-
-
- Inertial Control +
- 1. Inertial Control -
- Integral Force Feedback +
- 2. Integral Force Feedback -
- Direct Velocity Feedback +
- 3. Direct Velocity Feedback
-
-
- Inertial Control (proportional feedback of the absolute velocity): Section No description for this link -
- Integral Force Feedback: Section No description for this link -
- Direct feedback of the relative velocity of each strut: Section No description for this link +
- Inertial Control (proportional feedback of the absolute velocity): Section 1 +
- Integral Force Feedback: Section 2 +
- Direct feedback of the relative velocity of each strut: Section 3
Inertial Control
-1 Inertial Control
+Identification of the Dynamics
-1.1 Identification of the Dynamics
+stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); @@ -362,9 +362,9 @@ The transfer function from actuator forces to force sensors is shown in Figure <
Effect of the Flexible Joint stiffness on the Dynamics
-1.2 Effect of the Flexible Joint stiffness on the Dynamics
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
@@ -388,9 +388,9 @@ The new dynamics from force actuator to force sensor is shown in FigureObtained Damping
-1.3 Obtained Damping
+The control is a performed in a decentralized manner. The \(6 \times 6\) control is a diagonal matrix with pure proportional action on the diagonal: @@ -421,9 +421,9 @@ The root locus is shown in figure 3 and the obtained p
Conclusion
-1.4 Conclusion
+Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors. @@ -435,16 +435,16 @@ Joint stiffness does increase the resonance frequencies of the system but does n
Integral Force Feedback
-2 Integral Force Feedback
+ -Identification of the Dynamics with perfect Joints
-2.1 Identification of the Dynamics with perfect Joints
+We first initialize the Stewart platform without joint stiffness.
@@ -498,9 +498,9 @@ The transfer function from actuator forces to force sensors is shown in Figure <Effect of the Flexible Joint stiffness on the Dynamics
-2.2 Effect of the Flexible Joint stiffness on the Dynamics
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
@@ -524,9 +524,9 @@ The new dynamics from force actuator to force sensor is shown in FigureObtained Damping
-2.3 Obtained Damping
+The control is a performed in a decentralized manner. The \(6 \times 6\) control is a diagonal matrix with pure integration action on the diagonal: @@ -557,9 +557,9 @@ The root locus is shown in figure 7 and the obtained p
Conclusion
-2.4 Conclusion
+The joint stiffness has a huge impact on the attainable active damping performance when using force sensors. @@ -572,16 +572,16 @@ Thus, if Integral Force Feedback is to be used in a Stewart platform with flexib
Direct Velocity Feedback
-3 Direct Velocity Feedback
+ -Identification of the Dynamics with perfect Joints
-3.1 Identification of the Dynamics with perfect Joints
+We first initialize the Stewart platform without joint stiffness.
@@ -635,9 +635,9 @@ The transfer function from actuator forces to relative motion sensors is shown iEffect of the Flexible Joint stiffness on the Dynamics
-3.2 Effect of the Flexible Joint stiffness on the Dynamics
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
@@ -661,9 +661,9 @@ The new dynamics from force actuator to relative motion sensor is shown in FigurObtained Damping
-3.3 Obtained Damping
+The control is a performed in a decentralized manner. The \(6 \times 6\) control is a diagonal matrix with pure derivative action on the diagonal: @@ -694,9 +694,9 @@ The root locus is shown in figure 11 and the obtained
Conclusion
-3.4 Conclusion
+Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors. @@ -709,7 +709,7 @@ Joint stiffness does increase the resonance frequencies of the system but does n
Created: 2020-02-11 mar. 15:26
+Created: 2020-02-11 mar. 15:50
Table of Contents
First Control Architecture
-1 First Control Architecture
+Control Schematic
-\begin{tikzpicture} - % Blocs - \node[block] (J) at (0, 0) {$J$}; - \node[addb={+}{}{}{}{-}, right=1 of J] (subr) {}; - \node[block, right=0.8 of subr] (K) {$K_{L}$}; - \node[block, right=1 of K] (G) {$G_{L}$}; - - % Connections and labels - \draw[<-] (J.west)node[above left]{$\bm{r}_{n}$} -- ++(-1, 0); - \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{L}$}; - \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{L}$}; - \draw[->] (K.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) node[above right]{$\bm{L}$} -| ($(G.east)+(1, -1)$) -| (subr.south); -\end{tikzpicture} --
1.1 Control Schematic
+
@@ -316,8 +297,8 @@ for the JavaScript code in this tag.
