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#+TITLE: Stewart Platform - Simscape Model
:DRAWER:
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#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs
:END:
* Introduction :ignore:
Stewart platforms are generated in multiple steps.
We define 4 important *frames*:
- $\{F\}$: Frame fixed to the *Fixed* base and located at the center of its bottom surface.
This is used to fix the Stewart platform to some support.
- $\{M\}$: Frame fixed to the *Moving* platform and located at the center of its top surface.
This is used to place things on top of the Stewart platform.
- $\{A\}$: Frame fixed to the fixed base.
It defined the center of rotation of the moving platform.
- $\{B\}$: Frame fixed to the moving platform.
The motion of the moving platforms and forces applied to it are defined with respect to this frame $\{B\}$.
Then, we define the *location of the spherical joints*:
- $\bm{a}_{i}$ are the position of the spherical joints fixed to the fixed base
- $\bm{b}_{i}$ are the position of the spherical joints fixed to the moving platform
We define the *rest position* of the Stewart platform:
- For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position.
- Thus, to define the rest position of the Stewart platform, we just have to defined its total height $H$.
$H$ corresponds to the distance from the bottom of the fixed base to the top of the moving platform.
From $\bm{a}_{i}$ and $\bm{b}_{i}$, we can determine the *length and orientation of each strut*:
- $l_{i}$ is the length of the strut
- ${}^{A}\hat{\bm{s}}_{i}$ is the unit vector align with the strut
The position of the Spherical joints can be computed using various methods:
- Cubic configuration
- Circular configuration
- Arbitrary position
- These methods should be easily scriptable and corresponds to specific functions that returns ${}^{F}\bm{a}_{i}$ and ${}^{M}\bm{b}_{i}$.
The input of these functions are the parameters corresponding to the wanted geometry.
For Simscape, we need:
- The position and orientation of each spherical joint fixed to the fixed base: ${}^{F}\bm{a}_{i}$ and ${}^{F}\bm{R}_{a_{i}}$
- The position and orientation of each spherical joint fixed to the moving platform: ${}^{M}\bm{b}_{i}$ and ${}^{M}\bm{R}_{b_{i}}$
- The rest length of each strut: $l_{i}$
- The stiffness and damping of each actuator: $k_{i}$ and $c_{i}$
- The position of the frame $\{A\}$ with respect to the frame $\{F\}$: ${}^{F}\bm{O}_{A}$
- The position of the frame $\{B\}$ with respect to the frame $\{M\}$: ${}^{M}\bm{O}_{B}$
* Procedure
The procedure to define the Stewart platform is the following:
1. Define the initial position of frames {A}, {B}, {F} and {M}.
We do that using the =initializeFramesPositions= function.
We have to specify the total height of the Stewart platform $H$ and the position ${}^{M}O_{B}$ of {B} with respect to {M}.
2. Compute the positions of joints ${}^{F}a_{i}$ and ${}^{M}b_{i}$.
We can do that using various methods depending on the wanted architecture:
- =generateCubicConfiguration= permits to generate a cubic configuration
3. Compute the position and orientation of the joints with respect to the fixed base and the moving platform.
This is done with the =computeJointsPose= function.
4. Define the dynamical properties of the Stewart platform.
The output are the stiffness and damping of each strut $k_{i}$ and $c_{i}$.
This can be done we simply choosing directly the stiffness and damping of each strut.
The stiffness and damping of each actuator can also be determine from the wanted stiffness of the Stewart platform for instance.
5. Define the mass and inertia of each element of the Stewart platform.
By following this procedure, we obtain a Matlab structure =stewart= that contains all the information for the Simscape model and for further analysis.
* Matlab Code
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
addpath('./src/')
#+end_src
** Simscape Model
#+begin_src matlab
open('stewart_platform.slx')
#+end_src
** Test the functions
#+begin_src matlab
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
% stewart = generateCubicConfiguration(stewart, 'Hc', 60e-3, 'FOc', 45e-3, 'FHa', 5e-3, 'MHb', 5e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1));
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
[Li, dLi] = inverseKinematics(stewart, 'AP', [0;0;0.00001], 'ARB', eye(3));
[P, R] = forwardKinematicsApprox(stewart, 'dL', dLi);
#+end_src
* =initializeFramesPositions=: Initialize the positions of frames {A}, {B}, {F} and {M}
:PROPERTIES:
:header-args:matlab+: :tangle src/initializeFramesPositions.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeFramesPositions>>
This Matlab function is accessible [[file:src/initializeFramesPositions.m][here]].
