stewart-simscape/simscape-model.org

#+TITLE: Stewart Platform - Simscape Model
:DRAWER:
#+STARTUP: overview

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#+LATEX_CLASS: cleanreport
#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
#+LaTeX_HEADER: \usepackage{svg}
#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}

#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments no
#+PROPERTY: header-args:matlab+ :exports bode
#+PROPERTY: header-args:matlab+ :eval no
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :tangle src/initializeHexapod.m
:END:

* Function description and arguments
The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
#+begin_src matlab
  function [stewart] = initializeHexapod(opts_param)
#+end_src

Default values for opts.
#+begin_src matlab
  opts = struct(...
      'height',  90,    ... % Height of the platform [mm]
      'density', 8000,  ... % Density of the material used for the hexapod [kg/m3]
      'k_ax',    1e8,   ... % Stiffness of each actuator [N/m]
      'c_ax',    1000,   ... % Damping of each actuator [N/(m/s)]
      'stroke',  50e-6, ... % Maximum stroke of each actuator [m]
      'name',    'stewart' ... % Name of the file
      );
#+end_src

Populate opts with input parameters
#+begin_src matlab
  if exist('opts_param','var')
      for opt = fieldnames(opts_param)'
          opts.(opt{1}) = opts_param.(opt{1});
      end
  end
#+end_src

* Initialization of the stewart structure
We initialize the Stewart structure
#+begin_src matlab
  stewart = struct();
#+end_src

And we defined its total height.
#+begin_src matlab
  stewart.H = opts.height; % [mm]
#+end_src

* Bottom Plate
#+name: fig:stewart_bottom_plate
#+caption: Schematic of the bottom plates with all the parameters
[[file:./figs/stewart_bottom_plate.png]]


The bottom plate structure is initialized.
#+begin_src matlab
  BP = struct();
#+end_src

We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
#+begin_src matlab
  BP.Rint = 0;   % Internal Radius [mm]
  BP.Rext = 150; % External Radius [mm]
#+end_src

We define its thickness.
#+begin_src matlab
  BP.H = 10; % Thickness of the Bottom Plate [mm]
#+end_src

At which radius legs will be fixed and with that angle offset.
#+begin_src matlab
  BP.Rleg  = 100; % Radius where the legs articulations are positionned [mm]
  BP.alpha = 10;  % Angle Offset [deg]
#+end_src

We defined the density of the material of the bottom plate.
#+begin_src matlab
  BP.density = opts.density; % Density of the material [kg/m3]
#+end_src

And its color.
#+begin_src matlab
  BP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src

Then the profile of the bottom plate is computed and will be used by Simscape
#+begin_src matlab
  BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
#+end_src

The structure is added to the stewart structure
#+begin_src matlab
  stewart.BP = BP;
#+end_src

* Top Plate
The top plate structure is initialized.
#+begin_src matlab
  TP = struct();
#+end_src

We defined the internal and external radius of the top plate.
#+begin_src matlab
  TP.Rint = 0;   % [mm]
  TP.Rext = 100; % [mm]
#+end_src

The thickness of the top plate.
#+begin_src matlab
  TP.H = 10; % [mm]
#+end_src

At which radius and angle are fixed the legs.
#+begin_src matlab
  TP.Rleg   = 100; % Radius where the legs articulations are positionned [mm]
  TP.alpha  = 20; % Angle [deg]
  TP.dalpha = 0; % Angle Offset from 0 position [deg]
#+end_src

The density of its material.
#+begin_src matlab
  TP.density = opts.density; % Density of the material [kg/m3]
#+end_src

Its color.
#+begin_src matlab
  TP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src

Then the shape of the top plate is computed
#+begin_src matlab
  TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
#+end_src

The structure is added to the stewart structure
#+begin_src matlab
  stewart.TP  = TP;
#+end_src

* Legs
#+name: fig:stewart_legs
#+caption: Schematic for the legs of the Stewart platform
[[file:./figs/stewart_legs.png]]


The leg structure is initialized.
#+begin_src matlab
  Leg = struct();
#+end_src

The maximum Stroke of each leg is defined.
#+begin_src matlab
  Leg.stroke = opts.stroke; % [m]
#+end_src

The stiffness and damping of each leg are defined
#+begin_src matlab
  Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
  Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
#+end_src

The radius of the legs are defined
#+begin_src matlab
  Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
  Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
#+end_src

The density of its material.
#+begin_src matlab
  Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
#+end_src

Its color.
#+begin_src matlab
  Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
#+end_src

The radius of spheres representing the ball joints are defined.
#+begin_src matlab
  Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
#+end_src

The structure is added to the stewart structure
#+begin_src matlab
  stewart.Leg = Leg;
#+end_src

* Ball Joints
#+name: fig:stewart_ball_joints
#+caption: Schematic of the support for the ball joints
[[file:./figs/stewart_ball_joints.png]]

=SP= is the structure representing the support for the ball joints at the extremity of each leg.