Initialize the Stewart platform
-1.2 Initialize the Stewart platform
+stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart); @@ -334,8 +315,8 @@ stewart = initializeStewartPose(stewart);
Identification of the plant
-1.3 Identification of the plant
+Let’s identify the transfer function from \(\bm{\tau}}\) to \(\bm{L}\).
@@ -362,8 +343,8 @@ G.OutputName = {'L1', 'Plant Analysis
-1.4 Plant Analysis
+Diagonal terms Compare to off-diagonal terms @@ -371,8 +352,8 @@ Compare to off-diagonal terms
Controller Design
-1.5 Controller Design
+One integrator should be present in the controller.
@@ -401,7 +382,7 @@ Kl = Kl * eye(6);Created: 2020-02-11 mar. 15:23
+Created: 2020-02-11 mar. 15:50
Table of Contents
-
-
- Configuration Analysis - Stiffness Matrix +
- 1. Configuration Analysis - Stiffness Matrix
-
-
- Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center -
- Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center -
- Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center -
- Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center -
- Conclusion -
- Having Cube’s center above the top platform +
- 1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center +
- 1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center +
- 1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center +
- 1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center +
- 1.5. Conclusion +
- 1.6. Having Cube’s center above the top platform
- - Functions +
- 2. Functions
-
-
generateCubicConfiguration: Generate a Cubic Configuration +- 2.1.
generateCubicConfiguration: Generate a Cubic Configuration- Function description
- Documentation @@ -315,7 +315,7 @@ According to No description for this link). +To generate and study the Cubic configuration,
- The cubic configuration permits to have \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\) @@ -771,8 +771,8 @@ stewart = initializeCylindricalPlatforms(stewart, 'Fpr'
- Some tests +
- 1. Some tests
-
-
- Simscape Model -
- test -
- Compare external forces and forces applied by the actuators -
- Comparison of the static transfer function and the Compliance matrix -
- Transfer function from forces applied in the legs to the displacement of the legs +
- 1.1. Simscape Model +
- 1.2. test +
- 1.3. Compare external forces and forces applied by the actuators +
- 1.4. Comparison of the static transfer function and the Compliance matrix +
- 1.5. Transfer function from forces applied in the legs to the displacement of the legs
- Identification +
- 1. Identification -
- States as the motion of the mobile platform +
- 2. States as the motion of the mobile platform
-
-
- Initialize the Stewart Platform -
- Identification -
- Coordinate transformation -
- Analysis -
- Visualizing the modes -
- Identification -
- Change of states +
- 2.1. Initialize the Stewart Platform +
- 2.2. Identification +
- 2.3. Coordinate transformation +
- 2.4. Analysis +
- 2.5. Visualizing the modes +
- 2.6. Identification +
- 2.7. Change of states
- - Simple Model without any sensor +
- 3. Simple Model without any sensor -
- Cartesian Plot -
- From a force to force sensor -
- From a force applied in the leg to the displacement of the leg -
- Transmissibility -
- Compliance -
- Inertial +
- 4. Cartesian Plot +
- 5. From a force to force sensor +
- 6. From a force applied in the leg to the displacement of the leg +
- 7. Transmissibility +
- 8. Compliance +
- 9. Inertial
- Simulink Project (link) -
- Stewart Platform Architecture Definition (link) -
- Simscape Model of the Stewart Platform (link) -
- Kinematic Analysis (link) -
- Identification of the Stewart Dynamics (link) -
- Active Damping (link) -
- Motion Control of the Stewart Platform (link) -
- Cubic Configuration (link) +
- 1. Simulink Project (link) +
- 2. Stewart Platform Architecture Definition (link) +
- 3. Simscape Model of the Stewart Platform (link) +
- 4. Kinematic Analysis (link) +
- 5. Identification of the Stewart Dynamics (link) +
- 6. Active Damping (link) +
- 7. Motion Control of the Stewart Platform (link) +
- 8. Cubic Configuration (link)
- Jacobian Analysis +
- 1. Jacobian Analysis -
- Stiffness Analysis +
- 2. Stiffness Analysis -
- Forward and Inverse Kinematics +
- 3. Forward and Inverse Kinematics
-
-
- Inverse Kinematics -
- Forward Kinematics -
- Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix -
- 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
- - Estimated required actuator stroke from specified platform mobility +
- 4. Estimated required actuator stroke from specified platform mobility
-
-
- Stewart architecture definition -
- Wanted translations and rotations -
- Needed stroke for “pure” rotations or translations -
- 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
- - Estimated platform mobility from specified actuator stroke +
- 5. Estimated platform mobility from specified actuator stroke -
- Functions +
- 6. Functions
-
-
computeJacobian: Compute the Jacobian Matrix +- 6.1.