** Function description
#+begin_src matlab
function [stewart] = initializeFramesPositions(args)
% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
%
% Syntax: [stewart] = initializeFramesPositions(args)
%
% Inputs:
% - args - Can have the following fields:
% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
%
% Outputs:
% - stewart - A structure with the following fields:
% - H [1x1] - Total Height of the Stewart Platform [m]
% - FO_M [3x1] - Position of {M} with respect to {F} [m]
% - MO_B [3x1] - Position of {B} with respect to {M} [m]
% - FO_A [3x1] - Position of {A} with respect to {F} [m]
#+end_src
** Documentation
#+name: fig:stewart-frames-position
#+caption: Definition of the position of the frames
[[file:figs/stewart-frames-position.png]]
** Optional Parameters
#+begin_src matlab
arguments
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
args.MO_B (1,1) double {mustBeNumeric} = 50e-3
end
#+end_src
** Initialize the Stewart structure
#+begin_src matlab
stewart = struct();
#+end_src
** Compute the position of each frame
#+begin_src matlab
stewart.H = args.H; % Total Height of the Stewart Platform [m]
stewart.FO_M = [0; 0; stewart.H]; % Position of {M} with respect to {F} [m]
stewart.MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]
stewart.FO_A = stewart.MO_B + stewart.FO_M; % Position of {A} with respect to {F} [m]
#+end_src
* Initialize the position of the Joints
** =generateCubicConfiguration=: Generate a Cubic Configuration
:PROPERTIES:
:header-args:matlab+: :tangle src/generateCubicConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:generateCubicConfiguration>>
This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here]].
*** Function description
#+begin_src matlab
function [stewart] = generateCubicConfiguration(stewart, args)
% generateCubicConfiguration - Generate a Cubic Configuration
%
% Syntax: [stewart] = generateCubicConfiguration(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - H [1x1] - Total height of the platform [m]
% - args - Can have the following fields:
% - Hc [1x1] - Height of the "useful" part of the cube [m]
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
#+end_src
*** Documentation
#+name: fig:cubic-configuration-definition
#+caption: Cubic Configuration
[[file:figs/cubic-configuration-definition.png]]
*** Optional Parameters
#+begin_src matlab
arguments
stewart
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
args.FOc (1,1) double {mustBeNumeric} = 50e-3
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
end
#+end_src
*** Position of the Cube
We define the useful points of the cube with respect to the Cube's center.
${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is
located at the center of the cube and aligned with {F} and {M}.
#+begin_src matlab
sx = [ 2; -1; -1];
sy = [ 0; 1; -1];
sz = [ 1; 1; 1];
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
L = args.Hc*sqrt(3);
Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];
CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
#+end_src
*** Compute the pose
We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).
#+begin_src matlab
CSi = (CCm - CCf)./vecnorm(CCm - CCf);
#+end_src
We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
#+begin_src matlab
stewart.Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
stewart.Mb = CCf + [0; 0; args.FOc-stewart.H] + ((stewart.H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
#+end_src
** =generateGeneralConfiguration=: Generate a Very General Configuration
:PROPERTIES:
:header-args:matlab+: :tangle src/generateGeneralConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:generateGeneralConfiguration>>
This Matlab function is accessible [[file:src/generateGeneralConfiguration.m][here]].
*** Function description
#+begin_src matlab
function [stewart] = generateGeneralConfiguration(stewart, args)
% generateGeneralConfiguration - Generate a Very General Configuration
%
% Syntax: [stewart] = generateGeneralConfiguration(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - H [1x1] - Total height of the platform [m]
% - args - Can have the following fields:
% - FH [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]
% - FR [1x1] - Radius of the position of the fixed joints in the X-Y [m]
% - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]
% - MH [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]
% - FR [1x1] - Radius of the position of the mobile joints in the X-Y [m]
% - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
#+end_src
*** Documentation
Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}.
The radius of the circles can be chosen as well as the angles where the joints are located.
*** Optional Parameters
#+begin_src matlab
arguments
stewart
args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 90e-3;
args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180);
args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 70e-3;
args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180);
end
#+end_src
*** Compute the pose
#+begin_src matlab
stewart.Fa = zeros(3,6);
stewart.Mb = zeros(3,6);
#+end_src
#+begin_src matlab
for i = 1:6
stewart.Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i)); args.FH];
stewart.Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH];
end
#+end_src
* =computeJointsPose=: Compute the Pose of the Joints
:PROPERTIES:
:header-args:matlab+: :tangle src/computeJointsPose.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeJointsPose>>
This Matlab function is accessible [[file:src/computeJointsPose.m][here]].
