}$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
+    \node[above] at (Gx.north) {$\bm{G}_{\{M\}}$};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+RESULTS:
+# [[file:figs/detail_control_decoupling_control_jacobian_CoM.png]]
+
+Analytical formula of the plant is eqref:eq:detail_control_decoupling_plant_CoM.
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_CoM}
   \frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^t \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^t \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1}
 \end{equation}
 
-At high frequency, converges towards the inverse of the mass matrix:
+At high frequency, converges towards the inverse of the mass matrix, which is a diagonal matrix eqref:eq:detail_control_decoupling_plant_CoM_high_freq.
 
-\begin{equation}
+\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq}
   \bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix}
     1/m & 0   & 0 \\
     0   & 1/m & 0 \\
@@ -2354,13 +2365,15 @@ At high frequency, converges towards the inverse of the mass matrix:
   \end{bmatrix}
 \end{equation}
 
-
+Plant is therefore well decoupled above the suspension mode with the highest frequency.
-
-Plant is well decoupled above the suspension mode with the highest frequency.
 Such strategy is usually applied on systems with low frequency suspension modes, such that the plant corresponds to decoupled mass lines.
 
 - [ ] Reference to some papers about vibration isolation or ASML?
 
+The coupling at low frequency can easily be understood physically.
+When a static (or with frequency lower than the suspension modes) force is applied at the center of mass, rotation is induced by the stiffness of the first actuator, not in line with the force application point.
+this is illustrated in Figure ref:fig:detail_control_model_test_CoM.
+
 #+begin_src matlab
 %% Jacobian Decoupling - Center of Mass
 G_CoM = pinv(J_CoM)*G_L*pinv(J_CoM');
@@ -2371,56 +2384,67 @@ G_CoM.OutputName  = {'Dx', 'Dy', 'Rz'};
 #+begin_src matlab :exports none
 figure;
 hold on;
-for i_in = 1:3
+plot(freqs, abs(squeeze(freqresp(G_CoM(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
-    for i_out = [i_in+1:3]
+     'DisplayName', '$D_{x,\{M\}}/M_{z,\{M\}}$');
-        plot(freqs, abs(squeeze(freqresp(G_CoM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+plot(freqs, abs(squeeze(freqresp(G_CoM(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
-             'HandleVisibility', 'off');
+     'DisplayName', '$R_{z,\{M\}}/F_{x,\{M\}}$');
-    end
+plot(freqs, abs(squeeze(freqresp(G_CoM(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{M\}}/F_{x,\{M\}}$');
-end
+plot(freqs, abs(squeeze(freqresp(G_CoM(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{M\}}/F_{y,\{M\}}$');
-plot(freqs, abs(squeeze(freqresp(G_CoM(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+plot(freqs, abs(squeeze(freqresp(G_CoM(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{M\}}/M_{z,\{M\}}$');
-     'DisplayName', '$G_{CoM}(i,j)\ i \neq j$');
-set(gca,'ColorOrderIndex',1)
-for i_in_out = 1:3
-    plot(freqs, abs(squeeze(freqresp(G_CoM(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{CoM}(%d,%d)$', i_in_out, i_in_out));
-end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Magnitude');
 ylim([1e-10, 1e-3]);
-leg = legend('location', 'northeast', 'FontSize', 8);
+leg = legend('location', 'southwest', 'FontSize', 8);
 leg.ItemTokenSize(1) = 18;
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/detail_control_jacobian_plant_CoM.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/detail_control_jacobian_plant_CoM.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
-#+name: fig:detail_control_jacobian_plant_CoM
+#+name: fig:detail_control_jacobian_plant_CoM_results
-#+caption: Plant decoupled using the Jacobian matrices $G_x(s)$
+#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoM}).
-#+RESULTS:
+#+attr_latex: :options [htbp]
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:detail_control_jacobian_plant_CoM}Dynamics at the CoM}
+#+attr_latex: :options {0.48\textwidth}
+#+begin_subfigure
+#+attr_latex: :width 0.95\linewidth
 [[file:figs/detail_control_jacobian_plant_CoM.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_CoM}Static force applied at the CoM}
+#+attr_latex: :options {0.48\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/detail_control_model_test_CoM.png]]
+#+end_subfigure
+#+end_figure
 
 **** Center Of Stiffness
 
 #+begin_src latex :file detail_control_decoupling_control_jacobian_CoK.pdf
 \begin{tikzpicture}
-  \node[block] (G) {$\bm{G}$};
+  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
   \node[block, left=0.6 of G] (Jt) {$J_{\{K\}}^{-T}$};
   \node[block, right=0.6 of G] (Ja) {$J_{\{K\}}^{-1}$};
 
   % Connections and labels
-  \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$};
+  \draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$};
   \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
   \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
-  \draw[->] (Ja.east) -- ++( 1.8, 0)  node[above left]{$\bm{\mathcal{X}}_{\{K\}}$};
+  \draw[->] (Ja.east) -- ++( 1.4, 0)  node[above left]{$\bm{\mathcal{X}}_{\{K\}}$};
 
   \begin{scope}[on background layer]
-    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {};
+    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
-    \node[below right] at (Gx.north west) {$\bm{G}_{\{K\}}$};
+    \node[above] at (Gx.north) {$\bm{G}_{\{K\}}$};
   \end{scope}
 \end{tikzpicture}
 #+end_src
 
+#+RESULTS:
+# [[file:figs/detail_control_decoupling_control_jacobian_CoK.png]]
+
 \begin{equation}
   J_{\{K\}} = \begin{bmatrix}
     1 & 0 &  0   \\
@@ -2436,8 +2460,8 @@ exportFig('figs/detail_control_jacobian_plant_CoM.pdf', 'width', 'wide', 'height
 Frame $\{K\}$ is chosen such that $\bm{J}_{\{K\}}^t \bm{\mathcal{K}} \bm{J}_{\{K\}}$ is diagonal.
 Typically, it can me made based on physical reasoning as is the case here.
 
-\begin{equation}
+\begin{equation}\label{eq:detail_control_decoupling_plant_CoK}
-  \frac{\bm{\mathcal{X}}_{\{F\}}}{\bm{\mathcal{F}}_{\{F\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^t \bm{J}_{\{M\}}^{-T} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^t \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^t \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1}
+  \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^t \bm{J}_{\{M\}}^{-T} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^t \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^t \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1}
 \end{equation}
 
 Plant is well decoupled below the suspension mode with the lowest frequency.
@@ -2447,6 +2471,11 @@ This is usually suited for systems which high stiffness.
   \bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-t}
 \end{equation}
 
