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Thomas Dehaeze 2022-02-15 14:19:04 +01:00
parent 5c609777f1
commit 431f8e66eb
19 changed files with 3892 additions and 1249 deletions
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dcm-simscape.org → dcm-simscape-model.org
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@ -44,7 +44,7 @@
#+begin_export html #+begin_export html
<hr> <hr>
<p>This report is also available as a <a href="./dcm-simscape.pdf">pdf</a>.</p> <p>This report is also available as a <a href="./dcm-simscape-model.pdf">pdf</a>.</p>
<hr> <hr>
#+end_export #+end_export
@ -107,7 +107,7 @@ d_z = \frac{d_{\text{off}}}{2 \cos \theta_b}
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Tested bragg angles %% Tested bragg angles
bragg = linspace(5, 80, 1000); % Bragg angle [deg] bragg = linspace(5, 80, 5000); % Bragg angle [deg]
d_off = 10.5e-3; % Wanted offset between x-rays [m] d_off = 10.5e-3; % Wanted offset between x-rays [m]
#+end_src #+end_src
@ -148,7 +148,7 @@ ans = 1e3*(dz(end) - dz(1))
*** Introduction :ignore: *** Introduction :ignore:
The reference frame is taken at the center of the 311 second crystal. The reference frame is taken at the center of the 311 second crystal.
*** Interferometers - 311 Crystal *** Interferometers - 311 secondary Crystal
Three interferometers are pointed to the bottom surface of the 311 crystal. Three interferometers are pointed to the bottom surface of the 311 crystal.
@ -284,6 +284,142 @@ data2orgtable(inv(J_s_311), {}, {}, ' %.2f ');
| 7.14 | 0.0 | -7.14 | | 7.14 | 0.0 | -7.14 |
| -16.67 | 33.33 | -16.67 | | -16.67 | 33.33 | -16.67 |
*** Interferometers - 311 primary Crystal
Three interferometers are pointed to the bottom surface of the 311 crystal.
The position of the measurement points are shown in Figure [[fig:sensor_311_crystal_points_primary]] as well as the origin where the motion of the crystal is computed.
#+begin_src latex :file sensor_311_crystal_points_primary.pdf
\begin{tikzpicture}
% Crystal
\draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2);
% Measurement Points
\node[branch] (a1) at (-7, -1.5){};
\node[branch] (a2) at ( 0, 1.5){};
\node[branch] (a3) at ( 7, -1.5){};
% Labels
\node[right] at (a1) {$\mathcal{O}_1 = (-0.07, -0.015)$};
\node[right] at (a2) {$\mathcal{O}_2 = ( 0, 0.015)$};
\node[left] at (a3) {$\mathcal{O}_3 = ( 0.07, -0.015)$};
% Origin
\draw[->] (0, 0) node[] -- ++(1, 0) node[right]{$x$};
\draw[->] (0, 0) -- ++(0, 1) node[right]{$y$};
\draw[fill, color=black] (0, 0) circle (0.05);
\node[left] at (0,0) {$\mathcal{O}_{311}$};
\end{tikzpicture}
#+end_src
#+name: fig:sensor_311_crystal_points_primary
#+caption: Top view of the primary crystal 311. Position of the measurement points.
#+RESULTS:
[[file:figs/sensor_311_crystal_points_primary.png]]
The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_311]]):
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,311}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_311.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js) {$\bm{J}_{s,311}$};
% Connections and labels
\draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Js.west);
\draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name: fig:schematic_sensor_jacobian_inverse_kinematics_311
#+caption: Inverse Kinematics - Interferometers
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_inverse_kinematics_311.png]]
From the Figure [[fig:sensor_311_crystal_points]], the inverse kinematics can be solved as follow (for small motion):
\begin{equation}
\bm{J}_{s,311}
=
\begin{bmatrix}
-1 & -0.07 & 0.015 \\
-1 & 0 & -0.015 \\
-1 & 0.07 & 0.015
\end{bmatrix}
\end{equation}
#+begin_src matlab
%% Sensor Jacobian matrix for 311 crystal
J_s_311_1 = [-1, 0.07, -0.015
-1, 0, 0.015
-1, -0.07, -0.015];
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(J_s_311_1, {}, {}, ' %.3f ');
#+end_src
#+name: tab:jacobian_sensor_311_1
#+caption: Sensor Jacobian - Primary Crystal - $\bm{J}_{s,311}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
| -1.0 | 0.07 | -0.015 |
| -1.0 | 0.0 | 0.015 |
| -1.0 | -0.07 | -0.015 |
The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_311]]).
