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nass-simscape/simscape_subsystems/index.org
Thomas Dehaeze c6a4e5343e Important change for the implementation of motion input 
Now we provide the first and second derivatives.
This permits to not have any motion error.

Also, many experiments (tomography, ty-scans) are simulated
2019-12-17 18:03:21 +01:00

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#+TITLE: Subsystems used for the Simscape Models
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/readtheorg.js"></script>
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:shell :eval no-export
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
:END:
* Introduction :ignore:
* General Subsystems
<<sec:helping functions>>
** Generate Reference Signals
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeReferences.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeReferences>>
This Matlab function is accessible [[file:../src/initializeInputs.m][here]].
#+begin_src matlab
function [ref] = initializeReferences(opts_param)
%% Default values for opts
opts = struct( ...
'Ts', 1e-3, ... % Sampling Frequency [s]
'Tmax', 100, ... % Maximum simulation time [s]
'Dy_type', 'constant', ... % Either "constant" / "triangular" / "sinusoidal"
'Dy_amplitude', 0, ... % Amplitude of the displacement [m]
'Dy_period', 1, ... % Period of the displacement [s]
'Ry_type', 'constant', ... % Either "constant" / "triangular" / "sinusoidal"
'Ry_amplitude', 0, ... % Amplitude [rad]
'Ry_period', 1, ... % Period of the displacement [s]
'Rz_type', 'constant', ... % Either "constant" / "rotating"
'Rz_amplitude', 0, ... % Initial angle [rad]
'Rz_period', 1, ... % Period of the rotating [s]
'Dh_type', 'constant', ... % For now, only constant is implemented
'Dh_pos', zeros(6, 1), ... % Initial position [m,m,m,rad,rad,rad] of the top platform (Pitch-Roll-Yaw Euler angles)
'Rm_type', 'constant', ... % For now, only constant is implemented
'Rm_pos', [0; pi], ... % Initial position of the two masses
'Dn_type', 'constant', ... % For now, only constant is implemented
'Dn_pos', zeros(6,1) ... % Initial position [m,m,m,rad,rad,rad] of the top platform
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%% Set Sampling Time
Ts = opts.Ts;
Tmax = opts.Tmax;
%% Low Pass Filter to filter out the references
s = zpk('s');
w0 = 2*pi*100;
xi = 1;
H_lpf = 1/(1 + 2*xi/w0*s + s^2/w0^2);
%% Translation stage - Dy
t = 0:Ts:Tmax; % Time Vector [s]
Dy = zeros(length(t), 1);
Dyd = zeros(length(t), 1);
Dydd = zeros(length(t), 1);
switch opts.Dy_type
case 'constant'
Dy(:) = opts.Dy_amplitude;
Dyd(:) = 0;
Dydd(:) = 0;
case 'triangular'
% This is done to unsure that we start with no displacement
Dy_raw = opts.Dy_amplitude*sawtooth(2*pi*t/opts.Dy_period,1/2);
i0 = find(t>=opts.Dy_period/4,1);
Dy(1:end-i0+1) = Dy_raw(i0:end);
Dy(end-i0+2:end) = Dy_raw(end); % we fix the last value
% The signal is filtered out
Dy = lsim(H_lpf, Dy, t);
Dyd = lsim(H_lpf*s, Dy, t);
Dydd = lsim(H_lpf*s^2, Dy, t);
case 'sinusoidal'
Dy(:) = opts.Dy_amplitude*sin(2*pi/opts.Dy_period*t);
Dyd = opts.Dy_amplitude*2*pi/opts.Dy_period*cos(2*pi/opts.Dy_period*t);
Dydd = -opts.Dy_amplitude*(2*pi/opts.Dy_period)^2*sin(2*pi/opts.Dy_period*t);
otherwise
warning('Dy_type is not set correctly');
end
Dy = struct('time', t, 'signals', struct('values', Dy), 'deriv', Dyd, 'dderiv', Dydd);
%% Tilt Stage - Ry
t = 0:Ts:Tmax; % Time Vector [s]
Ry = zeros(length(t), 1);
Ryd = zeros(length(t), 1);
Rydd = zeros(length(t), 1);
switch opts.Ry_type
case 'constant'
Ry(:) = opts.Ry_amplitude;
Ryd(:) = 0;
Rydd(:) = 0;
case 'triangular'
Ry_raw = opts.Ry_amplitude*sawtooth(2*pi*t/opts.Ry_period,1/2);
i0 = find(t>=opts.Ry_period/4,1);
Ry(1:end-i0+1) = Ry_raw(i0:end);
Ry(end-i0+2:end) = Ry_raw(end); % we fix the last value
% The signal is filtered out
Ry = lsim(H_lpf, Ry, t);
Ryd = lsim(H_lpf*s, Ry, t);
Rydd = lsim(H_lpf*s^2, Ry, t);
case 'sinusoidal'
Ry(:) = opts.Ry_amplitude*sin(2*pi/opts.Ry_period*t);
Ryd = opts.Ry_amplitude*2*pi/opts.Ry_period*cos(2*pi/opts.Ry_period*t);
Rydd = -opts.Ry_amplitude*(2*pi/opts.Ry_period)^2*sin(2*pi/opts.Ry_period*t);
otherwise
warning('Ry_type is not set correctly');
end
Ry = struct('time', t, 'signals', struct('values', Ry), 'deriv', Ryd, 'dderiv', Rydd);
%% Spindle - Rz
t = 0:Ts:Tmax; % Time Vector [s]
Rz = zeros(length(t), 1);
Rzd = zeros(length(t), 1);
Rzdd = zeros(length(t), 1);
switch opts.Rz_type
case 'constant'
Rz(:) = opts.Rz_amplitude;
Rzd(:) = 0;
Rzdd(:) = 0;
case 'rotating'
Rz(:) = opts.Rz_amplitude+2*pi/opts.Rz_period*t;
% The signal is filtered out
Rz = lsim(H_lpf, Rz, t);
Rzd = lsim(H_lpf*s, Rz, t);
Rzdd = lsim(H_lpf*s^2, Rz, t);
otherwise
warning('Rz_type is not set correctly');
end
Rz = struct('time', t, 'signals', struct('values', Rz), 'deriv', Rzd, 'dderiv', Rzdd);
%% Micro-Hexapod
t = [0, Ts];
Dh = zeros(length(t), 6);
Dhl = zeros(length(t), 6);
switch opts.Dh_type
case 'constant'
Dh = [opts.Dh_pos, opts.Dh_pos];
load('./mat/stages.mat', 'micro_hexapod');
AP = [opts.Dh_pos(1) ; opts.Dh_pos(2) ; opts.Dh_pos(3)];
tx = opts.Dh_pos(4);
ty = opts.Dh_pos(5);
tz = opts.Dh_pos(6);
ARB = [cos(tz) -sin(tz) 0;
sin(tz) cos(tz) 0;
0 0 1]*...
