1622 lines
67 KiB
Org Mode
1622 lines
67 KiB
Org Mode
#+TITLE: Cubic configuration for the Stewart Platform
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ./index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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* Introduction :ignore:
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The Cubic configuration for the Stewart platform was first proposed in cite:geng94_six_degree_of_freed_activ.
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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.
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This configuration is now widely used (cite:preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm).
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According to cite:preumont07_six_axis_singl_stage_activ, the cubic configuration offers the following advantages:
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#+begin_quote
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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).
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#+end_quote
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In this document, the cubic architecture is analyzed:
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- In section [[sec:cubic_conf_stiffness]], we study the *uniform stiffness* of such configuration and we find the conditions to obtain a diagonal stiffness matrix
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- In section [[sec:cubic_conf_above_platform]], we find cubic configurations where the cube's center is located above the mobile platform
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- In section [[sec:cubic_conf_size_analysis]], we study the effect of the cube's size on the Stewart platform properties
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- In section [[sec:cubic_conf_coupling_cartesian]], we study the dynamics of the cubic configuration in the cartesian frame
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- In section [[sec:cubic_conf_coupling_struts]], we study the dynamic *cross-coupling* of the cubic configuration from actuators to sensors of each strut
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- In section [[sec:functions]], function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function =generateCubicConfiguration= is used (described [[sec:generateCubicConfiguration][here]]).
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* Stiffness Matrix for the Cubic configuration
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:PROPERTIES:
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:header-args:matlab+: :tangle ../matlab/cubic_conf_stiffnessl.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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<<sec:cubic_conf_stiffness>>
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#+begin_note
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The Matlab script corresponding to this section is accessible [[file:../matlab/cubic_conf_stiffnessl.m][here]].
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To run the script, open the Simulink Project, and type =run cubic_conf_stiffness.m=.
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#+end_note
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** Introduction :ignore:
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First, we have to understand what is the physical meaning of the Stiffness matrix $\bm{K}$.
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The Stiffness matrix links forces $\bm{f}$ and torques $\bm{n}$ applied on the mobile platform at $\{B\}$ to the displacement $\Delta\bm{\mathcal{X}}$ of the mobile platform represented by $\{B\}$ with respect to $\{A\}$:
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\[ \bm{\mathcal{F}} = \bm{K} \Delta\bm{\mathcal{X}} \]
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with:
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- $\bm{\mathcal{F}} = [\bm{f}\ \bm{n}]^{T}$
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- $\Delta\bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_{x}, \delta \theta_{y}, \delta \theta_{z}]^{T}$
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If the stiffness matrix is inversible, its inverse is the compliance matrix: $\bm{C} = \bm{K}^{-1$ and:
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\[ \Delta \bm{\mathcal{X}} = C \bm{\mathcal{F}} \]
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Thus, if the stiffness matrix is diagonal, the compliance matrix is also diagonal and a force (resp. torque) $\bm{\mathcal{F}}_i$ applied on the mobile platform at $\{B\}$ will induce a pure translation (resp. rotation) of the mobile platform represented by $\{B\}$ with respect to $\{A\}$.
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One has to note that this is only valid in a static way.
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We here study what makes the Stiffness matrix diagonal when using a cubic configuration.
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
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** Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
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We create a cubic Stewart platform (figure [[fig:cubic_conf_centered_J_center]]) in such a way that the center of the cube (black star) is located at the center of the Stewart platform (blue dot).
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The Jacobian matrix is estimated at the location of the center of the cube.
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#+begin_src matlab
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H = 100e-3; % height of the Stewart platform [m]
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MO_B = -H/2; % Position {B} with respect to {M} [m]
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Hc = H; % Size of the useful part of the cube [m]
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FOc = H + MO_B; % Center of the cube with respect to {F}
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#+end_src
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
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stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
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#+end_src
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#+begin_src matlab :exports none
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displayArchitecture(stewart, 'labels', false);
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scatter3(0, 0, FOc, 200, 'kh');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/cubic_conf_centered_J_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:cubic_conf_centered_J_center
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#+caption: Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center ([[./figs/cubic_conf_centered_J_center.png][png]], [[./figs/cubic_conf_centered_J_center.pdf][pdf]])
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[[file:figs/cubic_conf_centered_J_center.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
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#+end_src
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#+name: tab:cubic_conf_centered_J_center
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#+caption: Stiffness Matrix
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#+RESULTS:
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| 2 | 0 | -2.5e-16 | 0 | 2.1e-17 | 0 |
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| 0 | 2 | 0 | -7.8e-19 | 0 | 0 |
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| -2.5e-16 | 0 | 2 | -2.4e-18 | -1.4e-17 | 0 |
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| 0 | -7.8e-19 | -2.4e-18 | 0.015 | -4.3e-19 | 1.7e-18 |
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| 1.8e-17 | 0 | -1.1e-17 | 0 | 0.015 | 0 |
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| 6.6e-18 | -3.3e-18 | 0 | 1.7e-18 | 0 | 0.06 |
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** Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
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We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure [[fig:cubic_conf_centered_J_not_center]]).
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The Jacobian matrix is not estimated at the location of the center of the cube.