+
+The physical reason for high frequency coupling is schematically shown in Figure ref:fig:detail_control_model_test_CoK.
+At high frequency, a force applied on a point which is not aligned with the center of mass.
+Therefore, it will induce some rotation around the center of mass.
+
 #+begin_src matlab
 %% Jacobian Decoupling - Center of Mass
 %  Location of the joints with respect to the center of stiffness
@@ -2467,42 +2496,60 @@ G_CoK.OutputName  = {'Dx', 'Dy', 'Rz'};
 #+begin_src matlab :exports none
 figure;
 hold on;
-for i_in = 1:3
+plot(freqs, abs(squeeze(freqresp(G_CoK(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{K\}}/F_{x,\{K\}}$');
-    for i_out = [i_in+1:3]
+plot(freqs, abs(squeeze(freqresp(G_CoK(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{K\}}/F_{y,\{K\}}$');
-        plot(freqs, abs(squeeze(freqresp(G_CoK(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+plot(freqs, abs(squeeze(freqresp(G_CoK(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{K\}}/M_{z,\{K\}}$');
-             'HandleVisibility', 'off');
+plot(freqs, abs(squeeze(freqresp(G_CoK(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
-    end
+     'DisplayName', '$D_{x,\{K\}}/M_{z,\{K\}}$');
-end
+plot(freqs, abs(squeeze(freqresp(G_CoK(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
-plot(freqs, abs(squeeze(freqresp(G_CoK(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$R_{z,\{K\}}/F_{x,\{K\}}$');
-     'DisplayName', '$G_{CoK}(i,j)\ i \neq j$');
-set(gca,'ColorOrderIndex',1)
-for i_in_out = 1:3
-    plot(freqs, abs(squeeze(freqresp(G_CoK(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{CoK}(%d,%d)$', i_in_out, i_in_out));
-end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Magnitude');
+xlabel('Frequency [Hz]'); ylabel('Kagnitude');
 ylim([1e-10, 1e-3]);
-leg = legend('location', 'northeast', 'FontSize', 8);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
 leg.ItemTokenSize(1) = 18;
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/detail_control_jacobian_plant_CoK.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/detail_control_jacobian_plant_CoK.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
-#+name: fig:detail_control_jacobian_plant_CoK
+#+name: fig:detail_control_jacobian_plant_CoK_results
-#+caption: Plant decoupled using the Jacobian matrices $G_x(s)$
+#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoK}).
-#+RESULTS:
+#+attr_latex: :options [htbp]
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:detail_control_jacobian_plant_CoK}Dynamics at the CoK}
+#+attr_latex: :options {0.48\textwidth}
+#+begin_subfigure
+#+attr_latex: :width 0.95\linewidth
 [[file:figs/detail_control_jacobian_plant_CoK.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_CoK}High frequency force applied at the CoK}
+#+attr_latex: :options {0.48\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/detail_control_model_test_CoK.png]]
+#+end_subfigure
+#+end_figure
 
 ** Modal Decoupling
 <<ssec:detail_control_comp_modal>>
 **** Theory                                                        :ignore:
 
+- A mechanical system consists of several modes:
+  - Modal decomposition [[cite:&rankers98_machin]]
+  #+begin_quote
+The physical interpretation of the above two equations is that any motion of the system can be regarded as a combination of the contribution of the various modes.
+  #+end_quote
+  - Mode superposition [[cite:&preumont94_random_vibrat_spect_analy;&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt. 2]]
+- The idea is to control the system in the "modal space"
+
+
+
 Let's consider a system with the following equations of motion:
 \begin{equation}
-M \bm{\ddot{x}} + C \bm{\dot{x}}  + K \bm{x} = \bm{\mathcal{F}}
+M \bm{\ddot{x}} + C \bm{\dot{x}}  + K \bm{x} = J^T \bm{\tau}
 \end{equation}
 
 And the measurement output is a combination of the motion variable $\bm{x}$:
@@ -2518,11 +2565,6 @@ with:
 - $\bm{x}_m$ the modal amplitudes
 - $\Phi$ a matrix whose columns are the modes shapes of the system
 
-And we map the actuator forces:
-\begin{equation}
-\bm{\mathcal{F}} = J^T \bm{\tau}
-\end{equation}
-
 The equations of motion become:
 \begin{equation}
 M \Phi \bm{\ddot{x}}_m + C \Phi \bm{\dot{x}}_m  + K \Phi \bm{x}_m = J^T \bm{\tau}
@@ -2567,18 +2609,13 @@ Let's note the "modal input":
 \end{equation}
 
 The transfer function from $\bm{\tau}_m$ to $\bm{x}_m$ is:
-\begin{equation} \label{eq:modal_eq}
+\begin{equation}\label{eq:detail_control_decoupling_plant_modal}
   \boxed{\frac{\bm{x}_m}{\bm{\tau}_m} = \left( I_n s^2 + 2 \Xi \Omega s + \Omega^2 \right)^{-1}}
 \end{equation}
 which is a *diagonal* transfer function matrix.
 We therefore have decoupling of the dynamics from $\bm{\tau}_m$ to $\bm{x}_m$.
 
 
-We now expressed the transfer function from input $\bm{\tau}$ to output $\bm{y}$ as a function of the "modal variables":
-\begin{equation}
-  \boxed{\frac{\bm{y}}{\bm{\tau}} = \underbrace{\left( C_{ox} + s C_{ov} \right) \Phi}_{C_m} \underbrace{\left( I_n s^2 + 2 \Xi \Omega s + \Omega^2 \right)^{-1}}_{\text{diagonal}} \underbrace{\left( \mu^{-1} \Phi^T J^T \right)}_{B_m}}
-\end{equation}
-
 By inverting $B_m$ and $C_m$ and using them as shown in Figure ref:fig:modal_decoupling_architecture, we can see that we control the system in the "modal space" in which it is decoupled.
 
 #+begin_src latex :file detail_control_decoupling_modal.pdf
@@ -2667,9 +2704,6 @@ data2orgtable(Bm, {}, {}, ' %.4f ');
 #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
 #+attr_latex: :center t :booktabs t :float t
 #+RESULTS:
-| -0.0004 | -0.0007 |  0.0007 |
-| -0.0151 |  0.0041 | -0.0041 |
-|     0.0 |  0.0025 |  0.0025 |
 
 #+begin_src matlab :exports results :results value table replace :tangle no
 data2orgtable(Cm, {}, {}, ' %.1f ');
@@ -2680,9 +2714,6 @@ data2orgtable(Cm, {}, {}, ' %.1f ');
 #+attr_latex: :environment tabularx :width 0.2\linewidth :align ccc
 #+attr_latex: :center t :booktabs t :float t
 #+RESULTS:
-| -0.1 | -1.8 | 0.0 |
-| -0.2 |  0.5 | 1.0 |
-|  0.2 | -0.5 | 1.0 |
 
 #+begin_src matlab :exports none
 figure;
@@ -2714,7 +2745,6 @@ exportFig('figs/detail_control_modal_plant.pdf', 'width', 'wide', 'height', 'nor
 #+name: fig:detail_control_modal_plant
 #+caption: Modal plant $G_m(s)$
 #+RESULTS:
-[[file:figs/detail_control_modal_plant.png]]
 
 ** SVD Decoupling
 <<ssec:detail_control_comp_svd>>
@@ -2724,10 +2754,26 @@ exportFig('figs/detail_control_modal_plant.pdf', 'width', 'wide', 'height', 'nor
 - Introduction to SVD [[cite:&brunton22_data]]
 - Applied to parallel manipulator?
 