** Function description
#+begin_src matlab
function [stewart] = computeJointsPose(stewart)
% computeJointsPose -
%
% Syntax: [stewart] = computeJointsPose(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
% - FO_A [3x1] - Position of {A} with respect to {F}
% - MO_B [3x1] - Position of {B} with respect to {M}
% - FO_M [3x1] - Position of {M} with respect to {F}
%
% Outputs:
% - stewart - A structure with the following added fields
% - Aa [3x6] - The i'th column is the position of ai with respect to {A}
% - Ab [3x6] - The i'th column is the position of bi with respect to {A}
% - Ba [3x6] - The i'th column is the position of ai with respect to {B}
% - Bb [3x6] - The i'th column is the position of bi with respect to {B}
% - l [6x1] - The i'th element is the initial length of strut i
% - As [3x6] - The i'th column is the unit vector of strut i expressed in {A}
% - Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}
% - FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
% - MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
#+end_src
** Documentation
#+name: fig:stewart-struts
#+caption: Position and orientation of the struts
[[file:figs/stewart-struts.png]]
** Compute the position of the Joints
#+begin_src matlab
stewart.Aa = stewart.Fa - repmat(stewart.FO_A, [1, 6]);
stewart.Bb = stewart.Mb - repmat(stewart.MO_B, [1, 6]);
stewart.Ab = stewart.Bb - repmat(-stewart.MO_B-stewart.FO_M+stewart.FO_A, [1, 6]);
stewart.Ba = stewart.Aa - repmat( stewart.MO_B+stewart.FO_M-stewart.FO_A, [1, 6]);
#+end_src
** Compute the strut length and orientation
#+begin_src matlab
stewart.As = (stewart.Ab - stewart.Aa)./vecnorm(stewart.Ab - stewart.Aa); % As_i is the i'th vector of As
stewart.l = vecnorm(stewart.Ab - stewart.Aa)';
#+end_src
#+begin_src matlab
stewart.Bs = (stewart.Bb - stewart.Ba)./vecnorm(stewart.Bb - stewart.Ba);
#+end_src
** Compute the orientation of the Joints
#+begin_src matlab
stewart.FRa = zeros(3,3,6);
stewart.MRb = zeros(3,3,6);
for i = 1:6
stewart.FRa(:,:,i) = [cross([0;1;0], stewart.As(:,i)) , cross(stewart.As(:,i), cross([0;1;0], stewart.As(:,i))) , stewart.As(:,i)];
stewart.FRa(:,:,i) = stewart.FRa(:,:,i)./vecnorm(stewart.FRa(:,:,i));
stewart.MRb(:,:,i) = [cross([0;1;0], stewart.Bs(:,i)) , cross(stewart.Bs(:,i), cross([0;1;0], stewart.Bs(:,i))) , stewart.Bs(:,i)];
stewart.MRb(:,:,i) = stewart.MRb(:,:,i)./vecnorm(stewart.MRb(:,:,i));
end
#+end_src
* =initializeStrutDynamics=: Add Stiffness and Damping properties of each strut
:PROPERTIES:
:header-args:matlab+: :tangle src/initializeStrutDynamics.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeStrutDynamics>>
This Matlab function is accessible [[file:src/initializeStrutDynamics.m][here]].
** Function description
#+begin_src matlab
function [stewart] = initializeStrutDynamics(stewart, args)
% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
%
% Syntax: [stewart] = initializeStrutDynamics(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Ki [6x1] - Stiffness of each strut [N/m]
% - Ci [6x1] - Damping of each strut [N/(m/s)]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - Ki [6x1] - Stiffness of each strut [N/m]
% - Ci [6x1] - Damping of each strut [N/(m/s)]
#+end_src
** Optional Parameters
#+begin_src matlab
arguments
stewart
args.Ki (6,1) double {mustBeNumeric, mustBePositive} = 1e6*ones(6,1)
args.Ci (6,1) double {mustBeNumeric, mustBePositive} = 1e3*ones(6,1)
end
#+end_src
** Add Stiffness and Damping properties of each strut
#+begin_src matlab
stewart.Ki = args.Ki;
stewart.Ci = args.Ci;
#+end_src
* =computeJacobian=: Compute the Jacobian Matrix
:PROPERTIES:
:header-args:matlab+: :tangle src/computeJacobian.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeJacobian>>
This Matlab function is accessible [[file:src/computeJacobian.m][here]].
