nass-simscape/org/stewart_platform.org

#+TITLE: Stewart Platform - Simscape Model
#+SETUPFILE: ./setup/org-setup-file.org

* 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 $\{M\}$: ${}^{M}\bm{O}_{B}$

* =initializeStewartPlatform=: Initialize the Stewart Platform structure
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeStewartPlatform.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeStewartPlatform>>

This Matlab function is accessible [[file:../src/initializeStewartPlatform.m][here]].

** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:stewart-frames-position
#+caption: Definition of the position of the frames
[[file:figs/stewart-frames-position.png]]

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = initializeStewartPlatform()
  % initializeStewartPlatform - Initialize the stewart structure
  %
  % Syntax: [stewart] = initializeStewartPlatform(args)
  %
  % Outputs:
  %    - stewart - A structure with the following sub-structures:
  %      - platform_F -
  %      - platform_M -
  %      - joints_F   -
  %      - joints_M   -
  %      - struts_F   -
  %      - struts_M   -
  %      - actuators  -
  %      - geometry   -
  %      - properties -
#+end_src

** Initialize the Stewart structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart = struct();
  stewart.platform_F = struct();
  stewart.platform_M = struct();
  stewart.joints_F   = struct();
  stewart.joints_M   = struct();
  stewart.struts_F   = struct();
  stewart.struts_M   = struct();
  stewart.actuators  = struct();
  stewart.sensors    = struct();
  stewart.sensors.inertial = struct();
  stewart.sensors.force    = struct();
  stewart.sensors.relative = struct();
  stewart.geometry   = struct();
  stewart.kinematics = struct();
#+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]].

** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:stewart-frames-position
#+caption: Definition of the position of the frames
[[file:figs/stewart-frames-position.png]]

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = initializeFramesPositions(stewart, args)
  % initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
  %
  % Syntax: [stewart] = initializeFramesPositions(stewart, 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:
  %        - geometry.H      [1x1] - Total Height of the Stewart Platform [m]
  %        - geometry.FO_M   [3x1] - Position of {M} with respect to {F} [m]
  %        - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m]
  %        - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m]
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.H    (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
      args.MO_B (1,1) double {mustBeNumeric} = 50e-3
  end
#+end_src

** Compute the position of each frame
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  H = args.H; % Total Height of the Stewart Platform [m]

  FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m]

  MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]

  FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.geometry.H      = H;
  stewart.geometry.FO_M   = FO_M;
  stewart.platform_M.MO_B = MO_B;
  stewart.platform_F.FO_A = FO_A;
#+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]].

** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:
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 (see Figure [[fig:joint_position_general]]).

#+begin_src latex :file stewart_bottom_plate.pdf
  \begin{tikzpicture}
    % Internal and external limit
    \draw[fill=white!80!black] (0, 0) circle [radius=3];
    % Circle where the joints are located
    \draw[dashed] (0, 0) circle [radius=2.5];

    % Bullets for the positions of the joints
    \node[] (J1) at ( 80:2.5){$\bullet$};
    \node[] (J2) at (100:2.5){$\bullet$};
    \node[] (J3) at (200:2.5){$\bullet$};
    \node[] (J4) at (220:2.5){$\bullet$};
    \node[] (J5) at (320:2.5){$\bullet$};
    \node[] (J6) at (340:2.5){$\bullet$};

    % Name of the points
    \node[above right] at (J1) {$a_{1}$};
    \node[above left]  at (J2) {$a_{2}$};
    \node[above left]  at (J3) {$a_{3}$};
    \node[right     ]  at (J4) {$a_{4}$};
    \node[left      ]  at (J5) {$a_{5}$};
    \node[above right] at (J6) {$a_{6}$};

    % First 2 angles
    \draw[dashed, ->] (0:1)   arc [start angle=0, end angle=80, radius=1]    node[below right]{$\theta_{1}$};
    \draw[dashed, ->] (0:1.5) arc [start angle=0, end angle=100, radius=1.5] node[left       ]{$\theta_{2}$};

    % Division of 360 degrees by 3
    \draw[dashed] (0, 0) -- ( 80:3.2);
    \draw[dashed] (0, 0) -- (100:3.2);
    \draw[dashed] (0, 0) -- (200:3.2);
    \draw[dashed] (0, 0) -- (220:3.2);
    \draw[dashed] (0, 0) -- (320:3.2);
    \draw[dashed] (0, 0) -- (340:3.2);

    % Radius for the position of the joints
    \draw[<->] (0, 0) --node[near end, above]{$R$} (180:2.5);

    \draw[->] (0, 0) -- ++(3.4, 0) node[above]{$x$};
    \draw[->] (0, 0) -- ++(0, 3.4) node[left]{$y$};
  \end{tikzpicture}
#+end_src

#+name: fig:joint_position_general
#+caption: Position of the joints
#+RESULTS:
[[file:figs/stewart_bottom_plate.png]]

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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:
  %        - platform_F.Fa  [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
  %        - platform_M.Mb  [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.FH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
      args.FR  (1,1) double {mustBeNumeric, mustBePositive} = 115e-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} = 90e-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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  Fa = zeros(3,6);
  Mb = zeros(3,6);
#+end_src

#+begin_src matlab
  for i = 1:6
    Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i));  args.FH];
    Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH];
  end
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.platform_F.Fa = Fa;
  stewart.platform_M.Mb = Mb;
#+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]].

** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:stewart-struts
#+caption: Position and orientation of the struts
[[file:figs/stewart-struts.png]]

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = computeJointsPose(stewart)
  % computeJointsPose -
  %
  % Syntax: [stewart] = computeJointsPose(stewart)
  %
  % Inputs:
  %    - stewart - A structure with the following fields
  %        - platform_F.Fa   [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
  %        - platform_M.Mb   [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
  %        - platform_F.FO_A [3x1] - Position of {A} with respect to {F}
  %        - platform_M.MO_B [3x1] - Position of {B} with respect to {M}
  %        - geometry.FO_M   [3x1] - Position of {M} with respect to {F}
  %
  % Outputs:
  %    - stewart - A structure with the following added fields
  %        - geometry.Aa    [3x6]   - The i'th column is the position of ai with respect to {A}
  %        - geometry.Ab    [3x6]   - The i'th column is the position of bi with respect to {A}
  %        - geometry.Ba    [3x6]   - The i'th column is the position of ai with respect to {B}
  %        - geometry.Bb    [3x6]   - The i'th column is the position of bi with respect to {B}
  %        - geometry.l     [6x1]   - The i'th element is the initial length of strut i
  %        - geometry.As    [3x6]   - The i'th column is the unit vector of strut i expressed in {A}
  %        - geometry.Bs    [3x6]   - The i'th column is the unit vector of strut i expressed in {B}
  %        - struts_F.l     [6x1]   - Length of the Fixed part of the i'th strut
  %        - struts_M.l     [6x1]   - Length of the Mobile part of the i'th strut
  %        - platform_F.FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
  %        - platform_M.MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
#+end_src

** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
  Fa = stewart.platform_F.Fa;

  assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
  Mb = stewart.platform_M.Mb;

  assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
  FO_A = stewart.platform_F.FO_A;

  assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
  MO_B = stewart.platform_M.MO_B;

  assert(isfield(stewart.geometry,   'FO_M'), 'stewart.geometry should have attribute FO_M')
  FO_M = stewart.geometry.FO_M;
#+end_src

** Compute the position of the Joints
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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]);
#+end_src

** Compute the strut length and orientation
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As

  l = vecnorm(Ab - Aa)';
#+end_src

#+begin_src matlab
  Bs = (Bb - Ba)./vecnorm(Bb - Ba);
#+end_src

** Compute the orientation of the Joints
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.geometry.Aa = Aa;
  stewart.geometry.Ab = Ab;
  stewart.geometry.Ba = Ba;
  stewart.geometry.Bb = Bb;
  stewart.geometry.As = As;
  stewart.geometry.Bs = Bs;
  stewart.geometry.l  = l;

  stewart.struts_F.l  = l/2;
  stewart.struts_M.l  = l/2;

  stewart.platform_F.FRa = FRa;
  stewart.platform_M.MRb = MRb;
#+end_src

* =initializeStewartPose=: Determine the initial stroke in each leg to have the wanted pose
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeStewartPose.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeStewartPose>>

This Matlab function is accessible [[file:../src/initializeStewartPose.m][here]].

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = initializeStewartPose(stewart, args)
  % initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose
  %                         It uses the inverse kinematic
  %
  % Syntax: [stewart] = initializeStewartPose(stewart, args)
  %
  % 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:
  %    - stewart - updated Stewart structure with the added fields:
  %      - actuators.Leq [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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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

** Use the Inverse Kinematic function
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  [Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB);
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.actuators.Leq = dLi;
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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:
  %      - platform_F [struct] - structure with the following fields:
  %        - type = 1
  %        - M [1x1] - Fixed Platform Mass [kg]
  %        - I [3x3] - Fixed Platform Inertia matrix [kg*m^2]
  %        - H [1x1] - Fixed Platform Height [m]
  %        - R [1x1] - Fixed Platform Radius [m]
  %      - platform_M [struct] - structure with the following fields:
  %        - M [1x1] - Mobile Platform Mass [kg]
  %        - I [3x3] - Mobile Platform Inertia matrix [kg*m^2]
  %        - H [1x1] - Mobile Platform Height [m]
  %        - R [1x1] - Mobile Platform Radius [m]
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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

** Compute the Inertia matrices of platforms
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  I_F = diag([1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
              1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
              1/2 *args.Fpm * args.Fpr^2]);
#+end_src

#+begin_src matlab
  I_M = diag([1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
              1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
              1/2 *args.Mpm * args.Mpr^2]);
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.platform_F.type = 1;

  stewart.platform_F.I = I_F;
  stewart.platform_F.M = args.Fpm;
  stewart.platform_F.R = args.Fpr;
  stewart.platform_F.H = args.Fph;
#+end_src

#+begin_src matlab
  stewart.platform_M.type = 1;

  stewart.platform_M.I = I_M;
  stewart.platform_M.M = args.Mpm;
  stewart.platform_M.R = args.Mpr;
  stewart.platform_M.H = args.Mph;
#+end_src

* =initializeCylindricalStruts=: Define the 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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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_F [struct] - structure with the following fields:
  %        - M [6x1]   - Mass of the Fixed part of the struts [kg]
  %        - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]
  %        - H [6x1]   - Height of cylinder for the Fixed part of the struts [m]
  %        - R [6x1]   - Radius of cylinder for the Fixed part of the struts [m]
  %      - struts_M [struct] - structure with the following fields:
  %        - M [6x1]   - Mass of the Mobile part of the struts [kg]
  %        - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]
  %        - H [6x1]   - Height of cylinder for the Mobile part of the struts [m]
  %        - R [6x1]   - Radius of cylinder for the Mobile part of the struts [m]
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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

