diff --git a/figs/detail_control_coupled_plant_bode.pdf b/figs/detail_control_coupled_plant_bode.pdf
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diff --git a/figs/detail_control_svd_plant.pdf b/figs/detail_control_svd_plant.pdf
new file mode 100644
index 0000000..c1d6cd7
Binary files /dev/null and b/figs/detail_control_svd_plant.pdf differ
diff --git a/figs/detail_control_svd_plant.png b/figs/detail_control_svd_plant.png
new file mode 100644
index 0000000..f9906f3
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diff --git a/figs/detail_control_weights_W1_W2.pdf b/figs/detail_control_weights_W1_W2.pdf
index 0bf8529..0317631 100644
Binary files a/figs/detail_control_weights_W1_W2.pdf and b/figs/detail_control_weights_W1_W2.pdf differ
diff --git a/figs/detail_control_weights_W1_W2.png b/figs/detail_control_weights_W1_W2.png
index ef8835a..5a2a1e7 100644
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diff --git a/nass-control.org b/nass-control.org
index 69f0797..eed7f12 100644
--- a/nass-control.org
+++ b/nass-control.org
@@ -153,6 +153,7 @@ file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org
 SCHEDULED: <2025-04-03 Thu>
 
 file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
+Especially [[file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org::*SVD / Jacobian / Model decoupling comparison][SVD / Jacobian / Model decoupling comparison]]
 
 - [ ] Copy Content
 - [ ] Copy Tikz figures
@@ -1128,8 +1129,6 @@ W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
 #+begin_src matlab :exports none :results none
 %% description
 figure;
-tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
-ax1 = nexttile();
 hold on;
 set(gca,'ColorOrderIndex',1)
 plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
@@ -1893,7 +1892,7 @@ Several examples were used to emphasize the simplicity and the effectiveness of
 However, the shaping of the complementary filters' magnitude does not allow to directly optimize the super sensor noise and dynamical characteristics.
 Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.
 
-* Decoupling Strategies
+* Decoupling
 :PROPERTIES:
 :HEADER-ARGS:matlab+: :tangle matlab/detail_control_2_decoupling.m
 :END:
@@ -1901,6 +1900,8 @@ Future work will aim at developing a complementary filter synthesis method that
 
 ** Introduction                                                      :ignore:
 
+- [ ] Add some citations about different methods
+
 # *This report is based on*:
 # - file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
 # - [X] Maybe not relevant, as it is experimental results based on the Stewart platform file:/home/thomas/Cloud/meetings/group-meetings-me/2021-08-16-Nano-Hexapod-Control/2021-08-16-Nano-Hexapod-Control.org
@@ -1915,10 +1916,15 @@ Assumptions:
 Review of decoupling strategies for Stewart platforms:
 - [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Decoupling Strategies][Decoupling Strategies]]
 
+The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators.
 
-- [ ] What example should be taken?
-  *3dof system*? stewart platform?
-  Maybe simpler.
+It is structured as follow:
+- Section ref:ssec:detail_control_decoupling_comp_model: the model used to compare/test decoupling strategies is presented
+- Section ref:ssec:detail_control_comp_jacobian: decoupling using Jacobian matrices is presented
+- Section ref:ssec:detail_control_comp_modal: modal decoupling is presented
+- Section ref:ssec:detail_control_comp_svd: SVD decoupling is presented
+- Section ref:ssec:detail_control_decoupling_comp: the three decoupling methods are applied on the test model and compared
+- Conclusions are drawn on the three decoupling methods
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -1941,28 +1947,839 @@ Review of decoupling strategies for Stewart platforms:
 <<m-init-other>>
 #+end_src
 
-** Interaction Analysis
+** Test Model
+<<ssec:detail_control_decoupling_comp_model>>
 
-** Decentralized Control (actuator frame)
+Let's consider a parallel manipulator with several collocated actuator/sensors pairs.
 
-** Center of Stiffness and center of Mass
+System in Figure ref:fig:detail_control_model_test_decoupling will serve as an example.
 
-- Example
-- Show
+We will note:
+- $b_i$: location of the joints on the top platform
+- $\hat{s}_i$: unit vector corresponding to the struts direction
+- $k_i$: stiffness of the struts
+- $\tau_i$: actuator forces
+- $O_M$: center of mass of the solid body
+- $\mathcal{L}_i$: relative displacement of the struts
+
+#+name: fig:detail_control_model_test_decoupling
+#+caption: Model use to compare decoupling techniques
+[[file:figs/detail_control_model_test_decoupling.png]]
+
+#+begin_src matlab
+%% System parameters
+l  = 1.0; % Length of the mass [m]
+h  = 2*1.7; % Height of the mass [m]
+
+la = l/2; % Position of Act. [m]
+ha = h/2; % Position of Act. [m]
+
+m = 400; % Mass [kg]
+I = 115; % Inertia [kg m^2]
+
+%% Actuator Damping [N/(m/s)]
+c1 = 2e1;
+c2 = 2e1;
+c3 = 2e1;
+
+%% Actuator Stiffness [N/m]
+k1 = 15e3;
+k2 = 15e3;
+k3 = 15e3;
+
+%% Unit vectors of the actuators
+s1 = [1;0];
+s2 = [0;1];
+s3 = [0;1];
+
+%%  Location of the joints
+Mb1 = [-l/2;-ha];
+Mb2 = [-la; -h/2];
+Mb3 = [ la; -h/2];
+
+%% Jacobian matrix
+J = [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)];
+
+%% Stiffnesss and Damping matrices of the struts
+Kr = diag([k1,k2,k3]);
+Cr = diag([c1,c2,c3]);
+
+%% Mass Matrix in frame {M}
+M = diag([m,m,I]);
+
+%% Stiffness Matrix in frame {M}
+K = J'*Kr*J;
+
+%% Damping Matrix in frame {M}
+C = J'*Cr*J;
+
+%% Plant in frame {M}
+G = J*inv(M*s^2 + C*s + K)*J';
+#+end_src
+
+The magnitude of the coupled plant $G$ is shown in Figure ref:fig:detail_control_coupled_plant_bode.
+
+#+begin_src matlab :exports none
+figure;
+tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+for out_i = 1:3
+    for in_i = 1:3
+        nexttile;
+        plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), 'k-', ...
+             'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i));
+        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+        xlim([1e-1, 2e1]); ylim([1e-6, 1e-2]);
+        legend('location', 'northeast', 'FontSize', 8);
+
+        if in_i == 1
+            ylabel('Mag. [m/N]')
+        else
+            set(gca, 'YTickLabel',[]);
+        end
+
+        if out_i == 3
+            xlabel('Frequency [Hz]')
+        else
+            set(gca, 'XTickLabel',[]);
+        end
+    end
+end
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_coupled_plant_bode.pdf', 'width', 'full', 'height', 'tall');
+#+end_src
+
+#+name: fig:detail_control_coupled_plant_bode
+#+caption: Magnitude of the coupled plant.
+#+RESULTS:
+[[file:figs/detail_control_coupled_plant_bode.png]]
+
+** Decentralized Plant / Control in the frame of the struts
+** Jacobian Decoupling
+<<ssec:detail_control_comp_jacobian>>
+
+The Jacobian matrix can be used to:
+- 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{equation}
+  \bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}
+\end{equation}
+
+The obtained plan corresponds to forces/torques applied on origin of frame $\{O\}$ to the translation/rotation of the payload expressed in frame $\{O\}$.
+
+#+begin_src latex :file detail_control_jacobian_decoupling_arch.pdf
+\begin{tikzpicture}
+  \node[block] (G) {$\bm{G}$};
+  \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.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]{$\bm{\mathcal{X}}_{\{O\}}$};
+
+  \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) {$\bm{G}_{\{O\}}$};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+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]]
+
+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:
+- $\bm{\mathcal{F}}_{\{O\}}$ 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 $\{O\}$
+
+It is then easy to include a reference tracking input that specify the wanted motion of the payload in the frame $\{O\}$.
 