+Singular value is used a lot for multivariable control [[cite:&skogestad07_multiv_feedb_contr]].
+Used to study directions in multivariable systems
+
 **** Control Architecture
 
-- [ ] Have notation for the measured FRF
+- [ ] SVD controllers described in [[cite:&skogestad07_multiv_feedb_contr, chapt. 3.5.4]]
-- [ ] And for the real approximation
+- [ ] *Check if inverse U and V should be used or just U and V matrices*, Use correct notations.
+- [ ] Have notation for the measured FRF and for the real approximation
+
+\begin{equation}
+\bm{G}(j\omega) = \begin{bmatrix}
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+\end{bmatrix} \xrightarrow[approximation]{real} \begin{bmatrix}
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+\end{bmatrix} \xrightarrow[SVD]{} U = , \ V =
+\end{equation}
 
 Procedure:
 - Identify the dynamics of the system from inputs to outputs (can be obtained experimentally)
@@ -2735,7 +2781,7 @@ Procedure:
 - Choose a frequency where we want to decouple the system (usually, the crossover frequency is a good choice)
 - Compute a real approximation of the system's response at that frequency
   As /real/ V and U matrices need to be obtained, a real approximation of the complex measured response needs to be computed.
-  - [ ] Find reference to do so.
+  [[cite:&kouvaritakis79_theor_pract_charac_locus_desig_method]]: real matrix that preserves the most orthogonality in directions with the input complex matrix
 - Perform a Singular Value Decomposition of the real approximation.
   Unitary U and V matrices are then obtained such that:
   V-t Greal U-1 is a diagonal matrix
@@ -2784,10 +2830,17 @@ The inputs and outputs are ordered from higher gain to lower gain at the chosen
 **** Example
 
 - [ ] Analytical formulas in this case?
+- [ ] Maybe show the complex and real response matrices.
+- [ ] At least, show the obtained matrices
 - [ ] Do we have something special when applying SVD to a collocated MIMO system?
 - *Verify why such a good decoupling is obtained!*
 # - When applying SVD on a non-collocated MIMO system, we obtained a decoupled plant looking like the one in Figure ref:fig:detail_control_gravimeter_svd_plant
 
+\begin{equation}\label{eq:detail_control_decoupling_plant_svd}
+\bm{G}_{SVD}(s) =
+\end{equation}
+
+
 #+begin_src matlab
 %% SVD Decoupling
 
@@ -3122,6 +3175,10 @@ Conclusion about NASS:
 - Prefer to use Jacobian decoupling as we get more physical interpretation
 - Also, it is possible to take into account different specifications in the different DoF
 
+When possible, having a design providing the same CoK and CoM is good.
+Often, it is not possible and we have to deal with that with control.
+Idea about using CoK at low frequency and CoM at high frequency ?
+Maybe with complementary filters?
 
 * Closed-Loop Shaping using Complementary Filters
 :PROPERTIES:

nass-control.tex

@@ -1,4 +1,4 @@
-% Created 2025-04-04 Fri 19:20
+% Created 2025-04-05 Sat 11:47
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -185,7 +185,7 @@ Based on that, this work introduces a new way to design complementary filters us
 Complementary filtering provides a framework for fusing signals from different sensors.
 As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
 These requirements are discussed in this section.
-\subsubsection{Sensor Fusion Architecture}
+\paragraph{Sensor Fusion Architecture}
 
 A general sensor fusion architecture using complementary filters is shown in Figure \ref{fig:detail_control_sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
 The two sensors output signals \(\hat{x}_1\) and \(\hat{x}_2\) are estimates of \(x\).
@@ -206,7 +206,7 @@ That is, unity magnitude and zero phase at all frequencies.
 \begin{equation}\label{eq:detail_control_comp_filter}
   H_1(s) + H_2(s) = 1
 \end{equation}
-\subsubsection{Sensor Models and Sensor Normalization}
+\paragraph{Sensor Models and Sensor Normalization}
 
 In order to study such sensor fusion architecture, a model for the sensors is required.
 Such model is shown in Figure \ref{fig:detail_control_sensor_model} and consists of a linear time invariant (LTI) system \(G_i(s)\) representing the sensor dynamics and an input \(n_i\) representing the sensor noise.
@@ -248,7 +248,7 @@ The super sensor output \(\hat{x}\) is therefore described by \eqref{eq:detail_c
 \includegraphics[scale=1]{figs/detail_control_fusion_super_sensor.png}
 \caption{\label{fig:detail_control_fusion_super_sensor}Sensor fusion architecture with two normalized sensors.}
 \end{figure}
-\subsubsection{Noise Sensor Filtering}
+\paragraph{Noise Sensor Filtering}
 
 First, suppose that all the sensors are perfectly normalized \eqref{eq:detail_control_perfect_dynamics}.
 The effect of a non-perfect normalization will be discussed afterwards.
@@ -284,7 +284,7 @@ This is the simplest form of sensor fusion with complementary filters.
 However, the two sensors have usually high noise levels over distinct frequency regions.
 In such case, to lower the noise of the super sensor, the norm \(|H_1(j\omega)|\) has to be small when \(\Phi_{n_1}(\omega)\) is larger than \(\Phi_{n_2}(\omega)\) and the norm \(|H_2(j\omega)|\) has to be small when \(\Phi_{n_2}(\omega)\) is larger than \(\Phi_{n_1}(\omega)\).
 Hence, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor.
-\subsubsection{Sensor Fusion Robustness}
+\paragraph{Sensor Fusion Robustness}
 
 In practical systems the sensor normalization is not perfect and condition \eqref{eq:detail_control_perfect_dynamics} is not verified.
 
@@ -350,7 +350,7 @@ Therefore, a synthesis method of complementary filters that allows to shape thei
 In this section, such synthesis is proposed by writing the synthesis objective as a standard \(\mathcal{H}_\infty\) optimization problem.
 As weighting functions are used to represent the wanted complementary filters' shape during the synthesis, their proper design is discussed.
 Finally, the synthesis method is validated on an simple example.
-\subsubsection{Synthesis Objective}
+\paragraph{Synthesis Objective}
 
 The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:detail_control_comp_filter}.
 This is equivalent as to finding proper and stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:detail_control_hinf_cond_complementarity}, \eqref{eq:detail_control_hinf_cond_h1} and \eqref{eq:detail_control_hinf_cond_h2} are satisfied.
@@ -363,7 +363,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
   & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:detail_control_hinf_cond_h2}
   \end{align}
 \end{subequations}
-\subsubsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
+\paragraph{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
 
 The synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimization problem and therefore solved using convenient tools readily available.
 Consider the generalized plant \(P(s)\) shown in Figure \ref{fig:detail_control_h_infinity_robust_fusion_plant} and mathematically described by \eqref{eq:detail_control_generalized_plant}.
@@ -408,7 +408,7 @@ Therefore, applying the \(\mathcal{H}_\infty\) synthesis to the standard plant \
 
 Note that there is only an implication between the \(\mathcal{H}_\infty\) norm condition \eqref{eq:detail_control_hinf_problem} and the initial synthesis objectives \eqref{eq:detail_control_hinf_cond_h1} and \eqref{eq:detail_control_hinf_cond_h2} and not an equivalence.
 Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see \cite[,Chap. 2.8.3]{skogestad07_multiv_feedb_contr}.
-\subsubsection{Weighting Functions Design}
+\paragraph{Weighting Functions Design}
 
 Weighting functions are used during the synthesis to specify the maximum allowed complementary filters' norm.
 The proper design of these weighting functions is of primary importance for the success of the presented \(\mathcal{H}_\infty\) synthesis of complementary filters.
@@ -439,7 +439,7 @@ The typical magnitude of a weighting function generated using \eqref{eq:detail_c
          }\right)^n
 \end{equation}
 \end{minipage}
-\subsubsection{Validation of the proposed synthesis method}
+\paragraph{Validation of the proposed synthesis method}
 
 The proposed methodology for the design of complementary filters is now applied on a simple example.
 Let's suppose two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
@@ -621,8 +621,9 @@ Experimental closed-loop control results using the hexapod have shown that contr
 \end{quote}
 
 \begin{itemize}
-\item[{$\square$}] Review of \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies} for stewart platforms
+\item[{$\boxtimes$}] Review of \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies} for stewart platforms
 \item[{$\square$}] Add some citations about different methods
+\item[{$\square$}] Maybe transform table into text
 \end{itemize}
 
 \begin{table}[htbp]
@@ -652,10 +653,6 @@ Voice Coil & Accelerometers & MIMO H-Infinity, active damping & \cite{jiao18_dyn
 \end{tabularx}
 \end{table}
 
-Assumptions:
-\begin{itemize}
-\item parallel manipulators
-\end{itemize}
 
 The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators.
 
