DCM - Dynamical Multi-Body Model

• System Kinematics
  • Introduction
  • Bragg Angle
  • Kinematics (111 Crystal)
    • Introduction
    • Interferometers - 111 Crystal
    • Piezo - 111 Crystal
  • Save Kinematics
• System Identification
  • Introduction
  • Identification
  • Plant in the frame of the fastjacks
  • Plant in the frame of the crystal
• Active Damping Plant (Strain gauge)
  • Introduction
  • Identification
• Active Damping Plant (Force Sensors)
  • Introduction
  • Identification
  • Controller - Root Locus
  • Damped Plant
  • Save
• HAC-LAC (IFF) architecture
  • Introduction
• Bibliography

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This report is also available as a pdf.

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\clearpage

System Kinematics

Introduction   ignore

Bragg Angle

%% Tested bragg angles
bragg = linspace(5, 80, 1000); % Bragg angle [deg]
d_off = 10.5e-3; % Wanted offset between x-rays [m]

%% Vertical Jack motion as a function of Bragg angle
dz = d_off./(2*cos(bragg*pi/180));

Jack motion as a function of Bragg angle

%% Required Jack stroke
ans = 1e3*(dz(end) - dz(1))

24.963

Kinematics (111 Crystal)

Introduction   ignore

The reference frame is taken at the center of the 111 second crystal.

Interferometers - 111 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 as well as the origin where the motion of the crystal is computed.

\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[below]{$y$};
  \draw[fill, color=black] (0, 0) circle (0.05);

  \node[left] at (0,0) {$\mathcal{O}_{111}$};
\end{tikzpicture}

Bottom view of the second crystal 111. Position of the measurement points.

The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure fig:schematic_sensor_jacobian_inverse_kinematics):

\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{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}

Inverse Kinematics - Interferometers

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}

%% Sensor Jacobian matrix for 111 crystal
J_s_111 = [1,  0.07, -0.015
           1,  0,     0.015
           1, -0.07, -0.015];

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).

\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{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}

Forward Kinematics - Interferometers

0.25	0.5	0.25
7.14	0.0	-7.14
-16.67	33.33	-16.67

Piezo - 111 Crystal

The location of the actuators with respect with the center of the 111 second crystal are shown in Figure fig:actuator_jacobian_111_points.

Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal's center.

\begin{equation} \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} = \bm{J}_{a,111} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} \end{equation}

\begin{tikzpicture}
  % Blocs
  \node[block] (Ja) {$\bm{J}_{a,111}$};

  % 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.east)  -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$};
\end{tikzpicture}

Inverse Kinematics - Actuators

Based on the geometry in Figure fig:actuator_jacobian_111_points, we obtain:

\begin{equation} \bm{J}_{a,111} = \begin{bmatrix} 1 & 0.14 & -0.1525 \\ 1 & 0.14 & 0.0675 \\ 1 & -0.14 & -0.0425 \end{bmatrix} \end{equation}

%% Actuator Jacobian - 111 crystal
J_a_111 = [1,  0.14, -0.1525
           1,  0.14,  0.0675
           1, -0.14, -0.0425];

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} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} = \bm{J}_{a,111}^{-1} \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} \end{equation}

\begin{tikzpicture}
  % Blocs
  \node[block] (Ja_inv) {$\bm{J}_{a,111}^{-1}$};

  % 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]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\end{tikzpicture}

Forward Kinematics - Actuators for 111 crystal

0.0568	0.4432	0.5
1.7857	1.7857	-3.5714
-4.5455	4.5455	0.0

Save Kinematics

save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')

System Identification

Introduction   ignore

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

It is schematically shown in Figure fig:schematic_system_inputs_outputs.

\begin{tikzpicture}
  % Blocs
  \node[block] (G) {$\bm{G}(s)$};

  % Connections and labels
  \draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} -- (G.west);
  \draw[->] (G.east)  -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
\end{tikzpicture}

Dynamical system with inputs and outputs

The system is identified from the Simscape model.

%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;

%% Outputs
% Interferometers {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;

%% Extraction of the dynamics
G = linearize(mdl, io);

size(G)

size(G)
State-space model with 3 outputs, 3 inputs, and 24 states.

