tdehaeze/stewart-simscape
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity
2914d01e8f
stewart-simscape/simscape-model.org

504 lines
15 KiB
Org Mode
Raw Blame History

#+TITLE: Stewart Platform - Simscape Model
:DRAWER:
#+STARTUP: overview
#+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: <script src="js/jquery.min.js"></script>
#+HTML_HEAD: <script src="js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
#+LATEX_CLASS: cleanreport
#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
#+LaTeX_HEADER: \usepackage{svg}
#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments no
#+PROPERTY: header-args:matlab+ :exports bode
#+PROPERTY: header-args:matlab+ :eval no
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :tangle src/initializeHexapod.m
:END:
* Function description and arguments
The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
#+begin_src matlab
function [stewart] = initializeHexapod(opts_param)
#+end_src
Default values for opts.
#+begin_src matlab
opts = struct(...
'height', 90, ... % Height of the platform [mm]
'density', 8000, ... % Density of the material used for the hexapod [kg/m3]
'k_ax', 1e8, ... % Stiffness of each actuator [N/m]
'c_ax', 1000, ... % Damping of each actuator [N/(m/s)]
'stroke', 50e-6, ... % Maximum stroke of each actuator [m]
'name', 'stewart' ... % Name of the file
);
#+end_src
Populate opts with input parameters
#+begin_src matlab
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
#+end_src
* Initialization of the stewart structure
We initialize the Stewart structure
#+begin_src matlab
stewart = struct();
#+end_src
And we defined its total height.
#+begin_src matlab
stewart.H = opts.height; % [mm]
#+end_src
* Bottom Plate
#+name: fig:stewart_bottom_plate
#+caption: Schematic of the bottom plates with all the parameters
[[file:./figs/stewart_bottom_plate.png]]
The bottom plate structure is initialized.
#+begin_src matlab
BP = struct();
#+end_src
We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
#+begin_src matlab
BP.Rint = 0; % Internal Radius [mm]
BP.Rext = 150; % External Radius [mm]
#+end_src
We define its thickness.
#+begin_src matlab
BP.H = 10; % Thickness of the Bottom Plate [mm]
#+end_src
At which radius legs will be fixed and with that angle offset.
#+begin_src matlab
BP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
BP.alpha = 10; % Angle Offset [deg]
#+end_src
We defined the density of the material of the bottom plate.
#+begin_src matlab
BP.density = opts.density; % Density of the material [kg/m3]
#+end_src
And its color.
#+begin_src matlab
BP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src
Then the profile of the bottom plate is computed and will be used by Simscape
#+begin_src matlab
BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.BP = BP;
#+end_src
* Top Plate
The top plate structure is initialized.
#+begin_src matlab
TP = struct();
#+end_src
We defined the internal and external radius of the top plate.
#+begin_src matlab
TP.Rint = 0; % [mm]
TP.Rext = 100; % [mm]
#+end_src
The thickness of the top plate.
#+begin_src matlab
TP.H = 10; % [mm]
#+end_src
At which radius and angle are fixed the legs.
#+begin_src matlab
TP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
TP.alpha = 20; % Angle [deg]
TP.dalpha = 0; % Angle Offset from 0 position [deg]
#+end_src
The density of its material.
#+begin_src matlab
TP.density = opts.density; % Density of the material [kg/m3]
#+end_src
Its color.
#+begin_src matlab
TP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src
Then the shape of the top plate is computed
#+begin_src matlab
TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.TP = TP;
#+end_src
* Legs
#+name: fig:stewart_legs
#+caption: Schematic for the legs of the Stewart platform
[[file:./figs/stewart_legs.png]]
The leg structure is initialized.
#+begin_src matlab
Leg = struct();
#+end_src
The maximum Stroke of each leg is defined.
