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create mode 100644 figs/sensor_ring_secondary_crystal_points.pdf create mode 100644 figs/sensor_ring_secondary_crystal_points.png create mode 100644 figs/sensor_ring_secondary_crystal_points.svg create mode 100755 matlab/coordinate_transform.slx create mode 100755 matlab/coordinate_transform.slx.r2019b create mode 100644 matlab/simscape_dcm.slx.r2021b create mode 100644 matlab/test.slx delete mode 120000 readme.org diff --git a/dcm-simscape-model.org b/dcm-simscape-model.org index c584057..ee23c61 100644 --- a/dcm-simscape-model.org +++ b/dcm-simscape-model.org @@ -68,6 +68,119 @@ It is structured as follow: <> ** Introduction :ignore: +In this section, the kinematics of the DCM is described. +The elements (crystals, fast jacks, metrology frame, ...) are all defined. +The associated frames in which the motion of those elements are expressed will also be defined. +Finally, the transformation necessary to compute the relative pose of the crystals from the metrology is computed. + +** General description of the system architecture - Terminology +In order to describe the motion of the different elements, reference frames are used. + +One global frame is first defined with: +- the $x$ axis is aligned with the x-ray +- the $y$ axis aligned with the Bragg axis +- the $z$ axis is pointing up aligned with the gravity +The axis are shown in Figure [[fig:dcm_schematic_overview]]. +The origin of this frame is usually taken at the intersection between the x-ray and the bragg axis. + +The Bragg axis is therefore rotating around the $y$ axis. +Two sets of two crystals are also rotating with the bragg axis. +One set is positioned on the "hall" side (i.e toward $+y$) while the other set is positioned on the "ring" side (i.e. toward $-y$). + +For each of these two sets, there is a "primary" crystal directly fixed to the bragg axis, and a "secondary" crystal fixed on top of three fastjacks. +There are therefore 4 crystals in total for each Double Crystal Monochromator. + +There is then a metrology frame that hold all the interferometers. + +Several reference frame are used (displayed in Figure [[fig:dcm_schematic_overview]] for the "ring" crystals): +- $\{\mathcal{F}_{1r}\}$ and $\{\mathcal{F}_{1h}\}$: frames fixed to the bottom surface of first crystals with their origin at the intersection between the X-ray and the Bragg axis +- $\{\mathcal{F}_{2r}\}$ and $\{\mathcal{F}_{2h)}\}$: frames fixed to the top surface of second crystals. Their origin is at the crystal's center +- $\{\mathcal{F}_{mf}\}$: frame fixed to the center of the top part of the metrology frame + +#+name: fig:dcm_schematic_overview +#+caption: Schematic of the DCM with one set of first and second crystals with the associated frames +[[file:figs/dcm_schematic_overview.png]] + +#+begin_note +Here are the different letters used in the notations and there meaning: +- =r=: "ring" (i.e. toward $-y$) +- =h=: "hall" (i.e. toward $+y$) +- =u=: "upstream" (i.e. towards $-x$) +- =d=: "downstream" (i.e. towards $+x$) +- =c=: "center" (i.e. centered at $x=0$) +- =1=: primary crystal +- =2=: secondary crystal +- =mf=: metrology frame +#+end_note + +** Forward and Inverse Kinematics problems + +Let's consider one plane initially orthogonal to the $z$ direction (Figure [[fig:schematic_measurement_plane]]). +The pose of the plane, expressed in a frame $\{\mathcal{F}\}$ with origin $\mathcal{O}$ is defined by: +- its vertical motion: $d_z$ +- its rotation around the $y$ axis: $r_y$ +- its rotation around the $x$ axis: $r_x$ + +#+name: fig:schematic_measurement_plane +#+caption: SChematic of a plane with three vertical interferometers +[[file:figs/schematic_measurement_plane.png]] + +In order to measure the pose of the crystal, three interferometers are measuring the motion of the plane at three distinct positions $\mathcal{O}_1 = [x_1, y_1, 0]$, $\mathcal{O}_2 = [x_2, y_2, 0]$ and $\mathcal{O}_3 = [x_3, y_3, 0]$ (See Figure [[fig:schematic_measurement_plane]]). +The interferometers can be pointed in the $z$ direction or in the opposite $z$ direction. +The measured motion are noted $z_1$, $z_2$ and $z_3$. + +The *inverse kinematic* problem consists of deriving the measured $[z_1, z_2, z_3]$ motions from the pose $[d_z, r_y, r_x]$ of the crystal. +For small motion, this problem can be easily solved as follows: +\begin{equation} +\boxed{ +\begin{bmatrix} +z_{1} \\ z_{2} \\ z_{3} +\end{bmatrix} += +\bm{J} +\begin{bmatrix} +d_{z} \\ r_{y} \\ r_{x} +\end{bmatrix} +} +\end{equation} +with $\bm{J}$ called the *Jacobian matrix*. + +The Jacobian matrix can be computed as follows in the case where the interferometers are pointing toward positive $z$: +\begin{equation} \label{eq:jacobian_3dof_positive_z} +\boxed{ +\bm{J} = \begin{bmatrix} +1 & -x_1 & y_1 \\ +1 & -x_2 & y_2 \\ +1 & -x_3 & y_3 +\end{bmatrix} +} +\end{equation} +and computed as follows if there are pointing torward negative $z$: +\begin{equation} \label{eq:jacobian_3dof_negative_z} +\boxed{ +\bm{J} = \begin{bmatrix} +-1 & x_1 & -y_1 \\ +-1 & x_2 & -y_2 \\ +-1 & x_3 & -y_3 +\end{bmatrix} +} +\end{equation} + +The *forward kinematic* problem is then solved by inverting the Jacobian matrix: +\begin{equation} +\boxed{ +\begin{bmatrix} +d_{z} \\ r_{y} \\ r_{x} +\end{bmatrix} += +\bm{J}^{-1} +\begin{bmatrix} +z_{1} \\ z_{2} \\ z_{3} +\end{bmatrix} +} +\end{equation} + +The same reasoning can be performed to convert the wanted motion of one plane to the motion of three vertical actuators fixed to the plane (tripod architecture). ** Matlab Init :noexport:ignore: #+begin_src matlab @@ -96,19 +209,20 @@ It is structured as follow: #+end_src ** Bragg Angle -There is a simple relation eqref:eq:bragg_angle_formula between: -- $d_{\text{off}}$ is the wanted offset between the incident x-ray and the output x-ray -- $\theta_b$ is the bragg angle -- $d_z$ is the corresponding distance between the first and second crystals - +<> +In order to have a *fix exit*, the following relation must be verified: \begin{equation} \label{eq:bragg_angle_formula} -d_z = \frac{d_{\text{off}}}{2 \cos \theta_b} +\boxed{d_z = \frac{d_{\text{off}}}{2 \cos \theta_b}} \end{equation} +with: +- $d_{\text{off}}$ the wanted offset between the incident x-ray and the output x-ray +- $\theta_b$ the bragg angle +- $d_z$ the corresponding distance between the first and second crystal #+begin_src matlab :exports none %% Tested bragg angles -bragg = linspace(5, 80, 5000); % Bragg angle [deg] -d_off = 10.5e-3; % Wanted offset between x-rays [m] +bragg = linspace(0, 80, 5000); % Bragg angle [deg] +d_off = 10e-3; % Wanted offset between x-rays [m] #+end_src #+begin_src matlab :exports none @@ -116,13 +230,13 @@ d_off = 10.5e-3; % Wanted offset between x-rays [m] dz = d_off./(2*cos(bragg*pi/180)); #+end_src -This relation is shown in Figure [[fig:jack_motion_bragg_angle]]. +This relation is shown in Figure [[fig:jack_motion_bragg_angle]] for a wanted fix exit offset of 10mm. #+begin_src matlab :exports none %% Jack motion as a function of Bragg angle figure; plot(bragg, 1e3*dz) -xlabel('Bragg angle [deg]'); ylabel('Jack Motion [mm]'); +xlabel('Bragg angle [deg]'); ylabel('Crystal Distance [mm]'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace @@ -130,31 +244,178 @@ exportFig('figs/jack_motion_bragg_angle.pdf', 'width', 'wide', 'height', 'normal #+end_src #+name: fig:jack_motion_bragg_angle -#+caption: Jack motion as a function of Bragg angle +#+caption: Jack motion as a function of Bragg angle ($d_z = 10\,mm$) #+RESULTS: [[file:figs/jack_motion_bragg_angle.png]] -The required jack stroke is approximately 25mm. +** Kinematics of "hall" Crystals +<> +*** Introduction :ignore: +There are two "hall" crystals, the primary one and the secondary one. -#+begin_src matlab :results value replace :exports both -%% Required Jack stroke -ans = 1e3*(dz(end) - dz(1)) +*** Interferometers - Primary "Hall" Crystal + +Three interferometers fixed to the top metrology frame are pointed to the top surface (i.e. toward negative $z$) of the primary "hall" crystal. +The measurements are performed along the $z$ direction. + +The notations are: +- $\{\mathcal{F}_{1h}\}$ is the frame in which motion of the crystal are expressed +- $\mathcal{O}_{1h,ur}$ measurement point for the "upstream-ring" interferometer +- $\mathcal{O}_{1h,ch}$ measurement point for the "center-hall" interferometer +- $\mathcal{O}_{1h,dr}$ measurement point for the "downstream-ring" interferometer + +The measured displacements by the interferometers are noted $[z_{1h,ur},\ z_{1h,ch},\ z_{1h,dr}]$. + +The corresponding motion of the crystal expressed in the frame as shown in Figure [[fig:sensor_hall_crystal_points_primary]] are: +- $d_{1h,z}$ motion in the $z$ direction +- $r_{1h,y}$ rotation around the $y$ axis +- $r_{1h,x}$ rotation around the $x$ axis + +#+begin_src latex :file sensor_hall_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) {$\mathcal{O}_{1h,ur} = (-36, -15)$}; + \node[right] at (a2) {$\mathcal{O}_{1h,ch} = ( 0, 15)$}; + \node[left] at (a3) {$\mathcal{O}_{1h,dr} = ( 36, -15)$}; + + % 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); + \draw[color=black] (0, 0) circle (0.15); + \node[above right] at (0, 0){$z$}; + + \node[left] at (0,0) {$\{\mathcal{F}_{1h}\}$}; +\end{tikzpicture} #+end_src +#+name: fig:sensor_hall_crystal_points_primary +#+caption: Top view of the primary crystal hall. Position of the measurement points. Units in [mm] #+RESULTS: -: 24.963 +[[file:figs/sensor_hall_crystal_points_primary.png]] -** Kinematics (311 Crystal) -*** Introduction :ignore: -The reference frame is taken at the center of the 311 second crystal. -*** Interferometers - 311 secondary Crystal +The inverse kinematics problem consists of deriving the measured displacement by the interferometer from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_hall_primary]]): +\begin{equation} +\boxed{ +\begin{bmatrix} +z_{1h,ur} \\ z_{1h,ch} \\ z_{1h,dr} +\end{bmatrix} += +\bm{J}_{1h,s} +\begin{bmatrix} +d_{1h,z} \\ r_{1h,y} \\ r_{1h,x} +\end{bmatrix} +} +\end{equation} -Three interferometers are pointed to the bottom surface of the 311 crystal. +#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_hall_primary.pdf +\begin{tikzpicture} + % Blocs + \node[block] (Js) {$\bm{J}_{1h,s}$}; -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. + % Connections and labels + \draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{1h,z} \\ r_{1h,y} \\ r_{1h,x} \end{bmatrix}$} -- (Js.west); + \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} z_{1h,ur} \\ z_{1h,ch} \\ z_{1h,dr} \end{bmatrix}$}; +\end{tikzpicture} +#+end_src -#+begin_src latex :file sensor_311_crystal_points.pdf +#+name: fig:schematic_sensor_jacobian_inverse_kinematics_hall_primary +#+caption: Inverse Kinematics - Interferometers +#+RESULTS: +[[file:figs/schematic_sensor_jacobian_inverse_kinematics_hall_primary.png]] + +From the measurement coordinates in Figure [[fig:sensor_hall_crystal_points_primary]] and equation eqref:eq:jacobian_3dof_negative_z, the inverse kinematics the folowing Jacobian matrix is obtained: +\begin{equation} +\boxed{\bm{J}_{1h,s} += +\begin{bmatrix} +-1 & -0.036 & 0.015 \\ +-1 & 0 & -0.015 \\ +-1 & 0.036 & 0.015 +\end{bmatrix}} +\end{equation} + +#+begin_src matlab :exports none +%% Sensor Jacobian matrix for hall crystal +J_1h_s = [-1, -0.036, 0.015 + -1, 0, -0.015 + -1, 0.036, 0.015]; +#+end_src + +The forward kinematics problem (computing the crystal motion from the 3 interferometers) is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_hall_primary]]). +\begin{equation} +\boxed{ +\begin{bmatrix} +d_{1h,z} \\ r_{1h,y} \\ r_{1h,x} +\end{bmatrix} += +\bm{J}_{1h,s}^{-1} +\begin{bmatrix} +z_{1h,ur} \\ z_{1h,ch} \\ z_{1h,dr} +\end{bmatrix} +} +\end{equation} + +#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_hall_primary.pdf +\begin{tikzpicture} + % Blocs + \node[block] (Js_inv) {$\bm{J}_{1h,s}^{-1}$}; + + % Connections and labels + \draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} z_{1h,ur} \\ z_{1h,ch} \\ z_{1h,dr} \end{bmatrix}$} -- (Js_inv.west); + \draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{1h,z} \\ r_{1h,y} \\ r_{1h,x} \end{bmatrix}$}; +\end{tikzpicture} +#+end_src + +#+name: fig:schematic_sensor_jacobian_forward_kinematics_hall_primary +#+caption: Forward Kinematics - Interferometers +#+RESULTS: +[[file:figs/schematic_sensor_jacobian_forward_kinematics_hall_primary.png]] + +The obtained matrix is displayed in Table [[tab:inverse_jacobian_sensor_hall_primary]]. + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(inv(J_1h_s), {}, {}, ' %.2f '); +#+end_src + +#+name: tab:inverse_jacobian_sensor_hall_primary +#+caption: Inverse of the sensor Jacobian $\bm{J}_{1h,s}^{-1}$ +#+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 | + +*** Interferometers - Secondary "Hall" Crystal + +Three interferometers are pointed to the bottom surface (i.e. toward positive $z$) of the secondary "hall" crystal. + +The position of the measurement points are shown in Figure [[fig:sensor_hall_secondary_crystal_points]] as well as the origin where the motion of the crystal is computed. + +The notations are: +- $\{\mathcal{F}_{2h}\}$ is the frame in which motion of the crystal are expressed +- $\mathcal{O}_{2h,ur}$ measurement point for the "upstream-ring" interferometer +- $\mathcal{O}_{2h,ch}$ measurement point for the "center-hall" interferometer +- $\mathcal{O}_{2h,dr}$ measurement point for the "downstream-ring" interferometer + +The measured displacements by the interferometers are noted $[z_{2h,ur},\ z_{2h,ch},\ z_{2h,dr}]$. + +The corresponding motion of the crystal expressed in the frame as shown in Figure [[fig:sensor_hall_secondary_crystal_points]] are: +- $d_{2h,z}$ motion in the $z$ direction +- $r_{2h,y}$ rotation around the $y$ axis +- $r_{2h,x}$ rotation around the $x$ axis + +#+begin_src latex :file sensor_hall_secondary_crystal_points.pdf \begin{tikzpicture} % Crystal \draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2); @@ -165,118 +426,114 @@ The position of the measurement points are shown in Figure [[fig:sensor_311_crys \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) {$\mathcal{O}_3 = ( 0.07, -0.015)$}; + \node[right] at (a1) {$\mathcal{O}_{2h,ur} = (-70, -15)$}; + \node[right] at (a2) {$\mathcal{O}_{2h,ch} = ( 0, 15)$}; + \node[left] at (a3) {$\mathcal{O}_{2h,dr} = ( 70, -15)$}; % Origin - \draw[->] (0, 0) node[] -- ++(1, 0) node[right]{$x$}; - \draw[->] (0, 0) -- ++(0, -1) node[below]{$y$}; + \draw[->] (0, 0) node[] -- ++(1, 0) node[below]{$x$}; + \draw[->] (0, 0) -- ++(0, -1) node[right]{$y$}; \draw[fill, color=black] (0, 0) circle (0.05); + \draw[color=black] (0, 0) circle (0.15); + \draw[] (-0.1, 0.1) -- ( 0.1, -0.1); + \draw[] (-0.1,-0.1) -- ( 0.1, 0.1); + \node[below right] at (0,0) {$z$}; - \node[left] at (0,0) {$\mathcal{O}_{311}$}; + \node[left] at (0,0) {$\{\mathcal{F}_{2h}\}$}; \end{tikzpicture} #+end_src -#+name: fig:sensor_311_crystal_points -#+caption: Bottom view of the second crystal 311. Position of the measurement points. +#+name: fig:sensor_hall_secondary_crystal_points +#+caption: Bottom view of the secondary "hall" crystal. Position of the measurement points. Coordinate values are in [mm] #+RESULTS: -[[file:figs/sensor_311_crystal_points.png]] +[[file:figs/sensor_hall_secondary_crystal_points.