phd-simscape-micro-station/matlab/ustation_1_kinematics.m

% Matlab Init                                              :noexport:ignore:

%% ustation_1_kinematics.m

%% Clear Workspace and Close figures
clear; close all; clc;

%% Intialize Laplace variable
s = zpk('s');

%% Path for functions, data and scripts
addpath('./mat/'); % Path for Data
addpath('./src/'); % Path for functions
addpath('./STEPS/'); % Path for STEPS
addpath('./subsystems/'); % Path for Subsystems Simulink files

% Simulink Model name
mdl = 'ustation_simscape';

load('nass_model_conf_simulink.mat');

%% Colors for the figures
colors = colororder;

%% Frequency Vector
freqs = logspace(log10(10), log10(2e3), 1000);

% Micro-Station Kinematics
% <<ssec:ustation_kinematics>>

% Each stage is described by two frames; one is attached to the fixed platform $\{A\}$ while the other is fixed to the mobile platform $\{B\}$.
% At "rest" position, the two have the same pose and coincide with the point of interest ($O_A = O_B$).
% An example of the tilt stage is shown in Figure ref:fig:ustation_stage_motion.
% The mobile frame of the translation stage is equal to the fixed frame of the tilt stage: $\{B_{D_y}\} = \{A_{R_y}\}$.
% Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the spindle: $\{B_{R_y}\} = \{A_{R_z}\}$.

% #+name: fig:ustation_stage_motion
% #+caption: Example of the motion induced by the tilt-stage $R_y$. "Rest" position in shown in blue while a arbitrary position in shown in red. Parasitic motions are here magnified for clarity.
% [[file:figs/ustation_stage_motion.png]]

% The motion induced by a positioning stage can be described by a homogeneous transformation matrix from frame $\{A\}$ to frame $\{B\}$ as explain in Section ref:ssec:ustation_kinematics.
% As any motion stage induces parasitic motion in all 6 DoF, the transformation matrix representing its induced motion can be written as in eqref:eq:ustation_translation_stage_errors.

% \begin{equation}\label{eq:ustation_translation_stage_errors}
% {}^A\mathbf{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) =
% \left[ \begin{array}{ccc|c}
%     &   &   & D_x \\
%     & \mathbf{R}_x(\theta_x) \mathbf{R}_y(\theta_y) \mathbf{R}_z(\theta_z) &   & D_y \\
%     &   &   & D_z \cr
%   \hline
%   0 & 0 & 0 & 1
% \end{array} \right]
% \end{equation}

% The homogeneous transformation matrix corresponding to the micro-station $\mathbf{T}_{\mu\text{-station}}$ is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation eqref:eq:ustation_transformation_station.

% \begin{equation}\label{eq:ustation_transformation_station}
% \mathbf{T}_{\mu\text{-station}} = \mathbf{T}_{D_y} \cdot \mathbf{T}_{R_y} \cdot \mathbf{T}_{R_z} \cdot \mathbf{T}_{\mu\text{-hexapod}}
% \end{equation}

% $\mathbf{T}_{\mu\text{-station}}$ represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite.

% If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), $\mathbf{T}_{\mu\text{-station}}$ then represent the pose setpoint of the sample with respect to the granite.
% The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation eqref:eq:ustation_transformation_matrices_stages.

