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#+TITLE: Stewart Platform - Simscape Model
:DRAWER:
#+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 $\{M\}$: ${}^{M}\bm{O}_{B}$
* =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:
% - 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
** =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
** =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
* 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
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