-
-
- 1. TODO Configuration Analysis - Stiffness Matrix +
- 1. Configuration Analysis - Stiffness Matrix
-
-
- 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 +
- 1.1. 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
- - 2. TODO Cubic size analysis -
- 3. Functions +
- 2. Functions
-
-
- 3.1.
generateCubicConfiguration: Generate a Cubic Configuration + - 2.1.
generateCubicConfiguration: Generate a Cubic Configuration
generateCubicConfigurationis used (description in section 2.1). The goal is to study the benefits of using a cubic configuration:-
@@ -322,48 +322,34 @@ The goal is to study the benefits of using a cubic configuration:
- Is the center of the cube an important point?
-1 TODO Configuration Analysis - Stiffness Matrix
++1 Configuration Analysis - Stiffness Matrix
--1.1 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). +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.
++-stewart = initializeFramesPositions('H', 100e-3, 'MO_B', -50e-3); +stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); +stewart = computeJointsPose(stewart); +stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); +stewart = computeJacobian(stewart); +
++ +-
Figure 1: Centered cubic configuration
-- -opts = struct(... - 'H_tot', 100, ... % 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, -50], ... % 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); - -save('./mat/stewart.mat', 'stewart'); -
---K = stewart.Jf'*stewart.Jf; --@@ -383,88 +369,76 @@ save('./mat/stewart.mat', 2 -
1.9e-18 --2.3e-17 -1.8e-18 -5.5e-17 --1.5e-17 +0 +-2.5e-16 +0 +2.1e-17 +0 - 1.9e-18 +0 2 -6.8e-18 --6.1e-17 --1.6e-18 -4.8e-18 +0 +-7.8e-19 +0 +0 - -2.3e-17 -6.8e-18 +-2.5e-16 +0 2 --6.7e-18 -4.9e-18 -5.3e-19 +-2.4e-18 +-1.4e-17 +0 - 1.8e-18 --6.1e-17 --6.7e-18 -0.0067 --2.3e-20 --6.1e-20 +0 +-7.8e-19 +-2.4e-18 +0.015 +-4.3e-19 +1.7e-18 - 5.5e-17 --1.6e-18 -4.9e-18 --2.3e-20 -0.0067 -1e-18 +1.8e-17 +0 +-1.1e-17 +0 +0.015 +0 - -1.5e-17 -4.8e-18 -5.3e-19 --6.1e-20 -1e-18 -0.027 +6.6e-18 +-3.3e-18 +0 +1.7e-18 +0 +0.06 -1.2 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). +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.
-- -opts = struct(... - 'H_tot', 100, ... % 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, 0], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm] - 'Jf_pos', [0, 0, 0] ... % Position of the Jacobian for force location from the top of the mobile platform [mm] - ); -stewart = computeGeometricalProperties(stewart, opts); -
--@@ -487,102 +461,83 @@ stewart = computeGeometricalProperties(stewart, opts);K = stewart.Jf'*stewart.Jf; +
stewart = initializeFramesPositions('H', 100e-3, 'MO_B', 0); +stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); +stewart = computeJointsPose(stewart); +stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); +stewart = computeJacobian(stewart);
2 -1.9e-18 --2.3e-17 -1.5e-18 +0 +-2.5e-16 +1.4e-17 -0.1 --1.5e-17 +0 - 1.9e-18 +0 2 -6.8e-18 +0 0.1 --1.6e-18 -4.8e-18 +0 +0 - -2.3e-17 -6.8e-18 +-2.5e-16 +0 2 --5.1e-19 --5.5e-18 -5.3e-19 +3.4e-18 +-1.4e-17 +0 - 1.5e-18 +1.4e-17 0.1 --5.1e-19 -0.012 --3e-19 -3.1e-19 +3.4e-18 +0.02 +1.1e-20 +3.4e-18 -0.1 --1.6e-18 --5.5e-18 --3e-19 -0.012 -1.9e-18 +0 +-1.4e-17 +1.4e-19 +0.02 +-1.7e-18 - -1.5e-17 -4.8e-18 -5.3e-19 -3.1e-19 -1.9e-18 -0.027 +6.6e-18 +-3.3e-18 +0 +3.6e-18 +-1.7e-18 +0.06 -1.3 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 2). +Here, the “center” of the Stewart platform is not at the cube center (figure 2). The Jacobian is estimated at the cube center.