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#+begin_src matlab
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H = 100e-3; % height of the Stewart platform [m]
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MO_B = 20e-3; % Position {B} with respect to {M} [m]
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Hc = H; % Size of the useful part of the cube [m]
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FOc = H/2; % Center of the cube with respect to {F}
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#+end_src
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
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stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
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#+end_src
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#+begin_src matlab :exports none
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displayArchitecture(stewart, 'labels', false);
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scatter3(0, 0, FOc, 200, 'kh');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/cubic_conf_centered_J_not_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:cubic_conf_centered_J_not_center
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#+caption: Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center ([[./figs/cubic_conf_centered_J_not_center.png][png]], [[./figs/cubic_conf_centered_J_not_center.pdf][pdf]])
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[[file:figs/cubic_conf_centered_J_not_center.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
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#+end_src
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#+name: tab:cubic_conf_centered_J_not_center
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#+caption: Stiffness Matrix
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#+RESULTS:
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| 2 | 0 | -2.5e-16 | 0 | -0.14 | 0 |
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| 0 | 2 | 0 | 0.14 | 0 | 0 |
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| -2.5e-16 | 0 | 2 | -5.3e-19 | 0 | 0 |
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| 0 | 0.14 | -5.3e-19 | 0.025 | 0 | 8.7e-19 |
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| -0.14 | 0 | 2.6e-18 | 1.6e-19 | 0.025 | 0 |
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| 6.6e-18 | -3.3e-18 | 0 | 8.9e-19 | 0 | 0.06 |
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** Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
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Here, the "center" of the Stewart platform is not at the cube center (figure [[fig:cubic_conf_not_centered_J_center]]).
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The Jacobian is estimated at the cube center.
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#+begin_src matlab
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H = 80e-3; % height of the Stewart platform [m]
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MO_B = -30e-3; % Position {B} with respect to {M} [m]
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Hc = 100e-3; % Size of the useful part of the cube [m]
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FOc = H + MO_B; % Center of the cube with respect to {F}
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#+end_src
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
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stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
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#+end_src
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#+begin_src matlab :exports none
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displayArchitecture(stewart, 'labels', false);
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scatter3(0, 0, FOc, 200, 'kh');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/cubic_conf_not_centered_J_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:cubic_conf_not_centered_J_center
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#+caption: Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center ([[./figs/cubic_conf_not_centered_J_center.png][png]], [[./figs/cubic_conf_not_centered_J_center.pdf][pdf]])
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[[file:figs/cubic_conf_not_centered_J_center.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
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#+end_src
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#+name: tab:cubic_conf_not_centered_J_center
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#+caption: Stiffness Matrix
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#+RESULTS:
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| 2 | 0 | -1.7e-16 | 0 | 4.9e-17 | 0 |
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| 0 | 2 | 0 | -2.2e-17 | 0 | 2.8e-17 |
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| -1.7e-16 | 0 | 2 | 1.1e-18 | -1.4e-17 | 1.4e-17 |
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| 0 | -2.2e-17 | 1.1e-18 | 0.015 | 0 | 3.5e-18 |
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| 4.4e-17 | 0 | -1.4e-17 | -5.7e-20 | 0.015 | -8.7e-19 |
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| 6.6e-18 | 2.5e-17 | 0 | 3.5e-18 | -8.7e-19 | 0.06 |
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We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.
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** Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center
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Here, the "center" of the Stewart platform is not at the cube center.
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The Jacobian is estimated at the center of the Stewart platform.
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The center of the cube is at $z = 110$.
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The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$.
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The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$.
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The center of the cube from the top platform is at $z = 110 - 175 = -65$.
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#+begin_src matlab
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H = 100e-3; % height of the Stewart platform [m]
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MO_B = -H/2; % Position {B} with respect to {M} [m]
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Hc = 1.5*H; % Size of the useful part of the cube [m]
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FOc = H/2 + 10e-3; % Center of the cube with respect to {F}
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#+end_src
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
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stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 215e-3, 'Mpr', 195e-3);
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#+end_src
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#+begin_src matlab :exports none
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displayArchitecture(stewart, 'labels', false);
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scatter3(0, 0, FOc, 200, 'kh');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/cubic_conf_not_centered_J_stewart_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:cubic_conf_not_centered_J_stewart_center
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#+caption: Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center ([[./figs/cubic_conf_not_centered_J_stewart_center.png][png]], [[./figs/cubic_conf_not_centered_J_stewart_center.pdf][pdf]])
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[[file:figs/cubic_conf_not_centered_J_stewart_center.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
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#+end_src
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#+name: tab:cubic_conf_not_centered_J_stewart_center
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#+caption: Stiffness Matrix
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#+RESULTS:
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| 2 | 0 | 1.5e-16 | 0 | 0.02 | 0 |
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| 0 | 2 | 0 | -0.02 | 0 | 0 |
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| 1.5e-16 | 0 | 2 | -3e-18 | -2.8e-17 | 0 |
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| 0 | -0.02 | -3e-18 | 0.034 | -8.7e-19 | 5.2e-18 |
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| 0.02 | 0 | -2.2e-17 | -4.4e-19 | 0.034 | 0 |
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| 5.9e-18 | -7.5e-18 | 0 | 3.5e-18 | 0 | 0.14 |
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** Conclusion
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#+begin_important
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Here are the conclusion about the Stiffness matrix for the Cubic configuration:
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- The cubic configuration permits to have $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$
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- The stiffness matrix $K$ is diagonal for the cubic configuration if the Jacobian is estimated at the cube center.
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#+end_important
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* Configuration with the Cube's center above the mobile platform
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:PROPERTIES:
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:header-args:matlab+: :tangle ../matlab/cubic_conf_above_platforml.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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<<sec:cubic_conf_above_platform>>
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#+begin_note
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The Matlab script corresponding to this section is accessible [[file:../matlab/cubic_conf_above_platforml.m][here]].
