1 Jacobian Analysis
++The kinematic analysis of a parallel manipulator is well described in taghirad13_paral: +
+++ ++Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces an torques that cause the motion. +In this analysis, the relation between the geometrical parameters of the manipulator with the final motion of the moving platform is derived and analyzed. +
+
+The current document is divided in the following sections: +
+-
+
- Section 1: The Jacobian matrix is derived from the geometry of the Stewart platform. Then it is shown that the Jacobian can link velocities and forces present in the system, and thus this matrix can be very useful for both analysis and control of the Stewart platform. +
- Section 2: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut. +
- Section 3: The Forward and Inverse kinematic problems are presented. +
+
1 Jacobian Analysis
+
+
-+From taghirad13_paral: +
+++The Jacobian matrix not only reveals the relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities, it also constructs the transformation needed to find the actuator forces from the forces and moments acting on the moving platform. +
+
-
+1.1 Jacobian Computation
-
+
1.1 Jacobian Computation
+
+
+If we note: +
+-
+
- \({}^A\hat{\bm{s}}_i\) the unit vector representing the direction of the i’th strut and expressed in frame \(\{A\}\) +
- \({}^A\bm{b}_i\) the position vector of the i’th joint fixed to the mobile platform and expressed in frame \(\{A\}\) +
-
+1.2 Velocity of the struts to the velocity of the mobile platform
-+Then, we can compute the Jacobian with the following equation (the superscript \(A\) is ignored): +
+\begin{equation*} + \bm{J} = \begin{bmatrix} + {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\ + {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\ + {\hat{\bm{s}}_3}^T & (\bm{b}_3 \times \hat{\bm{s}}_3)^T \\ + {\hat{\bm{s}}_4}^T & (\bm{b}_4 \times \hat{\bm{s}}_4)^T \\ + {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\ + {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T + \end{bmatrix} +\end{equation*} -
-
-1.3 Displacement of the struts to the displacement of the mobile platform
+
+The Jacobian matrix \(\bm{J}\) can be computed using the computeJacobian function (accessible here).
+For instance:
+
+
-
-stewart = computeJacobian(stewart); +
-
1.4 Force Transformation
+
+This will add three new matrix to the stewart structure:
+
-
+
Jthe Jacobian matrix
+Kthe stiffness matrix
+Cthe compliance matrix
+
-
+
+2 Forward and Inverse Kinematics
+
+
+
+1.2 Jacobian - Velocity loop closure
+
+
++The Jacobian matrix links the input joint rate \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T\) of each strut to the output twist vector of the mobile platform is denoted by \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\): +
+\begin{equation*} + \dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}} +\end{equation*} + ++The input joint rate \(\dot{\bm{\mathcal{L}}}\) can be measured by taking the derivative of the relative motion sensor in each strut. +The output twist vector can be measured with a “Transform Sensor” block measuring the relative velocity and relative angular velocity of frame \(\{B\}\) with respect to frame \(\{A\}\). +
+ ++If the Jacobian matrix is inversible, we can also compute \(\dot{\bm{\mathcal{X}}}\) from \(\dot{\bm{\mathcal{L}}}\). +
+\begin{equation*} + \dot{\bm{\mathcal{X}}} = \bm{J}^{-1} \dot{\bm{\mathcal{L}}} +\end{equation*} + ++The Jacobian matrix can also be used to approximate forward and inverse kinematics for small displacements. +This is explained in section 3.3. +
+
+
+1.3 Jacobian - Static Force Transformation
+
+
++If we note: +
+-
+
- \(\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T\): vector of actuator forces applied in each strut +
- \(\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T\): external force/torque action on the mobile platform at \(\bm{O}_B\) +
+We find that the transpose of the Jacobian matrix links the two by the following equation: +
+\begin{equation*} + \bm{\mathcal{F}} = \bm{J}^T \bm{\tau} +\end{equation*} + ++If the Jacobian matrix is inversible, we also have the following relation: +
+\begin{equation*} + \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}} +\end{equation*} +
+
2 Stiffness Analysis
+
+
-+Here, we focus on the deflections of the manipulator moving platform that are the result of the external applied wrench to the mobile platform. +The amount of these deflections are a function of the applied wrench as well as the manipulator structural stiffness. +
-
2.1 Inverse Kinematics
+
+
-
-2.1 Computation of the Stiffness and Compliance Matrix
+
+
-
-
-+As explain in this document, each Actuator is modeled by 3 elements in parallel: +
+-
+
- A spring with a stiffness \(k_{i}\) +
- A dashpot with a damping \(c_{i}\) +
+The stiffness of the actuator \(k_i\) links the applied actuator force \(\delta \tau_i\) and the corresponding small deflection \(\delta l_i\): +
+\begin{equation*} + \tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6 +\end{equation*} ++If we combine these 6 relations: +
+\begin{equation*} + \bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] +\end{equation*} + ++Substituting \(\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}\) and \(\delta \bm{\mathcal{L}} = \bm{J} \cdot \delta \bm{\mathcal{X}}\) gives +
+\begin{equation*} + \bm{\mathcal{F}} = \bm{J}^T \mathcal{K} \bm{J} \cdot \delta \bm{\mathcal{X}} +\end{equation*} ++And then we identify the stiffness matrix \(\bm{K}\): +
+\begin{equation*} + \bm{K} = \bm{J}^T \mathcal{K} \bm{J} +\end{equation*} + ++If the stiffness matrix \(\bm{K}\) is inversible, the compliance matrix of the manipulator is defined as +
+\begin{equation*} + \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1} +\end{equation*} + ++The compliance matrix of a manipulator shows the mapping of the moving platform wrench applied at \(\bm{O}_B\) to its small deflection by +
+\begin{equation*} + \delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}} +\end{equation*} + +
+The stiffness and compliance matrices are computed using the computeJacobian function (accessible here).