-+-
Figure 2: Not centered cubic configuration
-The center of the cube is at \(z = 110\). -The Stewart platform is from \(z = H_0 = 75\) to \(z = H_0 + H_{tot} = 175\). -The center height of the Stewart platform is then at \(z = \frac{175-75}{2} = 50\). -The center of the cube from the top platform is at \(z = 110 - 175 = -65\). -
--- -opts = struct(... - 'H_tot', 100, ... % Total height of the Hexapod [mm] - 'L', 220/sqrt(3), ... % Size of the Cube [mm] - 'H', 60, ... % Height between base joints and platform joints [mm] - 'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm] - ); -stewart = initializeCubicConfiguration(opts); -opts = struct(... - 'Jd_pos', [0, 0, -65], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm] - 'Jf_pos', [0, 0, -65] ... % Position of the Jacobian for force location from the top of the mobile platform [mm] - ); -stewart = computeGeometricalProperties(stewart, opts); -
--@@ -605,56 +560,56 @@ stewart = computeGeometricalProperties(stewart, opts);K = stewart.Jf'*stewart.Jf; +
stewart = initializeFramesPositions('H', 80e-3, 'MO_B', -40e-3); +stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); +stewart = computeJointsPose(stewart); +stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); +stewart = computeJacobian(stewart);
2 --1.8e-17 -2.6e-17 -3.3e-18 +0 +-1.5e-16 +0 0.04 -1.7e-19 +0 - -1.8e-17 +0 2 -1.9e-16 +0 -0.04 -2.2e-19 --5.3e-19 +0 +-2.8e-17 - 2.6e-17 -1.9e-16 +-1.5e-16 +0 2 --8.9e-18 -6.5e-19 --5.8e-19 +1.2e-18 +-1e-17 +0 - 3.3e-18 +0 -0.04 --8.9e-18 -0.0089 --9.3e-20 -9.8e-20 +1.2e-18 +0.016 +0 +8.7e-19 0.04 -2.2e-19 -6.5e-19 --9.3e-20 -0.0089 --2.4e-18 +0 +-6.2e-18 +-1.1e-19 +0.016 +8.7e-19 - @@ -665,8 +620,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff1.7e-19 --5.3e-19 --5.8e-19 -9.8e-20 --2.4e-18 -0.032 +-3.7e-19 +-2.5e-17 +0 +1.2e-18 +8.7e-19 +0.06 --1.4 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. @@ -681,23 +636,11 @@ The center of the cube from the top platform is at \(z = 110 - 175 = -65\).
-- -opts = struct(... - 'H_tot', 100, ... % Total height of the Hexapod [mm] - 'L', 220/sqrt(3), ... % Size of the Cube [mm] - 'H', 60, ... % Height between base joints and platform joints [mm] - 'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm] - ); -stewart = initializeCubicConfiguration(opts); -opts = struct(... - 'Jd_pos', [0, 0, -60], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm] - 'Jf_pos', [0, 0, -60] ... % Position of the Jacobian for force location from the top of the mobile platform [mm] - ); -stewart = computeGeometricalProperties(stewart, opts); -
--@@ -720,56 +663,177 @@ stewart = computeGeometricalProperties(stewart, opts);K = stewart.Jf'*stewart.Jf; +
stewart = initializeFramesPositions('H', 80e-3, 'MO_B', -30e-3); +stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); +stewart = computeJointsPose(stewart); +stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); +stewart = computeJacobian(stewart);
2 --1.8e-17 -2.6e-17 --5.7e-19 -0.03 -1.7e-19 +0 +-1.7e-16 +0 +4.9e-17 +0 - -1.8e-17 +0 2 -1.9e-16 --0.03 -2.2e-19 --5.3e-19 +0 +-2.2e-17 +0 +2.8e-17 - 2.6e-17 -1.9e-16 +-1.7e-16 +0 2 --1.5e-17 -6.5e-19 --5.8e-19 +1.1e-18 +-1.4e-17 +1.4e-17 - -5.7e-19 --0.03 --1.5e-17 -0.0085 -4.9e-20 -1.7e-19 +0 +-2.2e-17 +1.1e-18 +0.015 +0 +3.5e-18 - 0.03 -2.2e-19 -6.5e-19 -4.9e-20 -0.0085 --1.1e-18 +4.4e-17 +0 +-1.4e-17 +-5.7e-20 +0.015 +-8.7e-19 - + + + +1.7e-19 --5.3e-19 --5.8e-19 -1.7e-19 --1.1e-18 -0.032 +6.6e-18 +2.5e-17 +0 +3.5e-18 +-8.7e-19 +0.06 ++We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), and the Stiffness matrix is diagonal. +