[ cos(ty) 0 sin(ty);
0 1 0;
-sin(ty) 0 cos(ty)]*...
[1 0 0;
0 cos(tx) -sin(tx);
0 sin(tx) cos(tx)];
[Dhl] = inverseKinematicsHexapod(micro_hexapod, AP, ARB);
Dhl = [Dhl, Dhl];
otherwise
warning('Dh_type is not set correctly');
end
Dh = struct('time', t, 'signals', struct('values', Dh));
Dhl = struct('time', t, 'signals', struct('values', Dhl));
%% Axis Compensation - Rm
t = [0, Ts];
Rm = [opts.Rm_pos, opts.Rm_pos];
Rm = struct('time', t, 'signals', struct('values', Rm));
%% Nano-Hexapod
t = [0, Ts];
Dn = zeros(length(t), 6);
switch opts.Dn_type
case 'constant'
Dn = [opts.Dn_pos, opts.Dn_pos];
otherwise
warning('Dn_type is not set correctly');
end
Dn = struct('time', t, 'signals', struct('values', Dn));
%% Save
save('./mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'Rm', 'Dn', 'Ts');
end
#+end_src
** Initialize Disturbances
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initDisturbances.m
:header-args:matlab+: :comments none :mkdirp yes
:header-args:matlab+: :eval no :results none
:END:
<<sec:initDisturbances>>
This Matlab function is accessible [[file:src/initDisturbances.m][here]].
*** Function Declaration and Documentation
#+begin_src matlab
function [] = initDisturbances(opts_param)
% initDisturbances - Initialize the disturbances
%
% Syntax: [] = initDisturbances(opts_param)
%
% Inputs:
% - opts_param -
#+end_src
*** Default values for the Options
#+begin_src matlab
%% Default values for opts
opts = struct(...
'Dwx', true, ... % Ground Motion - X direction
'Dwy', true, ... % Ground Motion - Y direction
'Dwz', true, ... % Ground Motion - Z direction
'Fty_x', true, ... % Translation Stage - X direction
'Fty_z', true, ... % Translation Stage - Z direction
'Frz_z', true ... % Spindle - Z direction
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
#+end_src
*** Load Data
#+begin_src matlab
load('./disturbances/mat/dist_psd.mat', 'dist_f');
#+end_src
We remove the first frequency point that usually is very large.
#+begin_src matlab :exports none
dist_f.f = dist_f.f(2:end);
dist_f.psd_gm = dist_f.psd_gm(2:end);
dist_f.psd_ty = dist_f.psd_ty(2:end);
dist_f.psd_rz = dist_f.psd_rz(2:end);
#+end_src
*** Parameters
We define some parameters that will be used in the algorithm.
#+begin_src matlab
Fs = 2*dist_f.f(end); % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
N = 2*length(dist_f.f); % Number of Samples match the one of the wanted PSD
T0 = N/Fs; % Signal Duration [s]
df = 1/T0; % Frequency resolution of the DFT [Hz]
% Also equal to (dist_f.f(2)-dist_f.f(1))
t = linspace(0, T0, N+1)'; % Time Vector [s]
Ts = 1/Fs; % Sampling Time [s]
#+end_src
*** Ground Motion
#+begin_src matlab
phi = dist_f.psd_gm;
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(phi(i)*df);
end
#+end_src
#+begin_src matlab
if opts.Dwx
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dwx = N/sqrt(2)*ifft(Cx); % Ground Motion - x direction [m]
else
Dwx = zeros(length(t), 1);
end
#+end_src
#+begin_src matlab
if opts.Dwy
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dwy = N/sqrt(2)*ifft(Cx); % Ground Motion - y direction [m]
else
Dwy = zeros(length(t), 1);
end
#+end_src
#+begin_src matlab
if opts.Dwy
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dwz = N/sqrt(2)*ifft(Cx); % Ground Motion - z direction [m]
else
Dwz = zeros(length(t), 1);
end
#+end_src
*** Translation Stage - X direction
#+begin_src matlab
if opts.Fty_x
phi = dist_f.psd_ty; % TODO - we take here the vertical direction which is wrong but approximate
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(phi(i)*df);
end
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty x [N]
Fty_x = u;
else
Fty_x = zeros(length(t), 1);
end
#+end_src
*** Translation Stage - Z direction
#+begin_src matlab
if opts.Fty_z
phi = dist_f.psd_ty;
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(phi(i)*df);
end
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty z [N]
Fty_z = u;
else
Fty_z = zeros(length(t), 1);
end
#+end_src
*** Spindle - Z direction
#+begin_src matlab
if opts.Frz_z
phi = dist_f.psd_rz;
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(phi(i)*df);
end
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
u = N/sqrt(2)*ifft(Cx); % Disturbance Force Rz z [N]
Frz_z = u;
else
Frz_z = zeros(length(t), 1);
end
#+end_src
*** Direct Forces
#+begin_src matlab
u = zeros(length(t), 6);
Fd = u;
#+end_src
*** Set initial value to zero
#+begin_src matlab
Dwx = Dwx - Dwx(1);
Dwy = Dwy - Dwy(1);
Dwz = Dwz - Dwz(1);
Fty_x = Fty_x - Fty_x(1);
Fty_z = Fty_z - Fty_z(1);
Frz_z = Frz_z - Frz_z(1);
#+end_src
*** Save
#+begin_src matlab
save('./mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't');
#+end_src
* Initialize Elements
:PROPERTIES:
:ID: a0819dea-8d7a-4d55-b961-2b2ca2312344
:END:
<<sec:initialize_elements>>
** Ground
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeGround.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeGround>>
This Matlab function is accessible [[file:../src/initializeGround.m][here]].
