stewart-simscape/kinematic-study.org

#+TITLE: Kinematic Study of the Stewart Platform
:DRAWER:
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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :tangle matlab/kinematic_study.m
#+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:

#+begin_src matlab :results none :exports none :noweb yes
  <<matlab-init>>
  addpath('src');
  addpath('library');
#+end_src

* Needed Actuator Stroke
The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.

** Stewart architecture definition
We use a cubic architecture.

#+begin_src matlab :results silent
  opts = struct(...
      'H_tot', 90, ... % Total height of the Hexapod [mm]
      'L',     200/sqrt(3), ... % Size of the Cube [mm]
      'H',     60, ... % Height between base joints and platform joints [mm]
      'H0',    200/2-60/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
      );
  stewart = initializeCubicConfiguration(opts);
  opts = struct(...
      'Jd_pos', [0, 0, 100], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
      'Jf_pos', [0, 0, -50]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
      );
  stewart = computeGeometricalProperties(stewart, opts);
  opts = struct(...
      'stroke', 50e-6 ... % Maximum stroke of each actuator [m]
      );
  stewart = initializeMechanicalElements(stewart, opts);

  save('./mat/stewart.mat', 'stewart');
#+end_src

** Wanted translations and rotations
We define wanted translations and rotations
#+begin_src matlab :results silent
  Tx_max = 15e-6; % Translation [m]
  Ty_max = 15e-6; % Translation [m]
  Tz_max = 15e-6; % Translation [m]
  Rx_max = 30e-6; % Rotation [rad]
  Ry_max = 30e-6; % Rotation [rad]
#+end_src

** Needed stroke for "pure" rotations or translations
First, we estimate the needed actuator stroke for "pure" rotations and translation.
#+begin_src matlab :results silent
  LTx = stewart.Jd*[Tx_max 0 0 0 0 0]';
  LTy = stewart.Jd*[0 Ty_max 0 0 0 0]';
  LTz = stewart.Jd*[0 0 Tz_max 0 0 0]';
  LRx = stewart.Jd*[0 0 0 Rx_max 0 0]';
  LRy = stewart.Jd*[0 0 0 0 Ry_max 0]';
#+end_src

#+begin_src matlab :results value :exports results
  ans = sprintf('From %.2g[m] to %.2g[m]: Total stroke = %.1f[um]', min(min([LTx,LTy,LTz,LRx,LRy])), max(max([LTx,LTy,LTz,LRx,LRy])), 1e6*(max(max([LTx,LTy,LTz,LRx,LRy]))-min(min([LTx,LTy,LTz,LRx,LRy]))))
#+end_src

#+RESULTS:
: From -1.2e-05[m] to 1.1e-05[m]: Total stroke = 22.9[um]

** Needed stroke for combined translations and rotations
Now, we combine translations and rotations, and we try to find the worst case (that we suppose to happen at the border).
#+begin_src matlab :results none
  Lmax = 0;
  Lmin = 0;
  pos = [0, 0, 0, 0, 0];
  for Tx = [-Tx_max,Tx_max]
  for Ty = [-Ty_max,Ty_max]
  for Tz = [-Tz_max,Tz_max]
  for Rx = [-Rx_max,Rx_max]
  for Ry = [-Ry_max,Ry_max]
      lmax = max(stewart.Jd*[Tx Ty Tz Rx Ry 0]');
      lmin = min(stewart.Jd*[Tx Ty Tz Rx Ry 0]');
      if lmax > Lmax
          Lmax = lmax;
          pos = [Tx Ty Tz Rx Ry];
      end
      if lmin < Lmin
          Lmin = lmin;
      end
  end
  end
  end
  end
  end
#+end_src

We obtain a needed stroke shown below (almost two times the needed stroke for "pure" rotations and translations).
#+begin_src matlab :results value :exports results
  ans = sprintf('From %.2g[m] to %.2g[m]: Total stroke = %.1f[um]', Lmin, Lmax, 1e6*(Lmax-Lmin))
#+end_src

#+RESULTS:
: From -3.1e-05[m] to 3.1e-05[m]: Total stroke = 61.5[um]

* Maximum Stroke
From a specified actuator stroke, we try to estimate the available maneuverability of the Stewart platform.

#+begin_src matlab :results silent
  [X, Y, Z] = getMaxPositions(stewart);
#+end_src

#+begin_src matlab :results silent
  figure;
  plot3(X, Y, Z, 'k-')
#+end_src

* Functions
  :PROPERTIES:
  :HEADER-ARGS:matlab+: :exports code
  :HEADER-ARGS:matlab+: :comments no
  :HEADER-ARGS:matlab+: :mkdir yes
  :HEADER-ARGS:matlab+: :eval no
  :END:
** getMaxPositions
  :PROPERTIES:
  :HEADER-ARGS:matlab+: :tangle src/getMaxPositions.m
  :END:
#+begin_src matlab
  function [X, Y, Z] = getMaxPositions(stewart)
      Leg = stewart.Leg;
      J = stewart.Jd;
      theta = linspace(0, 2*pi, 100);
      phi = linspace(-pi/2 , pi/2, 100);
      dmax = zeros(length(theta), length(phi));

      for i = 1:length(theta)
          for j = 1:length(phi)
              L = J*[cos(phi(j))*cos(theta(i)) cos(phi(j))*sin(theta(i)) sin(phi(j)) 0 0 0]';
              dmax(i, j) = Leg.stroke/max(abs(L));
          end
      end

      X = dmax.*cos(repmat(phi,length(theta),1)).*cos(repmat(theta,length(phi),1))';
      Y = dmax.*cos(repmat(phi,length(theta),1)).*sin(repmat(theta,length(phi),1))';
      Z = dmax.*sin(repmat(phi,length(theta),1));
  end
#+end_src

** getMaxPureDisplacement
  :PROPERTIES:
  :HEADER-ARGS:matlab+: :tangle src/getMaxPureDisplacement.m
  :END:
#+begin_src matlab
  function [max_disp] = getMaxPureDisplacement(Leg, J)
      max_disp = zeros(6, 1);
      max_disp(1) = Leg.stroke/max(abs(J*[1 0 0 0 0 0]'));
      max_disp(2) = Leg.stroke/max(abs(J*[0 1 0 0 0 0]'));
      max_disp(3) = Leg.stroke/max(abs(J*[0 0 1 0 0 0]'));
      max_disp(4) = Leg.stroke/max(abs(J*[0 0 0 1 0 0]'));
      max_disp(5) = Leg.stroke/max(abs(J*[0 0 0 0 1 0]'));
      max_disp(6) = Leg.stroke/max(abs(J*[0 0 0 0 0 1]'));
  end
#+end_src