+++ +1.5 Conclusion
+++++-
+
- The cubic configuration permits to have \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\) +
- The stiffness matrix \(K\) is diagonal for the cubic configuration if the Jacobian is estimated at the cube center. +
+- -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. +This is possible, to do so: +
+-
+
- The position of the center of the cube should be positioned at A +
- The Height of the “useful” part of the cube should be at least equal to two times the distance from F to A. +It is possible to have small cube, but then to configuration is a little bit strange. +
++ +stewart = initializeFramesPositions('H', 100e-3, 'MO_B', 50e-3); +FOc = stewart.H + stewart.MO_B(3); +Hc = 2*(stewart.H + stewart.MO_B(3)); +stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 10e-3, 'MHb', 10e-3); +stewart = computeJointsPose(stewart); +stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); +stewart = initializeJointDynamics(stewart, 'disable', true); +stewart = initializeCylindricalPlatforms(stewart); +stewart = initializeCylindricalStruts(stewart); +stewart = computeJacobian(stewart); +stewart = initializeStewartPose(stewart); +
++ + +
@@ -779,131 +843,21 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff+ + ++ + + + + + + + + + + + + +2 +0 +-4.6e-16 +0 +4e-17 +0 ++ + +0 +2 +0 +-4.8e-17 +0 +-3.5e-17 ++ + +-4.6e-16 +0 +2 +1.5e-20 +4e-17 +0 ++ + +0 +-4.8e-17 +1.5e-20 +0.00034 +6.8e-21 +4.2e-19 ++ + +4e-17 +0 +4e-17 +-3e-21 +0.00034 +-2.7e-20 ++ -1.7e-19 +-3.6e-17 +0 +4.2e-19 +-2.7e-20 +0.0014 -1.5 Conclusion
------
-
- The cubic configuration permits to have \(k_x = k_y = k_z\) and \(k_{\theta\x} = k_{\theta_y}\) -
- The stiffness matrix \(K\) is diagonal for the cubic configuration if the Stewart platform and the cube are centered and the Jacobian is estimated at the cube center -
-2 TODO Cubic size analysis
++- -2 Functions
--We here study the effect of the size of the cube used for the Stewart configuration. -
- --We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform. -
- --We only vary the size of the cube. -
- --- -H_cubes = 250:20:350; -stewarts = {zeros(length(H_cubes), 1)}; -
--- - -for i = 1:length(H_cubes) - H_cube = H_cubes(i); - H_tot = 100; - H = 80; - - opts = struct(... - 'H_tot', H_tot, ... % Total height of the Hexapod [mm] - 'L', H_cube/sqrt(3), ... % Size of the Cube [mm] - 'H', H, ... % Height between base joints and platform joints [mm] - 'H0', H_cube/2-H/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, H_cube/2-opts.H0-opts.H_tot], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm] - 'Jf_pos', [0, 0, H_cube/2-opts.H0-opts.H_tot] ... % Position of the Jacobian for force location from the top of the mobile platform [mm] - ); - stewart = computeGeometricalProperties(stewart, opts); - stewarts(i) = {stewart}; -end -
--The Stiffness matrix is computed for all generated Stewart platforms. -
--- -Ks = zeros(6, 6, length(H_cube)); -for i = 1:length(H_cubes) - Ks(:, :, i) = stewarts{i}.Jd'*stewarts{i}.Jd; -end -
--The only elements of \(K\) that vary are \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\). -
- --Finally, we plot \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) -