 ** Modal Decoupling
+<<ssec:detail_control_comp_modal>>
 
-** Data Based Decoupling
+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}}
+\end{equation}
 
-- Static decoupling
-- SVD
+And the measurement output is a combination of the motion variable $\bm{x}$:
+\begin{equation}
+\bm{y} = C_{ox} \bm{x} + C_{ov} \dot{\bm{x}}
+\end{equation}
+
+Let's make a *change of variables*:
+\begin{equation}
+\boxed{\bm{x} = \Phi \bm{x}_m}
+\end{equation}
+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}
+\end{equation}
+And the measured output is:
+\begin{equation}
+\bm{y} = C_{ox} \Phi \bm{x}_m + C_{ov} \Phi \dot{\bm{x}}_m
+\end{equation}
+
+By pre-multiplying the EoM by $\Phi^T$:
+\begin{equation}
+\Phi^T M \Phi \bm{\ddot{x}}_m + \Phi^T C \Phi \bm{\dot{x}}_m  + \Phi^T K \Phi \bm{x}_m = \Phi^T J^T \bm{\tau}
+\end{equation}
+
+And we note:
+- $M_m = \Phi^T M \Phi = \text{diag}(\mu_i)$ the modal mass matrix
+- $C_m = \Phi^T C \Phi = \text{diag}(2 \xi_i \mu_i \omega_i)$ (classical damping)
+- $K_m = \Phi^T K \Phi = \text{diag}(\mu_i \omega_i^2)$ the modal stiffness matrix
+
+And we have:
+\begin{equation}
+  \ddot{\bm{x}}_m + 2 \Xi \Omega \dot{\bm{x}}_m + \Omega^2 \bm{x}_m = \mu^{-1} \Phi^T J^T \bm{\tau}
+\end{equation}
+with:
+- $\mu = \text{diag}(\mu_i)$
+- $\Omega = \text{diag}(\omega_i)$
+- $\Xi = \text{diag}(\xi_i)$
+
+And we call the *modal input matrix*:
+\begin{equation}
+  \boxed{B_m = \mu^{-1} \Phi^T J^T}
+\end{equation}
+And the *modal output matrices*:
+\begin{equation}
+  \boxed{C_m = C_{ox} \Phi + C_{ov} \Phi s}
+\end{equation}
+
+
+Let's note the "modal input":
+\begin{equation}
+\bm{\tau}_m = B_m \bm{\tau}
+\end{equation}
+
+The transfer function from $\bm{\tau}_m$ to $\bm{x}_m$ is:
+\begin{equation} \label{eq:modal_eq}
+\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
+\begin{tikzpicture}
+  \node[block] (G) {$\bm{G}$};
+  \node[block, left=0.6 of G] (Bm) {$B_m^{-1}$};
+  \node[block, right=0.6 of G] (Cm) {$C_m^{-1}$};
+
+  % Connections and labels
+  \draw[<-] (Bm.west) -- ++(-1.0, 0) node[above right]{$\bm{\tau}_m$};
+  \draw[->] (Bm.east) -- (G.west)  node[above left]{$\bm{\tau}$};
+  \draw[->] (G.east) -- (Cm.west)  node[above left]{$\bm{y}$};
+  \draw[->] (Cm.east) -- ++( 1.0, 0)  node[above left]{$\bm{x}_m$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(Bm.south west) (Cm.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gm) {};
+    \node[above] at (Gm.north) {$\bm{G}_m$};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:modal_decoupling_architecture
+#+caption: Modal Decoupling Architecture
+#+RESULTS:
+[[file:figs/detail_control_decoupling_modal.png]]
+
+The system $\bm{G}_m(s)$ shown in Figure ref:fig:modal_decoupling_architecture is diagonal eqref:eq:modal_eq.
+
+Modal decoupling requires to have the equations of motion of the system.
+From the equations of motion (and more precisely the mass and stiffness matrices), the mode shapes $\Phi$ are computed.
+
+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.
+
+** SVD Decoupling
+<<ssec:detail_control_comp_svd>>
+
+Procedure:
+- Identify the dynamics of the system from inputs to outputs (can be obtained experimentally)
+- Choose a frequency where we want to decouple the system (usually, the crossover frequency is a good choice)
+#+begin_src matlab :eval no
+%% Decoupling frequency [rad/s]
+wc = 2*pi*10;
+
+%% System's response at the decoupling frequency
+H1 = evalfr(G, j*wc);
+#+end_src
+- Compute a real approximation of the system's response at that frequency
+#+begin_src matlab :eval no
+%% Real approximation of G(j.wc)
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+#+end_src
+- Perform a Singular Value Decomposition of the real approximation
+#+begin_src matlab :eval no
+[U,S,V] = svd(H1);
+#+end_src
+- Use the singular input and output matrices to decouple the system as shown in Figure ref:fig:detail_control_decoupling_svd
+  \[ G_{svd}(s) = U^{-1} G(s) V^{-T} \]
+#+begin_src matlab :eval no
+Gsvd = inv(U)*G*inv(V');
+#+end_src
+
+#+begin_src latex :file detail_control_decoupling_svd.pdf
+\begin{tikzpicture}
+  \node[block] (G) {$\bm{G}$};
+
+  \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
+  \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
+
+  % Connections and labels
+  \draw[<-] (V.west) -- ++(-0.8, 0) node[above right]{$u$};
+  \draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
+  \draw[->] (G.east) -- (U.west) node[above left]{$a$};
+  \draw[->] (U.east) -- ++( 0.8, 0) node[above left]{$y$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gsvd) {};
+    \node[above] at (Gsvd.north) {$\bm{G}_{SVD}$};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:detail_control_decoupling_svd
+#+caption: Decoupled plant $\bm{G}_{SVD}$ using the Singular Value Decomposition
+#+RESULTS:
+[[file:figs/detail_control_decoupling_svd.png]]
+
+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.
+
+- [ ] Do we loose any physical meaning of the obtained inputs and outputs?
+- [ ] Can we take advantage of the fact that U and V are unitary?
+
+** Comparison
+<<ssec:detail_control_decoupling_comp>>
+*** Jacobian Decoupling
+Decoupling properties depends on the chosen frame $\{O\}$.
+
+Let's take the CoM as the decoupling frame.
+
+#+begin_src matlab
+Gx = pinv(J)*G*pinv(J');
+Gx.InputName  = {'Fx', 'Fy', 'Mz'};
+Gx.OutputName  = {'Dx', 'Dy', 'Rz'};
+#+end_src
+
+#+begin_src matlab :exports none
+freqs = logspace(-1, 2, 1000);
+figure;
+
+% Magnitude
+hold on;
+for i_in = 1:3
+    for i_out = [i_in+1:3]
+        plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$G_{x}(i,j)\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for i_in_out = 1:3
+    plot(freqs, abs(squeeze(freqresp(Gx(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{x}(%d,%d)$', i_in_out, i_in_out));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+ylim([1e-7, 1e-1]);
+legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_jacobian_plant.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:detail_control_jacobian_plant
+#+caption: Plant decoupled using the Jacobian matrices $G_x(s)$
+#+RESULTS:
+[[file:figs/detail_control_jacobian_plant.png]]
+
+*** 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} \\
+\bm{y} &= \mathcal{L} = J \bm{x}; \quad C_{ox} = J; \quad C_{ov} = 0 \\
+M &= \begin{bmatrix}
+m & 0 & 0 \\
+0 & m & 0 \\
+0 & 0 & I
+\end{bmatrix}; \quad K = J' \begin{bmatrix}
+k & 0 & 0 \\
+0 & k & 0 \\
+0 & 0 & k
+\end{bmatrix} J; \quad C = J' \begin{bmatrix}
+c & 0 & 0 \\
+0 & c & 0 \\
+0 & 0 & c
+\end{bmatrix} J
+\end{align}
+
+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_src matlab
+%% Modal Decomposition
+[V,D] = eig(M\K);
+
+%% Modal Mass Matrix
+mu = V'*M*V;
+
+%% Modal output matrix
+Cm = J*V;
+
+%% Modal input matrix
+Bm = inv(mu)*V'*J';
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+data2orgtable(Bm, {}, {}, ' %.4f ');
+#+end_src
+
+#+name: tab:modal_decoupling_Bm
+#+caption: $B_m$ matrix
+#+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 ');
+#+end_src
+
+#+name: tab:modal_decoupling_Cm
+#+caption: $C_m$ matrix
+#+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 |
+
+And the plant in the modal space is defined below and its magnitude is shown in Figure ref:fig:detail_control_modal_plant.
+#+begin_src matlab
+Gm = inv(Cm)*G*inv(Bm);
+#+end_src
+
+#+begin_src matlab :exports none
+freqs = logspace(-1, 2, 1000);
+figure;
+
+% Magnitude
+hold on;
+for i_in = 1:3
+    for i_out = [i_in+1:3]
+        plot(freqs, abs(squeeze(freqresp(Gm(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(Gm(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$G_m(i,j)\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for i_in_out = 1:3
+    plot(freqs, abs(squeeze(freqresp(Gm(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_m(%d,%d)$', i_in_out, i_in_out));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+ylim([1e-6, 1e2]);
+legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_modal_plant.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:detail_control_modal_plant
+#+caption: Modal plant $G_m(s)$
+#+RESULTS:
+[[file:figs/detail_control_modal_plant.png]]
+
+Let's now close one loop at a time and see how the transmissibility changes.
+
+*** SVD Decoupling
+#+begin_src matlab
