Study: estimation of the required joint stroke

org/kinematic-study.org

@@ -613,6 +613,64 @@ data2orgtable([1e6*L_min, 1e6*L_max, 1e6*(min(min(rs)))], {}, {'=L_min= [$\mu m$
 *** TODO Do that by slice                                         :noexport:
 using this function https://fr.mathworks.com/help/matlab/ref/contour3.html
 
+* Estimation of the Joint required Stroke
+** Introduction                                                      :ignore:
+
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+  <<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+  <<matlab-init>>
+#+end_src
+
+#+begin_src matlab :results none :exports none
+  simulinkproject('../');
+#+end_src
+
+** Example of the initialization of a Stewart Platform
+Let's first define the Stewart Platform Geometry.
+#+begin_src matlab
+  stewart = initializeStewartPlatform();
+  stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 150e-3);
+  stewart = generateGeneralConfiguration(stewart);
+  stewart = computeJointsPose(stewart);
+
+  As_init = stewart.geometry.As;
+#+end_src
+
+#+begin_src matlab
+  Tx_max = 50e-6; % Translation [m]
+  Ty_max = 50e-6; % Translation [m]
+  Tz_max = 50e-6; % Translation [m]
+#+end_src
+
+#+begin_src matlab
+  Ps = [2*(dec2bin(0:3^2-2,3)-'0')-1].*[Tx_max Ty_max Tz_max];
+#+end_src
+
+#+begin_src matlab
+  flex_ang = zeros(size(Ps, 1), 6);
+
+  for Ps_i = 1:size(Ps, 1)
+      stewart.geometry.FO_M = [0; 0; 90e-3] + Ps(Ps_i, :)';
+
+      stewart = generateGeneralConfiguration(stewart);
+      stewart = computeJointsPose(stewart);
+
+      flex_ang(Ps_i, :) = acos(sum(As_init.*stewart.geometry.As));
+  end
+#+end_src
+
+And the maximum bending of the flexible joints is: (in [mrad])
+#+begin_src matlab :results replace value
+  1e3*max(max(abs(flex_ang)))
+#+end_src
+
+#+RESULTS:
+: 0.90937
+
 * Functions
 <<sec:functions>>
 ** =computeJacobian=: Compute the Jacobian Matrix

org/stewart-architecture.org

@@ -951,6 +951,8 @@ This Matlab function is accessible [[file:../src/initializeCylindricalStruts.m][
 #+begin_src matlab
   arguments
       stewart
+      args.type_F    char   {mustBeMember(args.type_F,{'cylindrical', 'none'})} = 'cylindrical'
+      args.type_M    char   {mustBeMember(args.type_M,{'cylindrical', 'none'})} = 'cylindrical'
       args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
       args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
       args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
@@ -995,7 +997,12 @@ This Matlab function is accessible [[file:../src/initializeCylindricalStruts.m][
 :UNNUMBERED: t
 :END:
 #+begin_src matlab
+  switch args.type_M
+    case 'cylindrical'
       stewart.struts_M.type = 1;
+    case 'none'
+      stewart.struts_M.type = 2;
+  end
 
   stewart.struts_M.I = I_M;
   stewart.struts_M.M = Msm;
@@ -1004,7 +1011,12 @@ This Matlab function is accessible [[file:../src/initializeCylindricalStruts.m][
 #+end_src
 
 #+begin_src matlab
+  switch args.type_F
+    case 'cylindrical'
       stewart.struts_F.type = 1;
+    case 'none'
+      stewart.struts_F.type = 2;
+  end
 
   stewart.struts_F.I = I_F;
   stewart.struts_F.M = Fsm;
@@ -1237,10 +1249,83 @@ A simplistic model of such amplified actuator is shown in Figure [[fig:amplified
   stewart.actuators.Cr = args.Cr;
 
   stewart.actuators.K = K;
-  stewart.actuators.C = K;
+  stewart.actuators.C = C;
 #+end_src
 