+
-
-
-
-2.2 Forward Kinematics
-
-
2.3 Approximate Forward Kinematics
-
3 Stiffness Analysis
+
+
3 Forward and Inverse Kinematics
-
-
3.1 Computation of the Stiffness and Compliance Matrix
+
+
-3.1 Inverse Kinematics
+
+
+
+
++ ++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 \(\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T\). +
+
+This problem can be easily solved using the loop closures. +
+ ++The obtain joint variables are: +
+\begin{equation*} + \begin{aligned} + l_i = &\Big[ {}^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 + \dots\\ + &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 \Big]^{1/2} + \end{aligned} +\end{equation*} + ++If the position and orientation of the platform lie in the feasible workspace, the solution is unique. +Otherwise, the solution gives complex numbers. +
+ +
+This inverse kinematic solution can be obtained using the function inverseKinematics (described here).
+
-
4 Estimated required actuator stroke for specified platform mobility
-
+
+
+
+3.2 Forward Kinematics
+
+
+
+
-++ ++In forward kinematic analysis, it is assumed that the vector of limb lengths \(\bm{L}\) is given and the problem is to find the position \({}^A\bm{P}\) and the orientation \({}^A\bm{R}_B\). +
+
+This is a difficult problem that requires to solve nonlinear equations. +
+ ++In a next section, an approximate solution of the forward kinematics problem is proposed for small displacements. +
-
+
+4.1 Needed Actuator Stroke
+
+
+
+3.3 Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix
+
+
+
+
++For small displacements mobile platform displacement \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\) around \(\bm{\mathcal{X}}_0\), the associated joint displacement can be computed using the Jacobian: +
+\begin{equation*} + \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}} +\end{equation*} + ++Similarly, for small joint displacements \(\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T\) around \(\bm{\mathcal{L}}_0\), it is possible to find the induced small displacement of the mobile platform: +
+\begin{equation*} + \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} +\end{equation*} + ++These two relations solve the forward and inverse kinematic problems for small displacement in a approximate way. +As the inverse kinematic can be easily solved exactly this is not much useful, however, as the forward kinematic problem is difficult to solve, this approximation can be very useful for small displacements. +
+ +
+The function forwardKinematicsApprox (described here) can be used to solve the forward kinematic problem using the Jacobian matrix.
+
+
+3.4 Estimation of the range validity of the approximate inverse kinematics
+
+
+
++As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix. +For small displacements, the approximate solution is expected to work well. +We would like here to determine up to what displacement this approximation can be considered as correct. +
+ ++Then, we can determine the range for which the approximate inverse kinematic is valid. +This will also gives us the range for which the approximate forward kinematic is valid. +
+
+
+
+3.4.1 Stewart architecture definition
+
+
++We first define some general Stewart architecture. +
+
+
+stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3); +stewart = generateGeneralConfiguration(stewart); +stewart = computeJointsPose(stewart); +stewart = initializeStewartPose(stewart); +stewart = initializeCylindricalPlatforms(stewart); +stewart = initializeCylindricalStruts(stewart); +stewart = initializeStrutDynamics(stewart); +stewart = initializeJointDynamics(stewart); +stewart = computeJacobian(stewart); ++
+
+
+3.4.2 Comparison for “pure” translations
+
+
++Let’s first compare the perfect and approximate solution of the inverse for pure \(x\) translations. +
+ ++We compute the approximate and exact required strut stroke to have the wanted mobile platform \(x\) displacement. +The estimate required strut stroke for both the approximate and exact solutions are shown in Figure 1. +The relative strut length displacement is shown in Figure 2. +
+
+
+
+
+
+
+
+
+Xrs = logspace(-6, -1, 100); % Wanted X translation of the mobile platform [m] + +Ls_approx = zeros(6, length(Xrs)); +Ls_exact = zeros(6, length(Xrs)); + +for i = 1:length(Xrs) + Xr = Xrs(i); + L_approx(:, i) = stewart.J*[Xr; 0; 0; 0; 0; 0;]; + [~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xr; 0; 0]); +end ++
+
+
+
+3.4.3 Conclusion
+
+
++For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct. +
+
+
4 Estimated required actuator stroke from specified platform mobility
+
+
+
++Let’s say one want to design a Stewart platform with some specified mobility (position and orientation). +One may want to determine the required actuator stroke required to obtain the specified mobility. +This is what is analyzed in this section. +
+
+
-
-
4.1 Stewart architecture definition
-The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations. +Let’s first define the Stewart platform architecture that we want to study.
-
-
-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'); +stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3); +stewart = generateGeneralConfiguration(stewart); +stewart = computeJointsPose(stewart); +stewart = initializeStewartPose(stewart); +stewart = initializeCylindricalPlatforms(stewart); +stewart = initializeCylindricalStruts(stewart); +stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1)); +stewart = initializeJointDynamics(stewart); +stewart = computeJacobian(stewart);
-
4.1.2 Wanted translations and rotations
-
+
+
-4.2 Wanted translations and rotations
+-We define wanted translations and rotations +Let’s now define the wanted extreme translations and rotations.