+%% Decoupling frequency [rad/s]
+wc = 2*pi*10;
+
+%% System's response at the decoupling frequency
+H1 = evalfr(G, j*wc);
+
+%% Real approximation of G(j.wc)
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+
+[U,S,V] = svd(H1);
+
+Gsvd = inv(U)*G*inv(V');
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+data2orgtable(H1, {}, {}, ' %.2g ');
+#+end_src
+
+#+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$
+#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
+#+attr_latex: :center t :booktabs t :float t
+#+RESULTS:
+|   -8e-06 |  2.1e-06 | -2.1e-06 |
+|  2.1e-06 | -1.3e-06 | -2.5e-08 |
+| -2.1e-06 | -2.5e-08 | -1.3e-06 |
+
+- [ ] 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_src matlab :exports none
+freqs = logspace(-1, 2, 1000);
+figure;
+
+% Magnitude
+hold on;
+for i_in = 1:3
+    for i_out = [i_in+1:3]
+        plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$G_{svd}(i,j)\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for i_in_out = 1:3
+    plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd}(%d,%d)$', i_in_out, i_in_out));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+% ylim([1e-8, 1e-2]);
+legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_svd_plant.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:detail_control_svd_plant
+#+caption: Svd plant $G_m(s)$
+#+RESULTS:
+[[file:figs/detail_control_svd_plant.png]]
+
+** TODO Robustness of the decoupling strategies?                   :noexport:
+*** Introduction                                                    :ignore:
+What happens if we add an additional resonance in the system (Figure ref:fig:model_test_decoupling_spurious_res).
+
+Having less actuator than DoF (under-actuated system):
+- modal decoupling: can still control first $n$ modes?
+- SVD decoupling: does not matter
+- Jacobian decoupling: could give poor decoupling?
+
+#+name: fig:model_test_decoupling_spurious_res
+#+caption: Plant with spurious resonance (additional DoF)
+[[file:figs/model_test_decoupling_spurious_res.png]]
+
+*** Plant
+A multi body model of the system in Figure ref:fig:model_test_decoupling_spurious_res has been made using Simscape.
+
+Its parameters are defined below:
+#+begin_src matlab
+leq = 20e-3; % Equilibrium length of struts [m]
+mr = 5; % [kg]
+kr = (2*pi*10)^2*mr; % Stiffness [N/m]
+cr = 1e1; % Damping [N/(m/s)]
+
+m = 400 - mr; % Mass [kg]
+#+end_src
+
+The plant is then identified and shown in Figure ref:fig:detail_control_coupled_plant_bode_spurious.
+The added resonance only slightly modifies the plant around 10Hz.
+#+begin_src matlab :exports none
+%% Name of the Simulink File
+mdl = 'suspended_mass';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1;
+
+Gr = linearize(mdl, io);
+Gr.InputName  = {'F1', 'F2', 'F3'};
+Gr.OutputName = {'L1', 'L2', 'L3'};
+#+end_src
+
+#+begin_src matlab :exports none
+figure;
+tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+for out_i = 1:3
+    for in_i = 1:3
+        nexttile;
+        hold on;
+        plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), 'k-', ...
+             'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i));
+        plot(freqs, abs(squeeze(freqresp(Gr(out_i,in_i), freqs, 'Hz'))), 'k--', ...
+             'HandleVisibility', 'off');
+        hold off;
+        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+        xlim([1e-1, 2e1]); ylim([1e-6, 1e-2]);
+        legend('location', 'northeast', 'FontSize', 8);
+
+        if in_i == 1
+            ylabel('Mag. [m/N]')
+        else
+            set(gca, 'YTickLabel',[]);
+        end
+
+        if out_i == 3
+            xlabel('Frequency [Hz]')
+        else
+            set(gca, 'XTickLabel',[]);
+        end
+    end
+end
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_coupled_plant_bode_spurious.pdf', 'width', 'full', 'height', 'tall');
+#+end_src
+
+#+name: fig:detail_control_coupled_plant_bode_spurious
+#+caption: Magnitude of the coupled plant without additional mode (solid) and with the additional mode (dashed).
+#+RESULTS:
+[[file:figs/detail_control_coupled_plant_bode_spurious.png]]
+
+*** Jacobian Decoupling
+The obtained plant is decoupled using the Jacobian matrix.
+
+#+begin_src matlab
+Gxr = pinv(J)*Gr*pinv(J');
+Gxr.InputName  = {'Fx', 'Fy', 'Mz'};
+Gxr.OutputName  = {'Dx', 'Dy', 'Rz'};
+#+end_src
+
+The obtained plant is shown in Figure ref:fig:detail_control_jacobian_plant_spurious and is not much different than for the plant without the spurious resonance.
+#+begin_src matlab :exports none
+freqs = logspace(-1, 2, 1000);
+figure;
+
+% Magnitude
+hold on;
+for i_in = 1:3
+    for i_out = [i_in+1:3]
+        plot(freqs, abs(squeeze(freqresp(Gxr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(Gxr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$G_{x,r}(i,j)\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for i_in_out = 1:3
+    plot(freqs, abs(squeeze(freqresp(Gxr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{x,r}(%d,%d)$', i_in_out, i_in_out));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+ylim([1e-7, 1e-1]);
+legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_jacobian_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:detail_control_jacobian_plant_spurious
+#+caption: Plant with spurious resonance decoupled using the Jacobian matrices $G_{x,r}(s)$
+#+RESULTS:
+[[file:figs/detail_control_jacobian_plant_spurious.png]]
+
+*** Modal Decoupling
+The obtained plant is now decoupled using the modal matrices obtained with the plant not including the added resonance.
+#+begin_src matlab
+Gmr = inv(Cm)*Gr*inv(Bm);
+#+end_src
+
+The obtained decoupled plant is shown in Figure ref:fig:detail_control_modal_plant_spurious.
+Compare to the decoupled plant in Figure ref:fig:detail_control_modal_plant, the added resonance induces some coupling, especially around the frequency of the added spurious resonance.
+#+begin_src matlab :exports none
+freqs = logspace(-1, 2, 1000);
+figure;
+
+% Magnitude
+hold on;
+for i_in = 1:3
+    for i_out = [i_in+1:3]
+        plot(freqs, abs(squeeze(freqresp(Gmr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(Gmr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$G_{m,r}(i,j)\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for i_in_out = 1:3
+    plot(freqs, abs(squeeze(freqresp(Gmr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{m,r}(%d,%d)$', i_in_out, i_in_out));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+ylim([1e-6, 1e2]);
+legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_modal_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:detail_control_modal_plant_spurious
+#+caption: Modal plant including spurious resonance $G_{m,r}(s)$
+#+RESULTS:
+[[file:figs/detail_control_modal_plant_spurious.png]]
+
+*** SVD Decoupling
+The SVD decoupling is performed on the new obtained plant.
+The decoupling frequency is slightly shifted in order not to interfere too much with the spurious resonance.
+
+#+begin_src matlab
+%% Decoupling frequency [rad/s]
+wc = 2*pi*7;
+
+%% System's response at the decoupling frequency
+H1 = evalfr(Gr, j*wc);
+
+%% Real approximation of G(j.wc)
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+
+[U,S,V] = svd(H1);
+
+Gsvdr = inv(U)*Gr*inv(V');
+#+end_src
+
+The obtained plant is shown in Figure ref:fig:detail_control_svd_plant_spurious.
+#+begin_src matlab :exports none
+freqs = logspace(-1, 2, 1000);
+figure;
+
+% Magnitude
+hold on;
+for i_in = 1:3
+    for i_out = [i_in+1:3]
+        plot(freqs, abs(squeeze(freqresp(Gsvdr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(Gsvdr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$G_{svd,r}(i,j)\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for i_in_out = 1:3
+    plot(freqs, abs(squeeze(freqresp(Gsvdr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd,r}(%d,%d)$', i_in_out, i_in_out));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+ylim([1e-8, 1e-2]);
+legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/detail_control_svd_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:detail_control_svd_plant_spurious
+#+caption: SVD decoupled plant including a spurious resonance $G_{svd,r}(s)$
+#+RESULTS:
+[[file:figs/detail_control_svd_plant_spurious.png]]
 