Plant in the frame of the fastjacks

load('mat/dcm_kinematics.mat');

Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure fig:schematic_jacobian_frame_fastjack).

\begin{tikzpicture}
  % Blocs
  \node[block] (G) {$\bm{G}(s)$};
  \node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$};
  \node[block, right=1.5 of Js] (Ja) {$\bm{J}_{a}$};

  % Connections and labels
  \draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} -- (G.west);
  \draw[->] (G.east)  -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
  \draw[->] (Js.east) -- node[midway, above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} (Ja.west);
  \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix}$};

  \begin{scope}[on background layer]
    \node[fit={(G.south west) ($(Ja.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
    \node[above] at (system.north) {$\bm{G}_{\text{fj}}(s)$};
  \end{scope}
\end{tikzpicture}

Use of Jacobian matrices to obtain the system in the frame of the fastjacks

%% Compute the system in the frame of the fastjacks
G_pz = J_a_111*inv(J_s_111)*G;

The DC gain of the new system shows that the system is well decoupled at low frequency.

dcgain(G_pz)

4.4407e-09	2.7656e-12	1.0132e-12
2.7656e-12	4.4407e-09	1.0132e-12
1.0109e-12	1.0109e-12	4.4424e-09

The bode plot of $\bm{G}_{\text{fj}}(s)$ is shown in Figure fig:bode_plot_plant_fj.

Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks

Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).

Plant in the frame of the crystal

\begin{tikzpicture}
  % Blocs
  \node[block] (G) {$\bm{G}(s)$};
  \node[block, left=1.5 of G] (Ja) {$\bm{J}_{a}^{-T}$};
  \node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$};

  % Connections and labels
  \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} \mathcal{F}_{z} \\ \mathcal{M}_{y} \\ \mathcal{M}_{x} \end{bmatrix}$} -- (Ja.west);
  \draw[->] (Ja.east)  -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (G.west);
  \draw[->] (G.east) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
  \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};

  \begin{scope}[on background layer]
    \node[fit={(Ja.south west) ($(Js.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
    \node[above] at (system.north) {$\bm{G}_{\text{cr}}(s)$};
  \end{scope}
\end{tikzpicture}

Use of Jacobian matrices to obtain the system in the frame of the crystal

G_mr = inv(J_s_111)*G*inv(J_a_111');

dcgain(G_mr)

1.9978e-09	3.9657e-09	7.7944e-09
3.9656e-09	8.4979e-08	-1.5135e-17
7.7944e-09	-3.9252e-17	1.834e-07

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.

Active Damping Plant (Strain gauge)

Introduction   ignore

In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping.

Identification

%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/u'], 1, 'openinput');  io_i = io_i + 1;
% % Stepper Displacement {3x1} [m]
% io(io_i) = linio([mdl, '/d'], 1, 'openinput');  io_i = io_i + 1;

%% Outputs
% Strain Gauges {3x1} [m]
io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1;

%% Extraction of the dynamics
G_sg = linearize(mdl, io);

dcgain(G_sg)

-1.4113e-13	1.0339e-13	3.774e-14
1.0339e-13	-1.4113e-13	3.774e-14
3.7792e-14	3.7792e-14	-7.5585e-14

Active Damping Plant (Force Sensors)

Introduction   ignore

Force sensors are added above the piezoelectric actuators. They can consists of a simple piezoelectric ceramic stack. See for instance cite:fleming10_integ_strain_force_feedb_high.

Identification

%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;

%% Outputs
% Force Sensor {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 3, 'openoutput'); io_i = io_i + 1;

%% Extraction of the dynamics
G_fs = linearize(mdl, io);

dcgain(G_fs)

-1.4113e-13	1.0339e-13	3.774e-14
1.0339e-13	-1.4113e-13	3.774e-14
3.7792e-14	3.7792e-14	-7.5585e-14

Bode plot of IFF Plant

Controller - Root Locus

Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);

Root Locus plot for the IFF Control strategy

%% Integral Force Feedback Controller
Kiff = g*Kiff_g1;

Damped Plant

%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'input');  io_i = io_i + 1;

%% Outputs
% Force Sensor {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;

%% DCM Kinematics
load('mat/dcm_kinematics.mat');

%% 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.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_ol.OutputName = {'d_ur',  'd_uh',  'd_d'};

%% 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.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_dp.OutputName = {'d_ur',  'd_uh',  'd_d'};

Bode plot of both the open-loop plant and the damped plant using IFF

The Integral Force Feedback control strategy is very effective in damping the suspension modes of the DCM.

Save

save('mat/Kiff.mat', 'Kiff');

HAC-LAC (IFF) architecture

Introduction   ignore

Bibliography   ignore