#+begin_src matlab
Leg.stroke = opts.stroke; % [m]
#+end_src
The stiffness and damping of each leg are defined
#+begin_src matlab
Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
#+end_src
The radius of the legs are defined
#+begin_src matlab
Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
#+end_src
The density of its material.
#+begin_src matlab
Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
#+end_src
Its color.
#+begin_src matlab
Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
#+end_src
The radius of spheres representing the ball joints are defined.
#+begin_src matlab
Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.Leg = Leg;
#+end_src
* Ball Joints
#+name: fig:stewart_ball_joints
#+caption: Schematic of the support for the ball joints
[[file:./figs/stewart_ball_joints.png]]
=SP= is the structure representing the support for the ball joints at the extremity of each leg.
The =SP= structure is initialized.
#+begin_src matlab
SP = struct();
#+end_src
We can define its rotational stiffness and damping. For now, we use perfect joints.
#+begin_src matlab
SP.k = 0; % [N*m/deg]
SP.c = 0; % [N*m/deg]
#+end_src
Its height is defined
#+begin_src matlab
SP.H = 15; % [mm]
#+end_src
Its radius is based on the radius on the sphere at the end of the legs.
#+begin_src matlab
SP.R = Leg.R; % [mm]
#+end_src
#+begin_src matlab
SP.section = [0 SP.H-SP.R;
0 0;
SP.R 0;
SP.R SP.H];
#+end_src
The density of its material is defined.
#+begin_src matlab
SP.density = opts.density; % [kg/m^3]
#+end_src
Its color is defined.
#+begin_src matlab
SP.color = [0.7 0.7 0.7]; % [RGB]
#+end_src
The structure is added to the Hexapod structure
#+begin_src matlab
stewart.SP = SP;
#+end_src
* More parameters are initialized
#+begin_src matlab
stewart = initializeParameters(stewart);
#+end_src
* Save the Stewart Structure
#+begin_src matlab
save('./mat/stewart.mat', 'stewart')
#+end_src
* initializeParameters Function :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle no
:END:
#+begin_src matlab
function [stewart] = initializeParameters(stewart)
#+end_src
Computation of the position of the connection points on the base and moving platform
We first initialize =pos_base= corresponding to $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and =pos_top= corresponding to $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
stewart.pos_base = zeros(6, 3);
stewart.pos_top = zeros(6, 3);
#+end_src
We estimate the height between the ball joints of the bottom platform and of the top platform.
#+begin_src matlab
height = stewart.H - stewart.BP.H - stewart.TP.H - 2*stewart.SP.H; % [mm]
#+end_src
#+begin_src matlab
for i = 1:3
% base points
angle_m_b = 120*(i-1) - stewart.BP.alpha;
angle_p_b = 120*(i-1) + stewart.BP.alpha;
stewart.pos_base(2*i-1,:) = [stewart.BP.Rleg*cos(angle_m_b), stewart.BP.Rleg*sin(angle_m_b), 0.0];
stewart.pos_base(2*i,:) = [stewart.BP.Rleg*cos(angle_p_b), stewart.BP.Rleg*sin(angle_p_b), 0.0];
% top points
angle_m_t = 120*(i-1) - stewart.TP.alpha + stewart.TP.dalpha;
angle_p_t = 120*(i-1) + stewart.TP.alpha + stewart.TP.dalpha;
stewart.pos_top(2*i-1,:) = [stewart.TP.Rleg*cos(angle_m_t), stewart.TP.Rleg*sin(angle_m_t), height];
stewart.pos_top(2*i,:) = [stewart.TP.Rleg*cos(angle_p_t), stewart.TP.Rleg*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)];
#+end_src
leg vectors
#+begin_src matlab
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];
#+end_src
Calculate revolute and cylindrical axes
#+begin_src matlab
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
#+end_src
Coordinate systems
#+begin_src matlab
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
#+end_src
Position Matrix
#+begin_src matlab
