Kinematic Study of the Stewart Platform
Table of Contents
- 1. Jacobian Analysis
- 2. Forward and Inverse Kinematics
- 3. Stiffness Analysis
- 4. Estimated required actuator stroke for specified platform mobility
- 5. Estimated platform mobility from specified actuator stroke
- 6. Functions
1 Jacobian Analysis
1.1 Jacobian Computation
1.2 Velocity of the struts to the velocity of the mobile platform
1.3 Displacement of the struts to the displacement of the mobile platform
1.4 Force Transformation
2 Forward and Inverse Kinematics
2.1 Inverse Kinematics
2.2 Forward Kinematics
2.3 Approximate Forward Kinematics
3 Stiffness Analysis
3.1 Computation of the Stiffness and Compliance Matrix
4 Estimated required actuator stroke for specified platform mobility
4.1 Needed Actuator Stroke
The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.
4.1.1 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');
4.1.2 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]
4.1.3 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]
5 Estimated platform mobility from specified actuator stroke
6 Functions
6.1 computeJacobian: Compute the Jacobian Matrix
This Matlab function is accessible here.
6.1.1 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
6.1.2 Compute Jacobian Matrix
stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];
6.1.3 Compute Stiffness Matrix
stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;
6.1.4 Compute Compliance Matrix
stewart.C = inv(stewart.K);
6.2 inverseKinematics: Compute Inverse Kinematics
This Matlab function is accessible here.
6.2.1 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}
6.2.2 Optional Parameters
arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
6.2.3 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.
6.2.4 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;
6.3 forwardKinematicsApprox: Compute the Approximate Forward Kinematics
This Matlab function is accessible here.
6.3.1 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}
6.3.2 Optional Parameters
arguments
stewart
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
end
6.3.3 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)];