-
Tx_max = 15e-6; % Translation [m] -Ty_max = 15e-6; % Translation [m] -Tz_max = 15e-6; % Translation [m] +Tx_max = 50e-6; % Translation [m] +Ty_max = 50e-6; % Translation [m] +Tz_max = 50e-6; % Translation [m] Rx_max = 30e-6; % Rotation [rad] Ry_max = 30e-6; % Rotation [rad] +Rz_max = 0; % Rotation [rad]
-
-4.1.3 Needed stroke for “pure” rotations or translations
-
+
+
+
+
+4.3 Needed stroke for “pure” rotations or translations
+-First, we estimate the needed actuator stroke for “pure” rotations and translation. +As a first estimation, we estimate the needed actuator stroke for “pure” rotations and translation. +We do that using either the Inverse Kinematic solution or the Jacobian matrix as an approximation. +
+ +
+
+
+LTx = stewart.J*[Tx_max 0 0 0 0 0]'; +LTy = stewart.J*[0 Ty_max 0 0 0 0]'; +LTz = stewart.J*[0 0 Tz_max 0 0 0]'; +LRx = stewart.J*[0 0 0 Rx_max 0 0]'; +LRy = stewart.J*[0 0 0 0 Ry_max 0]'; +LRz = stewart.J*[0 0 0 0 0 Rz_max]'; ++
+The obtain required stroke is: +
++From -3.8e-05[m] to 3.8e-05[m]: Total stroke = 76.1[um] ++ + +
+This is surely a low estimation of the required stroke. +
+
+
4.4 Needed stroke for “combined” rotations or translations
+
+
+
+
+
+
+
+
+
++We know would like to have a more precise estimation. +
+ ++To do so, we may estimate the required actuator stroke for all possible combination of translation and rotation. +
+ ++Let’s first generate all the possible combination of maximum translation and rotations.
-
+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]'; +Ps = [2*(dec2bin(0:5^2-1,5)-'0')-1, zeros(5^2, 1)].*[Tx_max Ty_max Tz_max Rx_max Ry_max Rz_max];
| Tx [m] | +Ty [m] | +Tz [m] | +Rx [rad] | +Ry [rad] | +Rz [rad] | +
|---|---|---|---|---|---|
| -5.0e-05 | +-5.0e-05 | +-5.0e-05 | +-3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +-5.0e-05 | +-5.0e-05 | +-3.0e-05 | +3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +-5.0e-05 | +-5.0e-05 | +3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +-5.0e-05 | +-5.0e-05 | +3.0e-05 | +3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +-5.0e-05 | +5.0e-05 | +-3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +-5.0e-05 | +5.0e-05 | +-3.0e-05 | +3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +-5.0e-05 | +5.0e-05 | +3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +-5.0e-05 | +5.0e-05 | +3.0e-05 | +3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +-5.0e-05 | +-3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +-5.0e-05 | +-3.0e-05 | +3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +-5.0e-05 | +3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +-5.0e-05 | +3.0e-05 | +3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +5.0e-05 | +-3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +5.0e-05 | +-3.0e-05 | +3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +5.0e-05 | +3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| -5.0e-05 | +5.0e-05 | +5.0e-05 | +3.0e-05 | +3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +-5.0e-05 | +-3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +-5.0e-05 | +-3.0e-05 | +3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +-5.0e-05 | +3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +-5.0e-05 | +3.0e-05 | +3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +5.0e-05 | +-3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +5.0e-05 | +-3.0e-05 | +3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +5.0e-05 | +3.0e-05 | +-3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +-5.0e-05 | +5.0e-05 | +3.0e-05 | +3.0e-05 | +0.0e+00 | +
| 5.0e-05 | +5.0e-05 | +-5.0e-05 | +-3.0e-05 | +-3.0e-05 | +0.0e+00 | +
+For all possible combination, we compute the required actuator stroke using the inverse kinematic solution. +
+
+
+
+L_min = 0; +L_max = 0; + +for i = 1:size(Ps,1) + Rx = [1 0 0; + 0 cos(Ps(i, 4)) -sin(Ps(i, 4)); + 0 sin(Ps(i, 4)) cos(Ps(i, 4))]; + + Ry = [ cos(Ps(i, 5)) 0 sin(Ps(i, 5)); + 0 1 0; + -sin(Ps(i, 5)) 0 cos(Ps(i, 5))]; + + Rz = [cos(Ps(i, 6)) -sin(Ps(i, 6)) 0; + sin(Ps(i, 6)) cos(Ps(i, 6)) 0; + 0 0 1]; + + ARB = Rz*Ry*Rx; + [~, Ls] = inverseKinematics(stewart, 'AP', Ps(i, 1:3)', 'ARB', ARB); + + if min(Ls) < L_min + L_min = min(Ls) + end + if max(Ls) > L_max + L_max = max(Ls) + end +end ++
+We obtain the required actuator stroke: +
-From -1.2e-05[m] to 1.1e-05[m]: Total stroke = 22.9[um] +From -8.9e-05[m] to 8.9e-05[m]: Total stroke = 177.2[um]-
+This is probably a much realistic estimation of the required actuator stroke. +
-
5 Estimated platform mobility from specified actuator stroke
+
+
5 Estimated platform mobility from specified actuator stroke
+
+
+
-+Here, from some value of the actuator stroke, we would like to estimate the mobility of the Stewart platform. +
+ ++As explained in section 3, the forward kinematic problem of the Stewart platform is quite difficult to solve. +However, for small displacements, we can use the Jacobian as an approximate solution. +
-
+
+6 Functions
+
+
+
+5.1 Stewart architecture definition
+
+
++Let’s first define the Stewart platform architecture that we want to study. +
+
+
+
+stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3); +stewart = generateGeneralConfiguration(stewart); +stewart = computeJointsPose(stewart); +stewart = initializeStewartPose(stewart); +stewart = initializeCylindricalPlatforms(stewart); +stewart = initializeCylindricalStruts(stewart); +stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1)); +stewart = initializeJointDynamics(stewart); +stewart = computeJacobian(stewart); ++
+Let’s now define the actuator stroke. +
+
+
+L_min = -50e-6; % [m] +L_max = 50e-6; % [m] ++
+
+5.2 TODO Approximate Forward Dynamics
+
+
6 Functions
-
-6.1
+
6.1 computeJacobian: Compute the Jacobian Matrix
+
+6.1
-
6.1 computeJacobian: Compute the Jacobian Matrix
@@ -498,9 +1205,9 @@ This Matlab function is accessible here.