** Function description
#+begin_src matlab
function [stewart] = computeJacobian(stewart)
% computeJacobian -
%
% Syntax: [stewart] = computeJacobian(stewart)
%
% Inputs:
% - stewart - With at least the following fields:
% - As [3x6] - The 6 unit vectors for each strut expressed in {A}
% - Ab [3x6] - The 6 position of the joints bi expressed in {A}
%
% Outputs:
% - stewart - With the 3 added field:
% - J [6x6] - The Jacobian Matrix
% - K [6x6] - The Stiffness Matrix
% - C [6x6] - The Compliance Matrix
#+end_src
** Compute Jacobian Matrix
#+begin_src matlab
stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];
#+end_src
** Compute Stiffness Matrix
#+begin_src matlab
stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;
#+end_src
** Compute Compliance Matrix
#+begin_src matlab
stewart.C = inv(stewart.K);
#+end_src
* Initialize the Geometry of the Mechanical Elements
** =initializeCylindricalStruts=: Define the mass and moment of inertia of cylindrical struts
:PROPERTIES:
:header-args:matlab+: :tangle src/initializeCylindricalStruts.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeCylindricalStruts>>
This Matlab function is accessible [[file:src/initializeCylindricalStruts.m][here]].
*** Function description
#+begin_src matlab
function [stewart] = initializeCylindricalStruts(stewart, args)
% initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts
%
% Syntax: [stewart] = initializeCylindricalStruts(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Fsm [1x1] - Mass of the Fixed part of the struts [kg]
% - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]
% - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]
% - Msm [1x1] - Mass of the Mobile part of the struts [kg]
% - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]
% - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - struts [struct] - structure with the following fields:
% - Fsm [6x1] - Mass of the Fixed part of the struts [kg]
% - Fsi [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]
% - Msm [6x1] - Mass of the Mobile part of the struts [kg]
% - Msi [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]
% - Fsh [6x1] - Height of cylinder for the Fixed part of the struts [m]
% - Fsr [6x1] - Radius of cylinder for the Fixed part of the struts [m]
% - Msh [6x1] - Height of cylinder for the Mobile part of the struts [m]
% - Msr [6x1] - Radius of cylinder for the Mobile part of the struts [m]
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
stewart
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
end
#+end_src
*** Create the =struts= structure
#+begin_src matlab
struts = struct();
struts.Fsm = ones(6,1).*args.Fsm;
struts.Msm = ones(6,1).*args.Msm;
struts.Fsh = ones(6,1).*args.Fsh;
struts.Fsr = ones(6,1).*args.Fsr;
struts.Msh = ones(6,1).*args.Msh;
struts.Msr = ones(6,1).*args.Msr;
struts.Fsi = zeros(3, 3, 6);
struts.Msi = zeros(3, 3, 6);
for i = 1:6
struts.Fsi(:,:,i) = diag([1/12 * struts.Fsm(i) * (3*struts.Fsr(i)^2 + struts.Fsh(i)^2), ...
1/12 * struts.Fsm(i) * (3*struts.Fsr(i)^2 + struts.Fsh(i)^2), ...
1/2 * struts.Fsm(i) * struts.Fsr(i)^2]);
struts.Msi(:,:,i) = diag([1/12 * struts.Msm(i) * (3*struts.Msr(i)^2 + struts.Msh(i)^2), ...
1/12 * struts.Msm(i) * (3*struts.Msr(i)^2 + struts.Msh(i)^2), ...
1/2 * struts.Msm(i) * struts.Msr(i)^2]);
end
#+end_src
#+begin_src matlab
stewart.struts = struts;
#+end_src
** =initializeCylindricalPlatforms=: Initialize the geometry of the Fixed and Mobile Platforms
:PROPERTIES:
:header-args:matlab+: :tangle src/initializeCylindricalPlatforms.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeCylindricalPlatforms>>
This Matlab function is accessible [[file:src/initializeCylindricalPlatforms.m][here]].
*** Function description
#+begin_src matlab
function [stewart] = initializeCylindricalPlatforms(stewart, args)
% initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
%
% Syntax: [stewart] = initializeCylindricalPlatforms(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Fpm [1x1] - Fixed Platform Mass [kg]
% - Fph [1x1] - Fixed Platform Height [m]
% - Fpr [1x1] - Fixed Platform Radius [m]
% - Mpm [1x1] - Mobile Platform Mass [kg]
% - Mph [1x1] - Mobile Platform Height [m]
% - Mpr [1x1] - Mobile Platform Radius [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - platforms [struct] - structure with the following fields:
% - Fpm [1x1] - Fixed Platform Mass [kg]
% - Msi [3x3] - Mobile Platform Inertia matrix [kg*m^2]
% - Fph [1x1] - Fixed Platform Height [m]
% - Fpr [1x1] - Fixed Platform Radius [m]
% - Mpm [1x1] - Mobile Platform Mass [kg]
% - Fsi [3x3] - Fixed Platform Inertia matrix [kg*m^2]
% - Mph [1x1] - Mobile Platform Height [m]
% - Mpr [1x1] - Mobile Platform Radius [m]
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
stewart
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e-3
args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
end
#+end_src
*** Create the =platforms= struct
#+begin_src matlab
platforms = struct();
platforms.Fpm = args.Fpm;
platforms.Fph = args.Fph;
platforms.Fpr = args.Fpr;
platforms.Fpi = diag([1/12 * platforms.Fpm * (3*platforms.Fpr^2 + platforms.Fph^2), ...