computeJacobian: Compute the Jacobian Matrix
- inverseKinematics: Compute Inverse Kinematics +- 6.2.
inverseKinematics: Compute Inverse Kinematics
- forwardKinematicsApprox: Compute the Approximate Forward Kinematics +- 6.3.
forwardKinematicsApprox: Compute the Approximate Forward Kinematics
@@ -357,15 +357,15 @@ In this analysis, the relation between the geometrical parameters of the manipul
The current document is divided in the following sections:
- Section No description for this link: 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 No description for this link: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut. -
- Section No description for this link: The Forward and Inverse kinematic problems are presented. -
- Section No description for this link: The Inverse kinematic solution is used to estimate required actuator stroke from the wanted mobility of the Stewart platform. +
- 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.
-
-
-Jacobian Analysis
-+1 Jacobian Analysis
+-Jacobian Computation
-+1.1 Jacobian Computation
+If we note:
@@ -423,8 +423,8 @@ This will add three new matrix to thestewartstructure:-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 \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\):
@@ -446,14 +446,14 @@ 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 No description for this link. +This is explained in section 3.3.
-Jacobian - Static Force Transformation
-+1.3 Jacobian - Static Force Transformation
+If we note:
@@ -480,8 +480,8 @@ If the Jacobian matrix is inversible, we also have the following relation:-Stiffness Analysis
-+2 Stiffness Analysis
+-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:
@@ -549,15 +549,15 @@ The stiffness and compliance matrices are computed using thecomputeJacobi-Forward and Inverse Kinematics
-+3 Forward and Inverse Kinematics
+-Inverse Kinematics
-+3.1 Inverse Kinematics
+-Forward Kinematics
-+3.2 Forward Kinematics
+-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
+@@ -649,8 +649,8 @@ The functionforwardKinematicsApprox(described -Estimation of the range validity of the approximate inverse kinematics
-+3.4 Estimation of the range validity of the approximate inverse kinematics
+@@ -666,9 +666,9 @@ This will also gives us the range for which the approximate forward kinematic is--Stewart architecture definition
-++3.4.1 Stewart architecture definition
+We first define some general Stewart architecture.
@@ -689,8 +689,8 @@ stewart = computeJacobian(stewart);-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.
@@ -731,8 +731,8 @@ Ls_exact = zeros(6, length(Xrs));-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.
@@ -742,8 +742,8 @@ For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod-Estimated required actuator stroke from specified platform mobility
-+4 Estimated required actuator stroke from specified platform mobility
+@@ -753,9 +753,9 @@ One may want to determine the required actuator stroke required to obtain the sp This is what is analyzed in this section.--Stewart architecture definition
-++4.1 Stewart architecture definition
+Let’s first define the Stewart platform architecture that we want to study.
@@ -776,8 +776,8 @@ stewart = computeJacobian(stewart);-Wanted translations and rotations
-+4.2 Wanted translations and rotations
+Let’s now define the wanted extreme translations and rotations.