-
6.1.1 Function description
-
+
+
-
Function description
+function [stewart] = computeJacobian(stewart) % computeJacobian - @@ -522,9 +1229,9 @@ This Matlab function is accessible here.
-
6.1.2 Compute Jacobian Matrix
-
+
+
-
Compute Jacobian Matrix
+stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];@@ -532,9 +1239,9 @@ This Matlab function is accessible here.
-
6.1.3 Compute Stiffness Matrix
-
+
+
-
Compute Stiffness Matrix
+stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;@@ -542,9 +1249,9 @@ This Matlab function is accessible here.
-
6.1.4 Compute Compliance Matrix
-
+
+
-
Compute Compliance Matrix
+stewart.C = inv(stewart.K);@@ -553,11 +1260,11 @@ This Matlab function is accessible here.
-6.2
+
6.2 inverseKinematics: Compute Inverse Kinematics
+
+6.2
-
6.2 inverseKinematics: Compute Inverse Kinematics
@@ -565,9 +1272,9 @@ This Matlab function is accessible here.
-
6.2.1 Function description
-
+
+
-
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} @@ -590,9 +1297,9 @@ This Matlab function is accessible here.
-
6.2.2 Optional Parameters
-
+
+
-
Optional Parameters
+arguments
stewart
@@ -604,9 +1311,9 @@ This Matlab function is accessible here.
-
6.2.3 Theory
-
+
+
-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, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
@@ -640,9 +1347,9 @@ Otherwise, when the limbs’ lengths derived yield complex numbers, then the
-
6.2.4 Compute
-
+
+
-
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));@@ -656,11 +1363,11 @@ Otherwise, when the limbs’ lengths derived yield complex numbers, then the
-6.3
+
6.3 forwardKinematicsApprox: Compute the Approximate Forward Kinematics
+
+6.3
-
6.3 forwardKinematicsApprox: Compute the Approximate Forward Kinematics
@@ -668,9 +1375,9 @@ This Matlab function is accessible here<
-
6.3.1 Function description
-
+
+
-
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 @@ -692,9 +1399,9 @@ This Matlab function is accessible here<
-
6.3.2 Optional Parameters
-
+
+
->
** Introduction :ignore:
+From cite:taghirad13_paral:
+#+begin_quote
+The Jacobian matrix not only reveals the *relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities*, it also constructs the transformation needed to find the *actuator forces from the forces and moments acting on the moving platform*.
+#+end_quote
+
** Jacobian Computation
+If we note:
+- ${}^A\hat{\bm{s}}_i$ the unit vector representing the direction of the i'th strut and expressed in frame $\{A\}$
+- ${}^A\bm{b}_i$ the position vector of the i'th joint fixed to the mobile platform and expressed in frame $\{A\}$
-** Velocity of the struts to the velocity of the mobile platform
+Then, we can compute the Jacobian with the following equation (the superscript $A$ is ignored):
+\begin{equation*}
+ \bm{J} = \begin{bmatrix}
+ {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\
+ {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\
+ {\hat{\bm{s}}_3}^T & (\bm{b}_3 \times \hat{\bm{s}}_3)^T \\
+ {\hat{\bm{s}}_4}^T & (\bm{b}_4 \times \hat{\bm{s}}_4)^T \\
+ {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\
+ {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T
+ \end{bmatrix}
+\end{equation*}
-** Displacement of the struts to the displacement of the mobile platform
+The Jacobian matrix $\bm{J}$ can be computed using the =computeJacobian= function (accessible [[sec:computeJacobian][here]]).
+For instance:
+#+begin_src matlab :eval no
+ stewart = computeJacobian(stewart);
+#+end_src
+This will add three new matrix to the =stewart= structure:
+- =J= the Jacobian matrix
+- =K= the stiffness matrix
+- =C= the compliance matrix
-** Force Transformation
+** Jacobian - Velocity loop closure
+The Jacobian matrix links the input joint rate $\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T$ of each strut to the output twist vector of the mobile platform is denoted by $\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T$:
+\begin{equation*}
+ \dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}}
+\end{equation*}
-* Forward and Inverse Kinematics
-** Introduction :ignore:
+The input joint rate $\dot{\bm{\mathcal{L}}}$ can be measured by taking the derivative of the relative motion sensor in each strut.
+The output twist vector can be measured with a "Transform Sensor" block measuring the relative velocity and relative angular velocity of frame $\{B\}$ with respect to frame $\{A\}$.
-** Inverse Kinematics
+If the Jacobian matrix is inversible, we can also compute $\dot{\bm{\mathcal{X}}}$ from $\dot{\bm{\mathcal{L}}}$.