@@ -671,18 +668,17 @@ It is structured as follow:
 \section{Test Model}
 \label{ssec:detail_control_decoupling_comp_model}
 
-Let's consider a parallel manipulator with several collocated actuator/sensors pairs.
-
-System in Figure \ref{fig:detail_control_model_test_decoupling} will serve as an example.
-
-We will note:
 \begin{itemize}
-\item \(b_i\): location of the joints on the top platform
+\item Instead of comparing the decoupling strategies using the Stewart platform, a similar yet much simpler parallel manipulator is used instead
-\item \(\hat{s}_i\): unit vector corresponding to the struts direction
+\item to render the analysis simpler, the system of Figure \ref{fig:detail_control_model_test_decoupling_detail} is used
-\item \(k_i\): stiffness of the struts
+\item It has 3DoF, and has 3 parallels struts whose model is shown in Figure \ref{fig:detail_control_strut_model}
-\item \(\tau_i\): actuator forces
+\item It is quite similar to the Stewart platform (parallel architecture, as many struts as DoF)
-\item \(O_M\): center of mass of the solid body
+\end{itemize}
-\item \(\mathcal{L}_i\): relative displacement of the struts
+
+Two frames are defined:
+\begin{itemize}
+\item \(\{M\}\) with origin \(O_M\) at the Center of mass of the solid body
+\item \(\{K\}\) with origin \(O_K\) at the Center of mass of the parallel manipulator
 \end{itemize}
 
 \begin{figure}[htbp]
@@ -701,54 +697,265 @@ We will note:
 \caption{\label{fig:detail_control_model_test_decoupling_detail}3DoF model used to study decoupling strategies}
 \end{figure}
 
-The magnitude of the coupled plant \(G\) is shown in Figure \ref{fig:detail_control_coupled_plant_bode}.
+First, the equation of motion are derived.
+Expressing the second law of Newton on the suspended mass, expressed at its center of mass gives
+
+\begin{equation}
+  M_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t)
+\end{equation}
+
+with \(\bm{\mathcal{X}}_{\{M\}}\) the two translation and one rotation expressed with respect to the center of mass and \(\bm{\mathcal{F}}_{\{M\}}\) forces and torque applied at the center of mass.
+
+\begin{equation}
+  \bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix}
+    x \\
+    y \\
+    R_z
+  \end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix}
+    F_x \\
+    F_y \\
+    M_z
+  \end{bmatrix}
+\end{equation}
+
+In order to map the spring, damping and actuator forces to XY forces and Z torque expressed at the center of mass, the Jacobian matrix \(\bm{J}_{\{M\}}\) is used.
+
+\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM}
+  \bm{J}_{\{M\}} = \begin{bmatrix}
+    1 & 0 &  h_a \\
+    0 & 1 & -l_a \\
+    0 & 1 &  l_a \\
+  \end{bmatrix}
+\end{equation}
+
+Then, the equation of motion linking the actuator forces \(\tau\) to the motion of the mass \(\bm{\mathcal{X}}_{\{M\}}\) is obtained.
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_cartesian}
+  M_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^t \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^t \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^t \bm{\tau}(t)
+\end{equation}
+
+Matrices representing the payload inertia as well as the actuator stiffness and damping are shown in
+
+\begin{equation}
+  \bm{M}_{\{M\}} = \begin{bmatrix}
+    m & 0 & 0 \\
+    0 & m & 0 \\
+    0 & 0 & I
+  \end{bmatrix}, \quad
+  \bm{\mathcal{K}} = \begin{bmatrix}
+    k & 0 & 0 \\
+    0 & k & 0 \\
+    0 & 0 & k
+  \end{bmatrix}, \quad
+  \bm{\mathcal{C}} = \begin{bmatrix}
+    c & 0 & 0 \\
+    0 & c & 0 \\
+    0 & 0 & c
+  \end{bmatrix}
+\end{equation}
+
+Parameters used for the following analysis are summarized in table \ref{tab:detail_control_decoupling_test_model_params}.
+
+\begin{table}[htbp]
+\caption{\label{tab:detail_control_decoupling_test_model_params}Model parameters}
+\centering
+\scriptsize
+\begin{tabularx}{0.9\linewidth}{cXc}
+\toprule
+\textbf{Parameter} & \textbf{Description} & \textbf{Value}\\
+\midrule
+\(l_a\) &  & \(0.5\,m\)\\
+\(h_a\) &  & \(0.2\,m\)\\
+\(k\) & Actuator stiffness & \(10\,N/\mu m\)\\
+\(c\) & Actuator damping & \(200\,Ns/m\)\\
+\(m\) & Payload mass & \(40\,\text{kg}\)\\
+\(I\) & Payload rotational inertia & \(5\,\text{kg}m^2\)\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Control in the frame of the struts}
+
+Let's first study the obtained dynamics in the frame of the struts.
+The equation of motion linking actuator forces \(\bm{\mathcal{\tau}}\) to strut relative motion \(\bm{\mathcal{L}}\) is obtained from \eqref{eq:detail_control_decoupling_plant_cartesian} by mapping the cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix \(\bm{J}_{\{M\}}\) \eqref{eq:detail_control_decoupling_jacobian_CoM} .
+
+The transfer function from \(\bm{\mathcal{\tau}}\) to \(\bm{\mathcal{L}}\) is shown in equation \eqref{eq:detail_control_decoupling_plant_decentralized}.
+
+\begin{center}
+\includegraphics[scale=1]{figs/detail_control_decoupling_control_struts.png}
+\label{}
+\end{center}
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized}
+  \frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-t} M_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1}
+\end{equation}
+
+At low frequency the plant converges to a diagonal constant matrix whose diagonal elements are linked to the actuator stiffnesses \eqref{eq:detail_control_decoupling_plant_decentralized_low_freq}.
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq}
+  \bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}}
+\end{equation}
+
+At high frequency, the plant converges to the mass matrix mapped in the frame of the struts, which is in general highly non-diagonal.
+
+The magnitude of the coupled plant \(\bm{G}_{\mathcal{L}}\) is shown in Figure \ref{fig:detail_control_coupled_plant_bode}.
+This confirms that at low frequency (below the first suspension mode), the plant is well decoupled.
+Depending on the symmetry in the system, some diagonal elements may be equal (such as for struts 2 and 3 in this example).
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_coupled_plant_bode.png}
 \caption{\label{fig:detail_control_coupled_plant_bode}Magnitude of the coupled plant.}
 \end{figure}
-\section{Decentralized Plant / Control in the frame of the struts}
 \section{Jacobian Decoupling}
 \label{ssec:detail_control_comp_jacobian}
+\paragraph{Jacobian Matrix}
 
-The Jacobian matrix can be used to:
+As already explained, the Jacobian matrix can be used to both convert strut velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\) and Convert actuators forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\) \eqref{eq:detail_control_decoupling_jacobian}.
-\begin{itemize}
-\item Convert joints velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\):
-\[ \dot{\bm{\mathcal{X}}}_{\{O\}} = J_{\{O\}} \dot{\bm{\mathcal{L}}} \]
-\item Convert actuators forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\):
-\[ \bm{\mathcal{F}}_{\{O\}} = J_{\{O\}}^T \bm{\tau} \]
-\end{itemize}
-with \(\{O\}\) any chosen frame.
 