** Compute the properties of the cylindrical struts
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  Fsm = ones(6,1).*args.Fsm;
  Fsh = ones(6,1).*args.Fsh;
  Fsr = ones(6,1).*args.Fsr;

  Msm = ones(6,1).*args.Msm;
  Msh = ones(6,1).*args.Msh;
  Msr = ones(6,1).*args.Msr;
#+end_src

#+begin_src matlab
  I_F = zeros(3, 3, 6); % Inertia of the "fixed" part of the strut
  I_M = zeros(3, 3, 6); % Inertia of the "mobile" part of the strut

  for i = 1:6
    I_F(:,:,i) = diag([1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
                       1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
                       1/2  * Fsm(i) * Fsr(i)^2]);

    I_M(:,:,i) = diag([1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
                       1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
                       1/2  * Msm(i) * Msr(i)^2]);
  end
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.struts_M.type = 1;

  stewart.struts_M.I = I_M;
  stewart.struts_M.M = Msm;
  stewart.struts_M.R = Msr;
  stewart.struts_M.H = Msh;
#+end_src

#+begin_src matlab
  stewart.struts_F.type = 1;

  stewart.struts_F.I = I_F;
  stewart.struts_F.M = Fsm;
  stewart.struts_F.R = Fsr;
  stewart.struts_F.H = Fsh;
#+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]].

** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:piezoelectric_stack
#+attr_html: :width 500px
#+caption: Example of a piezoelectric stach actuator (PI)
[[file:figs/piezoelectric_stack.jpg]]

A simplistic model of such amplified actuator is shown in Figure [[fig:actuator_model_simple]] where:
- $K$ represent the vertical stiffness of the actuator
- $C$ represent the vertical damping of the actuator
- $F$ represents the force applied by the actuator
- $F_{m}$ represents the total measured force
- $v_{m}$ represents the absolute velocity of the top part of the actuator
- $d_{m}$ represents the total relative displacement of the actuator

#+begin_src latex :file actuator_model_simple.pdf
  \begin{tikzpicture}
    \draw (-1, 0) -- (1, 0);

    % Spring, Damper, and Actuator
    \draw[spring]   (-1, 0) -- (-1, 1.5) node[midway, left=0.1]{$K$};
    \draw[damper]   ( 0, 0) -- ( 0, 1.5) node[midway, left=0.2]{$C$};
    \draw[actuator] ( 1, 0) -- ( 1, 1.5) node[midway, left=0.1](F){$F$};

    \node[forcesensor={2}{0.2}] (fsens) at (0, 1.5){};

    \node[left] at (fsens.west) {$F_{m}$};

    \draw[dashed] (1, 0) -- ++(0.4, 0);
    \draw[dashed] (1, 1.7) -- ++(0.4, 0);

    \draw[->] (0, 1.7)node[]{$\bullet$} -- ++(0, 0.5) node[right]{$v_{m}$};

    \draw[<->] (1.4, 0) -- ++(0, 1.7) node[midway, right]{$d_{m}$};
  \end{tikzpicture}
#+end_src

#+name: fig:actuator_model_simple
#+caption: Simple model of an Actuator
#+RESULTS:
[[file:figs/actuator_model_simple.png]]

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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:
  %        - K [6x1] - Stiffness of each strut [N/m]
  %        - C [6x1] - Damping of each strut [N/(m/s)]
  %
  % Outputs:
  %    - stewart - updated Stewart structure with the added fields:
  %      - actuators.type = 1
  %      - actuators.K [6x1] - Stiffness of each strut [N/m]
  %      - actuators.C [6x1] - Damping of each strut [N/(m/s)]
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = 20e6*ones(6,1)
      args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e1*ones(6,1)
  end
#+end_src

** Add Stiffness and Damping properties of each strut
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.actuators.type = 1;

  stewart.actuators.K = args.K;
  stewart.actuators.C = args.C;
#+end_src

* =initializeAmplifiedStrutDynamics=: Add Stiffness and Damping properties of each strut for an amplified piezoelectric actuator
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeAmplifiedStrutDynamics.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeAmplifiedStrutDynamics>>

This Matlab function is accessible [[file:../src/initializeAmplifiedStrutDynamics.m][here]].

** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

An amplified piezoelectric actuator is shown in Figure [[fig:cedrat_apa95ml]].

#+name: fig:cedrat_apa95ml
#+attr_html: :width 500px
#+caption: Example of an Amplified piezoelectric actuator with an integrated displacement sensor (Cedrat Technologies)
[[file:figs/amplified_piezo_with_displacement_sensor.jpg]]

A simplistic model of such amplified actuator is shown in Figure [[fig:amplified_piezo_model]] where:
- $K_{r}$ represent the vertical stiffness when the piezoelectric stack is removed
- $K_{a}$ is the vertical stiffness contribution of the piezoelectric stack
- $F_{i}$ represents the part of the piezoelectric stack that is used as a force actuator
- $F_{m,i}$ represents the remaining part of the piezoelectric stack that is used as a force sensor
- $v_{m,i}$ represents the absolute velocity of the top part of the actuator
- $d_{m,i}$ represents the total relative displacement of the actuator

#+begin_src latex :file iff_1dof.pdf
  \begin{tikzpicture}
    % Ground
    \draw (-1.2, 0) -- (1, 0);

    % Mass
    \draw (-1.2, 1.4) -- ++(2.2, 0);
    \node[forcesensor={0.4}{0.4}] (fsensn) at (0, 1){};
    \draw[] (-0.4, 1) -- (0.4, 1);
    \node[right] at (fsensn.east) {$F_{m}$};