 ** Conclusion
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
 
-Table that compares all the strategies.
+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:
+- DC decoupling: pre-multiply the plant by $G(0)^{-1}$
+- full decoupling: pre-multiply the plant by $G(s)^{-1}$
+
+
+#+name: tab:detail_control_decoupling_strategies_comp
+#+caption: Comparison of decoupling strategies
+#+attr_latex: :environment tabularx :width \linewidth :align lXXX
+#+attr_latex: :center t :booktabs t :font \scriptsize
+|                           | *Jacobian*                                                                             | *Modal*                                                            | *SVD*                                                  |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Philosophy*              | Topology Driven                                                                        | Physics Driven                                                     | Data Driven                                            |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Requirements*            | Known geometry                                                                         | Known equations of motion                                          | Identified FRF                                         |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Decoupling Matrices*     | Decoupling using $J$ obtained from geometry                                            | Decoupling using $\Phi$ obtained from modal decomposition          | Decoupling using $U$ and $V$ obtained from SVD         |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Decoupled Plant*         | $\bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}$                                | $\bm{G}_m = C_m^{-1} \bm{G} B_m^{-1}$                              | $\bm{G}_{svd}(s) = U^{-1} \bm{G}(s) V^{-T}$            |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Implemented Controller*  | $\bm{K}_{\{O\}} = J_{\{O\}}^{-T} \bm{K}_{d}(s) J_{\{O\}}^{-1}$                         | $\bm{K}_m = B_m^{-1} \bm{K}_{d}(s) C_m^{-1}$                       | $\bm{K}_{svd}(s) = V^{-T} \bm{K}_{d}(s) U^{-1}$        |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Physical Interpretation* | Forces/Torques to Displacement/Rotation in chosen frame                                | Inputs to excite individual modes                                  | Directions of max to min controllability/observability |
+|                           |                                                                                        | Output to sense individual modes                                   |                                                        |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Decoupling Properties*   | Decoupling at low or high frequency depending on the chosen frame                      | Good decoupling at all frequencies                                 | Good decoupling near the chosen frequency              |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Pros*                    | Physical inputs / outputs                                                              | Target specific modes                                              | Good Decoupling near the crossover                     |
+|                           | Good decoupling at High frequency (diagonal mass matrix if Jacobian taken at the CoM)  | 2nd order diagonal plant                                           | Very General                                           |
+|                           | Good decoupling at Low frequency (if Jacobian taken at specific point)                 |                                                                    |                                                        |
+|                           | Easy integration of meaningful reference inputs                                        |                                                                    |                                                        |
+|                           |                                                                                        |                                                                    |                                                        |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Cons*                    | Coupling between force/rotation may be high at low frequency (non diagonal terms in K) | Need analytical equations                                          | Loose the physical meaning of inputs /outputs          |
+|                           | Limited to parallel mechanisms (?)                                                     |                                                                    | Decoupling depends on the real approximation validity  |
+|                           | If good decoupling at all frequencies => requires specific mechanical architecture     |                                                                    | Diagonal plants may not be easy to control             |
+|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
+| *Applicability*           | Parallel Mechanisms                                                                    | Systems whose dynamics that can be expressed with M and K matrices | Very general                                           |
+|                           | Only small motion for the Jacobian matrix to stay constant                             |                                                                    | Need FRF data (either experimentally or analytically)  |
 
