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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
<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>
#+end_export
@ -107,7 +107,7 @@ d_z = \frac{d_{\text{off}}}{2 \cos \theta_b}
#+begin_src matlab :exports none
%% 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]
#+end_src
@ -148,7 +148,7 @@ ans = 1e3*(dz(end) - dz(1))
*** Introduction :ignore:
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.
@ -284,6 +284,142 @@ data2orgtable(inv(J_s_311), {}, {}, ' %.2f ');
| 7.14 | 0.0 | -7.14 |
| -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
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:
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.
@ -533,6 +669,142 @@ data2orgtable(inv(J_s_111), {}, {}, ' %.2f ');
| 7.14 | 0.0 | -7.14 |
| 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
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}
=
\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
\end{bmatrix}
\end{equation}
#+begin_src matlab
%% Actuator Jacobian - 111 crystal
J_a_111 = [1, 0.14, -0.1525
1, 0.14, 0.0675
J_a_111 = [1, 0.14, -0.0675
1, 0.14, 0.1525
1, -0.14, 0.0425];
#+end_src
@ -642,6 +914,121 @@ data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
| 0.4058 | 3.1656 | -3.5714 |
| -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:
Disturbances:
@ -786,7 +1173,7 @@ Using the forward and inverse kinematics, we can computed the dynamics from piez
#+begin_src matlab
%% 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
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]].
#+begin_src matlab
G_pz = diag(1./diag(dcgain(G_pz)))*G_pz;
#+end_src
#+begin_src matlab :exports none
%% Bode plot for the plant
freqs = logspace(0,3,1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
@ -828,7 +1220,7 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
ylim([1e-13, 1e-6]);
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
@ -843,11 +1235,11 @@ yticks(-360:90:360);
ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
xlim([1, 1e3]);
#+end_src
#+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
#+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.
#+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
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_hac_iff.m

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@ -0,0 +1,448 @@
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