-By wisely choosing frame \(\{O\}\), we can obtain nice decoupling for plant:
+\begin{subequations}\label{eq:detail_control_decoupling_jacobian}
-\begin{equation}
+  \begin{align}
-  \bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}
+    \dot{\bm{\mathcal{X}}}_{\{O\}} &= J_{\{O\}} \dot{\bm{\mathcal{L}}}, \quad \dot{\bm{\mathcal{L}}} = J_{\{O\}}^{-1} \dot{\bm{\mathcal{X}}}_{\{O\}} \\
-\end{equation}
+    \bm{\mathcal{F}}_{\{O\}}       &= J_{\{O\}}^t \bm{\tau},            \quad \bm{\tau} = J_{\{O\}}^{-t} \bm{\mathcal{F}}_{\{O\}}
+  \end{align}
+\end{subequations}
 
-The obtained plan corresponds to forces/torques applied on origin of frame \(\{O\}\) to the translation/rotation of the payload expressed in frame \(\{O\}\).
+The obtained plan (Figure \ref{fig:detail_control_jacobian_decoupling_arch}) has inputs and outputs that have physical meaning:
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/detail_control_jacobian_decoupling_arch.png}
-\caption{\label{fig:detail_control_jacobian_decoupling_arch}Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{O\}}\) to \(\bm{\mathcal{X}}_{\{O\}}\)}
-\end{figure}
-
-The Jacobian matrix is only based on the geometry of the system and does not depend on the physical properties such as mass and stiffness.
-
-The inputs and outputs of the decoupled plant \(\bm{G}_{\{O\}}\) have physical meaning:
 \begin{itemize}
 \item \(\bm{\mathcal{F}}_{\{O\}}\) are forces/torques applied on the payload at the origin of frame \(\{O\}\)
 \item \(\bm{\mathcal{X}}_{\{O\}}\) are translations/rotation of the payload expressed in frame \(\{O\}\)
 \end{itemize}
 
-It is then easy to include a reference tracking input that specify the wanted motion of the payload in the frame \(\{O\}\).
+\begin{figure}[htbp]
-\subsubsection{Center Of Mass}
+\centering
+\includegraphics[scale=1]{figs/detail_control_decoupling_control_jacobian.png}
+\caption{\label{fig:detail_control_jacobian_decoupling_arch}Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{O\}}\) to \(\bm{\mathcal{X}}_{\{O\}}\)}
+\end{figure}
 
-\subsubsection{Center Of Stiffness}
+\begin{equation}\label{eq:detail_control_decoupling_plant_jacobian}
+  \frac{\bm{\mathcal{X}}_{\{O\}}}{\bm{\mathcal{F}}_{\{O\}}}(s) = \bm{G}_{\{O\}}(s) = \left( \bm{J}_{\{O\}}^t \bm{J}_{\{M\}}^{-T} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{O\}} s^2 + \bm{J}_{\{O\}}^t \bm{\mathcal{C}} \bm{J}_{\{O\}} s + \bm{J}_{\{O\}}^t \bm{\mathcal{K}} \bm{J}_{\{O\}} \right)^{-1}
+\end{equation}
+
+The frame \(\{O\}\) can be any chosen frame, but the decoupling properties depends on the chosen frame \(\{O\}\).
+There are two natural choices: the center of mass \(\{M\}\) and the center of stiffness \(\{K\}\).
+Note that the Jacobian matrix is only based on the geometry of the system and does not depend on the physical properties such as mass and stiffness.
+\paragraph{Center Of Mass}
+
+If the center of mass is chosen as the decoupling frame.
+The Jacobian matrix and its inverse are expressed in \eqref{eq:detail_control_decoupling_jacobian_CoM_inverse}.
+
+\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse}
+  J_{\{M\}} = \begin{bmatrix}
+    1 & 0 &  h_a \\
+    0 & 1 & -l_a \\
+    0 & 1 &  l_a \\
+  \end{bmatrix}, \quad J_{\{M\}}^{-1} = \begin{bmatrix}
+    1 & \frac{h_a}{2 l_a} & \frac{-h_a}{2 l_a} \\
+    0 & \frac{1}{2} & \frac{1}{2} \\
+    0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} \\
+  \end{bmatrix}
+\end{equation}
+
+\begin{center}
+\includegraphics[scale=1]{figs/detail_control_decoupling_control_jacobian_CoM.png}
+\label{}
+\end{center}
+
+Analytical formula of the plant is \eqref{eq:detail_control_decoupling_plant_CoM}.
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_CoM}
+  \frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^t \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^t \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1}
+\end{equation}
+
+At high frequency, converges towards the inverse of the mass matrix, which is a diagonal matrix \eqref{eq:detail_control_decoupling_plant_CoM_high_freq}.
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq}
+  \bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix}
+    1/m & 0   & 0 \\
+    0   & 1/m & 0 \\
+    0   & 0   & 1/I
+  \end{bmatrix}
+\end{equation}
+
+Plant is therefore well decoupled above the suspension mode with the highest frequency.
+Such strategy is usually applied on systems with low frequency suspension modes, such that the plant corresponds to decoupled mass lines.
+
+\begin{itemize}
+\item[{$\square$}] Reference to some papers about vibration isolation or ASML?
+\end{itemize}
+
+The coupling at low frequency can easily be understood physically.
+When a static (or with frequency lower than the suspension modes) force is applied at the center of mass, rotation is induced by the stiffness of the first actuator, not in line with the force application point.
+this is illustrated in Figure \ref{fig:detail_control_model_test_CoM}.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_jacobian_plant_CoM.png}
+\end{center}
+\subcaption{\label{fig:detail_control_jacobian_plant_CoM}Dynamics at the CoM}
+\end{subfigure}
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_CoM.png}
+\end{center}
+\subcaption{\label{fig:detail_control_model_test_CoM}Static force applied at the CoM}
+\end{subfigure}
+\caption{\label{fig:detail_control_jacobian_plant_CoM_results}Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoM}).}
+\end{figure}
+\paragraph{Center Of Stiffness}
+
+\begin{center}
+\includegraphics[scale=1]{figs/detail_control_decoupling_control_jacobian_CoK.png}
+\label{}
+\end{center}
+
+\begin{equation}
+  J_{\{K\}} = \begin{bmatrix}
+    1 & 0 &  0   \\
+    0 & 1 & -l_a \\
+    0 & 1 &  l_a
+  \end{bmatrix}, \quad J_{\{K\}}^{-1} = \begin{bmatrix}
+    1 & 0                & 0   \\
+    0 & \frac{1}{2}      & \frac{1}{2} \\
+    0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a}
+  \end{bmatrix}
+\end{equation}
+
+Frame \(\{K\}\) is chosen such that \(\bm{J}_{\{K\}}^t \bm{\mathcal{K}} \bm{J}_{\{K\}}\) is diagonal.
+Typically, it can me made based on physical reasoning as is the case here.
+
+\begin{equation}\label{eq:detail_control_decoupling_plant_CoK}
+  \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^t \bm{J}_{\{M\}}^{-T} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^t \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^t \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1}
+\end{equation}
+
+Plant is well decoupled below the suspension mode with the lowest frequency.
+This is usually suited for systems which high stiffness.
+
+\begin{equation}
+  \bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-t}
+\end{equation}
+
+\begin{itemize}
+\item[{$\square$}] Make a schematic where the thing is deformed at high frequency rotating about the center of mass
+\end{itemize}
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_jacobian_plant_CoK.png}
+\end{center}
+\subcaption{\label{fig:detail_control_jacobian_plant_CoK}Dynamics at the CoK}
+\end{subfigure}
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_CoK.png}
+\end{center}
+\subcaption{\label{fig:detail_control_model_test_CoK}High frequency force applied at the CoK}
+\end{subfigure}
+\caption{\label{fig:detail_control_jacobian_plant_CoK_results}Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_jacobian_plant_CoK}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoK}).}
+\end{figure}
 \section{Modal Decoupling}
 \label{ssec:detail_control_comp_modal}
-
 Let's consider a system with the following equations of motion:
 \begin{equation}
 M \bm{\ddot{x}} + C \bm{\dot{x}}  + K \bm{x} = \bm{\mathcal{F}}
@@ -822,7 +1029,7 @@ Let's note the ``modal input'':
 \end{equation}
 
