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Kinematic Study of the Stewart Platform

Table of Contents

1 Needed Actuator Stroke

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

1.1 Stewart architecture definition

We use a cubic architecture. 

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');

1.2 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]

1.3 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]';

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

1.4 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;
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

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

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

2 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-')

3 Functions

3.1 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

3.2 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

3.3 computeJacobian: Compute the Jacobian Matrix

This Matlab function is accessible here.

3.3.1 Function description

function [stewart] = computeJacobian(stewart)
% computeJacobian -
%
% Syntax: [stewart] = computeJacobian(stewart)
%
% Inputs:
%    - stewart - With at least the following fields:
%        - As [3x6] - The 6 unit vectors for each strut expressed in {A}
%        - Ab [3x6] - The 6 position of the joints bi expressed in {A}
%
% Outputs:
%    - stewart - With the 3 added field:
%        - J [6x6] - The Jacobian Matrix
%        - K [6x6] - The Stiffness Matrix
%        - C [6x6] - The Compliance Matrix

3.3.2 Compute Jacobian Matrix

stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];

3.3.3 Compute Stiffness Matrix

stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;

3.3.4 Compute Compliance Matrix

stewart.C = inv(stewart.K);

3.4 inverseKinematics: Compute Inverse Kinematics

This Matlab function is accessible here.

3.4.1 Function description

function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
%    - stewart - A structure with the following fields
%        - Aa   [3x6] - The positions ai expressed in {A}
%        - Bb   [3x6] - The positions bi expressed in {B}
%    - args - Can have the following fields:
%        - AP   [3x1] - The wanted position of {B} with respect to {A}
%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
%    - Li   [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
%    - dLi  [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}

3.4.2 Optional Parameters

arguments
    stewart
    args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
    args.ARB (3,3) double {mustBeNumeric} = eye(3)
end

3.4.3 Theory

For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).

From the geometry of the manipulator, the loop closure for each limb, \(i = 1, 2, \dots, 6\) can be written as

\begin{align*} l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\ &= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i \end{align*}

To obtain the length of each actuator and eliminate \(\hat{\bm{s}}_i\), it is sufficient to dot multiply each side by itself:

\begin{equation} l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right] \end{equation}

Hence, for \(i = 1, 2, \dots, 6\), each limb length can be uniquely determined by:

\begin{equation} l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i} \end{equation}

If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation. Otherwise, when the limbs’ lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.

3.4.4 Compute

Li = sqrt(args.AP'*args.AP + diag(stewart.Bb'*stewart.Bb) + diag(stewart.Aa'*stewart.Aa) - (2*args.AP'*stewart.Aa)' + (2*args.AP'*(args.ARB*stewart.Bb))' - diag(2*(args.ARB*stewart.Bb)'*stewart.Aa));

dLi = Li-stewart.l;

3.5 forwardKinematicsApprox: Compute the Approximate Forward Kinematics

This Matlab function is accessible here.

3.5.1 Function description

function [P, R] = forwardKinematicsApprox(stewart, args)
% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using
%                           the Jacobian Matrix
%
% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)
%
% Inputs:
%    - stewart - A structure with the following fields
%        - J  [6x6] - The Jacobian Matrix
%    - args - Can have the following fields:
%        - dL [6x1] - Displacement of each strut [m]
%
% Outputs:
%    - P  [3x1] - The estimated position of {B} with respect to {A}
%    - R  [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}

3.5.2 Optional Parameters

arguments
    stewart
    args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
end

3.5.3 Computation

From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the position and orientation of {B} with respect to {A} using the following formula: \[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \] 

X = stewart.J\args.dL;

The position vector corresponds to the first 3 elements. 

P = X(1:3);

The next 3 elements are the orientation of {B} with respect to {A} expressed using the screw axis. 

theta = norm(X(4:6));
s = X(4:6)/theta;

We then compute the corresponding rotation matrix. 

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)];

Author: Dehaeze Thomas

Created: 2020-01-28 mar. 17:38