Add 311 and 111 kinematics. Correct crystal positions
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dcm-simscape.org
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dcm-simscape.org
@ -144,17 +144,17 @@ ans = 1e3*(dz(end) - dz(1))
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#+RESULTS:
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: 24.963
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** Kinematics (111 Crystal)
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** Kinematics (311 Crystal)
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*** Introduction :ignore:
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The reference frame is taken at the center of the 111 second crystal.
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The reference frame is taken at the center of the 311 second crystal.
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*** Interferometers - 111 Crystal
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*** Interferometers - 311 Crystal
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Three interferometers are pointed to the bottom surface of the 111 crystal.
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Three interferometers are pointed to the bottom surface of the 311 crystal.
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The position of the measurement points are shown in Figure [[fig:sensor_111_crystal_points]] as well as the origin where the motion of the crystal is computed.
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The position of the measurement points are shown in Figure [[fig:sensor_311_crystal_points]] as well as the origin where the motion of the crystal is computed.
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#+begin_src latex :file sensor_111_crystal_points.pdf
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#+begin_src latex :file sensor_311_crystal_points.pdf
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\begin{tikzpicture}
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% Crystal
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\draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2);
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@ -174,6 +174,255 @@ The position of the measurement points are shown in Figure [[fig:sensor_111_crys
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\draw[->] (0, 0) -- ++(0, -1) node[below]{$y$};
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\draw[fill, color=black] (0, 0) circle (0.05);
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\node[left] at (0,0) {$\mathcal{O}_{311}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:sensor_311_crystal_points
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#+caption: Bottom view of the second crystal 311. Position of the measurement points.
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#+RESULTS:
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[[file:figs/sensor_311_crystal_points.png]]
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The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_311]]):
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\begin{equation}
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\begin{bmatrix}
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x_1 \\ x_2 \\ x_3
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\end{bmatrix}
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=
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\bm{J}_{s,311}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_311.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Js) {$\bm{J}_{s,311}$};
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% Connections and labels
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\draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Js.west);
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\draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_inverse_kinematics_311
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#+caption: Inverse Kinematics - Interferometers
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#+RESULTS:
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[[file:figs/schematic_sensor_jacobian_inverse_kinematics_311.png]]
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From the Figure [[fig:sensor_311_crystal_points]], the inverse kinematics can be solved as follow (for small motion):
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\begin{equation}
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\bm{J}_{s,311}
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=
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\begin{bmatrix}
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1 & 0.07 & -0.015 \\
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1 & 0 & 0.015 \\
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1 & -0.07 & -0.015
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\end{bmatrix}
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\end{equation}
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#+begin_src matlab
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%% Sensor Jacobian matrix for 311 crystal
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J_s_311 = [1, 0.07, -0.015
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1, 0, 0.015
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1, -0.07, -0.015];
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(J_s_311, {}, {}, ' %.3f ');
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#+end_src
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#+name: tab:jacobian_sensor_311
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#+caption: Sensor Jacobian $\bm{J}_{s,311}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 1.0 | 0.07 | -0.015 |
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| 1.0 | 0.0 | 0.015 |
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| 1.0 | -0.07 | -0.015 |
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The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_311]]).
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\begin{equation}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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=
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\bm{J}_{s,311}^{-1}
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\begin{bmatrix}
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x_1 \\ x_2 \\ x_3
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_311.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Js_inv) {$\bm{J}_{s,311}^{-1}$};
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% Connections and labels
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\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} -- (Js_inv.west);
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\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_forward_kinematics_311
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#+caption: Forward Kinematics - Interferometers
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#+RESULTS:
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[[file:figs/schematic_sensor_jacobian_forward_kinematics_311.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(inv(J_s_311), {}, {}, ' %.2f ');
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#+end_src
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#+name: tab:inverse_jacobian_sensor_311
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#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,311}^{-1}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 0.25 | 0.5 | 0.25 |
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| 7.14 | 0.0 | -7.14 |
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| -16.67 | 33.33 | -16.67 |
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*** Piezo - 311 Crystal
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The location of the actuators with respect with the center of the 311 second crystal are shown in Figure [[fig:actuator_jacobian_311_points]].
