-
-
- 1. Jacobian Analysis +
- 1. Jacobian Analysis -
- 2. Stiffness Analysis +
- 2. Stiffness Analysis -
- 3. Forward and Inverse Kinematics +
- 3. Forward and Inverse Kinematics
-
-
- 3.1. Inverse Kinematics -
- 3.2. Forward Kinematics -
- 3.3. Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix -
- 3.4. Estimation of the range validity of the approximate inverse kinematics +
- 3.1. Inverse Kinematics +
- 3.2. Forward Kinematics +
- 3.3. Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix +
- 3.4. Estimation of the range validity of the approximate inverse kinematics
- - 4. Estimated required actuator stroke from specified platform mobility +
- 4. Estimated required actuator stroke from specified platform mobility
-
-
- 4.1. Stewart architecture definition -
- 4.2. Wanted translations and rotations -
- 4.3. Needed stroke for “pure” rotations or translations -
- 4.4. Needed stroke for “combined” rotations or translations +
- 4.1. Stewart architecture definition +
- 4.2. Wanted translations and rotations +
- 4.3. Needed stroke for “pure” rotations or translations +
- 4.4. Needed stroke for “combined” rotations or translations
- - 5. Estimated platform mobility from specified actuator stroke +
- 5. Estimated platform mobility from specified actuator stroke -
- 6. Functions +
- 6. Functions
-
-
- 6.1.
computeJacobian: Compute the Jacobian Matrix + - 6.1.
computeJacobian: Compute the Jacobian Matrix
- - 6.2.
inverseKinematics: Compute Inverse Kinematics + - 6.2.
inverseKinematics: Compute Inverse Kinematics
- - 6.3.
forwardKinematicsApprox: Compute the Approximate Forward Kinematics + - 6.3.
forwardKinematicsApprox: Compute the Approximate Forward Kinematics
-
-
- 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. +
- 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. +
- Section 4: The Inverse kinematic solution is used to estimate required actuator stroke from the wanted mobility of the Stewart platform.
-1 Jacobian Analysis
++1 Jacobian Analysis
-From taghirad13_paral: @@ -373,8 +374,8 @@ The Jacobian matrix not only reveals the relation between the joint variable
-1.1 Jacobian Computation
++1.1 Jacobian Computation
-If we note: @@ -399,7 +400,7 @@ Then, we can compute the Jacobian with the following equation (the superscript \ \end{equation*}
-The Jacobian matrix \(\bm{J}\) can be computed using the
computeJacobianfunction (accessible here). +The Jacobian matrix \(\bm{J}\) can be computed using thecomputeJacobianfunction (accessible here). For instance:@@ -417,8 +418,8 @@ This will add three new matrix to thestewartstructure:-1.2 Jacobian - Velocity loop closure
++-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 \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\): @@ -441,13 +442,13 @@ If the Jacobian matrix is inversible, we can also compute \(\dot{\bm{\mathcal{X}
The Jacobian matrix can also be used to approximate forward and inverse kinematics for small displacements. -This is explained in section 3.3. +This is explained in section 3.3.