+\begin{equation*}
+ \dot{\bm{\mathcal{X}}} = \bm{J}^{-1} \dot{\bm{\mathcal{L}}}
+\end{equation*}
+The Jacobian matrix can also be used to approximate forward and inverse kinematics for small displacements.
+This is explained in section [[sec:approximate_forward_inverse_kinematics]].
-** Forward Kinematics
+** Jacobian - Static Force Transformation
+If we note:
+- $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$: vector of actuator forces applied in each strut
+- $\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T$: external force/torque action on the mobile platform at $\bm{O}_B$
+We find that the transpose of the Jacobian matrix links the two by the following equation:
+\begin{equation*}
+ \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}
+\end{equation*}
-** Approximate Forward Kinematics
-
+If the Jacobian matrix is inversible, we also have the following relation:
+\begin{equation*}
+ \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}
+\end{equation*}
* Stiffness Analysis
+<>
** Introduction :ignore:
+Here, we focus on the deflections of the manipulator moving platform that are the result of the external applied wrench to the mobile platform.
+The amount of these deflections are a function of the applied wrench as well as the manipulator *structural stiffness*.
** Computation of the Stiffness and Compliance Matrix
+As explain in [[file:stewart-architecture.org][this]] document, each Actuator is modeled by 3 elements in parallel:
+- A spring with a stiffness $k_{i}$
+- A dashpot with a damping $c_{i}$
-* Estimated required actuator stroke for specified platform mobility
+The stiffness of the actuator $k_i$ links the applied actuator force $\delta \tau_i$ and the corresponding small deflection $\delta l_i$:
+\begin{equation*}
+ \tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6
+\end{equation*}
+If we combine these 6 relations:
+\begin{equation*}
+ \bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right]
+\end{equation*}
+
+Substituting $\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}$ and $\delta \bm{\mathcal{L}} = \bm{J} \cdot \delta \bm{\mathcal{X}}$ gives
+\begin{equation*}
+ \bm{\mathcal{F}} = \bm{J}^T \mathcal{K} \bm{J} \cdot \delta \bm{\mathcal{X}}
+\end{equation*}
+And then we identify the stiffness matrix $\bm{K}$:
+\begin{equation*}
+ \bm{K} = \bm{J}^T \mathcal{K} \bm{J}
+\end{equation*}
+
+If the stiffness matrix $\bm{K}$ is inversible, the *compliance matrix* of the manipulator is defined as
+\begin{equation*}
+ \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1}
+\end{equation*}
+
+The compliance matrix of a manipulator shows the mapping of the moving platform wrench applied at $\bm{O}_B$ to its small deflection by
+\begin{equation*}
+ \delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}}
+\end{equation*}
+
+The stiffness and compliance matrices are computed using the =computeJacobian= function (accessible [[sec:computeJacobian][here]]).
+
+* Forward and Inverse Kinematics
+<>
+** Inverse Kinematics
+<>
+
+#+begin_quote
+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 $\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T$.
+#+end_quote
+
+This problem can be easily solved using the loop closures.
+
+The obtain joint variables are:
+\begin{equation*}
+ \begin{aligned}
+ l_i = &\Big[ {}^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 + \dots\\
+ &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 \Big]^{1/2}
+ \end{aligned}
+\end{equation*}
+
+If the position and orientation of the platform lie in the feasible workspace, the solution is unique.
+Otherwise, the solution gives complex numbers.
+
+This inverse kinematic solution can be obtained using the function =inverseKinematics= (described [[sec:inverseKinematics][here]]).
+
+** Forward Kinematics
+<>
+
+#+begin_quote
+In *forward kinematic analysis*, it is assumed that the vector of limb lengths $\bm{L}$ is given and the problem is to find the position ${}^A\bm{P}$ and the orientation ${}^A\bm{R}_B$.
+#+end_quote
+
+This is a difficult problem that requires to solve nonlinear equations.
+
+In a next section, an approximate solution of the forward kinematics problem is proposed for small displacements.
+
+** Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix
+<>
+
+For small displacements mobile platform displacement $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T$ around $\bm{\mathcal{X}}_0$, the associated joint displacement can be computed using the Jacobian:
+\begin{equation*}
+ \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}
+\end{equation*}
+
+Similarly, for small joint displacements $\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T$ around $\bm{\mathcal{L}}_0$, it is possible to find the induced small displacement of the mobile platform:
+\begin{equation*}
+ \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}}
+\end{equation*}
+
+These two relations solve the forward and inverse kinematic problems for small displacement in a *approximate* way.
+As the inverse kinematic can be easily solved exactly this is not much useful, however, as the forward kinematic problem is difficult to solve, this approximation can be very useful for small displacements.
+
+The function =forwardKinematicsApprox= (described [[sec:forwardKinematicsApprox][here]]) can be used to solve the forward kinematic problem using the Jacobian matrix.
+
+** Estimation of the range validity of the approximate inverse kinematics
+*** Introduction :ignore:
+As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix.
+For small displacements, the approximate solution is expected to work well.
+We would like here to determine up to what displacement this approximation can be considered as correct.
+
+Then, we can determine the range for which the approximate inverse kinematic is valid.
+This will also gives us the range for which the approximate forward kinematic is valid.
+
+*** Matlab Init :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+ <>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+ <>
+#+end_src
+
+#+begin_src matlab :results none :exports none
+ simulinkproject('./');
+#+end_src
+
+*** Stewart architecture definition
+We first define some general Stewart architecture.