@@ -794,8 +794,8 @@ Rz_max = 0; % Rotation [rad]-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. We do that using either the Inverse Kinematic solution or the Jacobian matrix as an approximation. @@ -826,8 +826,8 @@ This is surely a low estimation of the required stroke.
-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.
@@ -1147,8 +1147,8 @@ This is probably a much realistic estimation of the required actuator stroke.-Estimated platform mobility from specified actuator stroke
-+5 Estimated platform mobility from specified actuator stroke
+@@ -1157,13 +1157,13 @@ Here, from some value of the actuator stroke, we would like to estimate the mobi--As explained in section No description for this link, 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.
-Stewart architecture definition
-++5.1 Stewart architecture definition
+Let’s first define the Stewart platform architecture that we want to study.
@@ -1193,8 +1193,8 @@ L_max = 50e-6; % [m]-Pure translations
-+5.2 Pure translations
+Let’s first estimate the mobility in translation when the orientation of the Stewart platform stays the same.
@@ -1275,15 +1275,15 @@ We can also approximate the mobility by a sphere with a radius equal to the mini-Functions
-+6 Functions
+-
-computeJacobian: Compute the Jacobian Matrix+6.1
+computeJacobian: Compute the Jacobian Matrix@@ -1293,9 +1293,9 @@ This Matlab function is accessible here.--Function description
-++Function description
+-function [stewart] = computeJacobian(stewart) % computeJacobian - @@ -1318,9 +1318,9 @@ This Matlab function is accessible here.
-Check the
-stewartstructure elements++Check the
+stewartstructure elementsassert(isfield(stewart.geometry, 'As'), 'stewart.geometry should have attribute As') As = stewart.geometry.As; @@ -1381,8 +1381,8 @@ stewart.kinematics.C = C;
-
-inverseKinematics: Compute Inverse Kinematics+-6.2
+inverseKinematics: Compute Inverse Kinematics-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} @@ -1454,9 +1454,9 @@ Otherwise, when the limbs’ lengths derived yield complex numbers, then the
-Optional Parameters
-++Optional Parameters
+-arguments stewart @@ -1468,9 +1468,9 @@ Otherwise, when the limbs’ lengths derived yield complex numbers, then the-Check the
-stewartstructure elements++Check the
+stewartstructure elementsassert(isfield(stewart.geometry, 'Aa'), 'stewart.geometry should have attribute Aa') Aa = stewart.geometry.Aa; @@ -1503,8 +1503,8 @@ l = stewart.geometry.l;
-
-forwardKinematicsApprox: Compute the Approximate Forward Kinematics+6.3
+forwardKinematicsApprox: Compute the Approximate Forward Kinematics@@ -1514,9 +1514,9 @@ This Matlab function is accessible here<--Function description
-++Function description
+
generateCubicConfigurationis used (description in section 2.1). The goal is to study the benefits of using a cubic configuration:-
@@ -325,12 +325,12 @@ The goal is to study the benefits of using a cubic configuration:
-Configuration Analysis - Stiffness Matrix
-+1 Configuration Analysis - Stiffness Matrix
+-Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
-+1.1 Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
+We create a cubic Stewart platform (figure 1) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot). The Jacobian matrix is estimated at the location of the center of the cube. @@ -437,8 +437,8 @@ stewart = initializeCylindricalPlatforms(stewart, 'Fpr'
-Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
-+1.2 Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
+We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure 1). The Jacobian matrix is not estimated at the location of the center of the cube. @@ -538,8 +538,8 @@ stewart = initializeCylindricalPlatforms(stewart, 'Fpr'
-Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
-+1.3 Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
+Here, the “center” of the Stewart platform is not at the cube center (figure 4). The Jacobian is estimated at the cube center. @@ -650,8 +650,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
-Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center
-+1.4 Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center
+Here, the “center” of the Stewart platform is not at the cube center. The Jacobian is estimated at the center of the Stewart platform. @@ -758,8 +758,8 @@ stewart = initializeCylindricalPlatforms(stewart, 'Fpr'
-Conclusion
-+1.5 Conclusion
+-Having Cube’s center above the top platform
-+1.6 Having Cube’s center above the top platform
+Let’s say we want to have a decouple dynamics above the top platform. Thus, we want the cube’s center to be located above the top center. @@ -881,16 +881,16 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
-Functions
-+2 Functions
+-
-generateCubicConfiguration: Generate a Cubic Configuration+2.1
+generateCubicConfiguration: Generate a Cubic Configuration-diff --git a/docs/dynamics-study.html b/docs/dynamics-study.html index fc82af2..b454573 100644 --- a/docs/dynamics-study.html +++ b/docs/dynamics-study.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - +Created: 2020-02-11 mar. 15:26
+Created: 2020-02-11 mar. 15:50
Stewart Platform - Dynamics Study @@ -268,13 +268,13 @@ for the JavaScript code in this tag.Table of Contents
-
-
-Some tests
-+1 Some tests
+-Simscape Model
-+1.1 Simscape Model
+open('stewart_platform_dynamics.slx')@@ -296,8 +296,8 @@ for the JavaScript code in this tag.-test
-+1.2 test
+stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart); @@ -397,8 +397,8 @@ bode(Gd, G)
-Compare external forces and forces applied by the actuators
-+1.3 Compare external forces and forces applied by the actuators
+Initialization of the Stewart platform.