-** =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints
+** TODO =initializeFlexibleStrutDynamics=: Model each strut with a flexible element
+:PROPERTIES:
+:header-args:matlab+: :tangle ../src/initializeFlexibleStrutDynamics.m
+:header-args:matlab+: :comments none :mkdirp yes :eval no
+:END:
+<<sec:initializeFlexibleStrutDynamics>>
+
+This Matlab function is accessible [[file:../src/initializeFlexibleStrutDynamics.m][here]].
+
+*** Function description
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+#+begin_src matlab
+  function [stewart] = initializeFlexibleStrutDynamics(stewart, args)
+  % initializeFlexibleStrutDynamics - Add Stiffness and Damping properties of each strut
+  %
+  % Syntax: [stewart] = initializeFlexibleStrutDynamics(args)
+  %
+  % Inputs:
+  %    - args - Structure with the following fields:
+  %        - K [nxn] - Vertical stiffness contribution of the piezoelectric stack [N/m]
+  %        - M [nxn] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
+  %        - xi        [1x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
+  %        - step_file [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
+  %
+  % Outputs:
+  %    - stewart - updated Stewart structure with the added fields:
+#+end_src
+
+*** Optional Parameters
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+#+begin_src matlab
+  arguments
+      stewart
+      args.K        double {mustBeNumeric} = zeros(6,6)
+      args.M        double {mustBeNumeric} = zeros(6,6)
+      args.H        double {mustBeNumeric} = 0
+      args.n_xyz    double {mustBeNumeric} = zeros(2,3)
+      args.xi       double {mustBeNumeric} = 0.1
+      args.step_file char {} = ''
+  end
+#+end_src
+
+*** Compute the axial offset
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+#+begin_src matlab
+  stewart.actuators.ax_off = (stewart.geometry.l(1) - args.H)/2; % Axial Offset at the ends of the actuator
+#+end_src
+
+*** Populate the =stewart= structure
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+#+begin_src matlab
+  stewart.actuators.type = 3;
+
+  stewart.actuators.Km = args.K;
+  stewart.actuators.Mm = args.M;
+
+  stewart.actuators.n_xyz = args.n_xyz;
+  stewart.actuators.xi = args.xi;
+
+  stewart.actuators.step_file = args.step_file;
+
+  stewart.actuators.K = args.K(3,3); % Axial Stiffness
+#+end_src
+
+** TODO =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints
 :PROPERTIES:
 :header-args:matlab+: :tangle ../src/initializeJointDynamics.m
 :header-args:matlab+: :comments none :mkdirp yes :eval no
@@ -1291,8 +1376,8 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
 #+begin_src matlab
   arguments
       stewart
-      args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'universal'
+      args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'universal'
-      args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'spherical'
+      args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'spherical'
       args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
       args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
       args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
@@ -1301,6 +1386,16 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
       args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
       args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
       args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
+      args.K_M        double {mustBeNumeric} = zeros(6,6)
+      args.M_M        double {mustBeNumeric} = zeros(6,6)
+      args.n_xyz_M    double {mustBeNumeric} = zeros(2,3)
+      args.xi_M       double {mustBeNumeric} = 0.1
+      args.step_file_M char {} = ''
+      args.K_F        double {mustBeNumeric} = zeros(6,6)
+      args.M_F        double {mustBeNumeric} = zeros(6,6)
+      args.n_xyz_F    double {mustBeNumeric} = zeros(2,3)
+      args.xi_F       double {mustBeNumeric} = 0.1
+      args.step_file_F char {} = ''
   end
 #+end_src
 
@@ -1318,6 +1413,8 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
       stewart.joints_F.type = 3;
     case 'spherical_p'
       stewart.joints_F.type = 4;
+    case 'flexible'
+      stewart.joints_F.type = 5;
   end
 
   switch args.type_M
@@ -1329,6 +1426,8 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
       stewart.joints_M.type = 3;
     case 'spherical_p'
       stewart.joints_M.type = 4;
+    case 'flexible'
+      stewart.joints_M.type = 5;
   end
 #+end_src
 
@@ -1371,7 +1470,6 @@ Rotational Stiffness
   stewart.joints_F.Kt = args.Kf_F;
 #+end_src
 
-
 Rotational Damping
 #+begin_src matlab
   stewart.joints_M.Cf = args.Cf_M;
@@ -1381,6 +1479,26 @@ Rotational Damping
   stewart.joints_F.Ct = args.Cf_F;
 #+end_src
 