1/12 * platforms.Fpm * (3*platforms.Fpr^2 + platforms.Fph^2), ...
1/2 * platforms.Fpm * platforms.Fpr^2]);
platforms.Mpm = args.Mpm;
platforms.Mph = args.Mph;
platforms.Mpr = args.Mpr;
platforms.Mpi = diag([1/12 * platforms.Mpm * (3*platforms.Mpr^2 + platforms.Mph^2), ...
1/12 * platforms.Mpm * (3*platforms.Mpr^2 + platforms.Mph^2), ...
1/2 * platforms.Mpm * platforms.Mpr^2]);
#+end_src
*** Save the =platforms= struct
#+begin_src matlab
stewart.platforms = platforms;
#+end_src
* Utility Functions
** =inverseKinematics=: Compute Inverse Kinematics
:PROPERTIES:
:header-args:matlab+: :tangle src/inverseKinematics.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:inverseKinematics>>
This Matlab function is accessible [[file:src/inverseKinematics.m][here]].
*** Function description
#+begin_src matlab
function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - Aa [3x6] - The positions ai expressed in {A}
% - Bb [3x6] - The positions bi expressed in {B}
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
#+end_src
*** Theory
For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.
From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as
\begin{align*}
l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
&= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
\end{align*}
To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself:
\begin{equation}
l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
\end{equation}
Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by:
\begin{equation}
l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
\end{equation}
If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
Otherwise, when the limbs' lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
*** Compute
#+begin_src matlab
Li = sqrt(args.AP'*args.AP + diag(stewart.Bb'*stewart.Bb) + diag(stewart.Aa'*stewart.Aa) - (2*args.AP'*stewart.Aa)' + (2*args.AP'*(args.ARB*stewart.Bb))' - diag(2*(args.ARB*stewart.Bb)'*stewart.Aa));
#+end_src
#+begin_src matlab
dLi = Li-stewart.l;
#+end_src
** =forwardKinematicsApprox=: Compute the Forward Kinematics
:PROPERTIES:
:header-args:matlab+: :tangle src/forwardKinematicsApprox.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:forwardKinematicsApprox>>
This Matlab function is accessible [[file:src/forwardKinematicsApprox.m][here]].
*** Function description
#+begin_src matlab
function [P, R] = forwardKinematicsApprox(stewart, args)
% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using
% the Jacobian Matrix
%
% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - J [6x6] - The Jacobian Matrix
% - args - Can have the following fields:
% - dL [6x1] - Displacement of each strut [m]
%
% Outputs:
% - P [3x1] - The estimated position of {B} with respect to {A}
% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
stewart
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
end
#+end_src
*** Computation
From a small displacement of each strut $d\bm{\mathcal{L}}$, we can compute the
position and orientation of {B} with respect to {A} using the following formula:
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
#+begin_src matlab
X = stewart.J\args.dL;
#+end_src
The position vector corresponds to the first 3 elements.
#+begin_src matlab
P = X(1:3);
#+end_src
The next 3 elements are the orientation of {B} with respect to {A} expressed
using the screw axis.
#+begin_src matlab
theta = norm(X(4:6));
s = X(4:6)/theta;
#+end_src
We then compute the corresponding rotation matrix.