    % Spring, Damper, and Actuator
    \draw[spring] (-0.4, 0) -- (-0.4, 1) node[midway, right=0.1]{$K_{a}$};
    \draw[actuator={0.4}{0.2}] (0.4, 0) -- (0.4, 1) node[midway, right=0.1]{$F$};

    \draw[spring] (-1, 0) -- (-1, 1.4) node[midway, left=0.1]{$K_{r}$};

    \draw[dashed] (1, 0) -- ++(0.4, 0);

    \draw[dashed] (1, 1.4) -- ++(0.4, 0);

    \draw[->] (0, 1.4)node[]{$\bullet$} -- ++(0, 0.5) node[right]{$v_{m}$};

    \draw[<->] (1.4, 0) -- ++(0, 1.4) node[midway, right]{$d_{m}$};
  \end{tikzpicture}
#+end_src

#+name: fig:amplified_piezo_model
#+caption: Model of an amplified actuator
#+RESULTS:
[[file:figs/iff_1dof.png]]

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = initializeAmplifiedStrutDynamics(stewart, args)
  % initializeAmplifiedStrutDynamics - Add Stiffness and Damping properties of each strut
  %
  % Syntax: [stewart] = initializeAmplifiedStrutDynamics(args)
  %
  % Inputs:
  %    - args - Structure with the following fields:
  %        - Ka [6x1] - Vertical stiffness contribution of the piezoelectric stack [N/m]
  %        - Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
  %        - Kr [6x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
  %        - Cr [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
  %
  % Outputs:
  %    - stewart - updated Stewart structure with the added fields:
  %      - actuators.type = 2
  %      - actuators.K   [6x1] - Total Stiffness of each strut [N/m]
  %      - actuators.C   [6x1] - Total Damping of each strut [N/(m/s)]
  %      - actuators.Ka [6x1] - Vertical stiffness contribution of the piezoelectric stack [N/m]
  %      - actuators.Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
  %      - actuators.Kr [6x1] - Vertical stiffness when the piezoelectric stack is removed [N/m]
  %      - actuators.Cr [6x1] - Vertical damping when the piezoelectric stack is removed [N/(m/s)]
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.Kr (6,1) double {mustBeNumeric, mustBeNonnegative} = 5e6*ones(6,1)
      args.Cr (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
      args.Ka (6,1) double {mustBeNumeric, mustBeNonnegative} = 15e6*ones(6,1)
      args.Ca (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
  end
#+end_src

** Compute the total stiffness and damping
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  K = args.Ka + args.Kr;
  C = args.Ca + args.Cr;
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.actuators.type = 2;

  stewart.actuators.Ka = args.Ka;
  stewart.actuators.Ca = args.Ca;

  stewart.actuators.Kr = args.Kr;
  stewart.actuators.Cr = args.Cr;

  stewart.actuators.K = K;
  stewart.actuators.C = K;
#+end_src

* =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeJointDynamics.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeJointDynamics>>

This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here]].

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = initializeJointDynamics(stewart, args)
  % initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints
  %
  % Syntax: [stewart] = initializeJointDynamics(args)
  %
  % Inputs:
  %    - args - Structure with the following fields:
  %        - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p'
  %        - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p'
  %        - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad]
  %        - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]
  %        - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]
  %        - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]
  %        - Kz_M [6x1] - Translation (Tz) Stiffness for each top joints [N/m]
  %        - Cz_M [6x1] - Translation (Tz) Damping of each top joint [N/m]
  %        - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]
  %        - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]
  %        - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]
  %        - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)]
  %        - Kz_F [6x1] - Translation (Tz) Stiffness for each bottom joints [N/m]
  %        - Cz_F [6x1] - Translation (Tz) Damping of each bottom joint [N/m]
  %
  % Outputs:
  %    - stewart - updated Stewart structure with the added fields:
  %      - stewart.joints_F and stewart.joints_M:
  %        - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect)
  %        - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m]
  %        - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad]
  %        - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad]
  %        - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)]
  %        - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)]
  %        - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)]
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof'})} = 'universal'
      args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'spherical_3dof'})} = 'spherical'
      args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
      args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
      args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
      args.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
      args.Kz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
      args.Cz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
      args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
      args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
      args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
      args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
      args.Kz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
      args.Cz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
  end
#+end_src

** Add Actuator Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type_F
    case 'universal'
      stewart.joints_F.type = 1;
    case 'spherical'
      stewart.joints_F.type = 2;
    case 'universal_p'
      stewart.joints_F.type = 3;
    case 'spherical_p'
      stewart.joints_F.type = 4;
    case 'universal_3dof'
      stewart.joints_F.type = 5;
  end

  switch args.type_M
    case 'universal'
      stewart.joints_M.type = 1;
    case 'spherical'
      stewart.joints_M.type = 2;
    case 'universal_p'
      stewart.joints_M.type = 3;
    case 'spherical_p'
      stewart.joints_M.type = 4;
    case 'spherical_3dof'
      stewart.joints_M.type = 6;
  end
#+end_src

** Add Stiffness and Damping in Translation of each strut
:PROPERTIES:
:UNNUMBERED: t
:END:
Translation Stiffness
#+begin_src matlab
  stewart.joints_M.Kx = zeros(6,1);
  stewart.joints_M.Ky = zeros(6,1);
  stewart.joints_M.Kz = args.Kz_M;

  stewart.joints_F.Kx = zeros(6,1);
  stewart.joints_F.Ky = zeros(6,1);
  stewart.joints_F.Kz = args.Kz_F;
#+end_src