#+begin_src matlab
function [ground] = initializeGround()
%%
ground = struct();
ground.shape = [2, 2, 0.5]; % [m]
ground.density = 2800; % [kg/m3]
ground.color = [0.5, 0.5, 0.5];
%% Save
save('./mat/stages.mat', 'ground', '-append');
end
#+end_src
** Granite
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeGranite.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeGranite>>
This Matlab function is accessible [[file:../src/initializeGranite.m][here]].
#+begin_src matlab
function [granite] = initializeGranite(opts_param)
%% Default values for opts
opts = struct('rigid', false);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%%
granite = struct();
%% Static Properties
granite.density = 2800; % [kg/m3]
granite.volume = 0.749; % [m3] TODO - should
granite.mass = granite.density*granite.volume; % [kg]
granite.color = [1 1 1];
granite.STEP = './STEPS/granite/granite.STEP';
granite.mass_top = 4000; % [kg] TODO
%% Dynamical Properties
if opts.rigid
granite.k.x = 1e12; % [N/m]
granite.k.y = 1e12; % [N/m]
granite.k.z = 1e12; % [N/m]
granite.k.rx = 1e10; % [N*m/deg]
granite.k.ry = 1e10; % [N*m/deg]
granite.k.rz = 1e10; % [N*m/deg]
else
granite.k.x = 4e9; % [N/m]
granite.k.y = 3e8; % [N/m]
granite.k.z = 8e8; % [N/m]
granite.k.rx = 1e4; % [N*m/deg]
granite.k.ry = 1e4; % [N*m/deg]
granite.k.rz = 1e6; % [N*m/deg]
end
granite.c.x = 0.1*sqrt(granite.mass_top*granite.k.x); % [N/(m/s)]
granite.c.y = 0.1*sqrt(granite.mass_top*granite.k.y); % [N/(m/s)]
granite.c.z = 0.5*sqrt(granite.mass_top*granite.k.z); % [N/(m/s)]
granite.c.rx = 0.1*sqrt(granite.mass_top*granite.k.rx); % [N*m/(deg/s)]
granite.c.ry = 0.1*sqrt(granite.mass_top*granite.k.ry); % [N*m/(deg/s)]
granite.c.rz = 0.1*sqrt(granite.mass_top*granite.k.rz); % [N*m/(deg/s)]
%% Positioning parameters
granite.sample_pos = 0.8; % Z-offset for the initial position of the sample [m]
%% Save
save('./mat/stages.mat', 'granite', '-append');
end
#+end_src
** Translation Stage
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeTy.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeTy>>
This Matlab function is accessible [[file:../src/initializeTy.m][here]].
#+begin_src matlab
function [ty] = initializeTy(opts_param)
%% Default values for opts
opts = struct('rigid', false);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%%
ty = struct();
%% Y-Translation - Static Properties
% Ty Granite frame
ty.granite_frame.density = 7800; % [kg/m3]
ty.granite_frame.color = [0.753 1 0.753];
ty.granite_frame.STEP = './STEPS/Ty/Ty_Granite_Frame.STEP';
% Guide Translation Ty
ty.guide.density = 7800; % [kg/m3]
ty.guide.color = [0.792 0.820 0.933];
ty.guide.STEP = './STEPS/ty/Ty_Guide.STEP';
% Ty - Guide_Translation12
ty.guide12.density = 7800; % [kg/m3]
ty.guide12.color = [0.792 0.820 0.933];
ty.guide12.STEP = './STEPS/Ty/Ty_Guide_12.STEP';
% Ty - Guide_Translation11
ty.guide11.density = 7800; % [kg/m3]
ty.guide11.color = [0.792 0.820 0.933];
ty.guide11.STEP = './STEPS/ty/Ty_Guide_11.STEP';
% Ty - Guide_Translation22
ty.guide22.density = 7800; % [kg/m3]
ty.guide22.color = [0.792 0.820 0.933];
ty.guide22.STEP = './STEPS/ty/Ty_Guide_22.STEP';
% Ty - Guide_Translation21
ty.guide21.density = 7800; % [kg/m3]
ty.guide21.color = [0.792 0.820 0.933];
ty.guide21.STEP = './STEPS/Ty/Ty_Guide_21.STEP';
% Ty - Plateau translation
ty.frame.density = 7800; % [kg/m3]
ty.frame.color = [0.792 0.820 0.933];
ty.frame.STEP = './STEPS/ty/Ty_Stage.STEP';
% Ty Stator Part
ty.stator.density = 5400; % [kg/m3]
ty.stator.color = [0.792 0.820 0.933];
ty.stator.STEP = './STEPS/ty/Ty_Motor_Stator.STEP';
% Ty Rotor Part
ty.rotor.density = 5400; % [kg/m3]
ty.rotor.color = [0.792 0.820 0.933];
ty.rotor.STEP = './STEPS/ty/Ty_Motor_Rotor.STEP';
ty.m = 1000; % TODO [kg]
%% Y-Translation - Dynamicals Properties
if opts.rigid
ty.k.ax = 1e12; % Axial Stiffness for each of the 4 guidance (y) [N/m]
ty.k.rad = 1e12; % Radial Stiffness for each of the 4 guidance (x-z) [N/m]
else
ty.k.ax = 5e8; % Axial Stiffness for each of the 4 guidance (y) [N/m]
ty.k.rad = 5e7; % Radial Stiffness for each of the 4 guidance (x-z) [N/m]
end
ty.c.ax = 0.1*sqrt(ty.k.ax*ty.m);
ty.c.rad = 0.1*sqrt(ty.k.rad*ty.m);
%% Save
save('./mat/stages.mat', 'ty', '-append');
end
#+end_src
** Tilt Stage
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeRy.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeRy>>
This Matlab function is accessible [[file:../src/initializeRy.m][here]].