--- - -figure; -hold on; -plot(H_cubes, squeeze(Ks(4, 4, :)), 'DisplayName', '$k_{\theta_x}$'); -plot(H_cubes, squeeze(Ks(6, 6, :)), 'DisplayName', '$k_{\theta_z}$'); -hold off; -legend('location', 'northwest'); -xlabel('Cube Size [mm]'); ylabel('Rotational stiffnes [normalized]'); -
--- - -
-
-Figure 3: \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube
--We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size. -
- ----In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible. -In that case, the legs will the further separated. Size of the cube is then limited by allowed space. -
- --3 Functions
- --3.1
-generateCubicConfiguration: Generate a Cubic Configuration++2.1
+generateCubicConfiguration: Generate a Cubic Configuration-@@ -911,9 +865,9 @@ This Matlab function is accessible he
-Function description
-++Function description
+-function [stewart] = generateCubicConfiguration(stewart, args) % generateCubicConfiguration - Generate a Cubic Configuration @@ -938,37 +892,37 @@ This Matlab function is accessible he
-Documentation
-++Documentation
+--+
-
Figure 4: Cubic Configuration
+Figure 3: Cubic Configuration
-Optional Parameters
-++-Optional Parameters
+arguments stewart args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 args.FOc (1,1) double {mustBeNumeric} = 50e-3 - args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 - args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 + args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 + args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 end-Position of the Cube
-++-Position of the Cube
+We define the useful points of the cube with respect to the Cube’s center. \({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is @@ -993,9 +947,9 @@ CCm = [Cc(:,2), Cc(:
-Compute the pose
-++Compute the pose
+diff --git a/cubic-configuration.org b/cubic-configuration.org index 9a92416..262caf2 100644 --- a/cubic-configuration.org +++ b/cubic-configuration.org @@ -21,7 +21,7 @@ #+PROPERTY: header-args:matlab+ :output-dir figs :END: -* Introduction :ignore: +* Introduction :ignore: The discovery of the Cubic configuration is done in cite:geng94_six_degree_of_freed_activ. Further analysis is conducted in @@ -37,7 +37,8 @@ The goal is to study the benefits of using a cubic configuration: - No coupling between the actuators? - Is the center of the cube an important point? -* Matlab Init :noexport:ignore: +* Configuration Analysis - Stiffness Matrix +** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
@@ -1028,7 +982,7 @@ stewart.Mb = CCf + [0; 0; args.FOc -Created: 2020-02-06 jeu. 17:29
+Created: 2020-02-06 jeu. 18:22
> #+end_src @@ -50,84 +51,57 @@ The goal is to study the benefits of using a cubic configuration: simulinkproject('./'); #+end_src -* TODO Configuration Analysis - Stiffness Matrix ** Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center We create a cubic Stewart platform (figure [[fig:3d-cubic-stewart-aligned]]) 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. +#+begin_src matlab + stewart = initializeFramesPositions('H', 100e-3, 'MO_B', -50e-3); + stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); + stewart = computeJacobian(stewart); +#+end_src + #+name: fig:3d-cubic-stewart-aligned #+caption: Centered cubic configuration [[file:./figs/3d-cubic-stewart-aligned.png]] -#+begin_src matlab :results silent - opts = struct(... - 'H_tot', 100, ... % 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, -50], ... % 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); - - save('./mat/stewart.mat', 'stewart'); -#+end_src - -#+begin_src matlab :results