@@ -482,8 +482,8 @@ Seems quite similar.-Comparison of the static transfer function and the Compliance matrix
-+1.4 Comparison of the static transfer function and the Compliance matrix
+Initialization of the Stewart platform.
@@ -685,8 +685,8 @@ The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathc-Transfer function from forces applied in the legs to the displacement of the legs
-+@@ -396,17 +396,17 @@ An important difference from basic Simulink models is that the states in a physi -1.5 Transfer function from forces applied in the legs to the displacement of the legs
+Initialization of the Stewart platform.
@@ -742,7 +742,7 @@ G.OutputName = {'L1', '-diff --git a/docs/identification.html b/docs/identification.html index 1e24e68..550b650 100644 --- a/docs/identification.html +++ b/docs/identification.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - +Created: 2020-02-11 mar. 15:27
+Created: 2020-02-11 mar. 15:50
Identification of the Stewart Platform using Simscape @@ -268,37 +268,37 @@ for the JavaScript code in this tag.Table of Contents
-
-
-Identification
-++1 Identification
+--Simscape Model
++-1.1 Simscape Model
-Initialize the Stewart Platform
-++1.2 Initialize the Stewart Platform
+-stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart); @@ -422,9 +422,9 @@ stewart = initializeStewartPose(stewart);
-Identification
-++1.3 Identification
+%% Options for Linearized options = linearizeOptions; @@ -458,12 +458,12 @@ G.OutputName = {'Xdx',
-States as the motion of the mobile platform
-+2 States as the motion of the mobile platform
+--Initialize the Stewart Platform
-++2.1 Initialize the Stewart Platform
+-stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart); @@ -479,9 +479,9 @@ stewart = initializeStewartPose(stewart);
-Identification
-++2.2 Identification
+%% Options for Linearized options = linearizeOptions; @@ -541,8 +541,8 @@ And indeed, we obtain 12 states.-Coordinate transformation
-+2.3 Coordinate transformation
+We can perform the following transformation using the
@@ -577,8 +577,8 @@ Gt = ss(At, Bt, Ct, Dt);ss2sscommand.-Analysis
-+2.4 Analysis
+[V,D] = eig(Gt.A);
@@ -643,8 +643,8 @@ Gt = ss(At, Bt, Ct, Dt);-Visualizing the modes
-+-2.5 Visualizing the modes
+To visualize the i’th mode, we may excite the system using the inputs \(U_i\) such that \(B U_i\) is co-linear to \(\xi_i\) (the mode we want to excite).