#+begin_src matlab
function [ry] = initializeRy(opts_param)
%% Default values for opts
opts = struct('rigid', false);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%%
ry = struct();
%% Tilt Stage - Static Properties
% Ry - Guide for the tilt stage
ry.guide.density = 7800; % [kg/m3]
ry.guide.color = [0.792 0.820 0.933];
ry.guide.STEP = './STEPS/ry/Tilt_Guide.STEP';
% Ry - Rotor of the motor
ry.rotor.density = 2400; % [kg/m3]
ry.rotor.color = [0.792 0.820 0.933];
ry.rotor.STEP = './STEPS/ry/Tilt_Motor_Axis.STEP';
% Ry - Motor
ry.motor.density = 3200; % [kg/m3]
ry.motor.color = [0.792 0.820 0.933];
ry.motor.STEP = './STEPS/ry/Tilt_Motor.STEP';
% Ry - Plateau Tilt
ry.stage.density = 7800; % [kg/m3]
ry.stage.color = [0.792 0.820 0.933];
ry.stage.STEP = './STEPS/ry/Tilt_Stage.STEP';
ry.m = 800; % TODO [kg]
%% Tilt Stage - Dynamical Properties
if opts.rigid
ry.k.tilt = 1e10; % Rotation stiffness around y [N*m/deg]
ry.k.h = 1e12; % Stiffness in the direction of the guidance [N/m]
ry.k.rad = 1e12; % Stiffness in the top direction [N/m]
ry.k.rrad = 1e12; % Stiffness in the side direction [N/m]
else
ry.k.tilt = 1e4; % Rotation stiffness around y [N*m/deg]
ry.k.h = 1e8; % Stiffness in the direction of the guidance [N/m]
ry.k.rad = 1e8; % Stiffness in the top direction [N/m]
ry.k.rrad = 1e8; % Stiffness in the side direction [N/m]
end
ry.c.h = 0.1*sqrt(ry.k.h*ry.m);
ry.c.rad = 0.1*sqrt(ry.k.rad*ry.m);
ry.c.rrad = 0.1*sqrt(ry.k.rrad*ry.m);
ry.c.tilt = 0.1*sqrt(ry.k.tilt*ry.m);
%% Positioning parameters
ry.z_offset = 0.58178; % Z-Offset so that the center of rotation matches the sample center [m]
%% Save
save('./mat/stages.mat', 'ry', '-append');
end
#+end_src
** Spindle
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeRz.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeRz>>
This Matlab function is accessible [[file:../src/initializeRz.m][here]].
#+begin_src matlab
function [rz] = initializeRz(opts_param)
%% Default values for opts
opts = struct('rigid', false);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%%
rz = struct();
%% Spindle - Static Properties
% Spindle - Slip Ring
rz.slipring.density = 7800; % [kg/m3]
rz.slipring.color = [0.792 0.820 0.933];
rz.slipring.STEP = './STEPS/rz/Spindle_Slip_Ring.STEP';
% Spindle - Rotor
rz.rotor.density = 7800; % [kg/m3]
rz.rotor.color = [0.792 0.820 0.933];
rz.rotor.STEP = './STEPS/rz/Spindle_Rotor.STEP';
% Spindle - Stator
rz.stator.density = 7800; % [kg/m3]
rz.stator.color = [0.792 0.820 0.933];
rz.stator.STEP = './STEPS/rz/Spindle_Stator.STEP';
% Estimated mass of the mooving part
rz.m = 250; % [kg]
%% Spindle - Dynamical Properties
if opts.rigid
rz.k.rot = 1e10; % Rotational Stiffness (Rz) [N*m/deg]
rz.k.tilt = 1e10; % Rotational Stiffness (Rx, Ry) [N*m/deg]
rz.k.ax = 1e12; % Axial Stiffness (Z) [N/m]
rz.k.rad = 1e12; % Radial Stiffness (X, Y) [N/m]
else
rz.k.rot = 1e6; % TODO - Rotational Stiffness (Rz) [N*m/deg]
rz.k.tilt = 1e6; % Rotational Stiffness (Rx, Ry) [N*m/deg]
rz.k.ax = 2e9; % Axial Stiffness (Z) [N/m]
rz.k.rad = 7e8; % Radial Stiffness (X, Y) [N/m]
end
% Damping
rz.c.ax = 0.1*sqrt(rz.k.ax*rz.m);
rz.c.rad = 0.1*sqrt(rz.k.rad*rz.m);
rz.c.tilt = 0.1*sqrt(rz.k.tilt*rz.m);
rz.c.rot = 0.1*sqrt(rz.k.rot*rz.m);
%% Save
save('./mat/stages.mat', 'rz', '-append');
end
#+end_src
** Micro Hexapod
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeMicroHexapod.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeMicroHexapod>>
This Matlab function is accessible [[file:../src/initializeMicroHexapod.m][here]].