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#+name: fig:actuator_jacobian_311_points
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#+caption: Location of actuators with respect to the center of the 311 second crystal (bottom view)
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#+attr_latex: :width \linewidth
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[[file:figs/actuator_jacobian_311_points.png]]
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Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal's center.
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\begin{equation}
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\begin{bmatrix}
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d_{u_r} \\ d_{u_h} \\ d_{d}
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\end{bmatrix}
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=
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\bm{J}_{a,311}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics_311.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Ja) {$\bm{J}_{a,311}$};
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% Connections and labels
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\draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Ja.west);
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\draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_inverse_kinematics_311
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#+caption: Inverse Kinematics - Actuators
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#+RESULTS:
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[[file:figs/schematic_actuator_jacobian_inverse_kinematics_311.png]]
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Based on the geometry in Figure [[fig:actuator_jacobian_311_points]], we obtain:
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\begin{equation}
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\bm{J}_{a,311}
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=
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\begin{bmatrix}
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1 & 0.14 & -0.1525 \\
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1 & 0.14 & 0.0675 \\
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1 & -0.14 & -0.0425
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\end{bmatrix}
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\end{equation}
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#+begin_src matlab
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%% Actuator Jacobian - 311 crystal
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J_a_311 = [1, 0.14, -0.1525
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1, 0.14, 0.0675
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1, -0.14, -0.0425];
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(J_a_311, {}, {}, ' %.4f ');
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#+end_src
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#+name: tab:jacobian_actuator_311
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#+caption: Actuator Jacobian $\bm{J}_{a,311}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 1.0 | 0.14 | -0.1525 |
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| 1.0 | 0.14 | 0.0675 |
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| 1.0 | -0.14 | -0.0425 |
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The forward Kinematics is solved by inverting the Jacobian matrix:
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\begin{equation}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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=
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\bm{J}_{a,311}^{-1}
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\begin{bmatrix}
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d_{u_r} \\ d_{u_h} \\ d_{d}
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics_311.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Ja_inv) {$\bm{J}_{a,311}^{-1}$};
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% Connections and labels
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\draw[->] ($(Ja_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$} -- (Ja_inv.west);
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\draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_actuator_jacobian_forward_kinematics_311
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#+caption: Forward Kinematics - Actuators for 311 crystal
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#+RESULTS:
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[[file:figs/schematic_actuator_jacobian_forward_kinematics_311.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(inv(J_a_311), {}, {}, ' %.4f ');
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#+end_src
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#+name: tab:inverse_jacobian_actuator_311
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#+caption: Inverse of the actuator Jacobian $\bm{J}_{a,311}^{-1}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 0.0568 | 0.4432 | 0.5 |
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| 1.7857 | 1.7857 | -3.5714 |
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| -4.5455 | 4.5455 | 0.0 |
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** Kinematics (111 Crystal)
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*** Introduction :ignore:
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The reference frame is taken at the center of the 111 second crystal.
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*** Interferometers - 111 Crystal
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Three interferometers are pointed to the bottom surface of the 111 crystal.
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The position of the measurement points are shown in Figure [[fig:sensor_111_crystal_points]] as well as the origin where the motion of the crystal is computed.