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.h+stewart.Leg.sphere.top+stewart.SP.h.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
#+end_src
Compute Jacobian Matrix
#+begin_src matlab
% aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.h - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
bb = stewart.pos_top_tranform - (stewart.TP.h + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
stewart.J = getJacobianMatrix(leg_vectors', bb');
stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
end
#+end_src
* initializeParameters Function
#+begin_src matlab
function [stewart] = initializeParameters(stewart)
#+end_src
We first compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
stewart.Aa = zeros(6, 3); % [mm]
stewart.Ab = zeros(6, 3); % [mm]
stewart.Bb = zeros(6, 3); % [mm]
#+end_src
#+begin_src matlab
for i = 1:3
stewart.Aa(2*i-1,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
stewart.BP.Rleg*sin( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
stewart.BP.H+stewart.SP.H];
stewart.Aa(2*i,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
stewart.BP.Rleg*sin( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
stewart.BP.H+stewart.SP.H];
stewart.Ab(2*i-1,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
stewart.H - stewart.TP.H - stewart.SP.H];
stewart.Ab(2*i,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
stewart.H - stewart.TP.H - stewart.SP.H];
end
stewart.Bb = stewart.Ab - stewart.H*[0,0,1];
#+end_src
Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
\[ b_i - a_i = l_i \hat{s}_i \]
We initialize $l_i$ and $\hat{s}_i$
#+begin_src matlab
leg_length = zeros(6, 1); % [mm]
leg_vectors = zeros(6, 3);
#+end_src
We compute $b_i - a_i$, and then:
\begin{align*}
l_i &= \left|b_i - a_i\right| \\
\hat{s}_i &= \frac{b_i - a_i}{l_i}
\end{align*}
#+begin_src matlab
legs = stewart.Ab - stewart.Aa;
for i = 1:6
leg_length(i) = norm(legs(i,:));
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
#+end_src
Then the shape of the bottom leg is estimated
#+begin_src matlab
stewart.Leg.lenght = leg_length(1)/1.5;
stewart.Leg.shape.bot = ...
[0 0; ...
stewart.Leg.Rbot 0; ...
stewart.Leg.Rbot stewart.Leg.lenght; ...
stewart.Leg.Rtop stewart.Leg.lenght; ...
stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
#+end_src
We compute rotation matrices to have the orientation of the legs.
The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
#+begin_src matlab
stewart.Rm = struct('R', eye(3));
for i = 1:6
sx = cross(leg_vectors(i,:), [1 0 0]);
sx = sx/norm(sx);
sy = -cross(sx, leg_vectors(i,:));
sy = sy/norm(sy);
sz = leg_vectors(i,:);
sz = sz/norm(sz);
stewart.Rm(i).R = [sx', sy', sz'];
end
#+end_src
Compute Jacobian Matrix
#+begin_src matlab
J = zeros(6);
for i = 1:6
J(i, 1:3) = leg_vectors(i, :);
J(i, 4:6) = cross(0.001*(stewart.Ab(i, :)- stewart.H*[0,0,1]), leg_vectors(i, :));
end
stewart.J = J;
stewart.Jinv = inv(J);
#+end_src
#+begin_src matlab
stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;
#+end_src
#+begin_src matlab
end
end
#+end_src
* initializeSample
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/initializeSample.m
:END:
#+begin_src matlab
function [] = initializeSample(opts_param)
%% Default values for opts
sample = struct( ...
'radius', 100, ... % radius of the cylinder [mm]
'height', 100, ... % height of the cylinder [mm]
'mass', 10, ... % mass of the cylinder [kg]
'measheight', 50, ... % measurement point z-offset [mm]
'offset', [0, 0, 0], ... % offset position of the sample [mm]
'color', [0.9 0.1 0.1] ...
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
sample.(opt{1}) = opts_param.(opt{1});
end
end
%% Save
save('./mat/sample.mat', 'sample');
end
#+end_src
Reference in New Issue View Git Blame Copy Permalink