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To run the script, open the Simulink Project, and type =run cubic_conf_above_platform.m=.
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#+end_note
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** Introduction :ignore:
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We saw in section [[sec:cubic_conf_stiffness]] that in order to have a diagonal stiffness matrix, we need the cube's center to be located at frames $\{A\}$ and $\{B\}$.
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Or, we usually want to have $\{A\}$ and $\{B\}$ located above the top platform where forces are applied and where displacements are expressed.
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We here see if the cubic configuration can provide a diagonal stiffness matrix when $\{A\}$ and $\{B\}$ are above the mobile platform.
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
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** Having Cube's center above the top platform
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Let's say we want to have a diagonal stiffness matrix when $\{A\}$ and $\{B\}$ are located above the top platform.
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Thus, we want the cube's center to be located above the top center.
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Let's fix the Height of the Stewart platform and the position of frames $\{A\}$ and $\{B\}$:
|
|
#+begin_src matlab
|
|
H = 100e-3; % height of the Stewart platform [m]
|
|
MO_B = 20e-3; % Position {B} with respect to {M} [m]
|
|
#+end_src
|
|
|
|
We find the several Cubic configuration for the Stewart platform where the center of the cube is located at frame $\{A\}$.
|
|
The differences between the configuration are the cube's size:
|
|
- Small Cube Size in Figure [[fig:stewart_cubic_conf_type_1]]
|
|
- Medium Cube Size in Figure [[fig:stewart_cubic_conf_type_2]]
|
|
- Large Cube Size in Figure [[fig:stewart_cubic_conf_type_3]]
|
|
|
|
For each of the configuration, the Stiffness matrix is diagonal with $k_x = k_y = k_y = 2k$ with $k$ is the stiffness of each strut.
|
|
However, the rotational stiffnesses are increasing with the cube's size but the required size of the platform is also increasing, so there is a trade-off here.
|
|
|
|
#+begin_src matlab
|
|
Hc = 0.4*H; % Size of the useful part of the cube [m]
|
|
FOc = H + MO_B; % Center of the cube with respect to {F}
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
|
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), 'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
|
displayArchitecture(stewart, 'labels', false);
|
|
scatter3(0, 0, FOc, 200, 'kh');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_cubic_conf_type_1.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_cubic_conf_type_1
|
|
#+caption: Cubic Configuration for the Stewart Platform - Small Cube Size ([[./figs/stewart_cubic_conf_type_1.png][png]], [[./figs/stewart_cubic_conf_type_1.pdf][pdf]])
|
|
[[file:figs/stewart_cubic_conf_type_1.png]]
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
|
#+end_src
|
|
|
|
#+name: tab:stewart_cubic_conf_type_1
|
|
#+caption: Stiffness Matrix
|
|
#+RESULTS:
|
|
| 2 | 0 | -2.8e-16 | 0 | 2.4e-17 | 0 |
|
|
| 0 | 2 | 0 | -2.3e-17 | 0 | 0 |
|
|
| -2.8e-16 | 0 | 2 | -2.1e-19 | 0 | 0 |
|
|
| 0 | -2.3e-17 | -2.1e-19 | 0.0024 | -5.4e-20 | 6.5e-19 |
|
|
| 2.4e-17 | 0 | 4.9e-19 | -2.3e-20 | 0.0024 | 0 |
|
|
| -1.2e-18 | 1.1e-18 | 0 | 6.2e-19 | 0 | 0.0096 |
|
|
|
|
#+begin_src matlab
|
|
Hc = 1.5*H; % Size of the useful part of the cube [m]
|
|
FOc = H + MO_B; % Center of the cube with respect to {F}
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
|
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), 'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
|
displayArchitecture(stewart, 'labels', false);
|
|
scatter3(0, 0, FOc, 200, 'kh');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_cubic_conf_type_2.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_cubic_conf_type_2
|
|
#+caption: Cubic Configuration for the Stewart Platform - Medium Cube Size ([[./figs/stewart_cubic_conf_type_2.png][png]], [[./figs/stewart_cubic_conf_type_2.pdf][pdf]])
|
|
[[file:figs/stewart_cubic_conf_type_2.png]]
|
|
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
|
#+end_src
|
|
|
|
#+name: tab:stewart_cubic_conf_type_2
|
|
#+caption: Stiffness Matrix
|
|
#+RESULTS:
|
|
| 2 | 0 | -1.9e-16 | 0 | 5.6e-17 | 0 |
|
|
| 0 | 2 | 0 | -7.6e-17 | 0 | 0 |
|
|
| -1.9e-16 | 0 | 2 | 2.5e-18 | 2.8e-17 | 0 |
|
|
| 0 | -7.6e-17 | 2.5e-18 | 0.034 | 8.7e-19 | 8.7e-18 |
|
|
| 5.7e-17 | 0 | 3.2e-17 | 2.9e-19 | 0.034 | 0 |
|
|
| -1e-18 | -1.3e-17 | 5.6e-17 | 8.4e-18 | 0 | 0.14 |
|
|
|
|
#+begin_src matlab
|
|
Hc = 2.5*H; % Size of the useful part of the cube [m]
|
|
FOc = H + MO_B; % Center of the cube with respect to {F}
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
|
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), 'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
|
displayArchitecture(stewart, 'labels', false);
|
|
scatter3(0, 0, FOc, 200, 'kh');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_cubic_conf_type_3.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_cubic_conf_type_3
|
|
#+caption: Cubic Configuration for the Stewart Platform - Large Cube Size ([[./figs/stewart_cubic_conf_type_3.png][png]], [[./figs/stewart_cubic_conf_type_3.pdf][pdf]])
|
|
[[file:figs/stewart_cubic_conf_type_3.png]]
|
|
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
|
#+end_src
|
|
|
|
#+name: tab:stewart_cubic_conf_type_3
|
|
#+caption: Stiffness Matrix
|
|
#+RESULTS:
|
|
| 2 | 0 | -3e-16 | 0 | -8.3e-17 | 0 |
|
|
| 0 | 2 | 0 | -2.2e-17 | 0 | 5.6e-17 |
|
|
| -3e-16 | 0 | 2 | -9.3e-19 | -2.8e-17 | 0 |
|
|
| 0 | -2.2e-17 | -9.3e-19 | 0.094 | 0 | 2.1e-17 |
|
|
| -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 |
|
|
|
|
** Size of the platforms
|
|
The minimum size of the platforms depends on the cube's size and the height between the platform and the cube's center.