 The transfer function from \(\bm{\tau}_m\) to \(\bm{x}_m\) is:
-\begin{equation} \label{eq:modal_eq}
+\begin{equation}\label{eq:detail_control_decoupling_plant_modal}
   \boxed{\frac{\bm{x}_m}{\bm{\tau}_m} = \left( I_n s^2 + 2 \Xi \Omega s + \Omega^2 \right)^{-1}}
 \end{equation}
 which is a \textbf{diagonal} transfer function matrix.
@@ -851,51 +1058,8 @@ Then, the system can be decoupled in the modal space.
 The obtained system on the diagonal are second order resonant systems which can be easily controlled.
 
 Using this decoupling strategy, it is possible to control each mode individually.
-\section{SVD Decoupling}
+\paragraph{Example}
-\label{ssec:detail_control_comp_svd}
 
-Procedure:
-\begin{itemize}
-\item Identify the dynamics of the system from inputs to outputs (can be obtained experimentally)
-\item Choose a frequency where we want to decouple the system (usually, the crossover frequency is a good choice)
-\item Compute a real approximation of the system's response at that frequency
-\item Perform a Singular Value Decomposition of the real approximation
-\item Use the singular input and output matrices to decouple the system as shown in Figure \ref{fig:detail_control_decoupling_svd}
-\[ G_{svd}(s) = U^{-1} G(s) V^{-T} \]
-\end{itemize}
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/detail_control_decoupling_svd.png}
-\caption{\label{fig:detail_control_decoupling_svd}Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition}
-\end{figure}
-
-In order to apply the Singular Value Decomposition, we need to have the Frequency Response Function of the system, at least near the frequency where we wish to decouple the system.
-The FRF can be experimentally obtained or based from a model.
-
-This method ensure good decoupling near the chosen frequency, but no guaranteed decoupling away from this frequency.
-
-Also, it depends on how good the real approximation of the FRF is, therefore it might be less good for plants with high damping.
-
-This method is quite general and can be applied to any type of system.
-The inputs and outputs are ordered from higher gain to lower gain at the chosen frequency.
-
-\begin{itemize}
-\item[{$\square$}] Do we loose any physical meaning of the obtained inputs and outputs?
-\item[{$\square$}] Can we take advantage of the fact that U and V are unitary?
-\end{itemize}
-\section{Comparison}
-\label{ssec:detail_control_decoupling_comp}
-\subsubsection{Jacobian Decoupling}
-Decoupling properties depends on the chosen frame \(\{O\}\).
-
-Let's take the CoM as the decoupling frame.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/detail_control_jacobian_plant.png}
-\caption{\label{fig:detail_control_jacobian_plant}Plant decoupled using the Jacobian matrices \(G_x(s)\)}
-\end{figure}
-\subsubsection{Modal Decoupling}
 For the system in Figure \ref{fig:detail_control_model_test_decoupling}, we have:
 \begin{align}
 \bm{x} &= \begin{bmatrix} x \\ y \\ R_z \end{bmatrix} \\
@@ -917,39 +1081,96 @@ c & 0 & 0 \\
 
 In order to apply the architecture shown in Figure \ref{fig:modal_decoupling_architecture}, we need to compute \(C_{ox}\), \(C_{ov}\), \(\Phi\), \(\mu\) and \(J\).
 
-\begin{table}[htbp]
+\begin{itemize}
-\caption{\label{tab:modal_decoupling_Bm}\(B_m\) matrix}
+\item[{$\square$}] Is it possible to obtained the analytical formulas for decoupling matrices?
-\centering
+\end{itemize}
-\begin{tabularx}{0.3\linewidth}{ccc}
+\section{SVD Decoupling}
-\toprule
+\label{ssec:detail_control_comp_svd}
--0.0004 & -0.0007 & 0.0007\\
+\paragraph{Singular Value Decomposition}
--0.0151 & 0.0041 & -0.0041\\
-0.0 & 0.0025 & 0.0025\\
-\bottomrule
-\end{tabularx}
-\end{table}
 
-\begin{table}[htbp]
+\begin{itemize}
-\caption{\label{tab:modal_decoupling_Cm}\(C_m\) matrix}
+\item Introduction to SVD \cite{brunton22_data}
-\centering
+\item Applied to parallel manipulator?
-\begin{tabularx}{0.2\linewidth}{ccc}
+\end{itemize}
-\toprule
+
--0.1 & -1.8 & 0.0\\
+Singular value is used a lot for multivariable control \cite{skogestad07_multiv_feedb_contr}.
--0.2 & 0.5 & 1.0\\
+Used to study directions in multivariable systems
-0.2 & -0.5 & 1.0\\
+\paragraph{Control Architecture}
-\bottomrule
+
-\end{tabularx}
+\begin{itemize}
-\end{table}
+\item[{$\square$}] SVD controllers described in \cite[, chapt. 3.5.4]{skogestad07_multiv_feedb_contr}
+\item[{$\square$}] \textbf{Check if inverse U and V should be used or just U and V matrices}, Use correct notations.
+\item[{$\square$}] Have notation for the measured FRF and for the real approximation
+\end{itemize}
+
+\begin{equation}
+\bm{G}(j\omega) = \begin{bmatrix}
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+\end{bmatrix} \xrightarrow[approximation]{real} \begin{bmatrix}
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+\end{bmatrix} \xrightarrow[SVD]{} U = , \ V =
+\end{equation}
+
+Procedure:
+\begin{itemize}
+\item Identify the dynamics of the system from inputs to outputs (can be obtained experimentally)
+Frequency Response Function, which is a complex matrix obtained for several frequency points.
+\item Choose a frequency where we want to decouple the system (usually, the crossover frequency is a good choice)
+\item Compute a real approximation of the system's response at that frequency
+As \emph{real} V and U matrices need to be obtained, a real approximation of the complex measured response needs to be computed.
+\cite{kouvaritakis79_theor_pract_charac_locus_desig_method}: real matrix that preserves the most orthogonality in directions with the input complex matrix
+\item Perform a Singular Value Decomposition of the real approximation.
+Unitary U and V matrices are then obtained such that:
+V-t Greal U-1 is a diagonal matrix
+
+\item Use the singular input and output matrices to decouple the system as shown in Figure \ref{fig:detail_control_decoupling_svd}
+\[ G_{svd}(s) = U^{-1} G(s) V^{-T} \]
+\end{itemize}
 
