-
+
- 5.1.
generateCubicConfiguration: Generate a Cubic Configuration- Function description
- Documentation @@ -299,24 +316,13 @@ for the JavaScript code in this tag.
- Equal stiffness in all the degrees of freedom? -
- No coupling between the actuators? -
- Is the center of the cube an important point? +
- In section 1, we study the link between the Stiffness matrix and the cubic architecture and we find what are the conditions to obtain a diagonal stiffness matrix +
- In section 2, we study cubic configurations where the cube’s center is located above the mobile platform +
- In section 3, we study the effect of the cube’s size on the Stewart platform properties +
- In section 4, we study the dynamic coupling of the cubic configuration
- [geng94_six_degree_of_freed_activ] Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, IEEE Transactions on Control Systems Technology, 2(1), 45-53 (1994). link. doi. -
- [preumont07_six_axis_singl_stage_activ] Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, Journal of Sound and Vibration, 300(3-5), 644-661 (2007). link. doi. +
- [geng94_six_degree_of_freed_activ] Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, IEEE Transactions on Control Systems Technology, 2(1), 45-53 (1994). link. doi. +
- [preumont07_six_axis_singl_stage_activ] Preumont, Horodinca, Romanescu, de Marneffe, Avraam, Deraemaeker, Bossens & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, Journal of Sound and Vibration, 300(3-5), 644-661 (2007). link. doi.
- [jafari03_orthog_gough_stewar_platf_microm] Jafari & McInroy, Orthogonal Gough-Stewart Platforms for Micromanipulation, IEEE Transactions on Robotics and Automation, 19(4), 595-603 (2003). link. doi.
-The discovery of the Cubic configuration is done in geng94_six_degree_of_freed_activ. +The Cubic configuration for the Stewart platform was first proposed in geng94_six_degree_of_freed_activ. +This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube. +This configuration is now widely used (preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm).
-The specificity of the Cubic configuration is that each actuator is orthogonal with the others: -
--
- --the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube. -
--The cubic (or orthogonal) configuration of the Stewart platform is now widely used (preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm). -
- --According to preumont07_six_axis_singl_stage_activ: +According to preumont07_six_axis_singl_stage_activ, the cubic configuration offers the following advantages:
@@ -325,19 +331,26 @@ This topology provides a uniform control capability and a uniform stiffness in a
-To generate and study the Cubic configuration,
generateCubicConfigurationis used (description in section 3.1). -The goal is to study the benefits of using a cubic configuration: +In this document, the cubic architecture is analyzed:-
-
+To generate and study the Stewart platform with a Cubic configuration, the Matlab function
+generateCubicConfigurationis used (described here). +1 Configuration Analysis - Stiffness Matrix
++First, we have to understand what is the physical meaning of the Stiffness matrix \(\bm{K}\).
@@ -366,8 +379,11 @@ Thus, if the stiffness matrix is diagonal, the compliance matrix is also diagonaOne has to note that this is only valid in a static way.
-+We here study what makes the Stiffness matrix diagonal when using a cubic configuration. +
++ +1.1 Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
@@ -837,10 +853,26 @@ Here are the conclusion about the Stiffness matrix for the Cubic configuration:+2 Configuration with the Cube’s center above the mobile platform
++ ++We saw in section 1 that in order to have a diagonal stiffness matrix, we need the cube’s center to be located at frames \(\{A\}\) and \(\{B\}\). +Or, we usually want to have \(\{A\}\) and \(\{B\}\) located above the top platform where forces are applied and where displacements are expressed. +
+ ++We here see if the cubic configuration can provide a diagonal stiffness matrix when \(\{A\}\) and \(\{B\}\) are above the mobile platform. +
+-1.6 Having Cube’s center above the top platform
-++ +2.1 Having Cube’s center above the top platform
+Let’s say we want to have a diagonal stiffness matrix when \(\{A\}\) and \(\{B\}\) are located above the top platform. Thus, we want the cube’s center to be located above the top center. @@ -1131,11 +1163,27 @@ FOc = H + MO_B; % Cente
+2.2 Conclusion
++++++We found that we can have a diagonal stiffness matrix using the cubic architecture when \(\{A\}\) and \(\{B\}\) are located above the top platform. +Depending on the cube’s size, we obtain 3 different configurations. +
+ +-+ +2 Cubic size analysis
-++3 Cubic size analysis
+++We here study the effect of the size of the cube used for the Stewart Cubic configuration.