Translation Damping
#+begin_src matlab
  stewart.joints_M.Cx = zeros(6,1);
  stewart.joints_M.Cy = zeros(6,1);
  stewart.joints_M.Cz = args.Cz_M;

  stewart.joints_F.Cx = zeros(6,1);
  stewart.joints_F.Cy = zeros(6,1);
  stewart.joints_F.Cz = args.Cz_F;
#+end_src

** Add Stiffness and Damping in Rotation of each strut
:PROPERTIES:
:UNNUMBERED: t
:END:
Rotational Stiffness
#+begin_src matlab
  stewart.joints_M.Kf = args.Kf_M;
  stewart.joints_M.Kt = args.Kf_M;

  stewart.joints_F.Kf = args.Kf_F;
  stewart.joints_F.Kt = args.Kf_F;
#+end_src


Rotational Damping
#+begin_src matlab
  stewart.joints_M.Cf = args.Cf_M;
  stewart.joints_M.Ct = args.Cf_M;

  stewart.joints_F.Cf = args.Cf_F;
  stewart.joints_F.Ct = args.Cf_F;
#+end_src

* =initializeInertialSensor=: Initialize the inertial sensor in each strut
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeInertialSensor.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeInertialSensor>>

This Matlab function is accessible [[file:../src/initializeInertialSensor.m][here]].

** Geophone - Working Principle
:PROPERTIES:
:UNNUMBERED: t
:END:
From the schematic of the Z-axis geophone shown in Figure [[fig:z_axis_geophone]], we can write the transfer function from the support velocity $\dot{w}$ to the relative velocity of the inertial mass $\dot{d}$:
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
with:
- $\omega_0 = \sqrt{\frac{k}{m}}$
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$

#+name: fig:z_axis_geophone
#+caption: Schematic of a Z-Axis geophone
[[file:figs/inertial_sensor.png]]

We see that at frequencies above $\omega_0$:
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]

And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.

We generally want to have the smallest resonant frequency $\omega_0$ to measure low frequency absolute velocity, however there is a trade-off between $\omega_0$ and the mass of the inertial mass.

** Accelerometer - Working Principle
:PROPERTIES:
:UNNUMBERED: t
:END:
From the schematic of the Z-axis accelerometer shown in Figure [[fig:z_axis_accelerometer]], we can write the transfer function from the support acceleration $\ddot{w}$ to the relative position of the inertial mass $d$:
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
with:
- $\omega_0 = \sqrt{\frac{k}{m}}$
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$

#+name: fig:z_axis_accelerometer
#+caption: Schematic of a Z-Axis geophone
[[file:figs/inertial_sensor.png]]

We see that at frequencies below $\omega_0$:
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]

And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.

Note that there is trade-off between:
- the highest measurable acceleration $\omega_0$
- the sensitivity of the accelerometer which is equal to $-\frac{1}{{\omega_0}^2}$

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = initializeInertialSensor(stewart, args)
  % initializeInertialSensor - Initialize the inertial sensor in each strut
  %
  % Syntax: [stewart] = initializeInertialSensor(args)
  %
  % Inputs:
  %    - args - Structure with the following fields:
  %        - type       - 'geophone', 'accelerometer', 'none'
  %        - mass [1x1] - Weight of the inertial mass [kg]
  %        - freq [1x1] - Cutoff frequency [Hz]
  %
  % Outputs:
  %    - stewart - updated Stewart structure with the added fields:
  %      - stewart.sensors.inertial
  %        - type    - 1 (geophone), 2 (accelerometer), 3 (none)
  %        - K [1x1] - Stiffness [N/m]
  %        - C [1x1] - Damping [N/(m/s)]
  %        - M [1x1] - Inertial Mass [kg]
  %        - G [1x1] - Gain
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.type       char   {mustBeMember(args.type,{'geophone', 'accelerometer', 'none'})} = 'none'
      args.mass (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e-2
      args.freq (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e3
  end
#+end_src

** Compute the properties of the sensor
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  sensor = struct();

  switch args.type
    case 'geophone'
      sensor.type = 1;

      sensor.M = args.mass;
      sensor.K = sensor.M * (2*pi*args.freq)^2;
      sensor.C = 2*sqrt(sensor.M * sensor.K);
    case 'accelerometer'
      sensor.type = 2;

      sensor.M = args.mass;
      sensor.K = sensor.M * (2*pi*args.freq)^2;
      sensor.C = 2*sqrt(sensor.M * sensor.K);
      sensor.G = -sensor.K/sensor.M;
    case 'none'
      sensor.type = 3;
  end
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.sensors.inertial = sensor;
#+end_src

* =displayArchitecture=: 3D plot of the Stewart platform architecture
:PROPERTIES:
:header-args:matlab+: :tangle ../src/displayArchitecture.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:displayArchitecture>>

This Matlab function is accessible [[file:../src/displayArchitecture.m][here]].