+#+begin_src matlab
+ stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+ stewart = generateGeneralConfiguration(stewart);
+ stewart = computeJointsPose(stewart);
+ stewart = initializeStewartPose(stewart);
+ stewart = initializeCylindricalPlatforms(stewart);
+ stewart = initializeCylindricalStruts(stewart);
+ stewart = initializeStrutDynamics(stewart);
+ stewart = initializeJointDynamics(stewart);
+ stewart = computeJacobian(stewart);
+#+end_src
+
+*** Comparison for "pure" translations
+Let's first compare the perfect and approximate solution of the inverse for pure $x$ translations.
+
+We compute the approximate and exact required strut stroke to have the wanted mobile platform $x$ displacement.
+The estimate required strut stroke for both the approximate and exact solutions are shown in Figure [[fig:inverse_kinematics_approx_validity_x_translation]].
+The relative strut length displacement is shown in Figure [[fig:inverse_kinematics_approx_validity_x_translation_relative]].
+#+begin_src matlab
+ Xrs = logspace(-6, -1, 100); % Wanted X translation of the mobile platform [m]
+
+ Ls_approx = zeros(6, length(Xrs));
+ Ls_exact = zeros(6, length(Xrs));
+
+ for i = 1:length(Xrs)
+ Xr = Xrs(i);
+ L_approx(:, i) = stewart.J*[Xr; 0; 0; 0; 0; 0;];
+ [~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xr; 0; 0]);
+ end
+#+end_src
+
+#+begin_src matlab :exports none
+ figure;
+ hold on;
+ for i = 1:6
+ set(gca,'ColorOrderIndex',i);
+ plot(Xrs, abs(L_approx(i, :)));
+ set(gca,'ColorOrderIndex',i);
+ plot(Xrs, abs(L_exact(i, :)), '--');
+ end
+ hold off;
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ xlabel('Wanted $x$ displacement [m]');
+ ylabel('Estimated required stroke');
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/inverse_kinematics_approx_validity_x_translation.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
+#+end_src
+
+#+NAME: fig:inverse_kinematics_approx_validity_x_translation
+#+CAPTION: Comparison of the Approximate solution and True solution for the Inverse kinematic problem ([[./figs/inverse_kinematics_approx_validity_x_translation.png][png]], [[./figs/inverse_kinematics_approx_validity_x_translation.pdf][pdf]])
+[[file:figs/inverse_kinematics_approx_validity_x_translation.png]]
+
+#+begin_src matlab :exports none
+ figure;
+ hold on;
+ for i = 1:6
+ plot(Xrs, abs(L_approx(i, :) - L_exact(i, :))./abs(L_approx(i, :) + L_exact(i, :)), 'k-');
+ end
+ hold off;
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ xlabel('Wanted $x$ displacement [m]');
+ ylabel('Relative Stroke Error');
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/inverse_kinematics_approx_validity_x_translation_relative.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
+#+end_src
+
+#+NAME: fig:inverse_kinematics_approx_validity_x_translation_relative
+#+CAPTION: Relative length error by using the Approximate solution of the Inverse kinematic problem ([[./figs/inverse_kinematics_approx_validity_x_translation_relative.png][png]], [[./figs/inverse_kinematics_approx_validity_x_translation_relative.pdf][pdf]])
+[[file:figs/inverse_kinematics_approx_validity_x_translation_relative.png]]
+
+*** Conclusion
+For small wanted displacements (up to $\approx 1\%$ of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct.
+
+* Estimated required actuator stroke from specified platform mobility
+<>
** Introduction :ignore:
+Let's say one want to design a Stewart platform with some specified mobility (position and orientation).
+One may want to determine the required actuator stroke required to obtain the specified mobility.
+This is what is analyzed in this section.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -67,62 +317,144 @@
simulinkproject('./');
#+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');
+** Stewart architecture definition
+Let's first define the Stewart platform architecture that we want to study.
+#+begin_src matlab
+ stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+ stewart = generateGeneralConfiguration(stewart);
+ stewart = computeJointsPose(stewart);
+ stewart = initializeStewartPose(stewart);
+ stewart = initializeCylindricalPlatforms(stewart);
+ stewart = initializeCylindricalStruts(stewart);
+ stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1));
+ stewart = initializeJointDynamics(stewart);
+ stewart = computeJacobian(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]
+** Wanted translations and rotations
+Let's now define the wanted extreme translations and rotations.
+#+begin_src matlab
+ Tx_max = 50e-6; % Translation [m]
+ Ty_max = 50e-6; % Translation [m]
+ Tz_max = 50e-6; % Translation [m]
Rx_max = 30e-6; % Rotation [rad]
Ry_max = 30e-6; % Rotation [rad]
+ Rz_max = 0; % 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]';
+** Needed stroke for "pure" rotations or translations
+As a first estimation, we estimate the needed actuator stroke for "pure" rotations and translation.
+We do that using either the Inverse Kinematic solution or the Jacobian matrix as an approximation.
+
+#+begin_src matlab
+ LTx = stewart.J*[Tx_max 0 0 0 0 0]';
+ LTy = stewart.J*[0 Ty_max 0 0 0 0]';
+ LTz = stewart.J*[0 0 Tz_max 0 0 0]';
+ LRx = stewart.J*[0 0 0 Rx_max 0 0]';
+ LRy = stewart.J*[0 0 0 0 Ry_max 0]';
+ LRz = stewart.J*[0 0 0 0 0 Rz_max]';
#+end_src
-#+begin_src matlab :results value :exports results
+The obtain required stroke is:
+#+begin_src matlab :results value replace :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]
+: From -3.8e-05[m] to 3.8e-05[m]: Total stroke = 76.1[um]
+
+This is surely a low estimation of the required stroke.