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]]): +The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_hall_secondary]]): \begin{equation} +\boxed{ \begin{bmatrix} -x_1 \\ x_2 \\ x_3 +z_{2h,ur} \\ z_{2h,ch} \\ z_{2h,dr} \end{bmatrix} = -\bm{J}_{s,311} +\bm{J}_{2h,s} \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_311.pdf +#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_hall_secondary.pdf \begin{tikzpicture} % Blocs - \node[block] (Js) {$\bm{J}_{s,311}$}; + \node[block] (Js) {$\bm{J}_{2h,s}$}; % 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]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$}; + \draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix}$} -- (Js.west); + \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} z_{2h,ur} \\ z_{2h,ch} \\ z_{2h,dr} \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_sensor_jacobian_inverse_kinematics_311 +#+name: fig:schematic_sensor_jacobian_inverse_kinematics_hall_secondary #+caption: Inverse Kinematics - Interferometers #+RESULTS: -[[file:figs/schematic_sensor_jacobian_inverse_kinematics_311.png]] +[[file:figs/schematic_sensor_jacobian_inverse_kinematics_hall_secondary.png]] - -From the Figure [[fig:sensor_311_crystal_points]], the inverse kinematics can be solved as follow (for small motion): +From the measurement coordinates in Figure [[fig:sensor_hall_secondary_crystal_points]] and equation eqref:eq:jacobian_3dof_positive_z, the inverse kinematics the folowing Jacobian matrix is obtained: \begin{equation} -\bm{J}_{s,311} +\boxed{\bm{J}_{2h,s} = \begin{bmatrix} 1 & 0.07 & -0.015 \\ 1 & 0 & 0.015 \\ 1 & -0.07 & -0.015 -\end{bmatrix} +\end{bmatrix}} \end{equation} -#+begin_src matlab -%% Sensor Jacobian matrix for 311 crystal -J_s_311 = [1, 0.07, -0.015 - 1, 0, 0.015 - 1, -0.07, -0.015]; +#+begin_src matlab :exports none +%% Sensor Jacobian matrix for hall crystal +J_2h_s = [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, {}, {}, ' %.3f '); -#+end_src - -#+name: tab:jacobian_sensor_311 -#+caption: Sensor Jacobian $\bm{J}_{s,311}$ -#+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]]). +The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_hall_secondary]]). \begin{equation} +\boxed{ \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix} = -\bm{J}_{s,311}^{-1} +\bm{J}_{2h,s}^{-1} \begin{bmatrix} -x_1 \\ x_2 \\ x_3 +z_{2h,ur} \\ z_{2h,ch} \\ z_{2h,dr} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_311.pdf +#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_hall_secondary.pdf \begin{tikzpicture} % Blocs - \node[block] (Js_inv) {$\bm{J}_{s,311}^{-1}$}; + \node[block] (Js_inv) {$\bm{J}_{2h,s}^{-1}$}; % 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]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$}; + \draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} z_{2h,ur} \\ z_{2h,ch} \\ z_{2h,dr} \end{bmatrix}$} -- (Js_inv.west); + \draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_sensor_jacobian_forward_kinematics_311 +#+name: fig:schematic_sensor_jacobian_forward_kinematics_hall_secondary #+caption: Forward Kinematics - Interferometers #+RESULTS: -[[file:figs/schematic_sensor_jacobian_forward_kinematics_311.png]] +[[file:figs/schematic_sensor_jacobian_forward_kinematics_hall_secondary.png]] + +The results is shown in Table [[tab:inverse_jacobian_sensor_hall_secondary]]. #+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(inv(J_s_311), {}, {}, ' %.2f '); +data2orgtable(inv(J_2h_s), {}, {}, ' %.2f '); #+end_src -#+name: tab:inverse_jacobian_sensor_311 -#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,311}^{-1}$ +#+name: tab:inverse_jacobian_sensor_hall_secondary +#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,h}^{-1}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: @@ -284,244 +541,104 @@ data2orgtable(inv(J_s_311), {}, {}, ' %.2f '); | 7.14 | 0.0 | -7.14 | | -16.67 | 33.33 | -16.67 | -*** Interferometers - 311 primary Crystal +*** Piezo - hall Crystal -Three interferometers are pointed to the bottom surface of the 311 crystal. +The locations of the actuators with respect with the center of the secondary "hall" crystal are shown in Figure [[fig:actuator_jacobian_hall_points]]. -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]]. - -#+name: fig:actuator_jacobian_311_points -#+caption: Location of actuators with respect to the center of the 311 second crystal (bottom view) +#+name: fig:actuator_jacobian_hall_points +#+caption: Location of actuators with respect to the center of the hall second crystal (bottom view). Units are in [mm] #+attr_latex: :width \linewidth -[[file:figs/actuator_jacobian_311_points.png]] +[[file:figs/actuator_jacobian_hall_points.png]] -Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal's center. +Inverse Kinematics consist of deriving the corresponding z motion of the 3 actuators from the motion of the crystal expressed in $\{\mathcal{F}_{2h}\}$. +As in the previous section, the motion of the secondary "hall" crystal is expressed by $[d_{2h,z},\ r_{2h,y},\ r_{2h,x}]$. +The motion of the three fast jacks are expressed by $[d_{u_r},\ d_{u_h},\ d_{d}]$ are are corresponding to a motion along the $z$ axis toward positive values. + +The inverse kinematics can therefore be solved as follows: \begin{equation} +\boxed{ \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} = -\bm{J}_{a,311} +\bm{J}_{h,a} \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics_311.pdf +#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics_hall.pdf \begin{tikzpicture} % Blocs - \node[block] (Ja) {$\bm{J}_{a,311}$}; + \node[block] (Ja) {$\bm{J}_{h,a}$}; % Connections and labels - \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Ja.west); + \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix}$} -- (Ja.west); \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_sensor_jacobian_inverse_kinematics_311 +#+name: fig:schematic_sensor_jacobian_inverse_kinematics_hall #+caption: Inverse Kinematics - Actuators #+RESULTS: -[[file:figs/schematic_actuator_jacobian_inverse_kinematics_311.png]] +[[file:figs/schematic_actuator_jacobian_inverse_kinematics_hall.png]] -Based on the geometry in Figure [[fig:actuator_jacobian_311_points]], we obtain: +Based on the geometry in Figure [[fig:actuator_jacobian_hall_points]], the following Jacobian matrix is obtained (see Eq. eqref:eq:jacobian_3dof_positive_z): \begin{equation} -\bm{J}_{a,311} +\boxed{\bm{J}_{h,a} = \begin{bmatrix} 1 & 0.14 & -0.1525 \\ 1 & 0.14 & 0.0675 \\ 1 & -0.14 & -0.0425 -\end{bmatrix} +\end{bmatrix}} \end{equation} -#+begin_src matlab -%% Actuator Jacobian - 311 crystal -J_a_311 = [1, 0.14, -0.1525 - 1, 0.14, 0.0675 - 1, -0.14, -0.0425]; +#+begin_src matlab :exports none +%% Actuator Jacobian - hall crystal +J_h_a = [1, 0.14, -0.1525 + 1, 0.14, 0.0675 + 1, -0.14, -0.0425]; #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(J_a_311, {}, {}, ' %.4f '); -#+end_src - -#+name: tab:jacobian_actuator_311 -#+caption: Actuator Jacobian $\bm{J}_{a,311}$ -#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc -#+attr_latex: :center t :booktabs t -#+RESULTS: -| 1.0 | 0.14 | -0.1525 | -| 1.0 | 0.14 | 0.0675 | -| 1.0 | -0.14 | -0.0425 | - The forward Kinematics is solved by inverting the Jacobian matrix: \begin{equation} +\boxed{ \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix} = -\bm{J}_{a,311}^{-1} +\bm{J}_{h,a}^{-1} \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics_311.pdf +#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics_hall.pdf \begin{tikzpicture} % Blocs - \node[block] (Ja_inv) {$\bm{J}_{a,311}^{-1}$}; + \node[block] (Ja_inv) {$\bm{J}_{h,a}^{-1}$}; % Connections and labels \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); - \draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$}; + \draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_actuator_jacobian_forward_kinematics_311 -#+caption: Forward Kinematics - Actuators for 311 crystal +#+name: fig:schematic_actuator_jacobian_forward_kinematics_hall +#+caption: Forward Kinematics - Actuators for hall crystal #+RESULTS: -[[file:figs/schematic_actuator_jacobian_forward_kinematics_311.png]] +[[file:figs/schematic_actuator_jacobian_forward_kinematics_hall.png]] +The obtained inverse Jacobian matrix is shown in Table [[tab:inverse_jacobian_actuator_hall]]. #+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(inv(J_a_311), {}, {}, ' %.4f '); +data2orgtable(inv(J_h_a), {}, {}, ' %.4f '); #+end_src -#+name: tab:inverse_jacobian_actuator_311 -#+caption: Inverse of the actuator Jacobian $\bm{J}_{a,311}^{-1}$ +#+name: tab:inverse_jacobian_actuator_hall +#+caption: Inverse of the actuator Jacobian $\bm{J}_{a,h}^{-1}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: @@ -529,153 +646,30 @@ data2orgtable(inv(J_a_311), {}, {}, ' %.4f '); | 1.7857 | 1.7857 | -3.5714 | | -4.5455 | 4.5455 | 0.0 | -** Kinematics (111 Crystal) +** Kinematics of "ring" Crystals +<> *** Introduction :ignore: -The reference frame is taken at the center of the 111 second crystal. +The same reasoning is now done for the "ring" crystals. -*** Interferometers - 111 secondary Crystal +*** Interferometers - ring primary Crystal -Three interferometers are pointed to the bottom surface of the 111 crystal. +Three interferometers fixed to the top metrology frame are pointed to the top surface (i.e. toward negative $z$) of the primary "hall" crystal. +The measurements are perform along the $z$ direction. -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. +The notations are: +- $\{\mathcal{F}_{1r}\}$ is the frame in which motion of the crystal are expressed +- $\mathcal{O}_{1r,uh}$ measurement point for the "upstream-hall" interferometer +- $\mathcal{O}_{1r,cr}$ measurement point for the "center-ring" interferometer +- $\mathcal{O}_{1r,dh}$ measurement point for the "downstream-hall" interferometer -#+begin_src latex :file sensor_111_crystal_points.pdf -\begin{tikzpicture} - % Crystal - \draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2); +The measured displacements by the interferometers are noted $[z_{1r,ur},\ z_{1r,ch},\ z_{1r,dr}]$. - % Measurement Points - \node[branch] (a1) at (-7, -1.5){}; - \node[branch] (a2) at ( 0, 1.5){}; - \node[branch] (a3) at ( 7, -1.5){}; +The corresponding motion of the crystal expressed in the frame as shown in Figure [[fig:sensor_hall_crystal_points_primary]] are: +- $d_{1r,z}$ motion in the $z$ direction +- $r_{1r,y}$ rotation around the $y$ axis +- $r_{1r,x}$ rotation around the $x$ axis - % 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[below]{$y$}; - \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 -#+caption: Bottom view of the second crystal 111. Position of the measurement points. -#+RESULTS: -[[file:figs/sensor_111_crystal_points.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 & 0.07 & 0.015 \\ -1 & 0 & -0.015 \\ -1 & -0.07 & 0.015 -\end{bmatrix} -\end{equation} - -#+begin_src matlab -%% Sensor Jacobian matrix for 111 crystal -J_s_111 = [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_111, {}, {}, ' %.3f '); -#+end_src - -#+name: tab:jacobian_sensor_111 -#+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 | 0.07 | 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_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), {}, {}, ' %.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 | -| 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_src latex :file sensor_ring_crystal_points_primary.pdf \begin{tikzpicture} % Crystal \draw (-7.5/2, -3.5/2) rectangle (7.5/2, 3.5/2); @@ -686,118 +680,112 @@ The position of the measurement points are shown in Figure [[fig:sensor_111_crys \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) {$\mathcal{O}_3 = ( 0.036, 0.015)$}; + \node[right] at (a1) {$\mathcal{O}_{1r,uh} = (-36, 15)$}; + \node[right] at (a2) {$\mathcal{O}_{1r,cr} = ( 0, -15)$}; + \node[left] at (a3) {$\mathcal{O}_{1r,dh} = ( 36, 15)$}; % 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); + \draw[color=black] (0, 0) circle (0.15); + \node[above right] at (0, 0){$z$}; - \node[left] at (0,0) {$\mathcal{O}_{111}$}; + \node[left] at (0,0) {$\{\mathcal{F}_{1r}\}$}; \end{tikzpicture} #+end_src -#+name: fig:sensor_111_crystal_points_primary -#+caption: Top view of the primary crystal 111. Position of the measurement points. +#+name: fig:sensor_ring_crystal_points_primary +#+caption: Top view of the primary crystal ring. Position of the measurement points in [mm]. #+RESULTS: -[[file:figs/sensor_111_crystal_points_primary.png]] +[[file:figs/sensor_ring_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]]): +The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_ring_primary]]): \begin{equation} +\boxed{ \begin{bmatrix} -x_1 \\ x_2 \\ x_3 +z_{1r,ur} \\ z_{1r,ch} \\ z_{1r,dr} \end{bmatrix} = -\bm{J}_{s,111} +\bm{J}_{1r,s} \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{1r,z} \\ r_{1r,y} \\ r_{1r,x} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_111.pdf +#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_ring_primary.pdf \begin{tikzpicture} % Blocs - \node[block] (Js) {$\bm{J}_{s,111}$}; + \node[block] (Js) {$\bm{J}_{1r,s}$}; % 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]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$}; + \draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{1r,z} \\ r_{1r,y} \\ r_{1r,x} \end{bmatrix}$} -- (Js.west); + \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} z_{1r,ur} \\ z_{1r,ch} \\ z_{1r,dr} \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_sensor_jacobian_inverse_kinematics_111 -#+caption: Inverse Kinematics - Interferometers +#+name: fig:schematic_sensor_jacobian_inverse_kinematics_ring_primary +#+caption: Inverse Kinematics - Interferometers (primary "ring" crystal) #+RESULTS: -[[file:figs/schematic_sensor_jacobian_inverse_kinematics_111.png]] +[[file:figs/schematic_sensor_jacobian_inverse_kinematics_ring_primary.png]] - -From the Figure [[fig:sensor_111_crystal_points]], the inverse kinematics can be solved as follow (for small motion): +From the measurement coordinates in Figure [[fig:sensor_ring_crystal_points_primary]] and equation eqref:eq:jacobian_3dof_negative_z, the inverse kinematics the folowing Jacobian matrix is obtained: \begin{equation} -\bm{J}_{s,111} +\boxed{\bm{J}_{1r,s} = \begin{bmatrix} -1 & -0.036 & -0.015 \\ -1 & 0 & 0.015 \\ -1 & 0.036 & -0.015 -\end{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]; +#+begin_src matlab :exports none +%% Sensor Jacobian matrix for ring crystal +J_1r_s = [-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]]). +The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_ring_primary]]). \begin{equation} +\boxed{ \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{1r,z} \\ r_{1r,y} \\ r_{1r,x} \end{bmatrix} = -\bm{J}_{s,111}^{-1} +\bm{J}_{1r,s}^{-1} \begin{bmatrix} -x_1 \\ x_2 \\ x_3 +z_{1r,ur} \\ z_{1r,ch} \\ z_{1r,dr} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_111.pdf +#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_ring_primary.pdf \begin{tikzpicture} % Blocs - \node[block] (Js_inv) {$\bm{J}_{s,111}^{-1}$}; + \node[block] (Js_inv) {$\bm{J}_{1r,s}^{-1}$}; % 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]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$}; + \draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} z_{1r,ur} \\ z_{1r,ch} \\ z_{1r,dr} \end{bmatrix}$} -- (Js_inv.west); + \draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{1r,z} \\ r_{1r,y} \\ r_{1r,x} \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_sensor_jacobian_forward_kinematics_111 +#+name: fig:schematic_sensor_jacobian_forward_kinematics_ring_primary #+caption: Forward Kinematics - Interferometers #+RESULTS: -[[file:figs/schematic_sensor_jacobian_forward_kinematics_111.png]] +[[file:figs/schematic_sensor_jacobian_forward_kinematics_ring_primary.png]] + +The obtained inverse Jacobian matrix is shown in Table [[tab:inverse_jacobian_sensor_ring_primary]]. #+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(inv(J_s_111_1), {}, {}, ' %.2f '); +data2orgtable(inv(J_1r_s), {}, {}, ' %.2f '); #+end_src -#+name: tab:inverse_jacobian_sensor_111 -#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,111}^{-1}$ +#+name: tab:inverse_jacobian_sensor_ring_primary +#+caption: Inverse of the sensor Jacobian $\bm{J}_{1r,s}^{-1}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: @@ -805,244 +793,491 @@ data2orgtable(inv(J_s_111_1), {}, {}, ' %.2f '); | -13.89 | 0.0 | 13.89 | | -16.67 | 33.33 | -16.67 | -*** Piezo - 111 Crystal +*** Interferometers - ring secondary 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]]. +Three interferometers are pointed to the bottom surface (i.e. toward positive $z$) of the secondary "ring" crystal. -#+name: fig:actuator_jacobian_111_points -#+caption: Location of actuators with respect to the center of the 111 second crystal (bottom view) -#+attr_latex: :width \linewidth -[[file:figs/actuator_jacobian_111_points.png]] +The position of the measurement points are shown in Figure [[fig:sensor_ring_secondary_crystal_points]] as well as the origin where the motion of the crystal is computed. -Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal's center. +The notations are: +- $\{\mathcal{F}_{2r}\}$ is the frame in which motion of the crystal are expressed +- $\mathcal{O}_{2r,uh}$ measurement point for the "upstream-hall" interferometer +- $\mathcal{O}_{2r,cr}$ measurement point for the "center-ring" interferometer +- $\mathcal{O}_{2r,dh}$ measurement point for the "downstream-hall" interferometer + +The measured displacements by the interferometers are noted $[z_{2r,ur},\ z_{2r,ch},\ z_{2r,dr}]$. + +The corresponding motion of the crystal expressed in the frame as shown in Figure [[fig:sensor_hall_secondary_crystal_points]] are: +- $d_{2r,z}$ motion in the $z$ direction +- $r_{2r,y}$ rotation around the $y$ axis +- $r_{2r,x}$ rotation around the $x$ axis +The position of the measurement points are shown in Figure [[fig:sensor_ring_secondary_crystal_points]] as well as the origin where the motion of the crystal is computed. + +#+begin_src latex :file sensor_ring_secondary_crystal_points.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) {$\mathcal{O}_{2r,uh} = (-70, 15)$}; + \node[right] at (a2) {$\mathcal{O}_{2r,cr} = ( 0, -15)$}; + \node[left] at (a3) {$\mathcal{O}_{2r,dh} = ( 70, 15)$}; + + % Origin + \draw[->] (0, 0) node[] -- ++(1, 0) node[below]{$x$}; + \draw[->] (0, 0) -- ++(0, -1) node[right]{$y$}; + \draw[fill, color=black] (0, 0) circle (0.05); + \draw[color=black] (0, 0) circle (0.15); + \draw[] (-0.1, 0.1) -- ( 0.1, -0.1); + \draw[] (-0.1,-0.1) -- ( 0.1, 0.1); + \node[below right] at (0,0) {$z$}; + + \node[left] at (0,0) {$\{\mathcal{F}_{2r}\}$}; +\end{tikzpicture} +#+end_src + +#+name: fig:sensor_ring_secondary_crystal_points +#+caption: Bottom view of the second crystal ring. Position of the measurement points. +#+RESULTS: +[[file:figs/sensor_ring_secondary_crystal_points.png]] + +The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics_ring_secondary]]): \begin{equation} +\boxed{ +\begin{bmatrix} +z_{2r,uh} \\ z_{2r,cr} \\ z_{2r,dh} +\end{bmatrix} += +\bm{J}_{2r,s} +\begin{bmatrix} +d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} +\end{bmatrix} +} +\end{equation} + +#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_ring_secondary.pdf +\begin{tikzpicture} + % Blocs + \node[block] (Js) {$\bm{J}_{2r,s}$}; + + % Connections and labels + \draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} \end{bmatrix}$} -- (Js.west); + \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} z_{2r,uh} \\ z_{2r,cr} \\ z_{2r,dh} \end{bmatrix}$}; +\end{tikzpicture} +#+end_src + +#+name: fig:schematic_sensor_jacobian_inverse_kinematics_ring_secondary +#+caption: Inverse Kinematics - Interferometers +#+RESULTS: +[[file:figs/schematic_sensor_jacobian_inverse_kinematics_ring_secondary.png]] + + +From the measurement coordinates in Figure [[fig:sensor_ring_secondary_crystal_points]] and equation eqref:eq:jacobian_3dof_positive_z, the inverse kinematics the folowing Jacobian matrix is obtained: +\begin{equation} +\boxed{\bm{J}_{2r,s} += +\begin{bmatrix} +1 & 0.07 & 0.015 \\ +1 & 0 & -0.015 \\ +1 & -0.07 & 0.015 +\end{bmatrix}} +\end{equation} + +#+begin_src matlab :exports none +%% Sensor Jacobian matrix for ring crystal +J_2r_s = [1, 0.07, 0.015 + 1, 0, -0.015 + 1, -0.07, 0.015]; +#+end_src + +The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics_ring_secondary]]). +\begin{equation} +\boxed{ +\begin{bmatrix} +d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} +\end{bmatrix} += +\bm{J}_{2r,s}^{-1} +\begin{bmatrix} +z_{2r,uh} \\ z_{2r,cr} \\ z_{2r,dh} +\end{bmatrix} +} +\end{equation} + +#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_ring_secondary.pdf +\begin{tikzpicture} + % Blocs + \node[block] (Js_inv) {$\bm{J}_{s,r}^{-1}$}; + + % Connections and labels + \draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} z_{2r,uh} \\ z_{2r,cr} \\ z_{2r,dh} \end{bmatrix}$} -- (Js_inv.west); + \draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} \end{bmatrix}$}; +\end{tikzpicture} +#+end_src + +#+name: fig:schematic_sensor_jacobian_forward_kinematics_ring_secondary +#+caption: Forward Kinematics - Interferometers +#+RESULTS: +[[file:figs/schematic_sensor_jacobian_forward_kinematics_ring_secondary.png]] + +The obtained inverse Jacobian matrix is shown in Table [[tab:inverse_jacobian_sensor_ring_secondary]]. + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(inv(J_2r_s), {}, {}, ' %.2f '); +#+end_src + +#+name: tab:inverse_jacobian_sensor_ring_secondary +#+caption: Inverse of the sensor Jacobian $\bm{J}_{2r,s}^{-1}$ +#+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 - ring Crystal + +The location of the actuators with respect with the center of the secondary "ring" crystal are shown in Figure [[fig:actuator_jacobian_ring_points]]. + +#+name: fig:actuator_jacobian_ring_points +#+caption: Location of actuators with respect to the center of the ring second crystal (bottom view) +#+attr_latex: :width \linewidth +[[file:figs/actuator_jacobian_ring_points.png]] + + +Inverse Kinematics consist of deriving the corresponding z motion of the 3 actuators from the motion of the crystal expressed in $\{\mathcal{F}_{2r}\}$. +As in the previous section, the motion of the secondary "ring" crystal is expressed by $[d_{2r,z},\ r_{2r,y},\ r_{2r,x}]$. +The motion of the three fast jacks are expressed by $[d_{u_r},\ d_{u_h},\ d_{d}]$ are are corresponding to a motion along the $z$ axis toward positive values. + +The inverse kinematics can therefore be solved as follows: +\begin{equation} +\boxed{ \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} = -\bm{J}_{a,111} +\bm{J}_{h,a} \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics_111.pdf +#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics_ring.pdf \begin{tikzpicture} % Blocs - \node[block] (Ja) {$\bm{J}_{a,111}$}; + \node[block] (Ja) {$\bm{J}_{r,a}$}; % Connections and labels - \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Ja.west); + \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} \end{bmatrix}$} -- (Ja.west); \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_sensor_jacobian_inverse_kinematics_111 +#+name: fig:schematic_sensor_jacobian_inverse_kinematics_ring #+caption: Inverse Kinematics - Actuators #+RESULTS: -[[file:figs/schematic_actuator_jacobian_inverse_kinematics_111.png]] +[[file:figs/schematic_actuator_jacobian_inverse_kinematics_ring.png]] -Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain: +Based on the geometry in Figure [[fig:actuator_jacobian_ring_points]], we obtain (see Eq. eqref:eq:jacobian_3dof_positive_z): \begin{equation} -\bm{J}_{a,111} +\boxed{\bm{J}_{r,a} = \begin{bmatrix} 1 & 0.14 & -0.0675 \\ 1 & 0.14 & 0.1525 \\ 1 & -0.14 & 0.0425 -\end{bmatrix} +\end{bmatrix}} \end{equation} -#+begin_src matlab -%% Actuator Jacobian - 111 crystal -J_a_111 = [1, 0.14, -0.0675 - 1, 0.14, 0.1525 - 1, -0.14, 0.0425]; +#+begin_src matlab :exports none +%% Actuator Jacobian - ring crystal +J_r_a = [1, 0.14, -0.0675 + 1, 0.14, 0.1525 + 1, -0.14, 0.0425]; #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(J_a_111, {}, {}, ' %.4f '); -#+end_src - -#+name: tab:jacobian_actuator_111 -#+caption: Actuator Jacobian $\bm{J}_{a,111}$ -#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc -#+attr_latex: :center t :booktabs t -#+RESULTS: -| 1.0 | 0.14 | -0.1525 | -| 1.0 | 0.14 | 0.0675 | -| 1.0 | -0.14 | 0.0425 | - The forward Kinematics is solved by inverting the Jacobian matrix: \begin{equation} +\boxed{ \begin{bmatrix} -d_z \\ r_y \\ r_x +d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} \end{bmatrix} = -\bm{J}_{a,111}^{-1} +\bm{J}_{r,a}^{-1} \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} +} \end{equation} -#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics_111.pdf +#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics_ring.pdf \begin{tikzpicture} % Blocs - \node[block] (Ja_inv) {$\bm{J}_{a,111}^{-1}$}; + \node[block] (Ja_inv) {$\bm{J}_{r,a}^{-1}$}; % Connections and labels \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); - \draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$}; + \draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{2r,z} \\ r_{2r,y} \\ r_{2r,x} \end{bmatrix}$}; \end{tikzpicture} #+end_src -#+name: fig:schematic_actuator_jacobian_forward_kinematics_111 -#+caption: Forward Kinematics - Actuators for 111 crystal +#+name: fig:schematic_actuator_jacobian_forward_kinematics_ring +#+caption: Forward Kinematics - Actuators for ring crystal #+RESULTS: -[[file:figs/schematic_actuator_jacobian_forward_kinematics_111.png]] +[[file:figs/schematic_actuator_jacobian_forward_kinematics_ring.png]] + +The obtained inverse Jacobian matrix is shown in Table [[tab:inverse_jacobian_actuator_ring_secondary]]. #+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(inv(J_a_111), {}, {}, ' %.4f '); +data2orgtable(inv(J_r_a), {}, {}, ' %.4f '); #+end_src -#+name: tab:inverse_jacobian_actuator_111 -#+caption: Inverse of the actuator Jacobian $\bm{J}_{a,111}^{-1}$ +#+name: tab:inverse_jacobian_actuator_ring_secondary +#+caption: Inverse of the actuator Jacobian $\bm{J}_{r,a}^{-1}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: -| 0.25 | 0.25 | 0.5 | -| 0.4058 | 3.1656 | -3.5714 | +| 0.4432 | 0.0568 | 0.5 | +| 1.7857 | 1.7857 | -3.5714 | | -4.5455 | 4.5455 | 0.0 | -** Kinematics (Metrology Frame) +** Kinematics of the Metrology Frame +<> + +Three interferometers (aligned with the $z$ axis) are used to measure the relative motion between the bottom part and the top part of the metrology frame. + +The location of the three interferometers are shown in Figure [[fig:jacobian_metrology_frame]]. + +The measured displacements by the interferometers are are noted $[z_{mf,u},\ z_{mf,dh},\ z_{mf,dr}]$. + +We here suppose that the bottom part of the metrology frame is fixed and only the top part is moving. +The motion of the metrology top part expressed in the frame as shown in Figure [[fig:jacobian_metrology_frame]] are: +- $d_{mf,z}$ motion in the $z$ direction +- $r_{mf,y}$ rotation around the $y$ axis +- $r_{mf,x}$ rotation around the $x$ axis #+name: fig:jacobian_metrology_frame -#+caption: Top View - Top metrology frame +#+caption: Top View of the top part of the 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]; +The inverse kinematics consisting of deriving the interferometer measurements from the motion of the metrology top part: +\begin{equation} +\boxed{ +\begin{bmatrix} +z_{mf,u} \\ z_{mf,dh} \\ z_{mf,dr} +\end{bmatrix} += +\bm{J}_{mf,s} +\begin{bmatrix} +d_{mf,z} \\ r_{mf,y} \\ r_{mf,x} +\end{bmatrix} +} +\end{equation} + +From the coordinates in Figure [[fig:jacobian_metrology_frame]] and the equation eqref:eq:jacobian_3dof_positive_z, the following Jacobian matrix is obtained: +\begin{equation} +\boxed{\bm{J}_{mf,r} += +\begin{bmatrix} +1 & 0.102 & 0 \\ +1 & -0.088 & 0.1275 \\ +1 & -0.088 & -0.1275 +\end{bmatrix}} +\end{equation} + +#+begin_src matlab :exports none +%% Sensor Jacobian matrix for Metrology Frame +J_mf_s = [1, 0.102, 0 + 1, -0.088, 0.1275 + 1, -0.088, -0.1275]; #+end_src +The forward kinematics is solved by inverting the Jacobian matrix: +\begin{equation} +\boxed{ +\begin{bmatrix} +d_{mf,z} \\ r_{mf,y} \\ r_{mf,x} +\end{bmatrix} += +\bm{J}_{mf,s}^{-1} +\begin{bmatrix} +z_{mf,u} \\ z_{mf,dh} \\ z_{mf,dr} +\end{bmatrix} +} +\end{equation} + +The obtained inverse Jacobian matrix is shown in Table [[tab:inverse_jacobian_sensor_metrology_frame]]. + #+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 '); +data2orgtable(inv(J_mf_s), {}, {}, ' %.3f '); #+end_src +#+name: tab:inverse_jacobian_sensor_metrology_frame +#+caption: Inverse of the sensor Jacobian $\bm{J}_{mf,f}^{-1}$ +#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc +#+attr_latex: :center t :booktabs t #+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'; +** Computing relative position between Crystals -%% Input/Output definition -clear io; io_i = 1; +In the previous sections, the motion of the individual crystals have been computed. -%% Inputs -io(io_i) = linio([mdl, '/interferometers'], 1, 'openinput'); io_i = io_i + 1; +We wish to express the *relative pose* of the primary and secondary crystals $[d_{h,z},\ r_{h,y},\ r_{h,x}]$ or $[d_{r,z},\ r_{r,y},\ r_{r,x}]$. -%% Outputs -io(io_i) = linio([mdl, '/xtal_111'], 1, 'openoutput'); io_i = io_i + 1; -#+end_src +#+begin_note +The sign conventions for the relative crystal pose are: +- An increase of $d_{h,z}$ means the two crystals are further apart +- An increase of $r_{h,}$ means that the second crystals experiences a rotation around $y$ with respect to the primary crystal +- An increase of $r_{h,x}$ means that the second crystals experiences a rotation around $x$ with respect to the primary crystal +#+end_note -#+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 +We therefore have the following relation (same for "hall" and "ring" crystals): +\begin{equation} +\boxed{ +\begin{bmatrix} d_{h,z} \\ r_{h,y} \\ r_{h,x} \end{bmatrix} += +\begin{bmatrix} d_{2h,z} \\ r_{2h,y} \\ r_{2h,x} \end{bmatrix} +- +\begin{bmatrix} d_{1h,z} \\ r_{1h,y} \\ r_{1h,x} \end{bmatrix} +- +\begin{bmatrix} d_{mf,z} \\ r_{mf,y} \\ r_{mf,x} \end{bmatrix} +} +\end{equation} -#+begin_src matlab -mdl = 'coordinate_transform'; +The relative pose can be expressed as a function of the interferometers using the Jacobian matrices for the "hall" crystals: +\begin{equation} +\boxed{ +\begin{bmatrix} d_{h,z} \\ r_{h,y} \\ r_{h,x} \end{bmatrix} += +\bm{J}_{2h,s}^{-1} +\begin{bmatrix} z_{2h,ur} \\ z_{2h,ch} \\ z_{2h,dr} \end{bmatrix} +- +\bm{J}_{1h,s}^{-1} +\begin{bmatrix} z_{1h,ur} \\ z_{1h,ch} \\ z_{1h,dr} \end{bmatrix} +- +\bm{J}_{mf,s}^{-1} +\begin{bmatrix} z_{mf,u} \\ z_{mf,dh} \\ z_{mf,dr} \end{bmatrix} +} +\end{equation} -%% Input/Output definition -clear io; io_i = 1; +As well as for the "ring" crystals: +\begin{equation} +\boxed{ +\begin{bmatrix} d_{r,z} \\ r_{r,y} \\ r_{r,x} \end{bmatrix} += +\bm{J}_{2r,s}^{-1} +\begin{bmatrix} z_{2r,ur} \\ z_{2r,cr} \\ z_{2r,dr} \end{bmatrix} +- +\bm{J}_{1r,s}^{-1} +\begin{bmatrix} z_{1r,ur} \\ z_{1r,cr} \\ z_{1r,dr} \end{bmatrix} +- +\bm{J}_{mf,s}^{-1} +\begin{bmatrix} z_{mf,u} \\ z_{mf,dr} \\ z_{mf,dr} \end{bmatrix} +} +\end{equation} -%% Inputs -io(io_i) = linio([mdl, '/interferometers'], 1, 'openinput'); io_i = io_i + 1; +** Summary - List of notations -%% Outputs -io(io_i) = linio([mdl, '/xtal_111'], 1, 'openoutput'); io_i = io_i + 1; -#+end_src +There are 5 Jacobian matrices that are used to convert the 15 interferometers to the relative pose between the primary and secondary crystals as well as 2 Jacobian matrices for the actuators. +All Jacobian matrices are summarized in Table [[tab:jacobian_matrices]]. -#+begin_src matlab -G_111 = linearize(mdl, io); -G_111 = G_111(:,[4 5 6 10 11 12 13 14 15]); -#+end_src +#+name: tab:jacobian_matrices +#+caption: List of Jacobian matrices +#+attr_latex: :environment tabularx :width 0.9\linewidth :align ccl +#+attr_latex: :center t :booktabs t +| *Notation* | *Variable* | *Description* | +|-----------------+------------+------------------------------------------------------------| +| $\bm{J}_{1h,s}$ | =J_1h_s= | Converts interferometers to the first "hall" crystal pose | +| $\bm{J}_{2h,s}$ | =J_2h_s= | Converts interferometers to the second "hall" crystal pose | +| $\bm{J}_{1r,s}$ | =J_1r_s= | Converts interferometers to the first "ring" crystal pose | +| $\bm{J}_{2r,s}$ | =J_2r_s= | Converts interferometers to the second "ring" crystal pose | +| $\bm{J}_{mf,s}$ | =J_mf_s= | Converts interferometers to the first "hall" crystal pose | +| $\bm{J}_{h,a}$ | =J_h_a= | Converts Fastjack motion to the second "hall" crystal pose | +| $\bm{J}_{r,a}$ | =J_r_a= | Converts Fastjack motion to the second "ring" crystal pose | -#+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 +The 15 interferometer measurements are summarized in Table [[tab:interferometer_list]]. -#+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 +#+name: tab:interferometer_list +#+caption: List of Interferometer measurements +#+attr_latex: :environment tabularx :width 0.7\linewidth :align ccl +#+attr_latex: :center t :booktabs t +| *Notation* | *Variable* | *Interferometer* | +|-------------+------------+------------------------------------------| +| $z_{1h,ur}$ | =z_1h_ur= | First "Hall" Crystal, "upstream-ring" | +| $z_{1h,ch}$ | =z_1h_ch= | First "Hall" Crystal, "center-hall" | +| $z_{1h,dr}$ | =z_1h_dr= | First "Hall" Crystal, "downstream-ring" | +| $z_{2h,ur}$ | =z_2h_ur= | Second "Hall" Crystal, "upstream-ring" | +| $z_{2h,ch}$ | =z_2h_ch= | Second "Hall" Crystal, "center-hall" | +| $z_{2h,dr}$ | =z_2h_dr= | Second "Hall" Crystal, "downstream-ring" | +| $z_{1r,uh}$ | =z_1r_uh= | First "Hall" Crystal, "upstream-hall" | +| $z_{1r,cr}$ | =z_1r_cr= | First "Hall" Crystal, "center-ring" | +| $z_{1r,dh}$ | =z_1r_dh= | First "Hall" Crystal, "downstream-hall" | +| $z_{2r,uh}$ | =z_2r_uh= | Second "Hall" Crystal, "upstream-hall" | +| $z_{2r,cr}$ | =z_2r_cr= | Second "Hall" Crystal, "center-ring" | +| $z_{2r,dh}$ | =z_2r_dh= | Second "Hall" Crystal, "downstream-hall" | +| $z_{mf,u}$ | =z_mf_u= | Metrology Frame, "upstream" | +| $z_{mf,dh}$ | =z_mf_dh= | Metrology Frame, "downstream-hall" | +| $z_{mf,dr}$ | =z_mf_dr= | Metrology Frame, "downstream-ring" | -#+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 +From the 15 interferometers, several crystal pose are computed. +They are summarized in Table [[tab:crystal_pose_list]]. -#+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 +#+name: tab:crystal_pose_list +#+caption: List of crystal's pose +#+attr_latex: :environment tabularx :width 0.9\linewidth :align ccl +#+attr_latex: :center t :booktabs t +| *Notation* | *Variable* | *Description* | +|------------+------------+---------------------------------------------------------| +| $d_{1h,z}$ | =d_1h_z= | First "Hall" Crystal, z-translation | +| $r_{1h,y}$ | =r_1h_y= | First "Hall" Crystal, y-rotation | +| $r_{1h,x}$ | =r_1h_x= | First "Hall" Crystal, x-rotation | +| $d_{2h,z}$ | =d_2h_z= | Second "Hall" Crystal, z-translation | +| $r_{2h,y}$ | =r_2h_y= | Second "Hall" Crystal, y-rotation | +| $r_{2h,x}$ | =r_2h_x= | Second "Hall" Crystal, x-rotation | +| $d_{1r,z}$ | =d_1r_z= | First "Ring" Crystal, z-translation | +| $r_{1r,y}$ | =r_1r_y= | First "Ring" Crystal, y-rotation | +| $r_{1r,x}$ | =r_1r_x= | First "Ring" Crystal, x-rotation | +| $d_{2r,z}$ | =d_2r_z= | Second "Ring" Crystal, z-translation | +| $r_{2r,y}$ | =r_2r_y= | Second "Ring" Crystal, y-rotation | +| $r_{2r,x}$ | =r_2r_x= | Second "Ring" Crystal, x-rotation | +| $d_{mf,z}$ | =d_mf_z= | Metrology Frame, z-translation (top relative to bottom) | +| $r_{mf,y}$ | =r_mf_y= | Metrology Frame, y-rotation (top relative to bottom) | +| $r_{mf,x}$ | =r_mf_x= | Metrology Frame, x-rotation (top relative to bottom) | -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(G_111_t, {}, {}, ' %.3f '); -#+end_src +And finally, the relative pose between the first and second crystals are defined in Table [[tab:crystal_relative_pose_list]]. -#+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: -- Motion errors of the stepper motor -- Vibrations from the outside -- Vibrations from the cooling system directly applied on the crystals +#+name: tab:crystal_relative_pose_list +#+caption: Relative pose between the primary and secondary crystals +#+attr_latex: :environment tabularx :width 0.9\linewidth :align ccl +#+attr_latex: :center t :booktabs t +| *Notation* | *Variable* | *Description* | +|------------+------------+------------------------------------------------------------------| +| $d_{h,z}$ | =d_h_z= | Relative distance between the "hall" crystals | +| $r_{h,y}$ | =r_h_y= | Relative y-rotation of the 2nd "hall" crystal w.r.t. the primary | +| $r_{h,x}$ | =r_h_x= | Relative x-rotation of the 2nd "hall" crystal w.r.t. the primary | +| $d_{r,z}$ | =d_r_z= | Relative distance between the "ring" crystals | +| $r_{r,y}$ | =r_r_y= | Relative y-rotation of the 2nd "ring" crystal w.r.t. the primary | +| $r_{r,x}$ | =r_r_x= | Relative x-rotation of the 2nd "ring" crystal w.r.t. the primary | ** Save Kinematics + +All the Jacobian matrix are exported so that there can be easily imported for computation purposes. + #+begin_src matlab :exports none :tangle no -save('matlab/mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', 'J_a_111', 'J_s_111') +save('matlab/mat/dcm_kinematics.mat', 'J_1h_s', 'J_1r_s', 'J_2h_s', 'J_2r_s', 'J_mf_s', 'J_h_a', 'J_r_a') #+end_src #+begin_src matlab :eval no -save('mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', 'J_a_111', 'J_s_111') +save('mat/dcm_kinematics.mat', 'J_1h_s', 'J_1r_s', 'J_2h_s', 'J_2r_s', 'J_mf_s', 'J_h_a', 'J_r_a') #+end_src * Open Loop System Identification @@ -1084,7 +1319,7 @@ save('mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', 'J_a_111', 'J_s_111') ** Identification Let's considered the system $\bm{G}(s)$ with: - 3 inputs: force applied to the 3 fast jacks -- 3 outputs: measured displacement by the 3 interferometers pointing at the 111 second crystal +- 3 outputs: measured displacement by the 3 interferometers pointing at the ring second crystal It is schematically shown in Figure [[fig:schematic_system_inputs_outputs]]. @@ -1127,7 +1362,7 @@ G = linearize(mdl, io); #+begin_src matlab :exports none %% Input and Output names G.InputName = {'u_ur', 'u_uh', 'u_d'}; -G.OutputName = {'int_111_1', 'int_111_2', 'int_111_3'}; +G.OutputName = {'int_r_1', 'int_r_2', 'int_r_3'}; #+end_src #+begin_src matlab :results output replace :exports both :tangle no @@ -1173,7 +1408,7 @@ 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_311*inv(J_s_311)*G; +G_pz = J_a_h*inv(J_2h_s)*G; #+end_src The DC gain of the new system shows that the system is well decoupled at low frequency. @@ -1278,7 +1513,7 @@ Computing the system in the frame of the fastjack gives good decoupling at low f [[file:figs/schematic_jacobian_frame_crystal.png]] #+begin_src matlab -G_mr = inv(J_s_111)*G*inv(J_a_111'); +G_mr = inv(J_s_r)*G*inv(J_a_r'); #+end_src #+begin_src matlab :results value replace :exports both :tangle no @@ -1348,7 +1583,7 @@ xlim([freqs(1), freqs(end)]); #+end_src This results in a coupled system. -The main reason is that, as we map forces to the center of the 111 crystal and not at the center of mass/stiffness of the moving part, vertical forces will induce rotation and torques will induce vertical motion. +The main reason is that, as we map forces to the center of the ring crystal and not at the center of mass/stiffness of the moving part, vertical forces will induce rotation and torques will induce vertical motion. ** Plant at the center of stiffness :noexport: @@ -1697,7 +1932,7 @@ load('dcm_kinematics.mat'); #+begin_src matlab %% Identification of the Open Loop plant controller.type = 0; % Open Loop -G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io); +G_ol = J_a_r*inv(J_s_r)*linearize(mdl, io); G_ol.InputName = {'u_ur', 'u_uh', 'u_d'}; G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src @@ -1705,7 +1940,7 @@ G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+begin_src matlab %% Identification of the damped plant with Relative Active Damping controller.type = 2; % RAD -G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io); +G_dp = J_a_r*inv(J_s_r)*linearize(mdl, io); G_dp.InputName = {'u_ur', 'u_uh', 'u_d'}; G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src @@ -2014,7 +2249,7 @@ load('dcm_kinematics.mat'); #+begin_src matlab :exports none %% Identification of the Open Loop plant controller.type = 0; % Open Loop -G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io); +G_ol = J_a_r*inv(J_s_r)*linearize(mdl, io); G_ol.InputName = {'u_ur', 'u_uh', 'u_d'}; G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src @@ -2022,7 +2257,7 @@ G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+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_r*inv(J_s_r)*linearize(mdl, io); G_dp.InputName = {'u_ur', 'u_uh', 'u_d'}; G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src @@ -2229,7 +2464,7 @@ load('Kiff.mat'); #+begin_src matlab :exports none %% Identification of the damped plant with IFF controller.type = 1; % IFF -G_dp = J_a_311*inv(J_s_311)*linearize(mdl, io); +G_dp = J_a_h*inv(J_2h_s)*linearize(mdl, io); G_dp.InputName = {'u_ur', 'u_uh', 'u_d'}; G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src @@ -2427,7 +2662,7 @@ io(io_i) = linio([mdl, '/DCM'], 1, 'output'); io_i = io_i + 1; #+begin_src matlab :exports none %% Identification of the transmissibility of errors in open-loop controller.type = 0; % Open Loop -T_ol = inv(J_s_111)*linearize(mdl, io)*J_a_111; +T_ol = inv(J_s_r)*linearize(mdl, io)*J_a_r; T_ol.InputName = {'e_dz', 'e_ry', 'e_rx'}; T_ol.OutputName = {'dx', 'ry', 'rx'}; #+end_src @@ -2441,7 +2676,7 @@ load('Kiff.mat'); #+begin_src matlab :exports none %% Identification of the transmissibility of errors with HAC-IFF controller.type = 3; % IFF -T_hl = inv(J_s_111)*linearize(mdl, io)*J_a_111; +T_hl = inv(J_s_r)*linearize(mdl, io)*J_a_r; T_hl.InputName = {'e_dz', 'e_ry', 'e_rx'}; T_hl.OutputName = {'dx', 'ry', 'rx'}; #+end_src diff --git a/dcm-simscape-model.pdf b/dcm-simscape-model.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3aa8426ce33f88088cdc514ea4e4bf9ca08f5117 GIT binary patch literal 2006410 zcmce-W0Wo3(k5EAZF`ricG4D|o{XhYHcBl*`=3{Z5^`c4k!e@8I<9r0KGPy2U- zg0Z8mvxA|rBf;;#Jql2CQpPr>PG$rQzX|?p`+FC2D<@+I0y+^ZeJ5ifV?$dbV<;XT zC`Ts;V|{BVxAg`1X*r7NT_6DPFTh(UV;iIYlKegT=clv&z43oW1-5^e3he(e6&U_8 z&cDSB4F8z?KVt_5#($d<|2b!3__uiz<3Be3&sc)-AEWMGF#X%+1k*o2;6D?cnEsi5|CTwy{NHiY ze?b~XM&|!(NE02yZQD#joyODvB;$^Qq?G@jaD9u5@q7S^j_Xyl{ea2%2)w)E!D 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