#+begin_src matlab
function [micro_hexapod] = initializeMicroHexapod(opts_param)
%% Default values for opts
opts = struct(...
'rigid', false, ...
'AP', zeros(3, 1), ... % Wanted position in [m] of OB with respect to frame {A}
'ARB', eye(3) ... % Rotation Matrix that represent the wanted orientation of frame {B} with respect to frame {A}
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%% Stewart Object
micro_hexapod = struct();
micro_hexapod.h = 350; % Total height of the platform [mm]
micro_hexapod.jacobian = 270; % Distance from the top of the mobile platform to the Jacobian point [mm]
%% Bottom Plate - Mechanical Design
BP = struct();
BP.rad.int = 110; % Internal Radius [mm]
BP.rad.ext = 207.5; % External Radius [mm]
BP.thickness = 26; % Thickness [mm]
BP.leg.rad = 175.5; % Radius where the legs articulations are positionned [mm]
BP.leg.ang = 9.5; % Angle Offset [deg]
BP.density = 8000; % Density of the material [kg/m^3]
BP.color = [0.6 0.6 0.6]; % Color [rgb]
BP.shape = [BP.rad.int BP.thickness; BP.rad.int 0; BP.rad.ext 0; BP.rad.ext BP.thickness];
%% Top Plate - Mechanical Design
TP = struct();
TP.rad.int = 82; % Internal Radius [mm]
TP.rad.ext = 150; % Internal Radius [mm]
TP.thickness = 26; % Thickness [mm]
TP.leg.rad = 118; % Radius where the legs articulations are positionned [mm]
TP.leg.ang = 12.1; % Angle Offset [deg]
TP.density = 8000; % Density of the material [kg/m^3]
TP.color = [0.6 0.6 0.6]; % Color [rgb]
TP.shape = [TP.rad.int TP.thickness; TP.rad.int 0; TP.rad.ext 0; TP.rad.ext TP.thickness];
%% Struts
Leg = struct();
Leg.stroke = 10e-3; % Maximum Stroke of each leg [m]
if opts.rigid
Leg.k.ax = 1e12; % Stiffness of each leg [N/m]
else
Leg.k.ax = 2e7; % Stiffness of each leg [N/m]
end
Leg.ksi.ax = 0.1; % Modal damping ksi = 1/2*c/sqrt(km) []
Leg.rad.bottom = 25; % Radius of the cylinder of the bottom part [mm]
Leg.rad.top = 17; % Radius of the cylinder of the top part [mm]
Leg.density = 8000; % Density of the material [kg/m^3]
Leg.color.bottom = [0.5 0.5 0.5]; % Color [rgb]
Leg.color.top = [0.5 0.5 0.5]; % Color [rgb]
Leg.sphere.bottom = Leg.rad.bottom; % Size of the sphere at the end of the leg [mm]
Leg.sphere.top = Leg.rad.top; % Size of the sphere at the end of the leg [mm]
Leg.m = TP.density*((pi*(TP.rad.ext/1000)^2)*(TP.thickness/1000)-(pi*(TP.rad.int/1000^2))*(TP.thickness/1000))/6; % TODO [kg]
Leg = updateDamping(Leg);
%% Sphere
SP = struct();
SP.height.bottom = 27; % [mm]
SP.height.top = 27; % [mm]
SP.density.bottom = 8000; % [kg/m^3]
SP.density.top = 8000; % [kg/m^3]
SP.color.bottom = [0.6 0.6 0.6]; % [rgb]
SP.color.top = [0.6 0.6 0.6]; % [rgb]
SP.k.ax = 0; % [N*m/deg]
SP.ksi.ax = 10;
SP.thickness.bottom = SP.height.bottom-Leg.sphere.bottom; % [mm]
SP.thickness.top = SP.height.top-Leg.sphere.top; % [mm]
SP.rad.bottom = Leg.sphere.bottom; % [mm]
SP.rad.top = Leg.sphere.top; % [mm]
SP.m = SP.density.bottom*2*pi*((SP.rad.bottom*1e-3)^2)*(SP.height.bottom*1e-3); % TODO [kg]
SP = updateDamping(SP);
%%
Leg.support.bottom = [0 SP.thickness.bottom; 0 0; SP.rad.bottom 0; SP.rad.bottom SP.height.bottom];
Leg.support.top = [0 SP.thickness.top; 0 0; SP.rad.top 0; SP.rad.top SP.height.top];
%%
micro_hexapod.BP = BP;
micro_hexapod.TP = TP;
micro_hexapod.Leg = Leg;
micro_hexapod.SP = SP;
%%
micro_hexapod = initializeParameters(micro_hexapod);
%% Setup equilibrium position of each leg
micro_hexapod.L0 = inverseKinematicsHexapod(micro_hexapod, opts.AP, opts.ARB);
%% Save
save('./mat/stages.mat', 'micro_hexapod', '-append');
%%
function [element] = updateDamping(element)
field = fieldnames(element.k);
for i = 1:length(field)
element.c.(field{i}) = 2*element.ksi.(field{i})*sqrt(element.k.(field{i})*element.m);
end
end
%%
function [stewart] = initializeParameters(stewart)
%% Connection points on base and top plate w.r.t. World frame at the center of the base plate
stewart.pos_base = zeros(6, 3);
stewart.pos_top = zeros(6, 3);
alpha_b = stewart.BP.leg.ang*pi/180; % angle de décalage par rapport à 120 deg (pour positionner les supports bases)
alpha_t = stewart.TP.leg.ang*pi/180; % +- offset angle from 120 degree spacing on top
height = (stewart.h-stewart.BP.thickness-stewart.TP.thickness-stewart.Leg.sphere.bottom-stewart.Leg.sphere.top-stewart.SP.thickness.bottom-stewart.SP.thickness.top)*0.001; % TODO
radius_b = stewart.BP.leg.rad*0.001; % rayon emplacement support base
radius_t = stewart.TP.leg.rad*0.001; % top radius in meters
for i = 1:3