@@ -1147,7 +1195,13 @@ We fix the height of the Stewart platform, the center of the cube is at the centWe only vary the size of the cube.
- +++3.1 Analysis
++++We initialize the wanted cube’s size. +
-Hcs = 1e-3*[250:20:350]; % Heights for the Cube [m] Ks = zeros(6, 6, length(Hcs)); @@ -1171,43 +1225,23 @@ FOc = H + MO_B; % Cente
--stewart = initializeStewartPlatform(); -stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B); -for i = 1:length(Hcs) - Hc = Hcs(i); - stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0); - stewart = computeJointsPose(stewart); - stewart = initializeStrutDynamics(stewart, 'K', ones(6,1)); - stewart = computeJacobian(stewart); - Ks(:,:,i) = stewart.kinematics.K; -end -
-We find that for all the cube’s size, \(k_x = k_y = k_z = k\) where \(k\) is the strut stiffness. We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varying with the cube’s size (figure 9).
---figure; -hold on; -plot(Hcs, squeeze(Ks(4, 4, :)), 'DisplayName', '$k_{\theta_x} = k_{\theta_y}$'); -plot(Hcs, squeeze(Ks(6, 6, :)), 'DisplayName', '$k_{\theta_z}$'); -hold off; -legend('location', 'northwest'); -xlabel('Cube Size [m]'); ylabel('Rotational stiffnes [normalized]'); -
-+
Figure 9: \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube
+3.2 Conclusion
+We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
@@ -1220,18 +1254,39 @@ In order to maximize the rotational stiffness of the Stewart platform, the size+4 Dynamic Coupling
+ +++ +4.1 Cube’s center at the Center of Mass of the Payload
+++ +4.2 Dynamic decoupling between the actuators and sensors
+++4.3 Conclusion
+-3 Functions
-+5 Functions
+-3.1
-generateCubicConfiguration: Generate a Cubic Configuration+5.1
+generateCubicConfiguration: Generate a Cubic Configuration@@ -1372,15 +1427,15 @@ stewart.platform_M.Mb = Mb;Bibliography
--diff --git a/org/cubic-configuration.org b/org/cubic-configuration.org index ae3a98d..1ca357f 100644 --- a/org/cubic-configuration.org +++ b/org/cubic-configuration.org @@ -39,27 +39,25 @@ :END: * Introduction :ignore: -The discovery of the Cubic configuration is done in cite:geng94_six_degree_of_freed_activ. +The Cubic configuration for the Stewart platform was first proposed in cite:geng94_six_degree_of_freed_activ. +This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube. +This configuration is now widely used (cite:preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm). -The specificity of the Cubic configuration is that each actuator is orthogonal with the others: -#+begin_quote -the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube. -#+end_quote - -The cubic (or orthogonal) configuration of the Stewart platform is now widely used (cite:preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm). - -According to cite:preumont07_six_axis_singl_stage_activ: +According to cite:preumont07_six_axis_singl_stage_activ, the cubic configuration offers the following advantages: #+begin_quote This topology provides a uniform control capability and a uniform stiffness in all directions, and it minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other). #+end_quote -To generate and study the Cubic configuration, =generateCubicConfiguration= is used (description in section [[sec:generateCubicConfiguration]]). -The goal is to study the benefits of using a cubic configuration: -- Equal stiffness in all the degrees of freedom? -- No coupling between the actuators? -- Is the center of the cube an important point? +In this document, the cubic architecture is analyzed: +- In section [[sec:cubic_conf_stiffness]], we study the link between the Stiffness matrix and the cubic architecture and we find what are the conditions to obtain a diagonal stiffness matrix +- In section [[sec:cubic_conf_above_platform]], we study cubic configurations where the cube's center is located above the mobile platform +- In section [[sec:cubic_conf_size_analysis]], we study the effect of the cube's size on the Stewart platform properties +- In section [[sec:cubic_conf_coupling]], we study the dynamic coupling of the cubic configuration + +To generate and study the Stewart platform with a Cubic configuration, the Matlab function =generateCubicConfiguration= is used (described [[sec:generateCubicConfiguration][here]]). * Configuration Analysis - Stiffness Matrix +<Created: 2020-02-12 mer. 10:37
+Created: 2020-02-12 mer. 11:18
> @@ -296,6 +296,27 @@ Here are the conclusion about the Stiffness matrix for the Cubic configuration: - The stiffness matrix $K$ is diagonal for the cubic configuration if the Jacobian is estimated at the cube center. #+end_important +* Configuration with the Cube's center above the mobile platform +< > +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + < > +#+end_src + +#+begin_src matlab :results none :exports none + simulinkproject('../'); +#+end_src + ** Having Cube's center above the top platform Let's say we want to have a diagonal stiffness matrix when $\{A\}$ and $\{B\}$ are located above the top platform. Thus, we want the cube's center to be located above the top center. @@ -431,13 +452,36 @@ However, the rotational stiffnesses are increasing with the cube's size but the | -8e-17 | 0 | -3e-17 | -6.1e-19 | 0.094 | 0 | | -6.2e-18 | 7.2e-17 | 5.6e-17 | 2.3e-17 | 0 | 0.37 | +** Conclusion +#+begin_important + We found that we can have a diagonal stiffness matrix using the cubic architecture when $\{A\}$ and $\{B\}$ are located above the top platform. + Depending on the cube's size, we obtain 3 different configurations. +#+end_important + * Cubic size analysis +< > +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + < > +#+end_src + +#+begin_src matlab :results none :exports none + simulinkproject('../'); +#+end_src + +** Analysis +We initialize the wanted cube's size. #+begin_src matlab :results silent Hcs = 1e-3*[250:20:350]; % Heights for the Cube [m] Ks = zeros(6, 6, length(Hcs)); @@ -454,7 +498,7 @@ The frames $\{A\}$ and $\{B\}$ are positioned at the Stewart platform center as FOc = H + MO_B; % Center of the cube with respect to {F} #+end_src -#+begin_src matlab :results silent +#+begin_src matlab :exports none stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B); for i = 1:length(Hcs) @@ -470,7 +514,7 @@ The frames $\{A\}$ and $\{B\}$ are positioned at the Stewart platform center as We find that for all the cube's size, $k_x = k_y = k_z = k$ where $k$ is the strut stiffness. We also find that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ are varying with the cube's size (figure [[fig:stiffness_cube_size]]). -#+begin_src matlab :results none :exports code +#+begin_src matlab :exports none figure; hold on; plot(Hcs, squeeze(Ks(4, 4, :)), 'DisplayName', '$k_{\theta_x} = k_{\theta_y}$'); @@ -491,12 +535,38 @@ We also find that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ are varying w #+RESULTS: fig:stiffness_cube_size [[file:figs/stiffness_cube_size.png]] +** Conclusion We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linearly with the cube size. #+begin_important In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible. #+end_important +* Dynamic Coupling +< > +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + < > +#+end_src + +#+begin_src matlab :results none :exports none + simulinkproject('../'); +#+end_src + + + +** Cube's center at the Center of Mass of the Payload + +** Dynamic decoupling between the actuators and sensors + +** Conclusion + * Functions <