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{s,311}^{-1}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
\end{equation}
#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_311.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js_inv) {$\bm{J}_{s,311}^{-1}$};
% Connections and labels
\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} -- (Js_inv.west);
\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name: fig:schematic_sensor_jacobian_forward_kinematics_311
#+caption: Forward Kinematics - Interferometers
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_forward_kinematics_311.png]]
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(inv(J_s_311_1), {}, {}, ' %.2f ');
#+end_src
#+name: tab:inverse_jacobian_sensor_311
#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,311}^{-1}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
| -0.25 | -0.5 | -0.25 |
| 7.14 | 0.0 | -7.14 |
| -16.67 | 33.33 | -16.67 |
*** Piezo - 311 Crystal *** Piezo - 311 Crystal
The location of the actuators with respect with the center of the 311 second crystal are shown in Figure [[fig:actuator_jacobian_311_points]]. The location of the actuators with respect with the center of the 311 second crystal are shown in Figure [[fig:actuator_jacobian_311_points]].
@ -397,7 +533,7 @@ data2orgtable(inv(J_a_311), {}, {}, ' %.4f ');
*** Introduction :ignore: *** Introduction :ignore:
The reference frame is taken at the center of the 111 second crystal. The reference frame is taken at the center of the 111 second crystal.
*** Interferometers - 111 Crystal *** Interferometers - 111 secondary Crystal
Three interferometers are pointed to the bottom surface of the 111 crystal. Three interferometers are pointed to the bottom surface of the 111 crystal.
@ -533,6 +669,142 @@ data2orgtable(inv(J_s_111), {}, {}, ' %.2f ');
| 7.14 | 0.0 | -7.14 | | 7.14 | 0.0 | -7.14 |
| 16.67 | -33.33 | 16.67 | | 16.67 | -33.33 | 16.67 |
*** Interferometers - 111 primary Crystal
Three interferometers are pointed to the bottom surface of the 111 crystal.
The position of the measurement points are shown in Figure [[fig:sensor_111_crystal_points_primary]] as well as the origin where the motion of the crystal is computed.
#+begin_src latex :file sensor_111_crystal_points_primary.pdf
\begin{tikzpicture}
% Crystal
\draw (-7.5/2, -3.5/2) rectangle (7.5/2, 3.5/2);
% Measurement Points
\node[branch] (a1) at (-3.6, 1.5){};
\node[branch] (a2) at ( 0, -1.5){};
\node[branch] (a3) at ( 3.6, 1.5){};
% Labels
\node[right] at (a1) {$\mathcal{O}_1 = (-0.036, 0.015)$};
\node[right] at (a2) {$\mathcal{O}_2 = ( 0, -0.015)$};
\node[left] at (a3) {$\mathcal{O}_3 = ( 0.036, 0.015)$};
% Origin
\draw[->] (0, 0) node[] -- ++(1, 0) node[right]{$x$};
\draw[->] (0, 0) -- ++(0, 1) node[right]{$y$};
\draw[fill, color=black] (0, 0) circle (0.05);
\node[left] at (0,0) {$\mathcal{O}_{111}$};
\end{tikzpicture}
#+end_src
#+name: fig:sensor_111_crystal_points_primary
#+caption: Top view of the primary crystal 111. Position of the measurement points.
#+RESULTS:
[[file:figs/sensor_111_crystal_points_primary.png]]
The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_111]]):
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,111}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_111.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js) {$\bm{J}_{s,111}$};
% Connections and labels
\draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Js.west);
\draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name: fig:schematic_sensor_jacobian_inverse_kinematics_111
#+caption: Inverse Kinematics - Interferometers
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_inverse_kinematics_111.png]]
From the Figure [[fig:sensor_111_crystal_points]], the inverse kinematics can be solved as follow (for small motion):
\begin{equation}
\bm{J}_{s,111}
=
\begin{bmatrix}
1 & -0.036 & -0.015 \\
1 & 0 & 0.015 \\
1 & 0.036 & -0.015
\end{bmatrix}
\end{equation}
#+begin_src matlab
%% Sensor Jacobian matrix for 111 crystal
J_s_111_1 = [-1, -0.036, -0.015
-1, 0, 0.015
-1, 0.036, -0.015];
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(J_s_111_1, {}, {}, ' %.3f ');
#+end_src
#+name: tab:jacobian_sensor_111_primary
#+caption: Sensor Jacobian $\bm{J}_{s,111}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
| -1.0 | -0.036 | -0.015 |
| -1.0 | 0.0 | 0.015 |
| -1.0 | 0.036 | -0.015 |
The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_111]]).