#+begin_src matlab
R = [s(1)^2*(1-cos(theta)) + cos(theta) , s(1)*s(2)*(1-cos(theta)) - s(3)*sin(theta), s(1)*s(3)*(1-cos(theta)) + s(2)*sin(theta);
s(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);
s(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];
#+end_src
* OLD :noexport:
** Define the Height of the Platform :noexport:
#+begin_src matlab
%% 1. Height of the platform. Location of {F} and {M}
H = 90e-3; % [m]
FO_M = [0; 0; H];
#+end_src
** Define the location of {A} and {B} :noexport:
#+begin_src matlab
%% 2. Location of {A} and {B}
FO_A = [0; 0; 100e-3] + FO_M;% [m,m,m]
MO_B = [0; 0; 100e-3];% [m,m,m]
#+end_src
** Define the position of $a_{i}$ and $b_{i}$ :noexport:
#+begin_src matlab
%% 3. Position of ai and bi
Fa = zeros(3, 6); % Fa_i is the i'th vector of Fa
Mb = zeros(3, 6); % Mb_i is the i'th vector of Mb
#+end_src
#+begin_src matlab
Aa = Fa - repmat(FO_A, [1, 6]);
Bb = Mb - repmat(MO_B, [1, 6]);
Ab = Bb - repmat(-MO_B-FO_M+FO_A, [1, 6]);
Ba = Aa - repmat( MO_B+FO_M-FO_A, [1, 6]);
As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As
l = vecnorm(Ab - Aa);
Bs = (Bb - Ba)./vecnorm(Bb - Ba);
FRa = zeros(3,3,6);
MRb = zeros(3,3,6);
for i = 1:6
FRa(:,:,i) = [cross([0;1;0],As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)];
FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i));
MRb(:,:,i) = [cross([0;1;0],Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)];
MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i));
end
#+end_src
** Define the dynamical properties of each strut :noexport:
#+begin_src matlab
%% 4. Stiffness and Damping of each strut
Ki = 1e6*ones(6,1);
Ci = 1e2*ones(6,1);
#+end_src
** Old Introduction :noexport:
First, geometrical parameters are defined:
- ${}^A\bm{a}_i$ - Position of the joints fixed to the fixed base w.r.t $\{A\}$
- ${}^A\bm{b}_i$ - Position of the joints fixed to the mobile platform w.r.t $\{A\}$
- ${}^B\bm{b}_i$ - Position of the joints fixed to the mobile platform w.r.t $\{B\}$
- $H$ - Total height of the mobile platform
These parameter are enough to determine all the kinematic properties of the platform like the Jacobian, stroke, stiffness, ...
These geometrical parameters can be generated using different functions: =initializeCubicConfiguration= for cubic configuration or =initializeGeneralConfiguration= for more general configuration.
A function =computeGeometricalProperties= is then used to compute:
- $\bm{J}_f$ - Jacobian matrix for the force location
- $\bm{J}_d$ - Jacobian matrix for displacement estimation
- $\bm{R}_m$ - Rotation matrices to position the leg vectors
Then, geometrical parameters are computed for all the mechanical elements with the function =initializeMechanicalElements=:
- Shape of the platforms
- External Radius
- Internal Radius
- Density
- Thickness
- Shape of the Legs
- Radius
- Size of ball joint
- Density
Other Parameters are defined for the Simscape simulation:
- Sample mass, volume and position (=initializeSample= function)
- Location of the inertial sensor
- Location of the point for the differential measurements
- Location of the Jacobian point for velocity/displacement computation
** Cubic Configuration :noexport:
To define the cubic configuration, we need to define 4 parameters:
- The size of the cube
- The location of the cube
- The position of the plane joint the points $a_{i}$
- The position of the plane joint the points $b_{i}$
To do so, we specify the following parameters:
- $H_{C}$ the height of the useful part of the cube
- ${}^{F}O_{C}$ the position of the center of the cube with respect to $\{F\}$
- ${}^{F}H_{A}$: the height of the plane joining the points $a_{i}$ with respect to the frame $\{F\}$
- ${}^{M}H_{B}$: the height of the plane joining the points $b_{i}$ with respect to the frame $\{M\}$
We define the parameters
#+begin_src matlab
Hc = 60e-3; % [m]
FOc = 50e-3; % [m]
FHa = 15e-3; % [m]
MHb = 15e-3; % [m]
#+end_src
We define the useful points of the cube with respect to the Cube's center.
${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is located at the center of the cube and aligned with {F} and {M}.
#+begin_src matlab
sx = [ 2; -1; -1];
sy = [ 0; 1; -1];
sz = [ 1; 1; 1];
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
L = Hc*sqrt(3);
Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*Hc];
CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
#+end_src
We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).
#+begin_src matlab
CSi = (CCm - CCf)./vecnorm(CCm - CCf);
#+end_src
We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
#+begin_src matlab
Fa = zeros(3, 6); % Fa_i is the i'th vector of Fa
Mb = zeros(3, 6); % Mb_i is the i'th vector of Mb
#+end_src
#+begin_src matlab
Fa = CCf + [0; 0; FOc] + ((FHa-(FOc-Hc/2))./CSi(3,:)).*CSi;
Mb = CCf + [0; 0; FOc-H] + ((H-MHb-(FOc-Hc/2))./CSi(3,:)).*CSi; % TODO
#+end_src
** initializeGeneralConfiguration :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :eval no
:HEADER-ARGS:matlab+: :tangle src/initializeGeneralConfiguration.m
:END:
*** Function description
The =initializeGeneralConfiguration= function takes one structure that contains configurations for the hexapod and returns one structure representing the Hexapod.