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#+begin_src latex :file sensor_111_crystal_points.pdf
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\begin{tikzpicture}
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% Crystal
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\draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2);
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% Measurement Points
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\node[branch] (a1) at (-7, -1.5){};
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\node[branch] (a2) at ( 0, 1.5){};
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\node[branch] (a3) at ( 7, -1.5){};
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% Labels
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\node[right] at (a1) {$\mathcal{O}_1 = (-0.07, 0.015)$};
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\node[right] at (a2) {$\mathcal{O}_2 = (0, -0.015)$};
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\node[left] at (a3) {$\mathcal{O}_3 = ( 0.07, 0.015)$};
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% Origin
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\draw[->] (0, 0) node[] -- ++(1, 0) node[right]{$x$};
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\draw[->] (0, 0) -- ++(0, -1) node[below]{$y$};
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\draw[fill, color=black] (0, 0) circle (0.05);
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\node[left] at (0,0) {$\mathcal{O}_{111}$};
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\end{tikzpicture}
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#+end_src
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@ -183,7 +432,7 @@ The position of the measurement points are shown in Figure [[fig:sensor_111_crys
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#+RESULTS:
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[[file:figs/sensor_111_crystal_points.png]]
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The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics]]):
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The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_111]]):
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\begin{equation}
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\begin{bmatrix}
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x_1 \\ x_2 \\ x_3
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@ -195,7 +444,7 @@ d_z \\ r_y \\ r_x
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics.pdf
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#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_111.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Js) {$\bm{J}_{s,111}$};
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@ -206,10 +455,10 @@ d_z \\ r_y \\ r_x
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_inverse_kinematics
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#+name: fig:schematic_sensor_jacobian_inverse_kinematics_111
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#+caption: Inverse Kinematics - Interferometers
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#+RESULTS:
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[[file:figs/schematic_sensor_jacobian_inverse_kinematics.png]]
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[[file:figs/schematic_sensor_jacobian_inverse_kinematics_111.png]]
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From the Figure [[fig:sensor_111_crystal_points]], the inverse kinematics can be solved as follow (for small motion):
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@ -217,17 +466,17 @@ From the Figure [[fig:sensor_111_crystal_points]], the inverse kinematics can be
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\bm{J}_{s,111}
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=
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\begin{bmatrix}
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1 & 0.07 & -0.015 \\
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1 & 0 & 0.015 \\
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1 & -0.07 & -0.015
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1 & 0.07 & 0.015 \\
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1 & 0 & -0.015 \\
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1 & -0.07 & 0.015
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\end{bmatrix}
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\end{equation}
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#+begin_src matlab
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%% Sensor Jacobian matrix for 111 crystal
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J_s_111 = [1, 0.07, -0.015
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1, 0, 0.015
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1, -0.07, -0.015];
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J_s_111 = [1, 0.07, 0.015
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1, 0, -0.015
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1, -0.07, 0.015];
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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@ -239,11 +488,11 @@ data2orgtable(J_s_111, {}, {}, ' %.3f ');
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 1.0 | 0.07 | -0.015 |
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| 1.0 | 0.0 | 0.015 |
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| 1.0 | -0.07 | -0.015 |
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| 1.0 | 0.07 | 0.015 |
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| 1.0 | 0.0 | -0.015 |
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| 1.0 | -0.07 | 0.015 |
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The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics]]).
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The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_111]]).