-1.3 Jacobian - Static Force Transformation
++-1.3 Jacobian - Static Force Transformation
If we note: @@ -474,19 +475,19 @@ If the Jacobian matrix is inversible, we also have the following relation:
-2 Stiffness Analysis
++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 Computation of the Stiffness and Compliance Matrix
++2.1 Computation of the Stiffness and Compliance Matrix
As explain in this document, each Actuator is modeled by 3 elements in parallel: @@ -537,24 +538,24 @@ The compliance matrix of a manipulator shows the mapping of the moving platform \end{equation*}
-The stiffness and compliance matrices are computed using the
computeJacobianfunction (accessible here). +The stiffness and compliance matrices are computed using thecomputeJacobianfunction (accessible here).-3 Forward and Inverse Kinematics
++3 Forward and Inverse Kinematics
--3.1 Inverse Kinematics
++-3.1 Inverse Kinematics
-3.2 Forward Kinematics
++-3.2 Forward Kinematics
-3.3 Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix
++-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: +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 (approximate solution of the inverse kinematic problem):
\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: +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 (approximate solution of the forward kinematic problem):
\begin{equation*} \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} @@ -638,13 +639,13 @@ As the inverse kinematic can be easily solved exactly this is not much useful, h-The function
forwardKinematicsApprox(described here) can be used to solve the forward kinematic problem using the Jacobian matrix. +The functionforwardKinematicsApprox(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
++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. @@ -658,8 +659,8 @@ This will also gives us the range for which the approximate forward kinematic is
-3.4.1 Stewart architecture definition
++-3.4.1 Stewart architecture definition
We first define some general Stewart architecture. @@ -679,8 +680,8 @@ stewart = computeJacobian(stewart);
-3.4.2 Comparison for “pure” translations
++3.4.2 Comparison for “pure” translations
Let’s first compare the perfect and approximate solution of the inverse for pure \(x\) translations. @@ -688,8 +689,8 @@ Let’s first compare the perfect and approximate solution of the inverse fo
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. +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] @@ -706,14 +707,14 @@ Ls_exact = zeros(6, length(Xrs));
+-
Figure 1: Comparison of the Approximate solution and True solution for the Inverse kinematic problem (png, pdf)
+--3.4.3 Conclusion
++-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. @@ -732,11 +733,11 @@ For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod
-4 Estimated required actuator stroke from specified platform mobility
++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). @@ -744,8 +745,8 @@ One may want to determine the required actuator stroke required to obtain the sp This is what is analyzed in this section.
-4.1 Stewart architecture definition
++-4.1 Stewart architecture definition
Let’s first define the Stewart platform architecture that we want to study. @@ -765,8 +766,8 @@ stewart = computeJacobian(stewart);
-4.2 Wanted translations and rotations
++-4.2 Wanted translations and rotations
Let’s now define the wanted extreme translations and rotations. @@ -783,8 +784,8 @@ Rz_max = 0; % Rotation [rad]
-4.3 Needed stroke for “pure” rotations or translations
++-4.3 Needed stroke for “pure” rotations or translations
As a first estimation, we estimate the needed actuator stroke for “pure” rotations and translation. @@ -815,8 +816,8 @@ This is surely a low estimation of the required stroke.
-4.4 Needed stroke for “combined” rotations or translations
++-4.4 Needed stroke for “combined” rotations or translations
We know would like to have a more precise estimation. @@ -1136,23 +1137,23 @@ 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. +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.
-5.1 Stewart architecture definition
++5.1 Stewart architecture definition
-Let’s first define the Stewart platform architecture that we want to study. @@ -1181,23 +1182,100 @@ L_max = 50e-6; % [m]
--5.2 TODO Approximate Forward Dynamics
++5.2 Pure translations
+++Let’s first estimate the mobility in translation when the orientation of the Stewart platform stays the same. +
+ ++As shown previously, for such small stroke, we can use the approximate Forward Dynamics solution using the Jacobian matrix: +
+\begin{equation*} + \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}} +\end{equation*} + ++To obtain the mobility “volume” attainable by the Stewart platform when it’s orientation is set to zero, we use the spherical coordinate \((r, \theta, \phi)\). +
+ ++For each possible value of \((\theta, \phi)\), we compute the maximum radius \(r\) attainable with the constraint that the stroke of each actuator should be between
+L_minandL_max. +++ + +thetas = linspace(0, pi, 50); +phis = linspace(0, 2*pi, 50); +rs = zeros(length(thetas), length(phis)); + +for i = 1:length(thetas) + for j = 1:length(phis) + Tx = sin(thetas(i))*cos(phis(j)); + Ty = sin(thetas(i))*sin(phis(j)); + Tz = cos(thetas(i)); + + dL = stewart.J*[Tx; Ty; Tz; 0; 0; 0;]; % dL required for 1m displacement in theta/phi direction + + rs(i, j) = max([dL(dL<0)*L_min; dL(dL>0)*L_max]); + end +end +
++Now that we have found the corresponding radius \(r\), we plot the obtained mobility. +We can also approximate the mobility by a sphere with a radius equal to the minimum obtained value of \(r\), this is however a pessimistic estimation of the mobility. +
+ ++ + +
+ + + ++ + ++ + + + + + + + +L_min[\(\mu m\)]L_max[\(\mu m\)]R[\(\mu m\)]+ + +-50.0 +50.0 +31.5 +-6 Functions
++6 Functions
--6.1
+computeJacobian: Compute the Jacobian Matrix+6.1
computeJacobian: Compute the Jacobian Matrix-@@ -1205,9 +1283,9 @@ This Matlab function is accessible here.