#+begin_src matlab
function [stewart] = initializeGeneralConfiguration(opts_param)
#+end_src
*** Optional Parameters
Default values for opts.
#+begin_src matlab
opts = struct(...
'H_tot', 90, ... % Height of the platform [mm]
'H_joint', 15, ... % Height of the joints [mm]
'H_plate', 10, ... % Thickness of the fixed and mobile platforms [mm]
'R_bot', 100, ... % Radius where the legs articulations are positionned [mm]
'R_top', 80, ... % Radius where the legs articulations are positionned [mm]
'a_bot', 10, ... % Angle Offset [deg]
'a_top', 40, ... % Angle Offset [deg]
'da_top', 0 ... % Angle Offset from 0 position [deg]
);
#+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
*** Geometry Description
#+name: fig:stewart_bottom_plate
#+caption: Schematic of the bottom plates with all the parameters
[[file:./figs/stewart_bottom_plate.png]]
*** Compute Aa and Ab
We 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
Aa = zeros(6, 3); % [mm]
Ab = zeros(6, 3); % [mm]
Bb = zeros(6, 3); % [mm]
#+end_src
#+begin_src matlab
for i = 1:3
Aa(2*i-1,:) = [opts.R_bot*cos( pi/180*(120*(i-1) - opts.a_bot) ), ...
opts.R_bot*sin( pi/180*(120*(i-1) - opts.a_bot) ), ...
opts.H_plate+opts.H_joint];
Aa(2*i,:) = [opts.R_bot*cos( pi/180*(120*(i-1) + opts.a_bot) ), ...
opts.R_bot*sin( pi/180*(120*(i-1) + opts.a_bot) ), ...
opts.H_plate+opts.H_joint];
Ab(2*i-1,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
opts.H_tot - opts.H_plate - opts.H_joint];
Ab(2*i,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
opts.H_tot - opts.H_plate - opts.H_joint];
end
Bb = Ab - opts.H_tot*[0,0,1];
#+end_src
*** Returns Stewart Structure
#+begin_src matlab :results none
stewart = struct();
stewart.Aa = Aa;
stewart.Ab = Ab;
stewart.Bb = Bb;
stewart.H_tot = opts.H_tot;
end
#+end_src
** initializeCubicConfiguration :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :eval no
:HEADER-ARGS:matlab+: :tangle src/initializeCubicConfiguration.m
:END:
<<sec:initializeCubicConfiguration>>
*** Function description
#+begin_src matlab
function [stewart] = initializeCubicConfiguration(opts_param)
#+end_src
*** Optional Parameters
Default values for opts.
#+begin_src matlab
opts = struct(...
'H_tot', 90, ... % Total height of the Hexapod [mm]
'L', 110, ... % Size of the Cube [mm]
'H', 40, ... % Height between base joints and platform joints [mm]
'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm]
);
#+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
*** Cube Creation
#+begin_src matlab :results none
points = [0, 0, 0; ...
0, 0, 1; ...
0, 1, 0; ...
0, 1, 1; ...
1, 0, 0; ...
1, 0, 1; ...
1, 1, 0; ...
1, 1, 1];
points = opts.L*points;
#+end_src
We create the rotation matrix to rotate the cube
#+begin_src matlab :results none
sx = cross([1, 1, 1], [1 0 0]);
sx = sx/norm(sx);
sy = -cross(sx, [1, 1, 1]);
sy = sy/norm(sy);
sz = [1, 1, 1];
sz = sz/norm(sz);
R = [sx', sy', sz']';
#+end_src
We use to rotation matrix to rotate the cube
#+begin_src matlab :results none
cube = zeros(size(points));
for i = 1:size(points, 1)
cube(i, :) = R * points(i, :)';
end
#+end_src
*** Vectors of each leg
#+begin_src matlab :results none
leg_indices = [3, 4; ...
2, 4; ...
2, 6; ...
5, 6; ...
5, 7; ...
3, 7];
#+end_src
Vectors are:
#+begin_src matlab :results none
legs = zeros(6, 3);
legs_start = zeros(6, 3);
for i = 1:6
legs(i, :) = cube(leg_indices(i, 2), :) - cube(leg_indices(i, 1), :);
legs_start(i, :) = cube(leg_indices(i, 1), :);
end
#+end_src
*** Verification of Height of the Stewart Platform
If the Stewart platform is not contained in the cube, throw an error.