|
|
|
|
Let's denote:
|
|
- $H$ the height between the cube's center and the considered platform
|
|
- $D$ the size of the cube's edges
|
|
|
|
Let's denote by $a$ and $b$ the points of both ends of one of the cube's edge.
|
|
|
|
Initially, we have:
|
|
\begin{align}
|
|
a &= \frac{D}{2} \begin{bmatrix}-1 \\ -1 \\ 1\end{bmatrix} \\
|
|
b &= \frac{D}{2} \begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix}
|
|
\end{align}
|
|
|
|
We rotate the cube around its center (origin of the rotated frame) such that one of its diagonal is vertical.
|
|
\[ R = \begin{bmatrix}
|
|
\frac{2}{\sqrt{6}} & 0 & \frac{1}{\sqrt{3}} \\
|
|
\frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\
|
|
\frac{-1}{\sqrt{6}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{3}}
|
|
\end{bmatrix} \]
|
|
|
|
After rotation, the points $a$ and $b$ become:
|
|
\begin{align}
|
|
a &= \frac{D}{2} \begin{bmatrix}-\frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ -\frac{1}{\sqrt{3}}\end{bmatrix} \\
|
|
b &= \frac{D}{2} \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ \frac{1}{\sqrt{3}}\end{bmatrix}
|
|
\end{align}
|
|
|
|
Points $a$ and $b$ define a vector $u = b - a$ that gives the orientation of one of the Stewart platform strut:
|
|
\[ u = \frac{D}{\sqrt{3}} \begin{bmatrix} -\sqrt{2} \\ 0 \\ -1\end{bmatrix} \]
|
|
|
|
Then we want to find the intersection between the line that defines the strut with the plane defined by the height $H$ from the cube's center.
|
|
To do so, we first find $g$ such that:
|
|
\[ a_z + g u_z = -H \]
|
|
We obtain:
|
|
\begin{align}
|
|
g &= - \frac{H + a_z}{u_z} \\
|
|
&= \sqrt{3} \frac{H}{D} - \frac{1}{2}
|
|
\end{align}
|
|
|
|
Then, the intersection point $P$ is given by:
|
|
\begin{align}
|
|
P &= a + g u \\
|
|
&= \begin{bmatrix}
|
|
H \sqrt{2} \\
|
|
D \frac{1}{\sqrt{2}} \\
|
|
H
|
|
\end{bmatrix}
|
|
\end{align}
|
|
|
|
Finally, the circle can contains the intersection point has a radius $r$:
|
|
\begin{align}
|
|
r &= \sqrt{P_x^2 + P_y^2} \\
|
|
&= \sqrt{2 H^2 + \frac{1}{2}D^2}
|
|
\end{align}
|
|
|
|
By symmetry, we can show that all the other intersection points will also be on the circle with a radius $r$.
|
|
|
|
For a small cube:
|
|
\[ r \approx \sqrt{2} H \]
|
|
|
|
** 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.
|
|
|
|
| Cube's Size | Paper with the corresponding cubic architecture |
|
|
|-------------+--------------------------------------------------|
|
|
| Small | cite:furutani04_nanom_cuttin_machin_using_stewar |
|
|
| Medium | cite:yang19_dynam_model_decoup_contr_flexib |
|
|
| Large | |
|
|
#+end_important
|
|
|
|
* Cubic size analysis
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../matlab/cubic_conf_size_analysisl.m
|
|
:header-args:matlab+: :comments org :mkdirp yes
|
|
:END:
|
|
<<sec:cubic_conf_size_analysis>>
|
|
|
|
#+begin_note
|
|
The Matlab script corresponding to this section is accessible [[file:../matlab/cubic_conf_size_analysisl.m][here]].
|
|
|
|
To run the script, open the Simulink Project, and type =run cubic_conf_size_analysis.m=.
|
|
#+end_note
|
|
|
|
** Introduction :ignore:
|
|
We here study the effect of the size of the cube used for the Stewart Cubic configuration.
|
|
|
|
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform and the frames $\{A\}$ and $\{B\}$ are also taken at the center of the cube.
|
|
|
|
We only vary the size of the cube.