The =SP= structure is initialized.
#+begin_src matlab
  SP = struct();
#+end_src

We can define its rotational stiffness and damping. For now, we use perfect joints.
#+begin_src matlab
  SP.k = 0; % [N*m/deg]
  SP.c = 0; % [N*m/deg]
#+end_src

Its height is defined
#+begin_src matlab
  SP.H = 15; % [mm]
#+end_src

Its radius is based on the radius on the sphere at the end of the legs.
#+begin_src matlab
  SP.R = Leg.R; % [mm]
#+end_src

#+begin_src matlab
  SP.section = [0    SP.H-SP.R;
                0    0;
                SP.R 0;
                SP.R SP.H];
#+end_src

The density of its material is defined.
#+begin_src matlab
  SP.density = opts.density; % [kg/m^3]
#+end_src

Its color is defined.
#+begin_src matlab
  SP.color = [0.7 0.7 0.7]; % [RGB]
#+end_src

The structure is added to the Hexapod structure
#+begin_src matlab
  stewart.SP  = SP;
#+end_src

* More parameters are initialized
#+begin_src matlab
  stewart = initializeParameters(stewart);
#+end_src

* Save the Stewart Structure
#+begin_src matlab
  save('./mat/stewart.mat', 'stewart')
#+end_src

* initializeParameters Function                                    :noexport:
  :PROPERTIES:
  :HEADER-ARGS:matlab+: :tangle no
  :END:
#+begin_src matlab
  function [stewart] = initializeParameters(stewart)
#+end_src

Computation of the position of the connection points on the base and moving platform
We first initialize =pos_base= corresponding to $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and =pos_top= corresponding to $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
  stewart.pos_base = zeros(6, 3);
  stewart.pos_top = zeros(6, 3);
#+end_src

We estimate the height between the ball joints of the bottom platform and of the top platform.
#+begin_src matlab
  height = stewart.H - stewart.BP.H - stewart.TP.H - 2*stewart.SP.H; % [mm]
#+end_src

#+begin_src matlab
      for i = 1:3
          % base points
          angle_m_b = 120*(i-1) - stewart.BP.alpha;
          angle_p_b = 120*(i-1) + stewart.BP.alpha;

          stewart.pos_base(2*i-1,:) = [stewart.BP.Rleg*cos(angle_m_b), stewart.BP.Rleg*sin(angle_m_b), 0.0];
          stewart.pos_base(2*i,:)   = [stewart.BP.Rleg*cos(angle_p_b), stewart.BP.Rleg*sin(angle_p_b), 0.0];

          % top points
          angle_m_t = 120*(i-1) - stewart.TP.alpha + stewart.TP.dalpha;
          angle_p_t = 120*(i-1) + stewart.TP.alpha + stewart.TP.dalpha;

          stewart.pos_top(2*i-1,:) = [stewart.TP.Rleg*cos(angle_m_t), stewart.TP.Rleg*sin(angle_m_t), height];
          stewart.pos_top(2*i,:) = [stewart.TP.Rleg*cos(angle_p_t), stewart.TP.Rleg*sin(angle_p_t), height];
      end

      % permute pos_top points so that legs are end points of base and top points
      stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
      stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
#+end_src

leg vectors
#+begin_src matlab
      legs = stewart.pos_top - stewart.pos_base;
      leg_length = zeros(6, 1);
      leg_vectors = zeros(6, 3);
      for i = 1:6
          leg_length(i) = norm(legs(i,:));
          leg_vectors(i,:)  = legs(i,:) / leg_length(i);
      end

      stewart.Leg.lenght = 1000*leg_length(1)/1.5;
      stewart.Leg.shape.bot = [0 0; ...
                          stewart.Leg.rad.bottom 0; ...
                          stewart.Leg.rad.bottom stewart.Leg.lenght; ...
                          stewart.Leg.rad.top stewart.Leg.lenght; ...
                          stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
                          0 0.2*stewart.Leg.lenght];
#+end_src

Calculate revolute and cylindrical axes
#+begin_src matlab
      rev1 = zeros(6, 3);
      rev2 = zeros(6, 3);
      cyl1 = zeros(6, 3);
      for i = 1:6
          rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
          rev1(i,:) = rev1(i,:) / norm(rev1(i,:));

          rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
          rev2(i,:) = rev2(i,:) / norm(rev2(i,:));

          cyl1(i,:) = leg_vectors(i,:);
      end
#+end_src

Coordinate systems
#+begin_src matlab
      stewart.lower_leg = struct('rotation', eye(3));
      stewart.upper_leg = struct('rotation', eye(3));

      for i = 1:6
          stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
          stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
      end
#+end_src