#+begin_src matlab :results none
Hmax = cube(4, 3) - cube(2, 3);
if opts.H0 < cube(2, 3)
error(sprintf('H0 is not high enought. Minimum H0 = %.1f', cube(2, 3)));
else if opts.H0 + opts.H > cube(4, 3)
error(sprintf('H0+H is too high. Maximum H0+H = %.1f', cube(4, 3)));
error('H0+H is too high');
end
#+end_src
*** Determinate the location of the joints
We now determine the location of the joints on the fixed platform w.r.t the fixed frame $\{A\}$.
$\{A\}$ is fixed to the bottom of the base.
#+begin_src matlab :results none
Aa = zeros(6, 3);
for i = 1:6
t = (opts.H0-legs_start(i, 3))/(legs(i, 3));
Aa(i, :) = legs_start(i, :) + t*legs(i, :);
end
#+end_src
And the location of the joints on the mobile platform with respect to $\{A\}$.
#+begin_src matlab :results none
Ab = zeros(6, 3);
for i = 1:6
t = (opts.H0+opts.H-legs_start(i, 3))/(legs(i, 3));
Ab(i, :) = legs_start(i, :) + t*legs(i, :);
end
#+end_src
And the location of the joints on the mobile platform with respect to $\{B\}$.
#+begin_src matlab :results none
Bb = zeros(6, 3);
Bb = Ab - (opts.H0 + opts.H_tot/2 + opts.H/2)*[0, 0, 1];
#+end_src
#+begin_src matlab :results none
h = opts.H0 + opts.H/2 - opts.H_tot/2;
Aa = Aa - h*[0, 0, 1];
Ab = Ab - h*[0, 0, 1];
#+end_src
*** Returns Stewart Structure
#+begin_src matlab :results none
stewart = struct();
stewart.Aa = Aa;
stewart.Ab = Ab;
stewart.Bb = Bb;
stewart.H_tot = opts.H_tot;
end
#+end_src
** computeGeometricalProperties :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :eval no
:HEADER-ARGS:matlab+: :tangle src/computeGeometricalProperties.m
:END:
*** Function description
#+begin_src matlab
function [stewart] = computeGeometricalProperties(stewart, opts_param)
#+end_src
*** Optional Parameters
Default values for opts.
#+begin_src matlab
opts = struct(...
'Jd_pos', [0, 0, 30], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
'Jf_pos', [0, 0, 30] ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
);
#+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
*** Rotation matrices
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
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
*** Jacobian matrices
Compute Jacobian Matrix
#+begin_src matlab
Jd = zeros(6);
for i = 1:6
Jd(i, 1:3) = leg_vectors(i, :);
Jd(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jd_pos), leg_vectors(i, :));
end
stewart.Jd = Jd;
stewart.Jd_inv = inv(Jd);
#+end_src
#+begin_src matlab
Jf = zeros(6);
for i = 1:6
Jf(i, 1:3) = leg_vectors(i, :);
Jf(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jf_pos), leg_vectors(i, :));
end
stewart.Jf = Jf;
stewart.Jf_inv = inv(Jf);
#+end_src
#+begin_src matlab
end
#+end_src
** initializeMechanicalElements :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :eval no
:HEADER-ARGS:matlab+: :tangle src/initializeMechanicalElements.m
:END:
*** Function description
#+begin_src matlab
function [stewart] = initializeMechanicalElements(stewart, opts_param)
#+end_src
*** Optional Parameters
Default values for opts.
#+begin_src matlab
opts = struct(...
'thickness', 10, ... % Thickness of the base and platform [mm]
'density', 1000, ... % 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]
);
#+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
*** 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 = opts.thickness; % Thickness of the Bottom Plate [mm]
#+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
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
We estimate the length of the legs.
#+begin_src matlab
legs = stewart.Ab - stewart.Aa;
Leg.lenght = norm(legs(1,:))/1.5;
#+end_src
Then the shape of the bottom leg is estimated
#+begin_src matlab
Leg.shape.bot = ...
[0 0; ...
Leg.Rbot 0; ...
Leg.Rbot Leg.lenght; ...
Leg.Rtop Leg.lenght; ...
Leg.Rtop 0.2*Leg.lenght; ...
0 0.2*Leg.lenght];
#+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 = stewart.Aa(1, 3) - BP.H; % [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
** initializeSample :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :eval no
:HEADER-ARGS:matlab+: :tangle src/initializeSample.m
:END:
*** Function description
#+begin_src matlab
function [] = initializeSample(opts_param)
#+end_src
*** Optional Parameters
Default values for opts.
#+begin_src matlab
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] ...
);
#+end_src
Populate opts with input parameters
#+begin_src matlab
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
sample.(opt{1}) = opts_param.(opt{1});
end
end
#+end_src
*** Save the Sample structure
#+begin_src matlab
save('./mat/sample.mat', 'sample');
#+end_src
#+begin_src matlab
end
#+end_src
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