|
|
|
|
** 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
|
|
|
|
** 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));
|
|
#+end_src
|
|
|
|
The height of the Stewart platform is fixed:
|
|
#+begin_src matlab
|
|
H = 100e-3; % height of the Stewart platform [m]
|
|
#+end_src
|
|
|
|
The frames $\{A\}$ and $\{B\}$ are positioned at the Stewart platform center as well as the cube's center:
|
|
#+begin_src matlab
|
|
MO_B = -50e-3; % Position {B} with respect to {M} [m]
|
|
FOc = H + MO_B; % Center of the cube with respect to {F}
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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
|
|
#+end_src
|
|
|
|
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 :exports none
|
|
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]');
|
|
#+end_src
|
|
|
|
#+NAME: fig:stiffness_cube_size
|
|
#+HEADER: :tangle no :exports results :results raw :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stiffness_cube_size.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:stiffness_cube_size
|
|
#+CAPTION: $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ function of the size of the cube
|
|
#+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 in the Cartesian Frame
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../matlab/cubic_conf_coupling_cartesianl.m
|
|
:header-args:matlab+: :comments org :mkdirp yes
|
|
:END:
|
|
<<sec:cubic_conf_coupling_cartesian>>
|
|
|
|
#+begin_note
|
|
The Matlab script corresponding to this section is accessible [[file:../matlab/cubic_conf_coupling_cartesianl.m][here]].
|
|
|
|
To run the script, open the Simulink Project, and type =run cubic_conf_coupling_cartesian.m=.
|
|
#+end_note
|
|
|
|
** Introduction :ignore:
|
|
In this section, we study the dynamics of the platform in the cartesian frame.
|
|
|
|
We here suppose that there is one relative motion sensor in each strut ($\delta\bm{\mathcal{L}}$ is measured) and we would like to control the position of the top platform pose $\delta \bm{\mathcal{X}}$.
|
|
|
|
Thanks to the Jacobian matrix, we can use the "architecture" shown in Figure [[fig:local_to_cartesian_coordinates]] to obtain the dynamics of the system from forces/torques applied by the actuators on the top platform to translations/rotations of the top platform.
|
|
|
|
#+begin_src latex :file local_to_cartesian_coordinates.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (Jt) at (0, 0) {$\bm{J}^{-T}$};
|
|
\node[block, right= of Jt] (G) {$\bm{G}$};
|
|
\node[block, right= of G] (J) {$\bm{J}^{-1}$};
|
|
|
|
\draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}$};
|
|
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
|
\draw[->] (G.east) -- (J.west) node[above left]{$\delta\bm{\mathcal{L}}$};
|
|
\draw[->] (J.east) -- ++(0.8, 0) node[above left]{$\delta\bm{\mathcal{X}}$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:local_to_cartesian_coordinates
|
|
#+caption: From Strut coordinate to Cartesian coordinate using the Jacobian matrix
|
|
#+RESULTS:
|
|
[[file:figs/local_to_cartesian_coordinates.png]]
|
|
|
|
We here study the dynamics from $\bm{\mathcal{F}}$ to $\delta\bm{\mathcal{X}}$.
|
|
|
|
One has to note that when considering the static behavior:
|
|
\[ \bm{G}(s = 0) = \begin{bmatrix}
|
|
1/k_1 & & 0 \\
|
|
& \ddots & 0 \\
|
|
0 & & 1/k_6
|
|
\end{bmatrix}\]
|
|
|
|
And thus:
|
|
\[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \]
|
|
|
|
We conclude that the *static* behavior of the platform depends on the stiffness matrix.
|
|
For the cubic configuration, we have a diagonal stiffness matrix is the frames $\{A\}$ and $\{B\}$ are coincident with the cube's center.
|
|
|
|
** 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
|
|
|
|
** Cube's center at the Center of Mass of the mobile platform
|
|
Let's create a Cubic Stewart Platform where the *Center of Mass of the mobile platform is located at the center of the cube*.
|
|
|
|
We define the size of the Stewart platform and the position of frames $\{A\}$ and $\{B\}$.
|
|
#+begin_src matlab
|
|
H = 200e-3; % height of the Stewart platform [m]
|
|
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
|
#+end_src
|
|
|
|
Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
|
|
#+begin_src matlab
|
|
Hc = 2.5*H; % Size of the useful part of the cube [m]
|
|
FOc = H + MO_B; % Center of the cube with respect to {F}
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
|
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
#+end_src
|
|
|
|
Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with $\{A\}$ and $\{B\}$.
|
|
#+begin_src matlab
|
|
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
|
'Mpm', 10, ...
|
|
'Mph', 20e-3, ...
|
|
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
|
#+end_src
|
|
|
|
And we set small mass for the struts.
|
|
#+begin_src matlab
|
|
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
#+end_src
|
|
|
|
No flexibility below the Stewart platform and no payload.
|
|
#+begin_src matlab
|
|
ground = initializeGround('type', 'none');
|
|
payload = initializePayload('type', 'none');
|
|
controller = initializeController('type', 'open-loop');
|
|
#+end_src
|
|
|
|
The obtain geometry is shown in figure [[fig:stewart_cubic_conf_decouple_dynamics]].
|
|
|
|
#+begin_src matlab :exports none
|
|
displayArchitecture(stewart, 'labels', false, 'view', 'all');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_cubic_conf_decouple_dynamics.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_cubic_conf_decouple_dynamics
|
|
#+caption: Geometry used for the simulations - The cube's center, the frames $\{A\}$ and $\{B\}$ and the Center of mass of the mobile platform are coincident ([[./figs/stewart_cubic_conf_decouple_dynamics.png][png]], [[./figs/stewart_cubic_conf_decouple_dynamics.pdf][pdf]])
|
|
[[file:figs/stewart_cubic_conf_decouple_dynamics.png]]
|
|
|
|
We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
|
|
#+begin_src matlab
|
|
open('stewart_platform_model.slx')
|
|
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'stewart_platform_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
|
#+end_src
|
|
|
|
Now, thanks to the Jacobian (Figure [[fig:local_to_cartesian_coordinates]]), we compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
|
|
#+begin_src matlab
|
|
Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
|
|
Gc = inv(stewart.kinematics.J)*G*stewart.kinematics.J;
|
|
Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
|
#+end_src
|
|
|
|
The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_cubic_decoupled_dynamics_cartesian]].