-And the plant in the modal space is defined below and its magnitude is shown in Figure \ref{fig:detail_control_modal_plant}.
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/detail_control_modal_plant.png}
+\includegraphics[scale=1]{figs/detail_control_decoupling_svd.png}
-\caption{\label{fig:detail_control_modal_plant}Modal plant \(G_m(s)\)}
+\caption{\label{fig:detail_control_decoupling_svd}Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition}
+\end{figure}
+
+In order to apply the Singular Value Decomposition, we need to have the Frequency Response Function of the system, at least near the frequency where we wish to decouple the system.
+The FRF can be experimentally obtained or based from a model.
+
+This method ensure good decoupling near the chosen frequency, but no guaranteed decoupling away from this frequency.
+
+Also, it depends on how good the real approximation of the FRF is, therefore it might be less good for plants with high damping.
+
+This method is quite general and can be applied to any type of system.
+The inputs and outputs are ordered from higher gain to lower gain at the chosen frequency.
+
+\begin{itemize}
+\item[{$\square$}] Do we loose any physical meaning of the obtained inputs and outputs?
+\item[{$\square$}] Can we take advantage of the fact that U and V are unitary?
+\end{itemize}
+\paragraph{Example}
+
+\begin{itemize}
+\item[{$\square$}] Analytical formulas in this case?
+\item[{$\square$}] Maybe show the complex and real response matrices.
+\item[{$\square$}] At least, show the obtained matrices
+\item[{$\square$}] Do we have something special when applying SVD to a collocated MIMO system?
+\item \textbf{Verify why such a good decoupling is obtained!}
+\end{itemize}
+\begin{equation}\label{eq:detail_control_decoupling_plant_svd}
+\bm{G}_{SVD}(s) =
+\end{equation}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/detail_control_svd_plant.png}
+\caption{\label{fig:detail_control_svd_plant}Svd plant \(G_m(s)\)}
 \end{figure}
 
-Let's now close one loop at a time and see how the transmissibility changes.
-\subsubsection{SVD Decoupling}
 \begin{table}[htbp]
 \caption{\label{}Real approximate of \(G\) at the decoupling frequency \(\omega_c\)}
 \centering
@@ -961,31 +1182,8 @@ Let's now close one loop at a time and see how the transmissibility changes.
 \bottomrule
 \end{tabularx}
 \end{table}
-
+\section{Comparison}
-\begin{itemize}
+\label{ssec:detail_control_decoupling_comp}
-\item[{$\square$}] Do we have something special when applying SVD to a collocated MIMO system?
-\item \textbf{Verify why such a good decoupling is obtained!}
-\end{itemize}
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/detail_control_svd_plant.png}
-\caption{\label{fig:detail_control_svd_plant}Svd plant \(G_m(s)\)}
-\end{figure}
-\section*{Conclusion}
-The three proposed methods clearly have a lot in common as they all tend to make system more decoupled by pre and/or post multiplying by a constant matrix
-However, the three methods also differs by a number of points which are summarized in Table \ref{tab:detail_control_decoupling_strategies_comp}.
-
-Other decoupling strategies could be included in this study, such as:
-\begin{itemize}
-\item DC decoupling: pre-multiply the plant by \(G(0)^{-1}\)
-\item full decoupling: pre-multiply the plant by \(G(s)^{-1}\)
-\end{itemize}
-
-Conclusion about NASS:
-\begin{itemize}
-\item Prefer to use Jacobian decoupling as we get more physical interpretation
-\item Also, it is possible to take into account different specifications in the different DoF
-\end{itemize}
 
 \begin{table}[htbp]
 \caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies}
@@ -1025,6 +1223,21 @@ Conclusion about NASS:
 \bottomrule
 \end{tabularx}
 \end{table}
+\section*{Conclusion}
+The three proposed methods clearly have a lot in common as they all tend to make system more decoupled by pre and/or post multiplying by a constant matrix
+However, the three methods also differs by a number of points which are summarized in Table \ref{tab:detail_control_decoupling_strategies_comp}.
+
+Other decoupling strategies could be included in this study, such as:
+\begin{itemize}
+\item DC decoupling: pre-multiply the plant by \(G(0)^{-1}\)
+\item full decoupling: pre-multiply the plant by \(G(s)^{-1}\)
+\end{itemize}
+
+Conclusion about NASS:
+\begin{itemize}
+\item Prefer to use Jacobian decoupling as we get more physical interpretation
+\item Also, it is possible to take into account different specifications in the different DoF
+\end{itemize}
 \chapter{Closed-Loop Shaping using Complementary Filters}
 \label{sec:detail_control_optimization}
 
@@ -1088,7 +1301,7 @@ In this paper, we propose a new controller synthesis method
 \end{itemize}
 \section{Control Architecture}
 \label{ssec:detail_control_control_arch}
-\subsubsection{Virtual Sensor Fusion}
+\paragraph{Virtual Sensor Fusion}
 Let's consider the control architecture represented in Figure \ref{fig:detail_control_sf_arch} where \(G^\prime\) is the physical plant to control, \(G\) is a model of the plant, \(k\) is a gain, \(H_L\) and \(H_H\) are complementary filters (\(H_L + H_H = 1\) in the complex sense).
 The signals are the reference signal \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\).
 
@@ -1126,7 +1339,7 @@ y &=  \frac{1}{1+G^{\prime} K H_L}     dy &&+ \frac{G^{\prime} K}{1+G^{\prime} K
 u &= \frac{-K H_L}{1+G^{\prime} K H_L} dy &&+ \frac{K}{1+G^{\prime} K H_L}            r &&- \frac{K H_L}{1+G^{\prime} K H_L}   n
 \end{alignat}
 with \(K = \frac{k}{1 + H_H G k}\)
-\subsubsection{Asymptotic behavior}
+\paragraph{Asymptotic behavior}
 We now want to study the asymptotic system obtained when using very high value of \(k\)
 \begin{equation}
   \lim_{k\to\infty} K = \lim_{k\to\infty} \frac{k}{1+H_H G k} = \left( H_H G \right)^{-1}
@@ -1166,7 +1379,7 @@ The process of designing a controller \(K(s)\) in order to obtain the desired sh
 The equations \eqref{eq:detail_control_cl_system_y} and \eqref{eq:detail_control_cl_system_u} describing the dynamics of the studied feedback architecture are not written in terms of \(K\) but in terms of the complementary filters \(H_L\) and \(H_H\).
 
 In this section, we then translate the typical specifications into the desired shapes of the complementary filters \(H_L\) and \(H_H\).\\
-\subsubsection{Nominal Stability (NS)}
+\paragraph{Nominal Stability (NS)}
 The closed-loop system is stable if all its elements are stable (\(K\), \(G^\prime\) and \(H_L\)) and if the sensitivity function (\(S = \frac{1}{1 + G^\prime K H_L}\)) is stable.
 
 For the nominal system (\(G^\prime = G\)), we have \(S = H_H\).
@@ -1174,7 +1387,7 @@ For the nominal system (\(G^\prime = G\)), we have \(S = H_H\).
 Nominal stability is then guaranteed if \(H_L\), \(H_H\) and \(G\) are stable and if \(G\) and \(H_H\) are minimum phase (to have \(K\) stable).
 
 Thus we must design stable and minimum phase complementary filters.\\
-\subsubsection{Nominal Performance (NP)}
+\paragraph{Nominal Performance (NP)}
 Typical performance specifications can usually be translated into upper bounds on \(|S(j\omega)|\) and \(|T(j\omega)|\).
 
 Two performance weights \(w_H\) and \(w_L\) are defined in such a way that performance specifications are satisfied if
@@ -1197,13 +1410,13 @@ The translation of typical performance requirements on the shapes of the complem
 \end{itemize}
 
 We may have other requirements in terms of stability margins, maximum or minimum closed-loop bandwidth.\\
-\subsubsection{Closed-Loop Bandwidth}
+\paragraph{Closed-Loop Bandwidth}
 The closed-loop bandwidth \(\w_B\) can be defined as the frequency where \(\abs{S(j\w)}\) first crosses \(\frac{1}{\sqrt{2}}\) from below.
 
 If one wants the closed-loop bandwidth to be at least \(\w_B^*\) (e.g. to stabilize an unstable pole), one can required that \(|S(j\omega)| \le \frac{1}{\sqrt{2}}\) below \(\omega_B^*\) by designing \(w_H\) such that \(|w_H(j\omega)| \ge \sqrt{2}\) for \(\omega \le \omega_B^*\).
 