+*** Stiffness and Mass matrices for flexible joint
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+#+begin_src matlab
+  stewart.joints_F.M = args.M_F;
+  stewart.joints_F.K = args.K_F;
+  stewart.joints_F.n_xyz = args.n_xyz_F;
+  stewart.joints_F.xi = args.xi_F;
+  stewart.joints_F.xi = args.xi_F;
+  stewart.joints_F.step_file = args.step_file_F;
+
+  stewart.joints_M.M = args.M_M;
+  stewart.joints_M.K = args.K_M;
+  stewart.joints_M.n_xyz = args.n_xyz_M;
+  stewart.joints_M.xi = args.xi_M;
+  stewart.joints_M.step_file = args.step_file_M;
+#+end_src
+
 ** =initializeInertialSensor=: Initialize the inertial sensor in each strut
 :PROPERTIES:
 :header-args:matlab+: :tangle ../src/initializeInertialSensor.m

src/initializeAmplifiedStrutDynamics.m

@@ -40,4 +40,4 @@ stewart.actuators.Kr = args.Kr;
 stewart.actuators.Cr = args.Cr;
 
 stewart.actuators.K = K;
-stewart.actuators.C = K;
+stewart.actuators.C = C;

src/initializeCylindricalStruts.m

@@ -27,6 +27,8 @@ function [stewart] = initializeCylindricalStruts(stewart, args)
 
 arguments
     stewart
+    args.type_F    char   {mustBeMember(args.type_F,{'cylindrical', 'none'})} = 'cylindrical'
+    args.type_M    char   {mustBeMember(args.type_M,{'cylindrical', 'none'})} = 'cylindrical'
     args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
     args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
     args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
@@ -56,14 +58,24 @@ for i = 1:6
                      1/2  * Msm(i) * Msr(i)^2]);
 end
 
-stewart.struts_M.type = 1;
+switch args.type_M
+  case 'cylindrical'
+    stewart.struts_M.type = 1;
+  case 'none'
+    stewart.struts_M.type = 2;
+end
 
 stewart.struts_M.I = I_M;
 stewart.struts_M.M = Msm;
 stewart.struts_M.R = Msr;
 stewart.struts_M.H = Msh;
 
-stewart.struts_F.type = 1;
+switch args.type_F
+  case 'cylindrical'
+    stewart.struts_F.type = 1;
+  case 'none'
+    stewart.struts_F.type = 2;
+end
 
 stewart.struts_F.I = I_F;
 stewart.struts_F.M = Fsm;

src/initializeJointDynamics.m

@@ -29,8 +29,8 @@ function [stewart] = initializeJointDynamics(stewart, args)
 
 arguments
     stewart
-    args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'universal'
+    args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'universal'
-    args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'spherical'
+    args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'spherical'
     args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
     args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
     args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
@@ -39,6 +39,16 @@ arguments
     args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
     args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
     args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
+    args.K_M        double {mustBeNumeric} = zeros(6,6)
+    args.M_M        double {mustBeNumeric} = zeros(6,6)
+    args.n_xyz_M    double {mustBeNumeric} = zeros(2,3)
+    args.xi_M       double {mustBeNumeric} = 0.1
+    args.step_file_M char {} = ''
+    args.K_F        double {mustBeNumeric} = zeros(6,6)
+    args.M_F        double {mustBeNumeric} = zeros(6,6)
+    args.n_xyz_F    double {mustBeNumeric} = zeros(2,3)
+    args.xi_F       double {mustBeNumeric} = 0.1
+    args.step_file_F char {} = ''
 end
 
 switch args.type_F
@@ -50,6 +60,8 @@ switch args.type_F
     stewart.joints_F.type = 3;
   case 'spherical_p'
     stewart.joints_F.type = 4;
+  case 'flexible'
+    stewart.joints_F.type = 5;
 end
 
 switch args.type_M
@@ -61,6 +73,8 @@ switch args.type_M
     stewart.joints_M.type = 3;
   case 'spherical_p'
     stewart.joints_M.type = 4;
+  case 'flexible'
+    stewart.joints_M.type = 5;
 end
 
 stewart.joints_M.Kx = zeros(6,1);
@@ -90,3 +104,16 @@ stewart.joints_M.Ct = args.Cf_M;
 
 stewart.joints_F.Cf = args.Cf_F;
 stewart.joints_F.Ct = args.Cf_F;
+
+stewart.joints_F.M = args.M_F;
+stewart.joints_F.K = args.K_F;
+stewart.joints_F.n_xyz = args.n_xyz_F;
+stewart.joints_F.xi = args.xi_F;
+stewart.joints_F.xi = args.xi_F;
+stewart.joints_F.step_file = args.step_file_F;
+
+stewart.joints_M.M = args.M_M;
+stewart.joints_M.K = args.K_M;
+stewart.joints_M.n_xyz = args.n_xyz_M;
+stewart.joints_M.xi = args.xi_M;
+stewart.joints_M.step_file = args.step_file_M;