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 500);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax3 = subplot(2, 2, 3);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
|
|
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
|
|
|
|
ax2 = subplot(2, 2, 2);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
|
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_cubic_decoupled_dynamics_cartesian.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_cubic_decoupled_dynamics_cartesian
|
|
#+caption: Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ ([[./figs/stewart_cubic_decoupled_dynamics_cartesian.png][png]], [[./figs/stewart_cubic_decoupled_dynamics_cartesian.pdf][pdf]])
|
|
[[file:figs/stewart_cubic_decoupled_dynamics_cartesian.png]]
|
|
|
|
It is interesting to note here that the system shown in Figure [[fig:local_to_cartesian_coordinates_bis]] also yield a decoupled system (explained in section 1.3.3 in cite:li01_simul_fault_vibrat_isolat_point).
|
|
|
|
#+begin_src latex :file local_to_cartesian_coordinates_bis.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (Jt) at (0, 0) {$\bm{J}$};
|
|
\node[block, right= of Jt] (G) {$\bm{G}$};
|
|
\node[block, right= of G] (J) {$\bm{J}^{-1}$};
|
|
|
|
\draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west);
|
|
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
|
\draw[->] (G.east) -- (J.west) node[above left]{$\delta\bm{\mathcal{L}}$};
|
|
\draw[->] (J.east) -- ++(0.8, 0) node[above left]{$\delta\bm{\mathcal{X}}$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:local_to_cartesian_coordinates_bis
|
|
#+caption: Alternative way to decouple the system
|
|
#+RESULTS:
|
|
[[file:figs/local_to_cartesian_coordinates_bis.png]]
|
|
|
|
#+begin_important
|
|
The dynamics is well decoupled at all frequencies.
|
|
|
|
We have the same dynamics for:
|
|
- $D_x/F_x$, $D_y/F_y$ and $D_z/F_z$
|
|
- $R_x/M_x$ and $D_y/F_y$
|
|
|
|
The Dynamics from $F_i$ to $D_i$ is just a 1-dof mass-spring-damper system.
|
|
|
|
This is because the Mass, Damping and Stiffness matrices are all diagonal.
|
|
#+end_important
|
|
|
|
** Cube's center not coincident with the Mass of the Mobile platform
|
|
Let's create a Stewart platform with a cubic architecture where the cube's center is at the center of the Stewart platform.
|
|
#+begin_src matlab
|
|
H = 200e-3; % height of the Stewart platform [m]
|
|
MO_B = -100e-3; % Position {B} with respect to {M} [m]
|
|
#+end_src
|
|
|
|
Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
|
|
#+begin_src matlab
|
|
Hc = 2.5*H; % Size of the useful part of the cube [m]
|
|
FOc = H + MO_B; % Center of the cube with respect to {F}
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
|
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
#+end_src
|
|
|
|
However, the Center of Mass of the mobile platform is *not* located at the cube's center.
|
|
#+begin_src matlab
|
|
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
|
'Mpm', 10, ...
|
|
'Mph', 20e-3, ...
|
|
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
|
#+end_src
|
|
|
|
And we set small mass for the struts.
|
|
#+begin_src matlab
|
|
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
#+end_src
|
|
|
|
No flexibility below the Stewart platform and no payload.
|
|
#+begin_src matlab
|
|
ground = initializeGround('type', 'none');
|
|
payload = initializePayload('type', 'none');
|
|
controller = initializeController('type', 'open-loop');
|
|
#+end_src
|
|
|
|
The obtain geometry is shown in figure [[fig:stewart_cubic_conf_mass_above]].
|
|
#+begin_src matlab :exports none
|
|
displayArchitecture(stewart, 'labels', false, 'view', 'all');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_cubic_conf_mass_above.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_cubic_conf_mass_above
|
|
#+caption: Geometry used for the simulations - The cube's center is coincident with the frames $\{A\}$ and $\{B\}$ but not with the Center of mass of the mobile platform ([[./figs/stewart_cubic_conf_mass_above.png][png]], [[./figs/stewart_cubic_conf_mass_above.pdf][pdf]])
|
|
[[file:figs/stewart_cubic_conf_mass_above.png]]
|
|
|
|
We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
|
|
#+begin_src matlab
|
|
open('stewart_platform_model.slx')
|
|
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'stewart_platform_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
|
#+end_src
|
|
|
|
And we use the Jacobian to compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
|
|
#+begin_src matlab
|
|
Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
|
|
Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
|
#+end_src
|
|
|
|
The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_conf_coupling_mass_matrix]].