 * Closed-Loop Shaping using Complementary Filters
 :PROPERTIES:
@@ -2618,7 +3435,7 @@ exportFig('figs/detail_control_spec_S_T.pdf', 'width', 'half', 'height', 'normal
 #+end_src
 
 #+name: fig:detail_control_spec_S_T_obtained_filters
-#+caption: Caption with reference to sub figure (\subref{fig:detail_control_spec_S_T}) (\subref{detail_control_hinf_filters_result_weights})
+#+caption: Caption with reference to sub figure (\subref{fig:detail_control_spec_S_T}) (\subref{fig:detail_control_hinf_filters_result_weights})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:detail_control_spec_S_T}Closed loop specifications}
diff --git a/nass-control.pdf b/nass-control.pdf
index da171a6..8ec6b2c 100644
Binary files a/nass-control.pdf and b/nass-control.pdf differ
diff --git a/nass-control.tex b/nass-control.tex
index 2b7a846..67371be 100644
--- a/nass-control.tex
+++ b/nass-control.tex
@@ -1,4 +1,4 @@
-% Created 2025-04-03 Thu 17:13
+% Created 2025-04-03 Thu 17:40
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -25,7 +25,7 @@ Several considerations:
 \item Section \ref{sec:detail_control_optimization}: How to design the controller
 \end{itemize}
 \chapter{Multiple Sensor Control}
-\label{sec:orgacbb166}
+\label{sec:orga9622a3}
 \label{sec:detail_control_multiple_sensor}
 
 \textbf{Look at what was done in the introduction \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/nass-introduction.org}{Stewart platforms: Control architecture}}
@@ -143,13 +143,13 @@ Although many design methods of complementary filters have been proposed in the
 Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters \cite{dehaeze19_compl_filter_shapin_using_synth}.
 Based on that, this paper introduces a new way to design complementary filters using the \(\mathcal{H}_\infty\) synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way.
 \section{Sensor Fusion and Complementary Filters Requirements}
-\label{sec:org9733630}
+\label{sec:org338cf90}
 \label{ssec:detail_control_sensor_fusion_requirements}
 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.
 \subsection{Sensor Fusion Architecture}
-\label{sec:org45e5dc5}
+\label{sec:org1a108b3}
 
 A general sensor fusion architecture using complementary filters is shown in Fig. \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 \(x\).
@@ -174,7 +174,7 @@ Therefore, a pair of complementary filter needs to satisfy the following conditi
 
 It will soon become clear why the complementary property is important for the sensor fusion architecture.
 \subsection{Sensor Models and Sensor Normalization}
-\label{sec:org117d609}
+\label{sec:orgf7e77fa}
 
 In order to study such sensor fusion architecture, a model for the sensors is required.
 Such model is shown in Fig. \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.
@@ -218,7 +218,7 @@ The super sensor output is therefore equal to:
 \caption{\label{fig:detail_control_fusion_super_sensor}Sensor fusion architecture with two normalized sensors.}
 \end{figure}
 \subsection{Noise Sensor Filtering}
-\label{sec:orgb1cc291}
+\label{sec:orgf16500e}
 
 In this section, it is supposed that all the sensors are perfectly normalized, such that:
 
@@ -257,7 +257,7 @@ However, the two sensors have usually high noise levels over distinct frequency
 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 reduce the noise of the super sensor.
 \subsection{Sensor Fusion Robustness}
-\label{sec:orgc1bccc3}
+\label{sec:orgfd6ea33}
 
 In practical systems the sensor normalization is not perfect and condition \eqref{eq:detail_control_perfect_dynamics} is not verified.
 
@@ -317,7 +317,7 @@ For instance, the phase \(\Delta\phi(\omega)\) added by the super sensor dynamic
 As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \(\Delta \phi\) is bounded to acceptable values.
 Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
 \section{Complementary Filters Shaping}
-\label{sec:org48106a8}
+\label{sec:orgd7c419a}
 \label{ssec:detail_control_hinf_method}
 As shown in Section \ref{ssec:detail_control_sensor_fusion_requirements}, the noise and robustness of the super sensor are a function of the complementary filters' norm.
 Therefore, a synthesis method of complementary filters that allows to shape their norm would be of great use.
@@ -325,7 +325,7 @@ In this section, such synthesis is proposed by writing the synthesis objective a
 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.
 \subsection{Synthesis Objective}
-\label{sec:org8a4a881}
+\label{sec:orgd2e0d7e}
 
 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.
@@ -340,7 +340,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
 
 \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are carefully chosen to specify the maximum wanted norm of the complementary filters during the synthesis.
 \subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
-\label{sec:org2122803}
+\label{sec:orgbfba454}
 
 In this section, it is shown that the synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimization problem and therefore solved using convenient tools readily available.
 
@@ -388,7 +388,7 @@ Note that there is only an implication between the \(\mathcal{H}_\infty\) norm c
 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}.
 In practice, this is however not an found to be an issue.
 \subsection{Weighting Functions Design}
-\label{sec:orgaaf1d4a}
+\label{sec:orga5537db}
 
 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.
@@ -434,7 +434,7 @@ The typical magnitude of a weighting function generated using \eqref{eq:detail_c
 \caption{\label{fig:detail_control_weight_formula}Magnitude of a weighting function generated using formula \eqref{eq:detail_control_weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).}
 \end{figure}
 \subsection{Validation of the proposed synthesis method}
-\label{sec:org6a4ff6e}
+\label{sec:orgd4c8c3e}
 
 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:
@@ -503,7 +503,7 @@ As expected, the obtained filters are of order \(5\), that is the sum of the wei
 This simple example illustrates the fact that the proposed methodology for complementary filters shaping is easy to use and effective.
 A more complex real life example is taken up in the next section.
 \section{``Closed-Loop'' complementary filters}
-\label{sec:org8c07343}
+\label{sec:orgc89b4c8}
 \label{ssec:detail_control_closed_loop_complementary_filters}
 
 An alternative way to implement complementary filters is by using a fundamental property of the classical feedback architecture shown in Fig. \ref{fig:detail_control_feedback_sensor_fusion}.
@@ -593,7 +593,7 @@ The obtained ``closed-loop'' complementary filters are indeed equal to the ones
 \caption{\label{fig:detail_control_hinf_filters_results_mixed_sensitivity}Bode plot of the obtained complementary filters after \(\mathcal{H}_\infty\) mixed-sensitivity synthesis}
 \end{figure}
 \section{Synthesis of a set of three complementary filters}
-\label{sec:org7f588fb}
+\label{sec:orgb190cc7}
 \label{sec:detail_control_hinf_three_comp_filters}
 