+
+** Needed stroke for "combined" rotations or translations
+We know would like to have a more precise estimation.
+
+To do so, we may estimate the required actuator stroke for all possible combination of translation and rotation.
+
+Let's first generate all the possible combination of maximum translation and rotations.
+#+begin_src matlab
+ Ps = [2*(dec2bin(0:5^2-1,5)-'0')-1, zeros(5^2, 1)].*[Tx_max Ty_max Tz_max Rx_max Ry_max Rz_max];
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+data2orgtable(Ps, {}, {'*Tx [m]*', '*Ty [m]*', '*Tz [m]*', '*Rx [rad]*', '*Ry [rad]*', '*Rz [rad]*'}, ' %.1e ');
+#+end_src
+
+#+RESULTS:
+| *Tx [m]* | *Ty [m]* | *Tz [m]* | *Rx [rad]* | *Ry [rad]* | *Rz [rad]* |
+|----------+----------+----------+------------+------------+------------|
+| -5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
+| -5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
+| -5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
+| -5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | -5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | -5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | -5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | 5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | 5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | 5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
+| -5.0e-05 | 5.0e-05 | 5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
+| 5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
+| 5.0e-05 | 5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
+
+For all possible combination, we compute the required actuator stroke using the inverse kinematic solution.
+#+begin_src matlab
+ L_min = 0;
+ L_max = 0;
+
+ for i = 1:size(Ps,1)
+ Rx = [1 0 0;
+ 0 cos(Ps(i, 4)) -sin(Ps(i, 4));
+ 0 sin(Ps(i, 4)) cos(Ps(i, 4))];
+
+ Ry = [ cos(Ps(i, 5)) 0 sin(Ps(i, 5));
+ 0 1 0;
+ -sin(Ps(i, 5)) 0 cos(Ps(i, 5))];
+
+ Rz = [cos(Ps(i, 6)) -sin(Ps(i, 6)) 0;
+ sin(Ps(i, 6)) cos(Ps(i, 6)) 0;
+ 0 0 1];
+
+ ARB = Rz*Ry*Rx;
+ [~, Ls] = inverseKinematics(stewart, 'AP', Ps(i, 1:3)', 'ARB', ARB);
+
+ if min(Ls) < L_min
+ L_min = min(Ls)
+ end
+ if max(Ls) > L_max
+ L_max = max(Ls)
+ end
+ end
+#+end_src
+
+We obtain the required actuator stroke:
+#+begin_src matlab :results value replace :exports results
+ ans = sprintf('From %.2g[m] to %.2g[m]: Total stroke = %.1f[um]', L_min, L_max, 1e6*(L_max-L_min))
+#+end_src
+
+#+RESULTS:
+: From -8.9e-05[m] to 8.9e-05[m]: Total stroke = 177.2[um]
+
+This is probably a much realistic estimation of the required actuator stroke.
* Estimated platform mobility from specified actuator stroke
+<>
** Introduction :ignore:
+Here, from some value of the actuator stroke, we would like to estimate the mobility of the Stewart platform.
+
+As explained in section [[sec:forward_inverse_kinematics]], the forward kinematic problem of the Stewart platform is quite difficult to solve.
+However, for small displacements, we can use the Jacobian as an approximate solution.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -137,8 +469,30 @@ First, we estimate the needed actuator stroke for "pure" rotations and translati
simulinkproject('./');
#+end_src
+** Stewart architecture definition
+Let's first define the Stewart platform architecture that we want to study.
+#+begin_src matlab
+ stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+ stewart = generateGeneralConfiguration(stewart);
+ stewart = computeJointsPose(stewart);
+ stewart = initializeStewartPose(stewart);
+ stewart = initializeCylindricalPlatforms(stewart);
+ stewart = initializeCylindricalStruts(stewart);
+ stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1));
+ stewart = initializeJointDynamics(stewart);
+ stewart = computeJacobian(stewart);
+#+end_src
+
+Let's now define the actuator stroke.
+#+begin_src matlab
+ L_min = -50e-6; % [m]
+ L_max = 50e-6; % [m]
+#+end_src
+
+** TODO Approximate Forward Dynamics
* Functions
+<>
** =computeJacobian=: Compute the Jacobian Matrix
:PROPERTIES:
:header-args:matlab+: :tangle src/computeJacobian.m
@@ -149,6 +503,9 @@ First, we estimate the needed actuator stroke for "pure" rotations and translati
This Matlab function is accessible [[file:src/computeJacobian.m][here]].
*** Function description
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
function [stewart] = computeJacobian(stewart)
% computeJacobian -
@@ -168,16 +525,25 @@ This Matlab function is accessible [[file:src/computeJacobian.m][here]].
#+end_src
*** Compute Jacobian Matrix
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];
#+end_src
*** Compute Stiffness Matrix
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;
#+end_src
*** Compute Compliance Matrix
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
stewart.C = inv(stewart.K);
#+end_src
@@ -192,6 +558,9 @@ This Matlab function is accessible [[file:src/computeJacobian.m][here]].
This Matlab function is accessible [[file:src/inverseKinematics.m][here]].
*** Function description
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
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}
@@ -212,6 +581,9 @@ This Matlab function is accessible [[file:src/inverseKinematics.m][here]].
#+end_src
*** Optional Parameters
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
arguments
stewart
@@ -221,6 +593,9 @@ This Matlab function is accessible [[file:src/inverseKinematics.m][here]].