% base points
angle_m_b = (2*pi/3)* (i-1) - alpha_b;
angle_p_b = (2*pi/3)* (i-1) + alpha_b;
stewart.pos_base(2*i-1,:) = [radius_b*cos(angle_m_b), radius_b*sin(angle_m_b), 0.0];
stewart.pos_base(2*i,:) = [radius_b*cos(angle_p_b), radius_b*sin(angle_p_b), 0.0];
% top points
% Top points are 60 degrees offset
angle_m_t = (2*pi/3)* (i-1) - alpha_t + 2*pi/6;
angle_p_t = (2*pi/3)* (i-1) + alpha_t + 2*pi/6;
stewart.pos_top(2*i-1,:) = [radius_t*cos(angle_m_t), radius_t*sin(angle_m_t), height];
stewart.pos_top(2*i,:) = [radius_t*cos(angle_p_t), radius_t*sin(angle_p_t), height];
end
% permute pos_top points so that legs are end points of base and top points
stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
%% leg vectors
legs = stewart.pos_top - stewart.pos_base;
leg_length = zeros(6, 1);
leg_vectors = zeros(6, 3);
for i = 1:6
leg_length(i) = norm(legs(i,:));
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
stewart.Leg.lenght = 1000*leg_length(1)/1.5;
stewart.Leg.shape.bot = [0 0; ...
stewart.Leg.rad.bottom 0; ...
stewart.Leg.rad.bottom stewart.Leg.lenght; ...
stewart.Leg.rad.top stewart.Leg.lenght; ...
stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
%% Calculate revolute and cylindrical axes
rev1 = zeros(6, 3);
rev2 = zeros(6, 3);
cyl1 = zeros(6, 3);
for i = 1:6
rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
cyl1(i,:) = leg_vectors(i,:);
end
%% Coordinate systems
stewart.lower_leg = struct('rotation', eye(3));
stewart.upper_leg = struct('rotation', eye(3));
for i = 1:6
stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
end
%% Position Matrix
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.thickness+stewart.Leg.sphere.top+stewart.SP.thickness.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
%% Compute Jacobian Matrix
aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
stewart.J = getJacobianMatrix(leg_vectors', aa');
end
%%
function J = getJacobianMatrix(RM, M_pos_base)
% RM: [3x6] unit vector of each leg in the fixed frame
% M_pos_base: [3x6] vector of the leg connection at the top platform location in the fixed frame
J = zeros(6);
J(:, 1:3) = RM';
J(:, 4:6) = cross(M_pos_base, RM)';
end
end
#+end_src
** Center of gravity compensation
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeAxisc.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeAxisc>>
This Matlab function is accessible [[file:../src/initializeAxisc.m][here]].
#+begin_src matlab
function [axisc] = initializeAxisc()
%%
axisc = struct();
%% Axis Compensator - Static Properties
% Structure
axisc.structure.density = 3400; % [kg/m3]
axisc.structure.color = [0.792 0.820 0.933];
axisc.structure.STEP = './STEPS/axisc/axisc_structure.STEP';
% Wheel
axisc.wheel.density = 2700; % [kg/m3]
axisc.wheel.color = [0.753 0.753 0.753];
axisc.wheel.STEP = './STEPS/axisc/axisc_wheel.STEP';
% Mass
axisc.mass.density = 7800; % [kg/m3]
axisc.mass.color = [0.792 0.820 0.933];
axisc.mass.STEP = './STEPS/axisc/axisc_mass.STEP';
% Gear
axisc.gear.density = 7800; % [kg/m3]
axisc.gear.color = [0.792 0.820 0.933];
axisc.gear.STEP = './STEPS/axisc/axisc_gear.STEP';
axisc.m = 40; % TODO [kg]
%% Axis Compensator - Dynamical Properties
% axisc.k.ax = 1; % TODO [N*m/deg)]
% axisc.c.ax = (1/1)*sqrt(axisc.k.ax/axisc.m);
%% Save
save('./mat/stages.mat', 'axisc', '-append');
end
#+end_src
** Mirror
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeMirror.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeMirror>>
This Matlab function is accessible [[file:../src/initializeMirror.m][here]].
#+begin_src matlab
function [] = initializeMirror(opts_param)
%% Default values for opts
opts = struct(...
'shape', 'spherical', ... % spherical or conical
'angle', 45 ...
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%%
mirror = struct();
mirror.h = 50; % height of the mirror [mm]
mirror.thickness = 25; % Thickness of the plate supporting the sample [mm]
mirror.hole_rad = 120; % radius of the hole in the mirror [mm]
mirror.support_rad = 100; % radius of the support plate [mm]
mirror.jacobian = 150; % point of interest offset in z (above the top surfave) [mm]
mirror.rad = 180; % radius of the mirror (at the bottom surface) [mm]
mirror.density = 2400; % Density of the mirror [kg/m3]
mirror.color = [0.4 1.0 1.0]; % Color of the mirror
mirror.cone_length = mirror.rad*tand(opts.angle)+mirror.h+mirror.jacobian; % Distance from Apex point of the cone to jacobian point
%% Shape
mirror.shape = [...