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\begin{equation}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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@ -255,7 +504,7 @@ x_1 \\ x_2 \\ x_3
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics.pdf
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#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_111.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Js_inv) {$\bm{J}_{s,111}^{-1}$};
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@ -266,10 +515,10 @@ x_1 \\ x_2 \\ x_3
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_forward_kinematics
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#+name: fig:schematic_sensor_jacobian_forward_kinematics_111
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#+caption: Forward Kinematics - Interferometers
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#+RESULTS:
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[[file:figs/schematic_sensor_jacobian_forward_kinematics.png]]
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[[file:figs/schematic_sensor_jacobian_forward_kinematics_111.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(inv(J_s_111), {}, {}, ' %.2f ');
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@ -280,9 +529,9 @@ data2orgtable(inv(J_s_111), {}, {}, ' %.2f ');
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 0.25 | 0.5 | 0.25 |
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| 7.14 | 0.0 | -7.14 |
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| -16.67 | 33.33 | -16.67 |
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| 0.25 | 0.5 | 0.25 |
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| 7.14 | 0.0 | -7.14 |
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| 16.67 | -33.33 | 16.67 |
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*** Piezo - 111 Crystal
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@ -305,7 +554,7 @@ d_z \\ r_y \\ r_x
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics.pdf
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#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics_111.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Ja) {$\bm{J}_{a,111}$};
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@ -316,10 +565,10 @@ d_z \\ r_y \\ r_x
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\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:schematic_sensor_jacobian_inverse_kinematics
|
||||
#+name: fig:schematic_sensor_jacobian_inverse_kinematics_111
|
||||
#+caption: Inverse Kinematics - Actuators
|
||||
#+RESULTS:
|
||||
[[file:figs/schematic_actuator_jacobian_inverse_kinematics.png]]
|
||||
[[file:figs/schematic_actuator_jacobian_inverse_kinematics_111.png]]
|
||||
|
||||
Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain:
|
||||
\begin{equation}
|
||||
@ -328,7 +577,7 @@ Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain:
|
||||
\begin{bmatrix}
|
||||
1 & 0.14 & -0.1525 \\
|
||||
1 & 0.14 & 0.0675 \\
|
||||
1 & -0.14 & -0.0425
|
||||
1 & -0.14 & 0.0425
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
@ -336,7 +585,7 @@ Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain:
|
||||
%% Actuator Jacobian - 111 crystal
|
||||
J_a_111 = [1, 0.14, -0.1525
|
||||
1, 0.14, 0.0675
|
||||
1, -0.14, -0.0425];
|
||||
1, -0.14, 0.0425];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no
|
||||
@ -350,7 +599,7 @@ data2orgtable(J_a_111, {}, {}, ' %.4f ');
|
||||
#+RESULTS:
|
||||
| 1.0 | 0.14 | -0.1525 |
|
||||
| 1.0 | 0.14 | 0.0675 |
|
||||
| 1.0 | -0.14 | -0.0425 |
|
||||
| 1.0 | -0.14 | 0.0425 |
|
||||
|
||||
The forward Kinematics is solved by inverting the Jacobian matrix:
|
||||
\begin{equation}
|
||||
@ -364,7 +613,7 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics.pdf
|
||||
#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics_111.pdf
|
||||
\begin{tikzpicture}
|
||||
% Blocs
|
||||
\node[block] (Ja_inv) {$\bm{J}_{a,111}^{-1}$};
|
||||
@ -375,10 +624,10 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:schematic_actuator_jacobian_forward_kinematics
|
||||
#+name: fig:schematic_actuator_jacobian_forward_kinematics_111
|
||||
#+caption: Forward Kinematics - Actuators for 111 crystal
|
||||
#+RESULTS:
|
||||
[[file:figs/schematic_actuator_jacobian_forward_kinematics.png]]
|
||||
[[file:figs/schematic_actuator_jacobian_forward_kinematics_111.png]]
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no
|
||||
data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
|
||||
@ -389,8 +638,8 @@ data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
|
||||
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
|
||||
#+attr_latex: :center t :booktabs t
|
||||
#+RESULTS:
|
||||
| 0.0568 | 0.4432 | 0.5 |
|
||||
| 1.7857 | 1.7857 | -3.5714 |
|
||||
| 0.25 | 0.25 | 0.5 |
|
||||
| 0.4058 | 3.1656 | -3.5714 |
|
||||
| -4.5455 | 4.5455 | 0.0 |
|
||||
|
||||
** TODO Inputs and Outputs :noexport:
|
||||
@ -402,11 +651,11 @@ Disturbances:
|
||||
|
||||
** Save Kinematics
|
||||
#+begin_src matlab :exports none :tangle no
|
||||
save('matlab/mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
|
||||
save('matlab/mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', 'J_a_111', 'J_s_111')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :eval no
|
||||
save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
|
||||
save('mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', 'J_a_111', 'J_s_111')
|
||||
#+end_src
|
||||
|
||||
* Open Loop System Identification
|
||||
@ -1560,7 +1809,7 @@ load('Kiff.mat');
|
||||
#+begin_src matlab :exports none
|
||||
%% Identification of the damped plant with IFF
|
||||
controller.type = 1; % IFF
|
||||
G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
|
||||
G_dp = J_a_311*inv(J_s_311)*linearize(mdl, io);
|
||||
G_dp.InputName = {'u_ur', 'u_uh', 'u_d'};
|
||||
G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'};
|
||||
#+end_src
|
||||
|
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Reference in New Issue
Block a user