none :exports code - K = stewart.Jf'*stewart.Jf; -#+end_src - -#+begin_src matlab :results value table :exports results - data = K; - data2orgtable(data, {}, {}, ' %.2g '); +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(stewart.K, {}, {}, ' %.2g '); #+end_src #+RESULTS: -| 2 | 1.9e-18 | -2.3e-17 | 1.8e-18 | 5.5e-17 | -1.5e-17 | -| 1.9e-18 | 2 | 6.8e-18 | -6.1e-17 | -1.6e-18 | 4.8e-18 | -| -2.3e-17 | 6.8e-18 | 2 | -6.7e-18 | 4.9e-18 | 5.3e-19 | -| 1.8e-18 | -6.1e-17 | -6.7e-18 | 0.0067 | -2.3e-20 | -6.1e-20 | -| 5.5e-17 | -1.6e-18 | 4.9e-18 | -2.3e-20 | 0.0067 | 1e-18 | -| -1.5e-17 | 4.8e-18 | 5.3e-19 | -6.1e-20 | 1e-18 | 0.027 | +| 2 | 0 | -2.5e-16 | 0 | 2.1e-17 | 0 | +| 0 | 2 | 0 | -7.8e-19 | 0 | 0 | +| -2.5e-16 | 0 | 2 | -2.4e-18 | -1.4e-17 | 0 | +| 0 | -7.8e-19 | -2.4e-18 | 0.015 | -4.3e-19 | 1.7e-18 | +| 1.8e-17 | 0 | -1.1e-17 | 0 | 0.015 | 0 | +| 6.6e-18 | -3.3e-18 | 0 | 1.7e-18 | 0 | 0.06 | ** 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 [[fig:3d-cubic-stewart-aligned]]). The Jacobian matrix is not estimated at the location of the center of the cube. -#+begin_src matlab :results silent - opts = struct(... - 'H_tot', 100, ... % 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, 0], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm] - 'Jf_pos', [0, 0, 0] ... % Position of the Jacobian for force location from the top of the mobile platform [mm] - ); - stewart = computeGeometricalProperties(stewart, opts); +#+begin_src matlab + stewart = initializeFramesPositions('H', 100e-3, 'MO_B', 0); + stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); + stewart = computeJacobian(stewart); #+end_src -#+begin_src matlab :results none :exports code - K = stewart.Jf'*stewart.Jf; -#+end_src - -#+begin_src matlab :results value table :exports results - data = K; - data2orgtable(data', {}, {}, ' %.2g '); +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(stewart.K, {}, {}, ' %.2g '); #+end_src #+RESULTS: -| 2 | 1.9e-18 | -2.3e-17 | 1.5e-18 | -0.1 | -1.5e-17 | -| 1.9e-18 | 2 | 6.8e-18 | 0.1 | -1.6e-18 | 4.8e-18 | -| -2.3e-17 | 6.8e-18 | 2 | -5.1e-19 | -5.5e-18 | 5.3e-19 | -| 1.5e-18 | 0.1 | -5.1e-19 | 0.012 | -3e-19 | 3.1e-19 | -| -0.1 | -1.6e-18 | -5.5e-18 | -3e-19 | 0.012 | 1.9e-18 | -| -1.5e-17 | 4.8e-18 | 5.3e-19 | 3.1e-19 | 1.9e-18 | 0.027 | +| 2 | 0 | -2.5e-16 | 1.4e-17 | -0.1 | 0 | +| 0 | 2 | 0 | 0.1 | 0 | 0 | +| -2.5e-16 | 0 | 2 | 3.4e-18 | -1.4e-17 | 0 | +| 1.4e-17 | 0.1 | 3.4e-18 | 0.02 | 1.1e-20 | 3.4e-18 | +| -0.1 | 0 | -1.4e-17 | 1.4e-19 | 0.02 | -1.7e-18 | +| 6.6e-18 | -3.3e-18 | 0 | 3.6e-18 | -1.7e-18 | 0.06 | ** 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 [[fig:3d-cubic-stewart-misaligned]]). @@ -137,42 +111,25 @@ The Jacobian is estimated at the cube center. #+caption: Not centered cubic configuration [[file:./figs/3d-cubic-stewart-misaligned.png]] -The center of the cube is at $z = 110$. -The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$. -The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$. -The center of the cube from the top platform is at $z = 110 - 175 = -65$. - -#+begin_src matlab :results silent - opts = struct(... - 'H_tot', 100, ... % Total height of the Hexapod [mm] - 'L', 220/sqrt(3), ... % Size of the Cube [mm] - 'H', 60, ... % Height between base joints and platform joints [mm] - 'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm] - ); - stewart = initializeCubicConfiguration(opts); - opts = struct(... - 'Jd_pos', [0, 