% \begin{equation}\label{eq:ustation_transformation_matrices_stages}
% \begin{align}
% \mathbf{T}_{D_y} &= \begin{bmatrix}
%     1 & 0 & 0 & 0 \\
%     0 & 1 & 0 & D_y \\
%     0 & 0 & 1 & 0 \\
%     0 & 0 & 0 & 1
% \end{bmatrix} \quad
% \mathbf{T}_{\mu\text{-hexapod}} =
% \left[ \begin{array}{ccc|c}
%     &   &   & D_{\mu x} \\
%     & \mathbf{R}_x(\theta_{\mu x}) \mathbf{R}_y(\theta_{\mu y}) \mathbf{R}_{z}(\theta_{\mu z}) &   & D_{\mu y} \\
%     &   &   & D_{\mu z} \cr
%   \hline
%   0 & 0 & 0 & 1
% \end{array} \right] \\
% \mathbf{T}_{R_z} &= \begin{bmatrix}
%     \cos(\theta_z) & -\sin(\theta_z) & 0 & 0 \\
%     \sin(\theta_z) & \cos(\theta_z) & 0 & 0 \\
%     0 & 0 & 1 & 0 \\
%     0 & 0 & 0 & 1
% \end{bmatrix} \quad
% \mathbf{T}_{R_y} = \begin{bmatrix}
%     \cos(\theta_y) & 0 & \sin(\theta_y) & 0 \\
%     0 & 1 & 0 & 0 \\
%     -\sin(\theta_y) & 0 & \cos(\theta_y) & 0 \\
%     0 & 0 & 0 & 1
% \end{bmatrix}
% \end{align}
% \end{equation}


%% Stage setpoints
Dy = 1e-3; % Translation Stage [m]
Ry = 3*pi/180; % Tilt Stage [rad]
Rz = 180*pi/180; % Spindle [rad]

%% Stage individual Homogeneous transformations
% Translation Stage
Rty = [1 0 0 0;
       0 1 0 Dy;
       0 0 1 0;
       0 0 0 1];

% Tilt Stage - Pure rotating aligned with Ob
Rry = [ cos(Ry) 0 sin(Ry) 0;
        0       1 0       0;
       -sin(Ry) 0 cos(Ry) 0;
        0       0 0       1];

% Spindle - Rotation along the Z axis
Rrz = [cos(Rz) -sin(Rz) 0 0 ;
       sin(Rz)  cos(Rz) 0 0 ;
       0        0       1 0 ;
       0        0       0 1 ];

% Micro-Station homogeneous transformation
Ttot = Rty*Rry*Rrz;

%% Compute translations and rotations (Euler angles) induced by the micro-station
ustation_dx = Ttot(1,4);
ustation_dy = Ttot(2,4);
ustation_dz = Ttot(3,4);
ustation_ry = atan2( Ttot(1, 3),                  sqrt(Ttot(1, 1)^2 + Ttot(1, 2)^2));
ustation_rx = atan2(-Ttot(2, 3)/cos(ustation_ry), Ttot(3, 3)/cos(ustation_ry));
ustation_rz = atan2(-Ttot(1, 2)/cos(ustation_ry), Ttot(1, 1)/cos(ustation_ry));

%% Verification using the Simscape model
% All stages are initialized as rigid bodies to avoid any guiding error
initializeGround(      'type', 'rigid');
initializeGranite(     'type', 'rigid');
initializeTy(          'type', 'rigid');
initializeRy(          'type', 'rigid');
initializeRz(          'type', 'rigid');
initializeMicroHexapod('type', 'rigid');

initializeLoggingConfiguration('log', 'all');

initializeReferences('Dy_amplitude', Dy, ...
                     'Ry_amplitude', Ry, ...
                     'Rz_amplitude', Rz);

initializeDisturbances('enable', false);

set_param(conf_simulink, 'StopTime', '0.5');

% Simulation is performed
sim(mdl);

% Sample's motion is computed from "external metrology"
T_sim = [simout.y.R.Data(:,:,end), [simout.y.x.Data(end); simout.y.y.Data(end); simout.y.z.Data(end)]; [0,0,0,1]];
sim_dx = T_sim(1,4);
sim_dy = T_sim(2,4);
sim_dz = T_sim(3,4);
sim_ry = atan2( T_sim(1, 3),             sqrt(T_sim(1, 1)^2 + T_sim(1, 2)^2));
sim_rx = atan2(-T_sim(2, 3)/cos(sim_ry), T_sim(3, 3)/cos(sim_ry));
sim_rz = atan2(-T_sim(1, 2)/cos(sim_ry), T_sim(1, 1)/cos(sim_ry));