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [] = displayArchitecture(stewart, args)
  % displayArchitecture - 3D plot of the Stewart platform architecture
  %
  % Syntax: [] = displayArchitecture(args)
  %
  % Inputs:
  %    - stewart
  %    - args - Structure with 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}
  %        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
  %        - F_color [color] - Color used for the Fixed elements
  %        - M_color [color] - Color used for the Mobile elements
  %        - L_color [color] - Color used for the Legs elements
  %        - frames    [true/false] - Display the Frames
  %        - legs      [true/false] - Display the Legs
  %        - joints    [true/false] - Display the Joints
  %        - labels    [true/false] - Display the Labels
  %        - platforms [true/false] - Display the Platforms
  %        - views     ['all', 'xy', 'yz', 'xz', 'default'] -
  %
  % Outputs:
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
      args.ARB (3,3) double {mustBeNumeric} = eye(3)
      args.F_color = [0 0.4470 0.7410]
      args.M_color = [0.8500 0.3250 0.0980]
      args.L_color = [0 0 0]
      args.frames    logical {mustBeNumericOrLogical} = true
      args.legs      logical {mustBeNumericOrLogical} = true
      args.joints    logical {mustBeNumericOrLogical} = true
      args.labels    logical {mustBeNumericOrLogical} = true
      args.platforms logical {mustBeNumericOrLogical} = true
      args.views     char    {mustBeMember(args.views,{'all', 'xy', 'xz', 'yz', 'default'})} = 'default'
  end
#+end_src

** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
  FO_A = stewart.platform_F.FO_A;

  assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
  MO_B = stewart.platform_M.MO_B;

  assert(isfield(stewart.geometry, 'H'),   'stewart.geometry should have attribute H')
  H = stewart.geometry.H;

  assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
  Fa = stewart.platform_F.Fa;

  assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
  Mb = stewart.platform_M.Mb;
#+end_src


** Figure Creation, Frames and Homogeneous transformations
:PROPERTIES:
:UNNUMBERED: t
:END:

The reference frame of the 3d plot corresponds to the frame $\{F\}$.
#+begin_src matlab
  if ~strcmp(args.views, 'all')
    figure;
  else
    f = figure('visible', 'off');
  end

  hold on;
#+end_src

We first compute homogeneous matrices that will be useful to position elements on the figure where the reference frame is $\{F\}$.
#+begin_src matlab
  FTa = [eye(3), FO_A; ...
         zeros(1,3), 1];
  ATb = [args.ARB, args.AP; ...
         zeros(1,3), 1];
  BTm = [eye(3), -MO_B; ...
         zeros(1,3), 1];

  FTm = FTa*ATb*BTm;
#+end_src

Let's define a parameter that define the length of the unit vectors used to display the frames.
#+begin_src matlab
  d_unit_vector = H/4;
#+end_src

Let's define a parameter used to position the labels with respect to the center of the element.
#+begin_src matlab
  d_label = H/20;
#+end_src

** Fixed Base elements
:PROPERTIES:
:UNNUMBERED: t
:END:
Let's first plot the frame $\{F\}$.
#+begin_src matlab
  Ff = [0, 0, 0];
  if args.frames
    quiver3(Ff(1)*ones(1,3), Ff(2)*ones(1,3), Ff(3)*ones(1,3), ...
            [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)

    if args.labels
      text(Ff(1) + d_label, ...
          Ff(2) + d_label, ...
          Ff(3) + d_label, '$\{F\}$', 'Color', args.F_color);
    end
  end
#+end_src

Now plot the frame $\{A\}$ fixed to the Base.
#+begin_src matlab
  if args.frames
    quiver3(FO_A(1)*ones(1,3), FO_A(2)*ones(1,3), FO_A(3)*ones(1,3), ...
            [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)

    if args.labels
      text(FO_A(1) + d_label, ...
           FO_A(2) + d_label, ...
           FO_A(3) + d_label, '$\{A\}$', 'Color', args.F_color);
    end
  end
#+end_src

Let's then plot the circle corresponding to the shape of the Fixed base.
#+begin_src matlab
  if args.platforms && stewart.platform_F.type == 1
    theta = [0:0.01:2*pi+0.01]; % Angles [rad]
    v = null([0; 0; 1]'); % Two vectors that are perpendicular to the circle normal
    center = [0; 0; 0]; % Center of the circle
    radius = stewart.platform_F.R; % Radius of the circle [m]

    points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));

    plot3(points(1,:), ...
          points(2,:), ...
          points(3,:), '-', 'Color', args.F_color);
  end
#+end_src

Let's now plot the position and labels of the Fixed Joints
#+begin_src matlab
  if args.joints
    scatter3(Fa(1,:), ...
             Fa(2,:), ...
             Fa(3,:), 'MarkerEdgeColor', args.F_color);
    if args.labels
      for i = 1:size(Fa,2)
        text(Fa(1,i) + d_label, ...
             Fa(2,i), ...
             Fa(3,i), sprintf('$a_{%i}$', i), 'Color', args.F_color);
      end
    end
  end
#+end_src

** Mobile Platform elements
:PROPERTIES:
:UNNUMBERED: t
:END:

Plot the frame $\{M\}$.
#+begin_src matlab
  Fm = FTm*[0; 0; 0; 1]; % Get the position of frame {M} w.r.t. {F}

  if args.frames
    FM_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
    quiver3(Fm(1)*ones(1,3), Fm(2)*ones(1,3), Fm(3)*ones(1,3), ...
            FM_uv(1,1:3), FM_uv(2,1:3), FM_uv(3,1:3), '-', 'Color', args.M_color)

    if args.labels
      text(Fm(1) + d_label, ...
           Fm(2) + d_label, ...
           Fm(3) + d_label, '$\{M\}$', 'Color', args.M_color);
    end
  end
#+end_src