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{s,111}^{-1}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
\end{equation}
#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_111.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js_inv) {$\bm{J}_{s,111}^{-1}$};
% Connections and labels
\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} -- (Js_inv.west);
\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name: fig:schematic_sensor_jacobian_forward_kinematics_111
#+caption: Forward Kinematics - Interferometers
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_forward_kinematics_111.png]]
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(inv(J_s_111_1), {}, {}, ' %.2f ');
#+end_src
#+name: tab:inverse_jacobian_sensor_111
#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,111}^{-1}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
| -0.25 | -0.5 | -0.25 |
| -13.89 | 0.0 | 13.89 |
| -16.67 | 33.33 | -16.67 |
*** Piezo - 111 Crystal *** Piezo - 111 Crystal
The location of the actuators with respect with the center of the 111 second crystal are shown in Figure [[fig:actuator_jacobian_111_points]]. The location of the actuators with respect with the center of the 111 second crystal are shown in Figure [[fig:actuator_jacobian_111_points]].
@ -575,16 +847,16 @@ Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain:
\bm{J}_{a,111} \bm{J}_{a,111}
= =
\begin{bmatrix} \begin{bmatrix}
1 & 0.14 & -0.1525 \\ 1 & 0.14 & -0.0675 \\
1 & 0.14 & 0.0675 \\ 1 & 0.14 & 0.1525 \\
1 & -0.14 & 0.0425 1 & -0.14 & 0.0425
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
#+begin_src matlab #+begin_src matlab
%% Actuator Jacobian - 111 crystal %% Actuator Jacobian - 111 crystal
J_a_111 = [1, 0.14, -0.1525 J_a_111 = [1, 0.14, -0.0675
1, 0.14, 0.0675 1, 0.14, 0.1525
1, -0.14, 0.0425]; 1, -0.14, 0.0425];
#+end_src #+end_src
@ -642,6 +914,121 @@ data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
| 0.4058 | 3.1656 | -3.5714 | | 0.4058 | 3.1656 | -3.5714 |
| -4.5455 | 4.5455 | 0.0 | | -4.5455 | 4.5455 | 0.0 |
** Kinematics (Metrology Frame)
#+name: fig:jacobian_metrology_frame
#+caption: Top View - Top metrology frame
[[file:figs/jacobian_metrology_frame.png]]
#+begin_src matlab
%% Sensor Jacobian matrix for 111 crystal
J_m = [1, 0.102, 0
1, -0.088, 0.1275
1, -0.088, -0.1275];
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(J_m, {}, {}, ' %.4f ');
#+end_src
#+RESULTS:
| 1.0 | 0.102 | 0.0 |
| 1.0 | -0.088 | 0.1275 |
| 1.0 | -0.088 | -0.1275 |
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(inv(J_m), {}, {}, ' %.3f ');
#+end_src
#+RESULTS:
| 0.463 | 0.268 | 0.268 |
| 5.263 | -2.632 | -2.632 |
| 0.0 | 3.922 | -3.922 |
** Verification
#+begin_src matlab
mdl = 'coordinate_transform';
%% Input/Output definition
clear io; io_i = 1;
%% Inputs
io(io_i) = linio([mdl, '/interferometers'], 1, 'openinput'); io_i = io_i + 1;
%% Outputs
io(io_i) = linio([mdl, '/xtal_111'], 1, 'openoutput'); io_i = io_i + 1;
#+end_src
#+begin_src matlab
%% Extraction of the dynamics
G_311 = linearize(mdl, io);
G_311 = G_311(:,[1 2 3 7 8 9 13 14 15]);
#+end_src
#+begin_src matlab
mdl = 'coordinate_transform';
%% Input/Output definition
clear io; io_i = 1;
%% Inputs
io(io_i) = linio([mdl, '/interferometers'], 1, 'openinput'); io_i = io_i + 1;
%% Outputs
io(io_i) = linio([mdl, '/xtal_111'], 1, 'openoutput'); io_i = io_i + 1;
#+end_src
#+begin_src matlab
G_111 = linearize(mdl, io);
G_111 = G_111(:,[4 5 6 10 11 12 13 14 15]);
#+end_src
#+begin_src matlab
%% Sensor Jacobian matrix for 1st 111 crystal
J_s_111_1 = [-1, -0.036, -0.015
-1, 0, 0.015
-1, 0.036, -0.015];
#+end_src
#+begin_src matlab
%% Sensor Jacobian matrix for 2nd 111 crystal
J_s_111_2 = [1, 0.07, 0.015
1, 0, -0.015
1, -0.07, 0.015];
#+end_src
#+begin_src matlab
%% Sensor Jacobian matrix for 111 crystal
J_m = [1, 0.102, 0
1, -0.088, 0.1275
1, -0.088, -0.1275];
#+end_src
#+begin_src matlab
G_111_t = [-inv(J_s_111_1), inv(J_s_111_2), -inv(J_m)]
G_111_t(1,:) = -G_111_t(1,:);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(G_111_t, {}, {}, ' %.3f ');