 Some applications may require to merge more than two sensors \cite{stoten01_fusion_kinet_data_using_compos_filter,fonseca15_compl}.
@@ -705,7 +705,7 @@ Figure \ref{fig:detail_control_three_complementary_filters_results} displays the
 \caption{\label{fig:detail_control_three_complementary_filters_results}Bode plot of the inverse weighting functions and of the three complementary filters obtained using the \(\mathcal{H}_\infty\) synthesis}
 \end{figure}
 \section*{Conclusion}
-\label{sec:org9b71274}
+\label{sec:orgfc547f6}
 A new method for designing complementary filters using the \(\mathcal{H}_\infty\) synthesis has been proposed.
 It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis.
 This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude.
@@ -714,10 +714,14 @@ Several examples were used to emphasize the simplicity and the effectiveness of
 
 However, the shaping of the complementary filters' magnitude does not allow to directly optimize the super sensor noise and dynamical characteristics.
 Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.
-\chapter{Decoupling Strategies}
-\label{sec:orgb61bade}
+\chapter{Decoupling}
+\label{sec:org55fd174}
 \label{sec:detail_control_decoupling}
 
+\begin{itemize}
+\item[{$\square$}] Add some citations about different methods
+\end{itemize}
+
 When dealing with MIMO systems, a typical strategy is to:
 \begin{itemize}
 \item first decouple the plant dynamics
@@ -734,40 +738,367 @@ Review of decoupling strategies for Stewart platforms:
 \item \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies}
 \end{itemize}
 
+The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators.
 
+It is structured as follow:
 \begin{itemize}
-\item[{$\square$}] What example should be taken?
-\textbf{3dof system}? stewart platform?
-Maybe simpler.
+\item Section \ref{ssec:detail_control_decoupling_comp_model}: the model used to compare/test decoupling strategies is presented
+\item Section \ref{ssec:detail_control_comp_jacobian}: decoupling using Jacobian matrices is presented
+\item Section \ref{ssec:detail_control_comp_modal}: modal decoupling is presented
+\item Section \ref{ssec:detail_control_comp_svd}: SVD decoupling is presented
+\item Section \ref{ssec:detail_control_decoupling_comp}: the three decoupling methods are applied on the test model and compared
+\item Conclusions are drawn on the three decoupling methods
 \end{itemize}
-\section{Interaction Analysis}
-\label{sec:org4e38e50}
+\section{Test Model}
+\label{sec:org07c14be}
+\label{ssec:detail_control_decoupling_comp_model}
+Let's consider a parallel manipulator with several collocated actuator/sensors pairs.
 
-\section{Decentralized Control (actuator frame)}
-\label{sec:orgbcd61c7}
-
-\section{Center of Stiffness and center of Mass}
-\label{sec:orgb312846}
+System in Figure \ref{fig:detail_control_model_test_decoupling} will serve as an example.
 
+We will note:
 \begin{itemize}
-\item Example
-\item Show
+\item \(b_i\): location of the joints on the top platform
+\item \(\hat{s}_i\): unit vector corresponding to the struts direction
+\item \(k_i\): stiffness of the struts
+\item \(\tau_i\): actuator forces
+\item \(O_M\): center of mass of the solid body
+\item \(\mathcal{L}_i\): relative displacement of the struts
 \end{itemize}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/detail_control_model_test_decoupling.png}
+\caption{\label{fig:detail_control_model_test_decoupling}Model use to compare decoupling techniques}
+\end{figure}
+
+The magnitude of the coupled plant \(G\) is shown in Figure \ref{fig:detail_control_coupled_plant_bode}.
+
+\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}
+\label{sec:orgc8c0b5f}
+\section{Jacobian Decoupling}
+\label{sec:org3e20680}
+\label{ssec:detail_control_comp_jacobian}
+
+The Jacobian matrix can be used to:
+\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{equation}
+  \bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}
+\end{equation}
+
+The obtained plan corresponds to forces/torques applied on origin of frame \(\{O\}\) to the translation/rotation of the payload expressed in frame \(\{O\}\).
+
+\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 \(\{O\}\).
 \section{Modal Decoupling}
-\label{sec:orgd3c8cf1}
+\label{sec:org08a0372}
+\label{ssec:detail_control_comp_modal}
 