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 500);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax3 = subplot(2, 2, 3);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
|
|
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
|
|
|
|
ax2 = subplot(2, 2, 2);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
|
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_conf_coupling_mass_matrix.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_conf_coupling_mass_matrix
|
|
#+caption: Obtained Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ ([[./figs/stewart_conf_coupling_mass_matrix.png][png]], [[./figs/stewart_conf_coupling_mass_matrix.pdf][pdf]])
|
|
[[file:figs/stewart_conf_coupling_mass_matrix.png]]
|
|
|
|
#+begin_important
|
|
The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is *not* decoupled at all frequencies.
|
|
|
|
This was expected as the mass matrix is not diagonal (the Center of Mass of the mobile platform not being coincident with the frame $\{B\}$).
|
|
#+end_important
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Some conclusions can be drawn from the above analysis:
|
|
- Static Decoupling <=> Diagonal Stiffness matrix <=> {A} and {B} at the cube's center
|
|
- Dynamic Decoupling <=> Static Decoupling + CoM of mobile platform coincident with {A} and {B}.
|
|
#+end_important
|
|
|
|
* Dynamic Coupling between actuators and sensors of each strut
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../matlab/cubic_conf_coupling_strutsl.m
|
|
:header-args:matlab+: :comments org :mkdirp yes
|
|
:END:
|
|
<<sec:cubic_conf_coupling_struts>>
|
|
|
|
#+begin_note
|
|
The Matlab script corresponding to this section is accessible [[file:../matlab/cubic_conf_coupling_strutsl.m][here]].
|
|
|
|
To run the script, open the Simulink Project, and type =run cubic_conf_coupling_struts.m=.
|
|
#+end_note
|
|
|
|
** Introduction :ignore:
|
|
From cite:preumont07_six_axis_singl_stage_activ, the cubic configuration "/minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other)/".
|
|
|
|
In this section, we wish to study such properties of the cubic architecture.
|
|
|
|
We will compare the transfer function from sensors to actuators in each strut for a cubic architecture and for a non-cubic architecture (where the struts are not orthogonal with each other).
|
|
|
|
** 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
|
|
|
|
** Coupling between the actuators and sensors - Cubic Architecture
|
|
Let's generate a Cubic architecture where the cube's center and the frames $\{A\}$ and $\{B\}$ are coincident.
|
|
|
|
#+begin_src matlab
|
|
H = 200e-3; % height of the Stewart platform [m]
|
|
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
|
Hc = 2.5*H; % Size of the useful part of the cube [m]
|
|
FOc = H + MO_B; % Center of the cube with respect to {F}
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
|
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
|
'Mpm', 10, ...
|
|
'Mph', 20e-3, ...
|
|
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
|
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
#+end_src
|
|
|
|
No flexibility below the Stewart platform and no payload.
|
|
#+begin_src matlab
|
|
ground = initializeGround('type', 'none');
|
|
payload = initializePayload('type', 'none');
|
|
controller = initializeController('type', 'open-loop');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
disturbances = initializeDisturbances();
|
|
references = initializeReferences(stewart);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
displayArchitecture(stewart, 'labels', false, 'view', 'all');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_architecture_coupling_struts_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_architecture_coupling_struts_cubic
|
|
#+caption: Geometry of the generated Stewart platform ([[./figs/stewart_architecture_coupling_struts_cubic.png][png]], [[./figs/stewart_architecture_coupling_struts_cubic.pdf][pdf]])
|
|
[[file:figs/stewart_architecture_coupling_struts_cubic.png]]
|
|
|
|
And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure [[fig:coupling_struts_relative_sensor_cubic]]) and to the force sensors $\tau_{m,i}$ (Figure [[fig:coupling_struts_force_sensor_cubic]]).
|
|
|
|
#+begin_src matlab :exports none
|
|
open('stewart_platform_model.slx')
|
|
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'stewart_platform_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax3 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2], {'$L_i/\tau_i$', '$L_i/\tau_j$'})
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/coupling_struts_relative_sensor_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:coupling_struts_relative_sensor_cubic
|
|
#+caption: Dynamics from the force actuators to the relative motion sensors ([[./figs/coupling_struts_relative_sensor_cubic.png][png]], [[./figs/coupling_struts_relative_sensor_cubic.pdf][pdf]])
|
|
[[file:figs/coupling_struts_relative_sensor_cubic.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensor Outputs [N]
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 500);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax3 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2], {'$F_{m,i}/\tau_i$', '$F_{m,i}/\tau_j$'})
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/coupling_struts_force_sensor_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:coupling_struts_force_sensor_cubic
|
|
#+caption: Dynamics from the force actuators to the force sensors ([[./figs/coupling_struts_force_sensor_cubic.png][png]], [[./figs/coupling_struts_force_sensor_cubic.pdf][pdf]])
|
|
[[file:figs/coupling_struts_force_sensor_cubic.png]]
|
|
|
|
** Coupling between the actuators and sensors - Non-Cubic Architecture
|
|
Now we generate a Stewart platform which is not cubic but with approximately the same size as the previous cubic architecture.
|
|
|
|
#+begin_src matlab
|
|
H = 200e-3; % height of the Stewart platform [m]
|
|
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
|
stewart = generateGeneralConfiguration(stewart, 'FR', 250e-3, 'MR', 150e-3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
|
'Mpm', 10, ...
|
|
'Mph', 20e-3, ...
|
|
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
|
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
#+end_src
|
|
|
|
No flexibility below the Stewart platform and no payload.