#+end_src
*** Theory
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
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
@@ -243,6 +618,9 @@ If the position and orientation of the moving platform lie in the feasible works
Otherwise, when the limbs' lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
*** Compute
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
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));
#+end_src
@@ -261,6 +639,9 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
This Matlab function is accessible [[file:src/forwardKinematicsApprox.m][here]].
*** Function description
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
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
@@ -280,6 +661,9 @@ This Matlab function is accessible [[file:src/forwardKinematicsApprox.m][here]].
#+end_src
*** Optional Parameters
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
#+begin_src matlab
arguments
stewart
@@ -288,6 +672,9 @@ This Matlab function is accessible [[file:src/forwardKinematicsApprox.m][here]].
#+end_src
*** Computation
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
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:
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
@@ -313,3 +700,6 @@ We then compute the corresponding rotation matrix.
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)];
#+end_src
+* Bibliography :ignore:
+bibliographystyle:unsrt
+bibliography:ref.bib
diff --git a/src/initializeJointDynamics.m b/src/initializeJointDynamics.m
index 586a469..a55aa17 100644
--- a/src/initializeJointDynamics.m
+++ b/src/initializeJointDynamics.m
@@ -5,19 +5,27 @@ function [stewart] = initializeJointDynamics(stewart, args)
%
% Inputs:
% - args - Structure with the following fields:
-% - Kri [6x1] - Rotational Stiffness for each spherical joints [N/rad]
-% - Cri [6x1] - Damping of each spherical joint [N/(rad/s)]
+% - Kbi [6x1] - Rotational Stiffness for each top spherical joints [N/rad]
+% - Cbi [6x1] - Damping of each top spherical joint [N/(rad/s)]
+% - Kti [6x1] - Rotational Stiffness for each bottom universal joints [N/rad]
+% - Cti [6x1] - Damping of each bottom universal joint [N/(rad/s)]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
-% - Kri [6x1] - Rotational Stiffness for each spherical joints [N/rad]
-% - Cri [6x1] - Damping of each spherical joint [N/(rad/s)]
+% - Kbi [6x1] - Rotational Stiffness for each top spherical joints [N/rad]
+% - Cbi [6x1] - Damping of each top spherical joint [N/(rad/s)]
+% - Kti [6x1] - Rotational Stiffness for each bottom universal joints [N/rad]
+% - Cti [6x1] - Damping of each bottom universal joint [N/(rad/s)]
arguments
stewart
- args.Kri (6,1) double {mustBeNumeric, mustBePositive} = zeros(6,1)
- args.Cri (6,1) double {mustBeNumeric, mustBePositive} = zeros(6,1)
+ args.Kti (6,1) double {mustBeNumeric, mustBeNonnegative} = zeros(6,1)
+ args.Cti (6,1) double {mustBeNumeric, mustBeNonnegative} = zeros(6,1)
+ args.Kbi (6,1) double {mustBeNumeric, mustBeNonnegative} = zeros(6,1)
+ args.Cbi (6,1) double {mustBeNumeric, mustBeNonnegative} = zeros(6,1)
end
-stewart.Kri = args.Kri;
-stewart.Cri = args.Cri;
+stewart.Kti = args.Kti;
+stewart.Cti = args.Cti;
+stewart.Kbi = args.Kbi;
+stewart.Cbi = args.Cbi;
diff --git a/src/initializeStrutDynamics.m b/src/initializeStrutDynamics.m
index 6863fd0..0bfd7cb 100644
--- a/src/initializeStrutDynamics.m
+++ b/src/initializeStrutDynamics.m
@@ -15,8 +15,8 @@ function [stewart] = initializeStrutDynamics(stewart, args)
arguments
stewart
- args.Ki (6,1) double {mustBeNumeric, mustBePositive} = 1e6*ones(6,1)
- args.Ci (6,1) double {mustBeNumeric, mustBePositive} = 1e1*ones(6,1)
+ args.Ki (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e6*ones(6,1)
+ args.Ci (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
end
stewart.Ki = args.Ki;
Optional Parameters
+arguments
stewart
@@ -705,9 +1412,9 @@ This Matlab function is accessible here<
-
6.3.3 Computation
-
+
+
+
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: @@ -749,10 +1456,16 @@ We then compute the corresponding rotation matrix.
+ +
Bibliography
+- [taghirad13_paral] Taghirad, Parallel robots : mechanics and control, CRC Press (2013). +
-
diff --git a/kinematic-study.org b/kinematic-study.org
index 93c3b5d..8eeb100 100644
--- a/kinematic-study.org
+++ b/kinematic-study.org
@@ -11,7 +11,6 @@
#+HTML_HEAD:
#+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
@@ -22,37 +21,288 @@
:END:
* Introduction :ignore:
+The kinematic analysis of a parallel manipulator is well described in cite:taghirad13_paral:
+#+begin_quote
+Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces an torques that cause the motion.
+In this analysis, the relation between the geometrical parameters of the manipulator with the final motion of the moving platform is derived and analyzed.
+#+end_quote
+
+The current document is divided in the following sections:
+- Section [[sec:jacobian_analysis]]: The Jacobian matrix is derived from the geometry of the Stewart platform. Then it is shown that the Jacobian can link velocities and forces present in the system, and thus this matrix can be very useful for both analysis and control of the Stewart platform.
+- Section [[sec:stiffness_analysis]]: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut.
+- Section [[sec:forward_inverse_kinematics]]: The Forward and Inverse kinematic problems are presented.
* Jacobian Analysis
+<Created: 2020-01-29 mer. 13:29
+Created: 2020-01-29 mer. 17:48