#+end_src
#+RESULTS:
| -0.25 | -0.5 | -0.25 | -0.25 | -0.5 | -0.25 | 0.463 | 0.268 | 0.268 |
| 13.889 | 0.0 | -13.889 | 7.143 | 0.0 | -7.143 | -5.263 | 2.632 | 2.632 |
| 16.667 | -33.333 | 16.667 | 16.667 | -33.333 | 16.667 | 0.0 | -3.922 | 3.922 |
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(dcgain(G_111), {}, {}, ' %.3f ');
#+end_src
#+RESULTS:
| -0.25 | -0.5 | -0.25 | -0.25 | -0.5 | -0.25 | 0.333 | 0.333 | 0.333 |
| 13.889 | 0.0 | -13.889 | 7.143 | 0.0 | -7.143 | -5.263 | 2.632 | 2.632 |
| 16.667 | -33.333 | 16.667 | 16.667 | -33.333 | 16.667 | 0.0 | -3.922 | 3.922 |
** TODO Inputs and Outputs :noexport: ** TODO Inputs and Outputs :noexport:
Disturbances: Disturbances:
@ -786,7 +1173,7 @@ Using the forward and inverse kinematics, we can computed the dynamics from piez
#+begin_src matlab #+begin_src matlab
%% Compute the system in the frame of the fastjacks %% Compute the system in the frame of the fastjacks
G_pz = J_a_111*inv(J_s_111)*G; G_pz = J_a_311*inv(J_s_311)*G;
#+end_src #+end_src
The DC gain of the new system shows that the system is well decoupled at low frequency. The DC gain of the new system shows that the system is well decoupled at low frequency.
@ -805,8 +1192,13 @@ dcgain(G_pz)
The bode plot of $\bm{G}_{\text{fj}}(s)$ is shown in Figure [[fig:bode_plot_plant_fj]]. The bode plot of $\bm{G}_{\text{fj}}(s)$ is shown in Figure [[fig:bode_plot_plant_fj]].
#+begin_src matlab
G_pz = diag(1./diag(dcgain(G_pz)))*G_pz;
#+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Bode plot for the plant %% Bode plot for the plant
freqs = logspace(0,3,1000);
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
@ -828,7 +1220,7 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
ylim([1e-13, 1e-6]); ylim([1e-2, 1e2]);
ax2 = nexttile; ax2 = nexttile;
hold on; hold on;
@ -843,11 +1235,11 @@ yticks(-360:90:360);
ylim([-180, 180]); ylim([-180, 180]);
linkaxes([ax1,ax2],'x'); linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]); xlim([1, 1e3]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_plant_fj.pdf', 'width', 'wide', 'height', 'tall'); exportFig('figs/bode_plot_plant_fj_pres.pdf', 'width', 'full', 'height', 'tall');
#+end_src #+end_src
#+name: fig:bode_plot_plant_fj #+name: fig:bode_plot_plant_fj
@ -1703,6 +2095,34 @@ exportFig('figs/comp_damped_undamped_plant_iff_bode_plot.pdf', 'width', 'wide',
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant. The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
#+end_important #+end_important
* Feedback Control
#+begin_src latex :file schematic_jacobian_frame_fastjack_feedback.pdf
\begin{tikzpicture}
% Blocs
\node[block={1.5cm}{1.5cm}] (G) at (0,0) {$\bm{G}(s)$};
\node[block, right=1.2 of G] (Js) {$\bm{J}_{s}^{-1}$};
\node[block, left=1.2 of G] (Khac) {$\bm{K}(s)$};
\node[block, left=1.2 of Khac] (Ja) {$\bm{J}_{a}$};
\node[addb={+}{}{}{}{-}, left=1.0 of Ja] (subL) {};
% Connections and labels
\draw[->] (Khac.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (G.west);
\draw[->] (G.east) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
\draw[->] (Js.east) -- ++(1.0, 0);
\draw[->] ($(subL.west) + (-1.0, 0)$) -- node[midway, above]{$\begin{bmatrix} r_{d_z} \\ r_{r_y} \\ r_{r_x} \end{bmatrix}$} (subL.west);
\draw[->] (subL.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_z} \\ \epsilon_{r_y} \\ \epsilon_{r_x} \end{bmatrix}$} (Ja.west);
\draw[->] (Ja.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_{u_r}} \\ \epsilon_{d_{u_h}} \\ \epsilon_{d_d} \end{bmatrix}$} (Khac.west);
\draw[->] ($(Js.east) + (0.6, 0)$)node[branch]{}node[above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- ++(0, -1.0) -| (subL.south);
\end{tikzpicture}
#+end_src
#+RESULTS:
[[file:figs/schematic_jacobian_frame_fastjack_feedback.png]]
* HAC-LAC (IFF) architecture * HAC-LAC (IFF) architecture
:PROPERTIES: :PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_hac_iff.m :header-args:matlab+: :tangle matlab/dcm_hac_iff.m

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