-Function description
-++Function description
+-function [stewart] = computeJacobian(stewart) % computeJacobian - @@ -1229,9 +1307,9 @@ This Matlab function is accessible here.
-Compute Jacobian Matrix
-++Compute Jacobian Matrix
+-stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];
@@ -1239,9 +1317,9 @@ This Matlab function is accessible here.-Compute Stiffness Matrix
-++Compute Stiffness Matrix
+-stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;
@@ -1249,9 +1327,9 @@ This Matlab function is accessible here.-Compute Compliance Matrix
-++Compute Compliance Matrix
+-stewart.C = inv(stewart.K);
@@ -1260,11 +1338,11 @@ This Matlab function is accessible here.-6.2
+inverseKinematics: Compute Inverse Kinematics+6.2
inverseKinematics: Compute Inverse Kinematics-@@ -1272,9 +1350,9 @@ This Matlab function is accessible here.
-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} @@ -1297,9 +1375,9 @@ This Matlab function is accessible here.
-Optional Parameters
-++Optional Parameters
+-arguments stewart @@ -1311,9 +1389,9 @@ This Matlab function is accessible here.-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, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
@@ -1347,9 +1425,9 @@ Otherwise, when the limbs’ lengths derived yield complex numbers, then the-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));
@@ -1363,11 +1441,11 @@ Otherwise, when the limbs’ lengths derived yield complex numbers, then the-6.3
+forwardKinematicsApprox: Compute the Approximate Forward Kinematics+6.3
forwardKinematicsApprox: Compute the Approximate Forward Kinematics-@@ -1375,9 +1453,9 @@ This Matlab function is accessible here<
-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 @@ -1399,9 +1477,9 @@ This Matlab function is accessible here<
-Optional Parameters
-++Optional Parameters
+-arguments stewart @@ -1412,9 +1490,9 @@ This Matlab function is accessible here<-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: @@ -1465,7 +1543,7 @@ We then compute the corresponding rotation matrix.
-diff --git a/kinematic-study.org b/kinematic-study.org index 8eeb100..bbf19ca 100644 --- a/kinematic-study.org +++ b/kinematic-study.org @@ -31,7 +31,8 @@ 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. - +- Section [[sec:required_actuator_stroke]]: The Inverse kinematic solution is used to estimate required actuator stroke from the wanted mobility of the Stewart platform. + * Jacobian Analysis <Created: 2020-01-29 mer. 17:48
+Created: 2020-01-29 mer. 20:22
> +#+end_src + +#+NAME: fig:mobility_translations_null_rotation +#+CAPTION: Obtain mobility of the Stewart platform for zero rotations ([[./figs/mobility_translations_null_rotation.png][png]], [[./figs/mobility_translations_null_rotation.pdf][pdf]]) +[[file:figs/mobility_translations_null_rotation.png]] + +*** TODO Do that by slice :noexport: +using this function https://fr.mathworks.com/help/matlab/ref/contour3.html * Functions < - 6.1.
+
