Kinematic Study of the Stewart Platform

Needed Actuator Stroke
Maximum Stroke
Functions
- getMaxPositions
- getMaxPureDisplacement

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.

  opts = struct(...
      'H_tot', 90, ... % Total height of the Hexapod [mm]
      'L',     180/sqrt(3), ... % Size of the Cube [mm]
      'H',     60, ... % Height between base joints and platform joints [mm]
      'H0',    180/2-60/2 ... % 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');

Wanted translations and rotations

We define wanted translations and rotations

  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]

Needed stroke for "pure" rotations or translations

First, we estimate the needed actuator stroke for "pure" rotations and translation.

  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]';

1.0607e-05

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).

  Lmax = 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]
      L = max(stewart.Jd*[Tx Ty Tz Rx Ry 0]');
      if L > Lmax
          Lmax = L;
          pos = [Tx Ty Tz Rx Ry];
      end
  end
  end
  end
  end
  end

We obtain a needed stroke shown below (almost two times the needed stroke for "pure" rotations and translations).

3.0927e-05

Maximum Stroke

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

  [X, Y, Z] = getMaxPositions(stewart);

  figure;
  plot3(X, Y, Z, 'k-')

Functions

getMaxPositions

  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

getMaxPureDisplacement

  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

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