@@ -745,9 +745,9 @@ Save the movie of the mode shape.-Identification
-++2.6 Identification
+%% Options for Linearized options = linearizeOptions; @@ -776,8 +776,8 @@ G = linearize(mdl, io, options);-Change of states
-+2.7 Change of states
+At = G.C*G.A*pinv(G.C); @@ -802,12 +802,12 @@ Dt = zeros(12, 6);
-Simple Model without any sensor
-+3 Simple Model without any sensor
+--Simscape Model
-++3.1 Simscape Model
+-open 'stewart_identification_simple.slx'@@ -816,9 +816,9 @@ Dt = zeros(12, 6);-Initialize the Stewart Platform
-++3.2 Initialize the Stewart Platform
+-stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart); @@ -834,9 +834,9 @@ stewart = initializeStewartPose(stewart);
-Identification
-++3.3 Identification
+stateorder = {... 'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_1_1_1',... @@ -923,8 +923,8 @@ G.OutputName = {'Xdx',-Cartesian Plot
-+4 Cartesian Plot
+From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
@@ -949,8 +949,8 @@ bode(G.G_cart, freqs);-From a force to force sensor
-+5 From a force to force sensor
+figure; hold on; @@ -981,8 +981,8 @@ legend('location', 'sou
-From a force applied in the leg to the displacement of the leg
-+6 From a force applied in the leg to the displacement of the leg
+figure; hold on; @@ -1012,8 +1012,8 @@ legend('location', 'nor
-Transmissibility
-+7 Transmissibility
+figure; hold on; @@ -1053,8 +1053,8 @@ xlabel('Frequency [Hz]'); ylabel( -
Compliance
-+8 Compliance
+From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.
@@ -1074,8 +1074,8 @@ xlabel('Frequency [Hz]'); ylabel( -Inertial
-+@@ -275,8 +275,8 @@ The project is divided into several section listed below.9 Inertial
+diff --git a/docs/index.html b/docs/index.html index 0512b44..2cb858e 100644 --- a/docs/index.html +++ b/docs/index.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - +From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.
@@ -1096,7 +1096,7 @@ xlabel('Frequency [Hz]'); ylabel( -Created: 2020-02-11 mar. 15:26
+Created: 2020-02-11 mar. 15:50
Stewart Platforms @@ -254,14 +254,14 @@ for the JavaScript code in this tag.Table of Contents
-
-
-Simulink Project (link)
-+1 Simulink Project (link)
+The project is managed with a Simulink Project. Such project is briefly presented here. @@ -285,8 +285,8 @@ Such project is briefly presented here.
-Stewart Platform Architecture Definition (link)
-+2 Stewart Platform Architecture Definition (link)
+The way the Stewart Platform is defined here.
@@ -311,8 +311,8 @@ Other parameters are also defined such as:-Simscape Model of the Stewart Platform (link)
-+3 Simscape Model of the Stewart Platform (link)
+The Stewart Platform is then modeled using Simscape.
@@ -324,8 +324,8 @@ The way to model is build and works is explained h-Kinematic Analysis (link)
-+4 Kinematic Analysis (link)
+From the defined geometry of the Stewart platform, we can perform static analysis such as:
@@ -345,8 +345,8 @@ All these analysis are described here.-Identification of the Stewart Dynamics (link)
-+5 Identification of the Stewart Dynamics (link)
+The Dynamics of the Stewart platform can be identified using the Simscape model.
@@ -367,8 +367,8 @@ The code that is used for identification is explained -Active Damping (link)
-+6 Active Damping (link)
+The use of different sensors are compared for active damping:
@@ -386,8 +386,8 @@ The result of the analysis is accessible here.-Motion Control of the Stewart Platform (link)
-+7 Motion Control of the Stewart Platform (link)
+Some control architecture for motion control of the Stewart platform are applied on the Simscape model and compared in this document.
@@ -395,8 +395,8 @@ Some control architecture for motion control of the Stewart platform are applied-Cubic Configuration (link)
-+8 Cubic Configuration (link)
+The cubic configuration is a special class of Stewart platform that has interesting properties.
@@ -409,7 +409,7 @@ These properties are studied in this docu-diff --git a/docs/kinematic-study.html b/docs/kinematic-study.html index a1a58a5..6e61920 100644 --- a/docs/kinematic-study.html +++ b/docs/kinematic-study.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - +Created: 2020-02-11 mar. 15:27
+Created: 2020-02-11 mar. 15:50
Kinematic Study of the Stewart Platform @@ -268,72 +268,72 @@ for the JavaScript code in this tag.Table of Contents
-
-