0, -65], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm] - 'Jf_pos', [0, 0, -65] ... % Position of the Jacobian for force location from the top of the mobile platform [mm] - ); - stewart = computeGeometricalProperties(stewart, opts); +#+begin_src matlab + stewart = initializeFramesPositions('H', 80e-3, 'MO_B', -40e-3); + stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); + stewart = computeJacobian(stewart); #+end_src -#+begin_src matlab :results none :exports code - K = stewart.Jf'*stewart.Jf; -#+end_src - -#+begin_src matlab :results value table :exports results - data = K; - data2orgtable(data', {}, {}, ' %.2g '); +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(stewart.K, {}, {}, ' %.2g '); #+end_src #+RESULTS: -| 2 | -1.8e-17 | 2.6e-17 | 3.3e-18 | 0.04 | 1.7e-19 | -| -1.8e-17 | 2 | 1.9e-16 | -0.04 | 2.2e-19 | -5.3e-19 | -| 2.6e-17 | 1.9e-16 | 2 | -8.9e-18 | 6.5e-19 | -5.8e-19 | -| 3.3e-18 | -0.04 | -8.9e-18 | 0.0089 | -9.3e-20 | 9.8e-20 | -| 0.04 | 2.2e-19 | 6.5e-19 | -9.3e-20 | 0.0089 | -2.4e-18 | -| 1.7e-19 | -5.3e-19 | -5.8e-19 | 9.8e-20 | -2.4e-18 | 0.032 | +| 2 | 0 | -1.5e-16 | 0 | 0.04 | 0 | +| 0 | 2 | 0 | -0.04 | 0 | -2.8e-17 | +| -1.5e-16 | 0 | 2 | 1.2e-18 | -1e-17 | 0 | +| 0 | -0.04 | 1.2e-18 | 0.016 | 0 | 8.7e-19 | +| 0.04 | 0 | -6.2e-18 | -1.1e-19 | 0.016 | 8.7e-19 | +| -3.7e-19 | -2.5e-17 | 0 | 1.2e-18 | 8.7e-19 | 0.06 | We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal. @@ -185,47 +142,71 @@ The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$. The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$. The center of the cube from the top platform is at $z = 110 - 175 = -65$. -#+begin_src matlab :results silent - opts = struct(... - 'H_tot', 100, ... % Total height of the Hexapod [mm] - 'L', 220/sqrt(3), ... % Size of the Cube [mm] - 'H', 60, ... % Height between base joints and platform joints [mm] - 'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm] - ); - stewart = initializeCubicConfiguration(opts); - opts = struct(... - 'Jd_pos', [0, 0, -60], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm] - 'Jf_pos', [0, 0, -60] ... % Position of the Jacobian for force location from the top of the mobile platform [mm] - ); - stewart = computeGeometricalProperties(stewart, opts); +#+begin_src matlab + stewart = initializeFramesPositions('H', 80e-3, 'MO_B', -30e-3); + stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); + stewart = computeJacobian(stewart); #+end_src -#+begin_src matlab :results none :exports code - K = stewart.Jf'*stewart.Jf; -#+end_src - -#+begin_src matlab :results value table :exports results - data = K; - data2orgtable(data', {}, {}, ' %.2g '); +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(stewart.K, {}, {}, ' %.2g '); #+end_src #+RESULTS: -| 2 | -1.8e-17 | 2.6e-17 | -5.7e-19 | 0.03 | 1.7e-19 | -| -1.8e-17 | 2 | 1.9e-16 | -0.03 | 2.2e-19 | -5.3e-19 | -| 2.6e-17 | 1.9e-16 | 2 | -1.5e-17 | 6.5e-19 | -5.8e-19 | -| -5.7e-19 | -0.03 | -1.5e-17 | 0.0085 | 4.9e-20 | 1.7e-19 | -| 0.03 | 2.2e-19 | 6.5e-19 | 4.9e-20 | 0.0085 | -1.1e-18 | -| 1.7e-19 | -5.3e-19 | -5.8e-19 | 1.7e-19 | -1.1e-18 | 0.032 | +| 2 | 0 | -1.7e-16 | 0 | 4.9e-17 | 0 | +| 0 | 2 | 0 | -2.2e-17 | 0 | 2.8e-17 | +| -1.7e-16 | 0 | 2 | 1.1e-18 | -1.4e-17 | 1.4e-17 | +| 0 | -2.2e-17 | 1.1e-18 | 0.015 | 0 | 3.5e-18 | +| 4.4e-17 | 0 | -1.4e-17 | -5.7e-20 | 0.015 | -8.7e-19 | +| 6.6e-18 | 2.5e-17 | 0 | 3.5e-18 | -8.7e-19 | 0.06 | -We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal. +We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, and