Plot the frame $\{B\}$.
#+begin_src matlab
  FB = FO_A + args.AP;

  if args.frames
    FB_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
    quiver3(FB(1)*ones(1,3), FB(2)*ones(1,3), FB(3)*ones(1,3), ...
            FB_uv(1,1:3), FB_uv(2,1:3), FB_uv(3,1:3), '-', 'Color', args.M_color)

    if args.labels
      text(FB(1) - d_label, ...
           FB(2) + d_label, ...
           FB(3) + d_label, '$\{B\}$', 'Color', args.M_color);
    end
  end
#+end_src

Let's then plot the circle corresponding to the shape of the Mobile platform.
#+begin_src matlab
  if args.platforms && stewart.platform_M.type == 1
    theta = [0:0.01:2*pi+0.01]; % Angles [rad]
    v = null((FTm(1:3,1:3)*[0;0;1])'); % Two vectors that are perpendicular to the circle normal
    center = Fm(1:3); % Center of the circle
    radius = stewart.platform_M.R; % Radius of the circle [m]

    points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));

    plot3(points(1,:), ...
          points(2,:), ...
          points(3,:), '-', 'Color', args.M_color);
  end
#+end_src

Plot the position and labels of the rotation joints fixed to the mobile platform.
#+begin_src matlab
  if args.joints
    Fb = FTm*[Mb;ones(1,6)];

    scatter3(Fb(1,:), ...
             Fb(2,:), ...
             Fb(3,:), 'MarkerEdgeColor', args.M_color);

    if args.labels
      for i = 1:size(Fb,2)
        text(Fb(1,i) + d_label, ...
             Fb(2,i), ...
             Fb(3,i), sprintf('$b_{%i}$', i), 'Color', args.M_color);
      end
    end
  end
#+end_src

** Legs
:PROPERTIES:
:UNNUMBERED: t
:END:
Plot the legs connecting the joints of the fixed base to the joints of the mobile platform.
#+begin_src matlab
  if args.legs
    for i = 1:6
      plot3([Fa(1,i), Fb(1,i)], ...
            [Fa(2,i), Fb(2,i)], ...
            [Fa(3,i), Fb(3,i)], '-', 'Color', args.L_color);

      if args.labels
        text((Fa(1,i)+Fb(1,i))/2 + d_label, ...
             (Fa(2,i)+Fb(2,i))/2, ...
             (Fa(3,i)+Fb(3,i))/2, sprintf('$%i$', i), 'Color', args.L_color);
      end
    end
  end
#+end_src

** Figure parameters
#+begin_src matlab
  switch args.views
    case 'default'
        view([1 -0.6 0.4]);
    case 'xy'
        view([0 0 1]);
    case 'xz'
        view([0 -1 0]);
    case 'yz'
        view([1 0 0]);
  end
  axis equal;
  axis off;
#+end_src

** Subplots
#+begin_src matlab
  if strcmp(args.views, 'all')
    hAx = findobj('type', 'axes');

    figure;
    s1 = subplot(2,2,1);
    copyobj(get(hAx(1), 'Children'), s1);
    view([0 0 1]);
    axis equal;
    axis off;
    title('Top')

    s2 = subplot(2,2,2);
    copyobj(get(hAx(1), 'Children'), s2);
    view([1 -0.6 0.4]);
    axis equal;
    axis off;

    s3 = subplot(2,2,3);
    copyobj(get(hAx(1), 'Children'), s3);
    view([1 0 0]);
    axis equal;
    axis off;
    title('Front')

    s4 = subplot(2,2,4);
    copyobj(get(hAx(1), 'Children'), s4);
    view([0 -1 0]);
    axis equal;
    axis off;
    title('Side')

    close(f);
  end
#+end_src


* =describeStewartPlatform=: Display some text describing the current defined Stewart Platform
:PROPERTIES:
:header-args:matlab+: :tangle ../src/describeStewartPlatform.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:describeStewartPlatform>>

This Matlab function is accessible [[file:../src/describeStewartPlatform.m][here]].

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [] = describeStewartPlatform(stewart)
  % describeStewartPlatform - Display some text describing the current defined Stewart Platform
  %
  % Syntax: [] = describeStewartPlatform(args)
  %
  % Inputs:
  %    - stewart
  %
  % Outputs:
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
  end
#+end_src

** Geometry
#+begin_src matlab
  fprintf('GEOMETRY:\n')
  fprintf('- The height between the fixed based and the top platform is %.3g [mm].\n', 1e3*stewart.geometry.H)

  if stewart.platform_M.MO_B(3) > 0
    fprintf('- Frame {A} is located %.3g [mm] above the top platform.\n',  1e3*stewart.platform_M.MO_B(3))
  else
    fprintf('- Frame {A} is located %.3g [mm] below the top platform.\n', - 1e3*stewart.platform_M.MO_B(3))
  end

  fprintf('- The initial length of the struts are:\n')
  fprintf('\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n', 1e3*stewart.geometry.l)
  fprintf('\n')
#+end_src

** Actuators
#+begin_src matlab
  fprintf('ACTUATORS:\n')
  if stewart.actuators.type == 1
      fprintf('- The actuators are classical.\n')
      fprintf('- The Stiffness and Damping of each actuators is:\n')
      fprintf('\t k = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.K(1), stewart.actuators.C(1))
  elseif stewart.actuators.type == 2
      fprintf('- The actuators are mechanicaly amplified.\n')
      fprintf('- The vertical stiffness and damping contribution of the piezoelectric stack is:\n')
      fprintf('\t ka = %.0e [N/m] \t ca = %.0e [N/(m/s)]\n', stewart.actuators.Ka(1), stewart.actuators.Ca(1))
      fprintf('- Vertical stiffness when the piezoelectric stack is removed is:\n')
      fprintf('\t kr = %.0e [N/m] \t cr = %.0e [N/(m/s)]\n', stewart.actuators.Kr(1), stewart.actuators.Cr(1))
  end
  fprintf('\n')
#+end_src