-\section{Data Based Decoupling}
-\label{sec:orgc72c6af}
+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}}
+\end{equation}
+
+And the measurement output is a combination of the motion variable \(\bm{x}\):
+\begin{equation}
+\bm{y} = C_{ox} \bm{x} + C_{ov} \dot{\bm{x}}
+\end{equation}
+
+Let's make a \textbf{change of variables}:
+\begin{equation}
+\boxed{\bm{x} = \Phi \bm{x}_m}
+\end{equation}
+with:
+\begin{itemize}
+\item \(\bm{x}_m\) the modal amplitudes
+\item \(\Phi\) a matrix whose columns are the modes shapes of the system
+\end{itemize}
+
+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}
+\end{equation}
+And the measured output is:
+\begin{equation}
+\bm{y} = C_{ox} \Phi \bm{x}_m + C_{ov} \Phi \dot{\bm{x}}_m
+\end{equation}
+
+By pre-multiplying the EoM by \(\Phi^T\):
+\begin{equation}
+\Phi^T M \Phi \bm{\ddot{x}}_m + \Phi^T C \Phi \bm{\dot{x}}_m  + \Phi^T K \Phi \bm{x}_m = \Phi^T J^T \bm{\tau}
+\end{equation}
+
+And we note:
+\begin{itemize}
+\item \(M_m = \Phi^T M \Phi = \text{diag}(\mu_i)\) the modal mass matrix
+\item \(C_m = \Phi^T C \Phi = \text{diag}(2 \xi_i \mu_i \omega_i)\) (classical damping)
+\item \(K_m = \Phi^T K \Phi = \text{diag}(\mu_i \omega_i^2)\) the modal stiffness matrix
+\end{itemize}
+
+And we have:
+\begin{equation}
+  \ddot{\bm{x}}_m + 2 \Xi \Omega \dot{\bm{x}}_m + \Omega^2 \bm{x}_m = \mu^{-1} \Phi^T J^T \bm{\tau}
+\end{equation}
+with:
+\begin{itemize}
+\item \(\mu = \text{diag}(\mu_i)\)
+\item \(\Omega = \text{diag}(\omega_i)\)
+\item \(\Xi = \text{diag}(\xi_i)\)
+\end{itemize}
+
+And we call the \textbf{modal input matrix}:
+\begin{equation}
+  \boxed{B_m = \mu^{-1} \Phi^T J^T}
+\end{equation}
+And the \textbf{modal output matrices}:
+\begin{equation}
+  \boxed{C_m = C_{ox} \Phi + C_{ov} \Phi s}
+\end{equation}
+
+
+Let's note the ``modal input'':
+\begin{equation}
+\bm{\tau}_m = B_m \bm{\tau}
+\end{equation}
+
+The transfer function from \(\bm{\tau}_m\) to \(\bm{x}_m\) is:
+\begin{equation} \label{eq:modal_eq}
+\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.
+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{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/detail_control_decoupling_modal.png}
+\caption{\label{fig:modal_decoupling_architecture}Modal Decoupling Architecture}
+\end{figure}
+
+The system \(\bm{G}_m(s)\) shown in Figure \ref{fig:modal_decoupling_architecture} is diagonal \eqref{eq:modal_eq}.
+
+Modal decoupling requires to have the equations of motion of the system.
+From the equations of motion (and more precisely the mass and stiffness matrices), the mode shapes \(\Phi\) are computed.
+
+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}
+\label{sec:orgd91e1be}
+\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 Static decoupling
-\item SVD
+\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{sec:org670a0b0}
+\label{ssec:detail_control_decoupling_comp}
+\subsection{Jacobian Decoupling}
+\label{sec:org2a50d56}
+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}
+\subsection{Modal Decoupling}
+\label{sec:org6cc56e8}
+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} \\
+\bm{y} &= \mathcal{L} = J \bm{x}; \quad C_{ox} = J; \quad C_{ov} = 0 \\
+M &= \begin{bmatrix}
+m & 0 & 0 \\
+0 & m & 0 \\
+0 & 0 & I
+\end{bmatrix}; \quad K = J' \begin{bmatrix}
+k & 0 & 0 \\
+0 & k & 0 \\
+0 & 0 & k
+\end{bmatrix} J; \quad C = J' \begin{bmatrix}
+c & 0 & 0 \\
+0 & c & 0 \\
+0 & 0 & c
+\end{bmatrix} J
+\end{align}
+
+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]
+\caption{\label{tab:modal_decoupling_Bm}\(B_m\) matrix}
+\centering
+\begin{tabularx}{0.3\linewidth}{ccc}
+\toprule
+-0.0004 & -0.0007 & 0.0007\\
+-0.0151 & 0.0041 & -0.0041\\
+0.0 & 0.0025 & 0.0025\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:modal_decoupling_Cm}\(C_m\) matrix}
+\centering
+\begin{tabularx}{0.2\linewidth}{ccc}
+\toprule
+-0.1 & -1.8 & 0.0\\
+-0.2 & 0.5 & 1.0\\
+0.2 & -0.5 & 1.0\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+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}
+\caption{\label{fig:detail_control_modal_plant}Modal plant \(G_m(s)\)}
+\end{figure}
+
+Let's now close one loop at a time and see how the transmissibility changes.
+\subsection{SVD Decoupling}
+\label{sec:org1bd92ef}
+\begin{table}[htbp]
+\caption{\label{}Real approximate of \(G\) at the decoupling frequency \(\omega_c\)}
+\centering
+\begin{tabularx}{0.3\linewidth}{ccc}
+\toprule
+-8e-06 & 2.1e-06 & -2.1e-06\\
+2.1e-06 & -1.3e-06 & -2.5e-08\\
+-2.1e-06 & -2.5e-08 & -1.3e-06\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{itemize}
+\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}
-\label{sec:org3802e66}
-Table that compares all the strategies.
+\label{sec:orge4184ce}
+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}
+
+
+\begin{table}[htbp]
+\caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies}
+\centering
+\scriptsize
+\begin{tabularx}{\linewidth}{lXXX}
+\toprule
+ & \textbf{Jacobian} & \textbf{Modal} & \textbf{SVD}\\
+\midrule
+\textbf{Philosophy} & Topology Driven & Physics Driven & Data Driven\\
+\midrule
+\textbf{Requirements} & Known geometry & Known equations of motion & Identified FRF\\
+\midrule
+\textbf{Decoupling Matrices} & Decoupling using \(J\) obtained from geometry & Decoupling using \(\Phi\) obtained from modal decomposition & Decoupling using \(U\) and \(V\) obtained from SVD\\
+\midrule
+\textbf{Decoupled Plant} & \(\bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}\) & \(\bm{G}_m = C_m^{-1} \bm{G} B_m^{-1}\) & \(\bm{G}_{svd}(s) = U^{-1} \bm{G}(s) V^{-T}\)\\
+\midrule
+\textbf{Implemented Controller} & \(\bm{K}_{\{O\}} = J_{\{O\}}^{-T} \bm{K}_{d}(s) J_{\{O\}}^{-1}\) & \(\bm{K}_m = B_m^{-1} \bm{K}_{d}(s) C_m^{-1}\) & \(\bm{K}_{svd}(s) = V^{-T} \bm{K}_{d}(s) U^{-1}\)\\
+\midrule
+\textbf{Physical Interpretation} & Forces/Torques to Displacement/Rotation in chosen frame & Inputs to excite individual modes & Directions of max to min controllability/observability\\
+ &  & Output to sense individual modes & \\
+\midrule
+\textbf{Decoupling Properties} & Decoupling at low or high frequency depending on the chosen frame & Good decoupling at all frequencies & Good decoupling near the chosen frequency\\
+\midrule
+\textbf{Pros} & Physical inputs / outputs & Target specific modes & Good Decoupling near the crossover\\
+ & Good decoupling at High frequency (diagonal mass matrix if Jacobian taken at the CoM) & 2nd order diagonal plant & Very General\\
+ & Good decoupling at Low frequency (if Jacobian taken at specific point) &  & \\
+ & Easy integration of meaningful reference inputs &  & \\
+ &  &  & \\
+\midrule
+\textbf{Cons} & Coupling between force/rotation may be high at low frequency (non diagonal terms in K) & Need analytical equations & Loose the physical meaning of inputs /outputs\\
+ & Limited to parallel mechanisms (?) &  & Decoupling depends on the real approximation validity\\
+ & If good decoupling at all frequencies => requires specific mechanical architecture &  & Diagonal plants may not be easy to control\\
+\midrule
+\textbf{Applicability} & Parallel Mechanisms & Systems whose dynamics that can be expressed with M and K matrices & Very general\\
+ & Only small motion for the Jacobian matrix to stay constant &  & Need FRF data (either experimentally or analytically)\\
+\bottomrule
+\end{tabularx}
+\end{table}
 \chapter{Closed-Loop Shaping using Complementary Filters}
-\label{sec:orgdd912f0}
+\label{sec:orga76ba90}
 \label{sec:detail_control_optimization}
 
 Performance of a feedback control is dictated by closed-loop transfer functions.
@@ -828,10 +1159,10 @@ In this paper, we propose a new controller synthesis method
 \item direct translation of requirements such as disturbance rejection and robustness to plant uncertainty
 \end{itemize}
 \section{Control Architecture}
-\label{sec:orgbad19e8}
+\label{sec:orgaae401b}
 \label{ssec:detail_control_control_arch}
 \paragraph{Virtual Sensor Fusion}
-\label{sec:org5db8fac}
+\label{sec:orgbe4fa57}
 Let's consider the control architecture represented in Fig. \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 \(u\).
 