0 mirror.h-mirror.thickness
mirror.hole_rad mirror.h-mirror.thickness; ...
mirror.hole_rad 0; ...
mirror.rad 0 ...
];
if strcmp(opts.shape, 'spherical')
mirror.sphere_radius = sqrt((mirror.jacobian+mirror.h)^2+mirror.rad^2); % Radius of the sphere [mm]
for z = linspace(0, mirror.h, 101)
mirror.shape = [mirror.shape; sqrt(mirror.sphere_radius^2-(z-mirror.jacobian-mirror.h)^2) z];
end
elseif strcmp(opts.shape, 'conical')
mirror.shape = [mirror.shape; mirror.rad+mirror.h/tand(opts.angle) mirror.h];
else
error('Shape should be either conical or spherical');
end
mirror.shape = [mirror.shape; 0 mirror.h];
%% Save
save('./mat/stages.mat', 'mirror', '-append');
end
#+end_src
** Nano Hexapod
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeNanoHexapod.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeNanoHexapod>>
This Matlab function is accessible [[file:../src/initializeNanoHexapod.m][here]].
#+begin_src matlab
function [nano_hexapod] = initializeNanoHexapod(opts_param)
%% Default values for opts
opts = struct('actuator', 'piezo');
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%% Stewart Object
nano_hexapod = struct();
nano_hexapod.h = 90; % Total height of the platform [mm]
nano_hexapod.jacobian = 175; % Point where the Jacobian is computed => Center of rotation [mm]
% nano_hexapod.jacobian = 174.26; % Point where the Jacobian is computed => Center of rotation [mm]
%% Bottom Plate
BP = struct();
BP.rad.int = 0; % Internal Radius [mm]
BP.rad.ext = 150; % External Radius [mm]
BP.thickness = 10; % Thickness [mm]
BP.leg.rad = 100; % Radius where the legs articulations are positionned [mm]
BP.leg.ang = 5; % Angle Offset [deg]
BP.density = 8000;% Density of the material [kg/m^3]
BP.color = [0.7 0.7 0.7]; % Color [rgb]
BP.shape = [BP.rad.int BP.thickness; BP.rad.int 0; BP.rad.ext 0; BP.rad.ext BP.thickness];
%% Top Plate
TP = struct();
TP.rad.int = 0; % Internal Radius [mm]
TP.rad.ext = 100; % Internal Radius [mm]
TP.thickness = 10; % Thickness [mm]
TP.leg.rad = 90; % Radius where the legs articulations are positionned [mm]
TP.leg.ang = 5; % Angle Offset [deg]
TP.density = 8000;% Density of the material [kg/m^3]
TP.color = [0.7 0.7 0.7]; % Color [rgb]
TP.shape = [TP.rad.int TP.thickness; TP.rad.int 0; TP.rad.ext 0; TP.rad.ext TP.thickness];
%% Leg
Leg = struct();
Leg.stroke = 80e-6; % Maximum Stroke of each leg [m]
if strcmp(opts.actuator, 'piezo')
Leg.k.ax = 1e7; % Stiffness of each leg [N/m]
elseif strcmp(opts.actuator, 'lorentz')
Leg.k.ax = 1e4; % Stiffness of each leg [N/m]
else
error('opts.actuator should be piezo or lorentz');
end
Leg.ksi.ax = 10; % Maximum amplification at resonance []
Leg.rad.bottom = 12; % Radius of the cylinder of the bottom part [mm]
Leg.rad.top = 10; % Radius of the cylinder of the top part [mm]
Leg.density = 8000; % Density of the material [kg/m^3]
Leg.color.bottom = [0.5 0.5 0.5]; % Color [rgb]
Leg.color.top = [0.5 0.5 0.5]; % Color [rgb]
Leg.sphere.bottom = Leg.rad.bottom; % Size of the sphere at the end of the leg [mm]
Leg.sphere.top = Leg.rad.top; % Size of the sphere at the end of the leg [mm]
Leg.m = TP.density*((pi*(TP.rad.ext/1000)^2)*(TP.thickness/1000)-(pi*(TP.rad.int/1000^2))*(TP.thickness/1000))/6; % TODO [kg]
Leg = updateDamping(Leg);
%% Sphere
SP = struct();
SP.height.bottom = 15; % [mm]
SP.height.top = 15; % [mm]
SP.density.bottom = 8000; % [kg/m^3]
SP.density.top = 8000; % [kg/m^3]
SP.color.bottom = [0.7 0.7 0.7]; % [rgb]
SP.color.top = [0.7 0.7 0.7]; % [rgb]
SP.k.ax = 0; % [N*m/deg]
SP.ksi.ax = 0;
SP.thickness.bottom = SP.height.bottom-Leg.sphere.bottom; % [mm]
SP.thickness.top = SP.height.top-Leg.sphere.top; % [mm]
SP.rad.bottom = Leg.sphere.bottom; % [mm]
SP.rad.top = Leg.sphere.top; % [mm]
SP.m = SP.density.bottom*2*pi*((SP.rad.bottom*1e-3)^2)*(SP.height.bottom*1e-3); % TODO [kg]
SP = updateDamping(SP);
%%
Leg.support.bottom = [0 SP.thickness.bottom; 0 0; SP.rad.bottom 0; SP.rad.bottom SP.height.bottom];
Leg.support.top = [0 SP.thickness.top; 0 0; SP.rad.top 0; SP.rad.top SP.height.top];
%%
nano_hexapod.BP = BP;
nano_hexapod.TP = TP;
nano_hexapod.Leg = Leg;
nano_hexapod.SP = SP;
%%
nano_hexapod = initializeParameters(nano_hexapod);
%% Save
save('./mat/stages.mat', 'nano_hexapod', '-append');
%%
function [element] = updateDamping(element)
field = fieldnames(element.k);
for i = 1:length(field)
if element.ksi.(field{i}) > 0
element.c.(field{i}) = 1/element.ksi.(field{i})*sqrt(element.k.(field{i})/element.m);
else
element.c.(field{i}) = 0;
end
end
end
%%
function [stewart] = initializeParameters(stewart)
%% Connection points on base and top plate w.r.t. World frame at the center of the base plate