** Joints
#+begin_src matlab
  fprintf('JOINTS:\n')
#+end_src

Type of the joints on the fixed base.
#+begin_src matlab
  switch stewart.joints_F.type
    case 1
      fprintf('- The joints on the fixed based are universal joints\n')
    case 2
      fprintf('- The joints on the fixed based are spherical joints\n')
    case 3
      fprintf('- The joints on the fixed based are perfect universal joints\n')
    case 4
      fprintf('- The joints on the fixed based are perfect spherical joints\n')
  end
#+end_src

Type of the joints on the mobile platform.
#+begin_src matlab
  switch stewart.joints_M.type
    case 1
      fprintf('- The joints on the mobile based are universal joints\n')
    case 2
      fprintf('- The joints on the mobile based are spherical joints\n')
    case 3
      fprintf('- The joints on the mobile based are perfect universal joints\n')
    case 4
      fprintf('- The joints on the mobile based are perfect spherical joints\n')
  end
#+end_src

Position of the fixed joints
#+begin_src matlab
  fprintf('- The position of the joints on the fixed based with respect to {F} are (in [mm]):\n')
  fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_F.Fa)
#+end_src

Position of the mobile joints
#+begin_src matlab
  fprintf('- The position of the joints on the mobile based with respect to {M} are (in [mm]):\n')
  fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_M.Mb)
  fprintf('\n')
#+end_src

** Kinematics
#+begin_src matlab
  fprintf('KINEMATICS:\n')

  if isfield(stewart.kinematics, 'K')
    fprintf('- The Stiffness matrix K is (in [N/m]):\n')
    fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.K)
  end

  if isfield(stewart.kinematics, 'C')
    fprintf('- The Damping matrix C is (in [m/N]):\n')
    fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.C)
  end
#+end_src

* =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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
  %        - geometry.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:
  %        - platform_F.Fa  [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
  %        - platform_M.Mb  [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
#+end_src

** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:
#+name: fig:cubic-configuration-definition
#+caption: Cubic Configuration
[[file:figs/cubic-configuration-definition.png]]

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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, mustBeNonnegative} = 15e-3
      args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
  end
#+end_src

** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  assert(isfield(stewart.geometry, 'H'),   'stewart.geometry should have attribute H')
  H = stewart.geometry.H;
#+end_src

** Position of the Cube
:PROPERTIES:
:UNNUMBERED: t
:END:
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
:PROPERTIES:
:UNNUMBERED: t
:END:
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 = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
  Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.platform_F.Fa = Fa;
  stewart.platform_M.Mb = Mb;
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = computeJacobian(stewart)
  % computeJacobian -
  %
  % Syntax: [stewart] = computeJacobian(stewart)
  %
  % Inputs:
  %    - stewart - With at least the following fields:
  %      - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}
  %      - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}
  %      - actuators.K [6x1] - Total stiffness of the actuators
  %
  % Outputs:
  %    - stewart - With the 3 added field:
  %        - kinematics.J [6x6] - The Jacobian Matrix
  %        - kinematics.K [6x6] - The Stiffness Matrix
  %        - kinematics.C [6x6] - The Compliance Matrix
#+end_src

** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  assert(isfield(stewart.geometry, 'As'),   'stewart.geometry should have attribute As')
  As = stewart.geometry.As;

  assert(isfield(stewart.geometry, 'Ab'),   'stewart.geometry should have attribute Ab')
  Ab = stewart.geometry.Ab;

  assert(isfield(stewart.actuators, 'K'),   'stewart.actuators should have attribute K')
  Ki = stewart.actuators.K;
#+end_src


** Compute Jacobian Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  J = [As' , cross(Ab, As)'];
#+end_src

** Compute Stiffness Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  K = J'*diag(Ki)*J;
#+end_src

** Compute Compliance Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  C = inv(K);
#+end_src

** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.kinematics.J = J;
  stewart.kinematics.K = K;
  stewart.kinematics.C = C;
#+end_src


* =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]].

** Theory
:PROPERTIES:
:UNNUMBERED: t
:END:
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.

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
  %        - geometry.Aa   [3x6] - The positions ai expressed in {A}
  %        - geometry.Bb   [3x6] - The positions bi expressed in {B}
  %        - geometry.l    [6x1] - Length of each strut
  %    - 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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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

** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  assert(isfield(stewart.geometry, 'Aa'),   'stewart.geometry should have attribute Aa')
  Aa = stewart.geometry.Aa;

  assert(isfield(stewart.geometry, 'Bb'),   'stewart.geometry should have attribute Bb')
  Bb = stewart.geometry.Bb;

  assert(isfield(stewart.geometry, 'l'),   'stewart.geometry should have attribute l')
  l = stewart.geometry.l;
#+end_src


** Compute
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));
#+end_src

#+begin_src matlab
  dLi = Li-l;
#+end_src

* =forwardKinematicsApprox=: Compute the Approximate 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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
  %        - kinematics.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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
  end
#+end_src

** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  assert(isfield(stewart.kinematics, 'J'),   'stewart.kinematics should have attribute J')
  J = stewart.kinematics.J;
#+end_src

** Computation
:PROPERTIES:
:UNNUMBERED: t
:END:
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 = 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