 Similarly, if one wants the closed-loop bandwidth to be less than \(\w_B^*\), one can approximately require that the magnitude of \(T\) is less than \(\frac{1}{\sqrt{2}}\) at frequencies above \(\w_B^*\) by designing \(w_L\) such that \(|w_L(j\omega)| \ge \sqrt{2}\) for \(\omega \ge \omega_B^*\).\\
-\subsubsection{Classical stability margins}
+\paragraph{Classical stability margins}
 Gain margin (GM) and phase margin (PM) are usual specifications on controlled system.
 Minimum GM and PM can be guaranteed by limiting the maximum magnitude of the sensibility function \(M_S = \max_{\omega} |S(j\omega)|\):
 \begin{equation}
@@ -1217,7 +1430,7 @@ For the nominal system \(M_S = \max_\omega |S| = \max_\omega |H_H|\), so one can
   |H_H(j\omega)| \le 2 \quad \forall\omega
 \end{equation}
 and thus obtain acceptable stability margins.\\
-\subsubsection{Response time to change of reference signal}
+\paragraph{Response time to change of reference signal}
 For the nominal system, the model is accurate and the transfer function from reference signal \(r\) to output \(y\) is \(1\) \eqref{eq:detail_control_cl_performance_y} and does not depends of the complementary filters.
 
 However, one can add a pre-filter as shown in Figure \ref{fig:detail_control_sf_arch_class_prefilter}.
@@ -1234,7 +1447,7 @@ Typically, \(K_r\) is a low pass filter of the form
   K_r(s) = \frac{1}{1 + \tau s}
 \end{equation}
 with \(\tau\) corresponding to the desired response time.\\
-\subsubsection{Input usage}
+\paragraph{Input usage}
 Input usage due to disturbances \(d_y\) and measurement noise \(n\) is determined by \(\big|\frac{u}{d_y}\big| = \big|\frac{u}{n}\big| = \big|G^{-1}H_L\big|\).
 Thus it can be limited by setting an upper bound on \(|H_L|\).
 
@@ -1242,7 +1455,7 @@ Thus it can be limited by setting an upper bound on \(|H_L|\).
 Input usage due to reference signal \(r\) is determined by \(\big|\frac{u}{r}\big| = \big|G^{-1} K_r\big|\) when using a pre-filter (Figure \ref{fig:detail_control_sf_arch_class_prefilter}) and \(\big|\frac{u}{r}\big| = \big|G^{-1}\big|\) otherwise.
 
 Proper choice of \(|K_r|\) is then useful to limit input usage due to change of reference signal.\\
-\subsubsection{Robust Stability (RS)}
+\paragraph{Robust Stability (RS)}
 Robustness stability represents the ability of the control system to remain stable even though there are differences between the actual system \(G^\prime\) and the model \(G\) that was used to design the controller.
 These differences can have various origins such as unmodelled dynamics or non-linearities.
 
@@ -1281,7 +1494,7 @@ Robust stability is then guaranteed by having the low pass filter \(H_L\) satisf
 \end{equation}
 
 To ensure robust stability condition \eqref{eq:detail_control_nominal_perf_hl} can be used if \(w_L\) is designed in such a way that \(|w_L| \ge |w_I| (2 + |w_I|)\).\\
-\subsubsection{Robust Performance (RP)}
+\paragraph{Robust Performance (RP)}
 Robust performance is a property for a controlled system to have its performance guaranteed even though the dynamics of the plant is changing within specified bounds.
 
 For robust performance, we then require to have the performance condition valid for all possible plants in the defined uncertainty set:
@@ -1321,7 +1534,7 @@ One should be aware than when looking for a robust performance condition, only t
 \label{ssec:detail_control_analytical_complementary_filters}
 \section{Numerical Example}
 \label{ssec:detail_control_simulations}
-\subsubsection{Procedure}
+\paragraph{Procedure}
 
 In order to apply this control technique, we propose the following procedure:
 \begin{enumerate}
@@ -1335,7 +1548,7 @@ If one does not want to use the \(\mathcal{H}_\infty\) synthesis, one can use pr
 \item Design a pre-filter \(K_r\) if requirements on input usage or response to reference change are not met
 \item Control implementation: Filter the measurement with \(H_L\), implement the controller \(K\) and the pre-filter \(K_r\) as shown on Figure \ref{fig:detail_control_sf_arch_class_prefilter}
 \end{enumerate}
-\subsubsection{Plant}
+\paragraph{Plant}
 Let's consider the problem of controlling an active vibration isolation system that consist of a mass \(m\) to be isolated, a piezoelectric actuator and a geophone.
 
 We represent this system by a mass-spring-damper system as shown Figure \ref{fig:detail_control_mech_sys_alone} where \(m\) typically represents the mass of the payload to be isolated, \(k\) and \(c\) represent respectively the stiffness and damping of the mount.
@@ -1361,7 +1574,7 @@ Its bode plot is shown on Figure \ref{fig:detail_control_bode_plot_mech_sys}.
 \includegraphics[scale=1]{figs/detail_control_bode_plot_mech_sys.png}
 \caption{\label{fig:detail_control_bode_plot_mech_sys}Bode plot of the transfer function \(G(s)\) from \(F\) to \(x\)}
 \end{figure}
-\subsubsection{Requirements}
+\paragraph{Requirements}
 The control objective is to isolate the displacement \(x\) of the mass from the ground motion \(w\).
 
 The disturbance rejection should be at least \(10\) at \(\SI{2}{\hertz}\) and with a slope of \(-2\) below \(\SI{2}{\hertz}\) until a rejection of \(10^4\).
@@ -1398,7 +1611,7 @@ All the requirements on \(H_L\) and \(H_H\) are represented on Figure \ref{fig:d
 \end{subfigure}
 \caption{\label{fig:detail_control_spec_S_T_obtained_filters}Caption with reference to sub figure (\subref{fig:detail_control_spec_S_T}) (\subref{fig:detail_control_hinf_filters_result_weights})}
 \end{figure}
-\subsubsection{Design of the filters}
+\paragraph{Design of the filters}
 
 \textbf{Or maybe use analytical formulas as proposed here: \href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Complementary filters using analytical formula}}
 
@@ -1418,7 +1631,7 @@ After the \(\hinf\text{-synthesis}\), we obtain \(H_L\) and \(H_H\), and we plot
     H_H &= \frac{0.9936 (s+111.1) (s^2 + 0.3988s + 0.08464)}{(s+57.99) (s^2 + 65.77s + 2981)}
   \end{align}
 \end{subequations}
-\subsubsection{Controller analysis}
+\paragraph{Controller analysis}
 The controller is \(K = \left( H_H G \right)^{-1}\).
 A low pass filter is added to \(K\) so that it is proper and implementable.
 
@@ -1447,7 +1660,7 @@ It is implemented as shown on Figure \ref{fig:detail_control_mech_sys_alone_ctrl
 \end{subfigure}
 \caption{\label{fig:detail_control_bode_Kfb_loop_gain}Caption with reference to sub figure (\subref{fig:detail_control_bode_Kfb}) (\subref{fig:detail_control_bode_plot_loop_gain_robustness})}
 \end{figure}
-\subsubsection{Robustness analysis}
+\paragraph{Robustness analysis}
 The robust stability can be access on the nyquist plot (Figure \ref{fig:detail_control_nyquist_robustness}).
 
 The robust performance is shown on Figure \ref{fig:detail_control_robust_perf}.