the Stiffness matrix is diagonal. ** Conclusion #+begin_important - - The cubic configuration permits to have $k_x = k_y = k_z$ and $k_{\theta\x} = k_{\theta_y}$ - - The stiffness matrix $K$ is diagonal for the cubic configuration if the Stewart platform and the cube are centered *and* the Jacobian is estimated at the cube center + - The cubic configuration permits to have $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$ + - The stiffness matrix $K$ is diagonal for the cubic configuration if the Jacobian is estimated at the cube center. #+end_important -* TODO Cubic size analysis +** 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. +This is possible, to do so: +- The position of the center of the cube should be positioned at A +- The Height of the "useful" part of the cube should be at least equal to two times the distance from F to A. + It is possible to have small cube, but then to configuration is a little bit strange. + +#+begin_src matlab + stewart = initializeFramesPositions('H', 100e-3, 'MO_B', 50e-3); + FOc = stewart.H + stewart.MO_B(3); + Hc = 2*(stewart.H + stewart.MO_B(3)); + stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 10e-3, 'MHb', 10e-3); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); + stewart = initializeJointDynamics(stewart, 'disable', true); + stewart = initializeCylindricalPlatforms(stewart); + stewart = initializeCylindricalStruts(stewart); + stewart = computeJacobian(stewart); + stewart = initializeStewartPose(stewart); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(stewart.K, {}, {}, ' %.2g '); +#+end_src + +#+RESULTS: +| 2 | 0 | -4.6e-16 | 0 | 4e-17 | 0 | +| 0 | 2 | 0 | -4.8e-17 | 0 | -3.5e-17 | +| -4.6e-16 | 0 | 2 | 1.5e-20 | 4e-17 | 0 | +| 0 | -4.8e-17 | 1.5e-20 | 0.00034 | 6.8e-21 | 4.2e-19 | +| 4e-17 | 0 | 4e-17 | -3e-21 | 0.00034 | -2.7e-20 | +| -1.7e-19 | -3.6e-17 | 0 | 4.2e-19 | -2.7e-20 | 0.0014 | + +We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal. + +* TODO Cubic size analysis :noexport: We here study the effect of the size of the cube used for the Stewart configuration. We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform. @@ -260,7 +241,6 @@ We only vary the size of the cube. end #+end_src - The Stiffness matrix is computed for all generated Stewart platforms. #+begin_src matlab :results none :exports code Ks = zeros(6, 6, length(H_cube)); @@ -355,8 +335,8 @@ This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here stewart args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 args.FOc (1,1) double {mustBeNumeric} = 50e-3 - args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 - args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 + args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 + args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 end #+end_src diff --git a/src/generateCubicConfiguration.m b/src/generateCubicConfiguration.m index 8bc48dc..920cfdf 100644 --- a/src/generateCubicConfiguration.m +++ b/src/generateCubicConfiguration.m @@ -21,8 +21,8 @@ arguments stewart args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 args.FOc (1,1) double {mustBeNumeric} = 50e-3 - args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 - args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 + args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 + args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 end sx = [ 2; -1; -1]; - 3.1.