|
|
#+begin_src matlab
|
|
ground = initializeGround('type', 'none');
|
|
payload = initializePayload('type', 'none');
|
|
controller = initializeController('type', 'open-loop');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
displayArchitecture(stewart, 'labels', false, 'view', 'all');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/stewart_architecture_coupling_struts_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:stewart_architecture_coupling_struts_non_cubic
|
|
#+caption: Geometry of the generated Stewart platform ([[./figs/stewart_architecture_coupling_struts_non_cubic.png][png]], [[./figs/stewart_architecture_coupling_struts_non_cubic.pdf][pdf]])
|
|
[[file:figs/stewart_architecture_coupling_struts_non_cubic.png]]
|
|
|
|
And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure [[fig:coupling_struts_relative_sensor_non_cubic]]) and to the force sensors $\tau_{m,i}$ (Figure [[fig:coupling_struts_force_sensor_non_cubic]]).
|
|
|
|
#+begin_src matlab :exports none
|
|
open('stewart_platform_model.slx')
|
|
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'stewart_platform_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax3 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2], {'$L_i/\tau_i$', '$L_i/\tau_j$'})
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/coupling_struts_relative_sensor_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:coupling_struts_relative_sensor_non_cubic
|
|
#+caption: Dynamics from the force actuators to the relative motion sensors ([[./figs/coupling_struts_relative_sensor_non_cubic.png][png]], [[./figs/coupling_struts_relative_sensor_non_cubic.pdf][pdf]])
|
|
[[file:figs/coupling_struts_relative_sensor_non_cubic.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensor Outputs [N]
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 500);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:6
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
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hold off;
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|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
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|
|
|
ax3 = subplot(2, 1, 2);
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|
hold on;
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|
for i = 1:6
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for j = i+1:6
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|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
|
end
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|
end
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|
set(gca,'ColorOrderIndex',1);
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|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
|
hold off;
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|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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|
ylim([-180, 180]);
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|
yticks([-180, -90, 0, 90, 180]);
|
|
legend([p1, p2], {'$F_{m,i}/\tau_i$', '$F_{m,i}/\tau_j$'})
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|
|
|
linkaxes([ax1,ax2],'x');
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|
#+end_src
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|
|
|
#+header: :tangle no :exports results :results none :noweb yes
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|
#+begin_src matlab :var filepath="figs/coupling_struts_force_sensor_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:coupling_struts_force_sensor_non_cubic
|
|
#+caption: Dynamics from the force actuators to the force sensors ([[./figs/coupling_struts_force_sensor_non_cubic.png][png]], [[./figs/coupling_struts_force_sensor_non_cubic.pdf][pdf]])
|
|
[[file:figs/coupling_struts_force_sensor_non_cubic.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut and thus provides no advantages for decentralized control.
|
|
#+end_important
|
|
|
|
* Functions
|
|
<<sec:functions>>
|
|
|
|
** =generateCubicConfiguration=: Generate a Cubic Configuration
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../src/generateCubicConfiguration.m
|
|
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
|
:END:
|
|
<<sec:generateCubicConfiguration>>
|
|
|
|
This Matlab function is accessible [[file:../src/generateCubicConfiguration.m][here]].
|
|
|
|
*** Function description
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
#+begin_src matlab
|
|
function [stewart] = generateCubicConfiguration(stewart, args)
|
|
% generateCubicConfiguration - Generate a Cubic Configuration
|
|
%
|
|
% Syntax: [stewart] = generateCubicConfiguration(stewart, args)
|
|
%
|
|
% Inputs:
|
|
% - stewart - A structure with the following fields
|
|
% - geometry.H [1x1] - Total height of the platform [m]
|
|
% - args - Can have the following fields:
|
|
% - Hc [1x1] - Height of the "useful" part of the cube [m]
|
|
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
|
|
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
|
|
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
|
|
%
|
|
% Outputs:
|
|
% - stewart - updated Stewart structure with the added fields:
|
|
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
|
|
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
|
|
#+end_src
|
|
|
|
*** Documentation
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
#+name: fig:cubic-configuration-definition
|
|
#+caption: Cubic Configuration
|
|
[[file:figs/cubic-configuration-definition.png]]
|
|
|
|
*** Optional Parameters
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
#+begin_src matlab
|
|
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, mustBeNonnegative} = 15e-3
|
|
args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
|
|
end
|
|
#+end_src
|
|
|
|
*** Check the =stewart= structure elements
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
#+begin_src matlab
|
|
assert(isfield(stewart.geometry, 'H'), 'stewart.geometry should have attribute H')
|
|
H = stewart.geometry.H;
|
|
#+end_src
|
|
|
|
*** Position of the Cube
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
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 located at the center of the cube and aligned with {F} and {M}.
|
|
|
|
#+begin_src matlab
|
|
sx = [ 2; -1; -1];
|
|
sy = [ 0; 1; -1];
|
|
sz = [ 1; 1; 1];
|
|
|
|
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
|
|
|
|
L = args.Hc*sqrt(3);
|
|
|
|
Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];
|
|
|
|
CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
|
|
CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
|
|
#+end_src
|
|
|
|
*** Compute the pose
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).
|
|
#+begin_src matlab
|
|
CSi = (CCm - CCf)./vecnorm(CCm - CCf);
|
|
#+end_src
|
|
|
|
We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
|
|
#+begin_src matlab
|
|
Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
|
|
Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
|
|
#+end_src
|
|
|
|
*** Populate the =stewart= structure
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
#+begin_src matlab
|
|
stewart.platform_F.Fa = Fa;
|
|
stewart.platform_M.Mb = Mb;
|
|
#+end_src
|
|
|
|
* Bibliography :ignore:
|
|
bibliographystyle:unsrtnat
|
|
bibliography:ref.bib
|