@@ -870,7 +1201,7 @@ u &= \frac{-K H_L}{1+G^{\prime} K H_L} dy &&+ \frac{K}{1+G^{\prime} K H_L}
 \end{alignat}
 with \(K = \frac{k}{1 + H_H G k}\)
 \paragraph{Asymptotic behavior}
-\label{sec:orgb964791}
+\label{sec:org100d48c}
 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}
@@ -903,7 +1234,7 @@ We obtain a sensitivity transfer function equals to the high pass filter \(S = \
 
 Assuming that we have a good model of the plant, we have then that the closed-loop behavior of the system converges to the designed complementary filters.
 \section{Translating the performance requirements into the shapes of the complementary filters}
-\label{sec:org1e41339}
+\label{sec:org30471b6}
 \label{ssec:detail_control_trans_perf}
  The required performance specifications in a feedback system can usually be translated into requirements on the upper bounds of \(\abs{S(j\w)}\) and \(|T(j\omega)|\) \cite{bibel92_guidel_h}.
 The process of designing a controller \(K(s)\) in order to obtain the desired shapes of \(\abs{S(j\w)}\) and \(\abs{T(j\w)}\) is called loop shaping.
@@ -912,7 +1243,7 @@ The equations \eqref{eq:detail_control_cl_system_y} and \eqref{eq:detail_control
 
 In this section, we then translate the typical specifications into the desired shapes of the complementary filters \(H_L\) and \(H_H\).\\
 \paragraph{Nominal Stability (NS)}
-\label{sec:org747119c}
+\label{sec:orgb61eb25}
 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 \(S = H_H\).
@@ -921,7 +1252,7 @@ Nominal stability is then guaranteed if \(H_L\), \(H_H\) and \(G\) are stable an
 
 Thus we must design stable and minimum phase complementary filters.\\
 \paragraph{Nominal Performance (NP)}
-\label{sec:org7e3c875}
+\label{sec:org4748252}
 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
@@ -945,14 +1276,14 @@ The translation of typical performance requirements on the shapes of the complem
 
 We may have other requirements in terms of stability margins, maximum or minimum closed-loop bandwidth.\\
 \paragraph{Closed-Loop Bandwidth}
-\label{sec:org91bb5c7}
+\label{sec:org20cf288}
 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 \(\omega \ge \omega_B^*\).\\
 \paragraph{Classical stability margins}
-\label{sec:org17bfb04}
+\label{sec:orgd6e8f52}
 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}
@@ -967,7 +1298,7 @@ For the nominal system \(M_S = \max_\omega |S| = \max_\omega |H_H|\), so one can
 \end{equation}
 and thus obtain acceptable stability margins.\\
 \paragraph{Response time to change of reference signal}
-\label{sec:org555bdc0}
+\label{sec:org8095473}
 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 Fig. \ref{fig:detail_control_sf_arch_class_prefilter}.
@@ -985,7 +1316,7 @@ Typically, \(K_r\) is a low pass filter of the form
 \end{equation}
 with \(\tau\) corresponding to the desired response time.\\
 \paragraph{Input usage}
-\label{sec:orgaed43be}
+\label{sec:org29193ac}
 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 \(|H_L|\).
 
@@ -994,7 +1325,7 @@ Input usage due to reference signal \(r\) is determined by \(\big|\frac{u}{r}\bi
 
 Proper choice of \(|K_r|\) is then useful to limit input usage due to change of reference signal.\\
 \paragraph{Robust Stability (RS)}
-\label{sec:org3210a2c}
+\label{sec:orgee16ad4}
 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.
 
@@ -1034,7 +1365,7 @@ Robust stability is then guaranteed by having the low pass filter \(H_L\) satisf
 
 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 \(|w_L| \ge |w_I| (2 + |w_I|)\).\\
 \paragraph{Robust Performance (RP)}
-\label{sec:org73ddee1}
+\label{sec:org289dea0}
 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:
@@ -1071,13 +1402,13 @@ Robust performance is then guaranteed if \eqref{eq:detail_control_robust_perf_a}
 
 One should be aware than when looking for a robust performance condition, only the worst case is evaluated and using the robust stability condition may lead to conservative control.
 \section{Analytical formulas for complementary filters?}
-\label{sec:org23bab6a}
+\label{sec:org9e830a6}
 \label{ssec:detail_control_analytical_complementary_filters}
 \section{Numerical Example}
-\label{sec:org082409f}
+\label{sec:orgafeeecf}
 \label{ssec:detail_control_simulations}
 \paragraph{Procedure}
-\label{sec:orgebbfce4}
+\label{sec:org267f5e8}
 
 In order to apply this control technique, we propose the following procedure:
 \begin{enumerate}
@@ -1092,7 +1423,7 @@ If one does not want to use the \(\mathcal{H}_\infty\) synthesis, one can use pr
 \item Control implementation: Filter the measurement with \(H_L\), implement the controller \(K\) and the pre-filter \(K_r\) as shown on Fig. \ref{fig:detail_control_sf_arch_class_prefilter}
 \end{enumerate}
 \paragraph{Plant}
-\label{sec:orgfc350fc}
+\label{sec:org4080126}
 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 Fig. \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.
@@ -1119,7 +1450,7 @@ Its bode plot is shown on Fig. \ref{fig:detail_control_bode_plot_mech_sys}.
 \caption{\label{fig:detail_control_bode_plot_mech_sys}Bode plot of the transfer function \(G(s)\) from \(F\) to \(x\)}
 \end{figure}
 \paragraph{Requirements}
-\label{sec:orgd766e8b}
+\label{sec:orgf3c8638}
 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 \(10^4\).
@@ -1154,10 +1485,10 @@ All the requirements on \(H_L\) and \(H_H\) are represented on Fig. \ref{fig:det
 \end{center}
 \subcaption{\label{fig:detail_control_hinf_filters_result_weights}Obtained complementary filters}
 \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{detail_control_hinf_filters_result_weights})}
+\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}
 \paragraph{Design of the filters}
-\label{sec:org11f2da3}
+\label{sec:org9e0d6e9}
 
 \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}}
 
@@ -1178,7 +1509,7 @@ After the \(\hinf\text{-synthesis}\), we obtain \(H_L\) and \(H_H\), and we plot
   \end{align}
 \end{subequations}
 \paragraph{Controller analysis}
-\label{sec:orgbd9d52a}
+\label{sec:org6b1f5a6}
 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.
 
@@ -1208,7 +1539,7 @@ It is implemented as shown on Fig. \ref{fig:detail_control_mech_sys_alone_ctrl}.
 \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}
 \paragraph{Robustness analysis}
-\label{sec:org85bb3dc}
+\label{sec:org6fc1bac}
 The robust stability can be access on the nyquist plot (Fig. \ref{fig:detail_control_nyquist_robustness}).
 
 The robust performance is shown on Fig. \ref{fig:detail_control_robust_perf}.
@@ -1229,12 +1560,12 @@ The robust performance is shown on Fig. \ref{fig:detail_control_robust_perf}.
 \caption{\label{fig:fig_label}Caption with reference to sub figure (\subref{fig:detail_control_nyquist_robustness}) (\subref{fig:detail_control_robust_perf})}
 \end{figure}
 \section{Experimental Validation?}
-\label{sec:org8a7211e}
+\label{sec:org7fb6422}
 \label{ssec:detail_control_exp_validation}
 
 \href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Experimental Validation}
 \section*{Conclusion}
-\label{sec:org40e7289}
+\label{sec:org8770e9e}
 \begin{itemize}
 \item[{$\square$}] Discuss how useful it is as the bandwidth can be changed in real time with analytical formulas of second order complementary filters.
 Maybe make a section about that.
@@ -1248,7 +1579,7 @@ Maybe give analytical formulas of second order complementary filters in the digi
 \end{itemize}
 \end{itemize}
 \chapter*{Conclusion}
-\label{sec:orgcbb5ce3}
+\label{sec:org64023b0}
 \label{sec:detail_control_conclusion}
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}