Position Matrix
#+begin_src matlab
      stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.h+stewart.Leg.sphere.top+stewart.SP.h.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
#+end_src

Compute Jacobian Matrix
#+begin_src matlab
      %         aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.h - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
      bb = stewart.pos_top_tranform - (stewart.TP.h + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
      bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
      stewart.J = getJacobianMatrix(leg_vectors', bb');

      stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
  end
#+end_src

* initializeParameters Function
#+begin_src matlab
  function [stewart] = initializeParameters(stewart)
#+end_src

We first compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
  stewart.Aa = zeros(6, 3); % [mm]
  stewart.Ab = zeros(6, 3); % [mm]
  stewart.Bb = zeros(6, 3); % [mm]
#+end_src

#+begin_src matlab
  for i = 1:3
      stewart.Aa(2*i-1,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
                             stewart.BP.Rleg*sin( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
                             stewart.BP.H+stewart.SP.H];
      stewart.Aa(2*i,:)   = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
                             stewart.BP.Rleg*sin( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
                             stewart.BP.H+stewart.SP.H];

      stewart.Ab(2*i-1,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
                             stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
                             stewart.H - stewart.TP.H - stewart.SP.H];
      stewart.Ab(2*i,:)   = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
                             stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
                             stewart.H - stewart.TP.H - stewart.SP.H];
  end
  stewart.Bb = stewart.Ab - stewart.H*[0,0,1];
#+end_src

Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
\[ b_i - a_i = l_i \hat{s}_i \]

We initialize $l_i$ and $\hat{s}_i$
#+begin_src matlab
  leg_length = zeros(6, 1); % [mm]
  leg_vectors = zeros(6, 3);
#+end_src

We compute $b_i - a_i$, and then:
\begin{align*}
  l_i       &= \left|b_i - a_i\right| \\
  \hat{s}_i &= \frac{b_i - a_i}{l_i}
\end{align*}

#+begin_src matlab
  legs = stewart.Ab - stewart.Aa;

  for i = 1:6
      leg_length(i) = norm(legs(i,:));
      leg_vectors(i,:) = legs(i,:) / leg_length(i);
  end
#+end_src

Then the shape of the bottom leg is estimated
#+begin_src matlab
  stewart.Leg.lenght = leg_length(1)/1.5;
  stewart.Leg.shape.bot = ...
      [0                0; ...
       stewart.Leg.Rbot 0; ...
       stewart.Leg.Rbot stewart.Leg.lenght; ...
       stewart.Leg.Rtop stewart.Leg.lenght; ...
       stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
       0                0.2*stewart.Leg.lenght];
#+end_src

We compute rotation matrices to have the orientation of the legs.
The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
#+begin_src matlab
  stewart.Rm = struct('R', eye(3));

  for i = 1:6
    sx = cross(leg_vectors(i,:), [1 0 0]);
    sx = sx/norm(sx);

    sy = -cross(sx, leg_vectors(i,:));
    sy = sy/norm(sy);

    sz = leg_vectors(i,:);
    sz = sz/norm(sz);

    stewart.Rm(i).R = [sx', sy', sz'];
  end
#+end_src

Compute Jacobian Matrix
#+begin_src matlab
  J = zeros(6);

  for i = 1:6
    J(i, 1:3) = leg_vectors(i, :);
    J(i, 4:6) = cross(0.001*(stewart.Ab(i, :)- stewart.H*[0,0,1]), leg_vectors(i, :));
  end

  stewart.J = J;
  stewart.Jinv = inv(J);
#+end_src

#+begin_src matlab
  stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;
#+end_src

#+begin_src matlab
    end
  end
#+end_src
* initializeSample
  :PROPERTIES:
  :HEADER-ARGS:matlab+: :tangle src/initializeSample.m
  :END:
#+begin_src matlab
  function [] = initializeSample(opts_param)
  %% Default values for opts
      sample = struct( ...
          'radius',     100, ... % radius of the cylinder [mm]
          'height',     100, ... % height of the cylinder [mm]
          'mass',       10,  ... % mass of the cylinder [kg]
          'measheight', 50, ... % measurement point z-offset [mm]
          'offset',     [0, 0, 0],   ... % offset position of the sample [mm]
          'color',      [0.9 0.1 0.1] ...
          );

      %% Populate opts with input parameters
      if exist('opts_param','var')
          for opt = fieldnames(opts_param)'
              sample.(opt{1}) = opts_param.(opt{1});
          end
      end

      %% Save
      save('./mat/sample.mat', 'sample');
  end
#+end_src