stewart.pos_base = zeros(6, 3);
stewart.pos_top = zeros(6, 3);
alpha_b = stewart.BP.leg.ang*pi/180; % angle de décalage par rapport à 120 deg (pour positionner les supports bases)
alpha_t = stewart.TP.leg.ang*pi/180; % +- offset angle from 120 degree spacing on top
height = (stewart.h-stewart.BP.thickness-stewart.TP.thickness-stewart.Leg.sphere.bottom-stewart.Leg.sphere.top-stewart.SP.thickness.bottom-stewart.SP.thickness.top)*0.001; % TODO
radius_b = stewart.BP.leg.rad*0.001; % rayon emplacement support base
radius_t = stewart.TP.leg.rad*0.001; % top radius in meters
for i = 1:3
% base points
angle_m_b = (2*pi/3)* (i-1) - alpha_b;
angle_p_b = (2*pi/3)* (i-1) + alpha_b;
stewart.pos_base(2*i-1,:) = [radius_b*cos(angle_m_b), radius_b*sin(angle_m_b), 0.0];
stewart.pos_base(2*i,:) = [radius_b*cos(angle_p_b), radius_b*sin(angle_p_b), 0.0];
% top points
% Top points are 60 degrees offset
angle_m_t = (2*pi/3)* (i-1) - alpha_t + 2*pi/6;
angle_p_t = (2*pi/3)* (i-1) + alpha_t + 2*pi/6;
stewart.pos_top(2*i-1,:) = [radius_t*cos(angle_m_t), radius_t*sin(angle_m_t), height];
stewart.pos_top(2*i,:) = [radius_t*cos(angle_p_t), radius_t*sin(angle_p_t), height];
end
% permute pos_top points so that legs are end points of base and top points
stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
%% leg vectors
legs = stewart.pos_top - stewart.pos_base;
leg_length = zeros(6, 1);
leg_vectors = zeros(6, 3);
for i = 1:6
leg_length(i) = norm(legs(i,:));
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
stewart.Leg.lenght = 1000*leg_length(1)/1.5;
stewart.Leg.shape.bot = [0 0; ...
stewart.Leg.rad.bottom 0; ...
stewart.Leg.rad.bottom stewart.Leg.lenght; ...
stewart.Leg.rad.top stewart.Leg.lenght; ...
stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
%% Calculate revolute and cylindrical axes
rev1 = zeros(6, 3);
rev2 = zeros(6, 3);
cyl1 = zeros(6, 3);
for i = 1:6
rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
cyl1(i,:) = leg_vectors(i,:);
end
%% Coordinate systems
stewart.lower_leg = struct('rotation', eye(3));
stewart.upper_leg = struct('rotation', eye(3));
for i = 1:6
stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
end
%% Position Matrix
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.thickness+stewart.Leg.sphere.top+stewart.SP.thickness.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
%% Compute Jacobian Matrix
aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
stewart.J = getJacobianMatrix(leg_vectors', aa');
end
function J = getJacobianMatrix(RM,M_pos_base)
% RM: [3x6] unit vector of each leg in the fixed frame
% M_pos_base: [3x6] vector of the leg connection at the top platform location in the fixed frame
J = zeros(6);
J(:, 1:3) = RM';
J(:, 4:6) = cross(M_pos_base, RM)';
end
end
#+end_src
** Cedrat Actuator
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeCedratPiezo.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeCedratPiezo>>
This Matlab function is accessible [[file:../src/initializeCedratPiezo.m][here]].
#+begin_src matlab
function [cedrat] = initializeCedratPiezo(opts_param)
%% Default values for opts
opts = struct();
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%% Stewart Object
cedrat = struct();
cedrat.k = 10e7; % Linear Stiffness of each "blade" [N/m]
cedrat.ka = 10e7; % Linear Stiffness of the stack [N/m]
cedrat.c = 0.1*sqrt(1*cedrat.k); % [N/(m/s)]
cedrat.ca = 0.1*sqrt(1*cedrat.ka); % [N/(m/s)]
cedrat.L = 80; % Total Width of the Actuator[mm]
cedrat.H = 45; % Total Height of the Actuator [mm]
cedrat.L2 = sqrt((cedrat.L/2)^2 + (cedrat.H/2)^2); % Length of the elipsoidal sections [mm]
cedrat.alpha = 180/pi*atan2(cedrat.L/2, cedrat.H/2); % [deg]
%% Save
save('./mat/stages.mat', 'cedrat', '-append');
end
#+end_src
** Sample
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeSample.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeSample>>
This Matlab function is accessible [[file:../src/initializeSample.m][here]].
#+begin_src matlab
function [sample] = initializeSample(opts_param)
%% Default values for opts
sample = struct('radius', 100, ...
'height', 300, ...
'mass', 50, ...
'offset', 0, ...
'color', [0.45, 0.45, 0.45] ...
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
sample.(opt{1}) = opts_param.(opt{1});
end
end
%%
sample.k.x = 1e8;
sample.c.x = 0.1*sqrt(sample.k.x*sample.mass);
sample.k.y = 1e8;
sample.c.y = 0.1*sqrt(sample.k.y*sample.mass);
sample.k.z = 1e8;
sample.c.z = 0.1*sqrt(sample.k.z*sample.mass);
%% Save
save('./mat/stages.mat', 'sample', '-append');
end
#+end_src
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