Add functions and documentation to initialize stewart platforms
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simscape-model.html
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simscape-model.html
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@ -46,29 +46,38 @@ From $\bm{a}_{i}$ and $\bm{b}_{i}$, we can determine the *length and orientation
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- $l_{i}$ is the length of the strut
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- ${}^{A}\hat{\bm{s}}_{i}$ is the unit vector align with the strut
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The position of the Spherical joints can be done using various methods:
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The position of the Spherical joints can be computed using various methods:
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- Cubic configuration
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- Geometrical
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- Definition them by hand
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- Circular configuration
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- Arbitrary position
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- These methods should be easily scriptable and corresponds to specific functions that returns ${}^{F}\bm{a}_{i}$ and ${}^{M}\bm{b}_{i}$.
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The input of these functions are the parameters corresponding to the wanted geometry.
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We need also to know the height of the platform.
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For Simscape, we need:
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- The position of the frame $\{A\}$ with respect to the frame $\{F\}$: ${}^{F}\bm{O}_{A}$
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- The position of the frame $\{B\}$ with respect to the frame $\{M\}$: ${}^{M}\bm{O}_{B}$
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- The position and orientation of each spherical joint fixed to the fixed base: ${}^{F}\bm{a}_{i}$ and ${}^{F}\bm{R}_{a_{i}}$
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- The position and orientation of each spherical joint fixed to the moving platform: ${}^{M}\bm{b}_{i}$ and ${}^{M}\bm{R}_{b_{i}}$
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- The rest length of each strut: $l_{i}$
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- The stiffness and damping of each actuator: $k_{i}$ and $c_{i}$
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- The position of the frame $\{A\}$ with respect to the frame $\{F\}$: ${}^{F}\bm{O}_{A}$
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- The position of the frame $\{B\}$ with respect to the frame $\{M\}$: ${}^{M}\bm{O}_{B}$
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* Procedure
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The procedure is the following:
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1. Choose $H$
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2. Choose ${}^{F}\bm{O}_{A}$ and ${}^{M}\bm{O}_{B}$
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3. Choose $\bm{a}_{i}$ and $\bm{b}_{i}$, probably by specifying ${}^{F}\bm{a}_{i}$ and ${}^{M}\bm{b}_{i}$
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4. Choose $k_{i}$ and $c_{i}$
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The procedure to define the Stewart platform is the following:
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1. Define the initial position of frames {A}, {B}, {F} and {M}.
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We do that using the =initializeFramesPositions= function.
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We have to specify the total height of the Stewart platform $H$ and the position ${}^{M}O_{B}$ of {B} with respect to {M}.
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2. Compute the positions of joints ${}^{F}a_{i}$ and ${}^{M}b_{i}$.
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We can do that using various methods depending on the wanted architecture:
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- =generateCubicConfiguration= permits to generate a cubic configuration
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3. Compute the position and orientation of the joints with respect to the fixed base and the moving platform.
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This is done with the =computeJointsPose= function.
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4. Define the dynamical properties of the Stewart platform.
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The output are the stiffness and damping of each strut $k_{i}$ and $c_{i}$.
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This can be done we simply choosing directly the stiffness and damping of each strut.
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The stiffness and damping of each actuator can also be determine from the wanted stiffness of the Stewart platform for instance.
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5. Define the mass and inertia of each element of the Stewart platform.
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By following this procedure, we obtain a Matlab structure =stewart= that contains all the information for the Simscape model and for further analysis.
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* Matlab Code
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** Matlab Init :noexport:ignore:
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@ -85,21 +94,294 @@ The procedure is the following:
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open('stewart_platform.slx')
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#+end_src
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** Define the Height of the Platform
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** Test the functions
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#+begin_src matlab
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stewart = initializeFramesPositions(struct('H', 90e-3, 'MO_B', 50e-3));
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stewart = generateCubicConfiguration(stewart, struct('Hc', 60e-3, 'FOc', 50e-3, 'FHa', 15e-3, 'MHb', 15e-3));
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, struct('Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1)));
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#+end_src
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* =initializeFramesPositions=: Initialize the positions of frames {A}, {B}, {F} and {M}
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:PROPERTIES:
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:header-args:matlab+: :tangle src/initializeFramesPositions.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:initializeFramesPositions>>
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This Matlab function is accessible [[file:src/initializeFramesPositions.m][here]].
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** Function description
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#+begin_src matlab
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function [stewart] = initializeFramesPositions(opts_param)
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% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
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%
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% Syntax: [stewart] = initializeFramesPositions(H, MO_B)
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%
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% Inputs:
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% - opts_param - Structure with the following fields:
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% - H [1x1] - Total Height of the Stewart Platform [m]
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% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
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%
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% Outputs:
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% - stewart - A structure with the following fields:
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% - H [1x1] - Total Height of the Stewart Platform [m]
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% - FO_M [3x1] - Position of {M} with respect to {F} [m]
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% - MO_B [3x1] - Position of {B} with respect to {M} [m]
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% - FO_A [3x1] - Position of {A} with respect to {F} [m]
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#+end_src
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** Optional Parameters
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Default values for opts.
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#+begin_src matlab
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opts = struct( ...
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'H', 90e-3, ... % [m]
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'MO_B', 50e-3 ... % [m]
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);
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#+end_src
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Populate opts with input parameters
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#+begin_src matlab
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if exist('opts_param','var')
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for opt = fieldnames(opts_param)'
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opts.(opt{1}) = opts_param.(opt{1});
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end
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end
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#+end_src
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** Initialize the Stewart structure
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#+begin_src matlab
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stewart = struct();
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#+end_src
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** Compute the position of each frame
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#+begin_src matlab
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stewart.H = opts.H; % Total Height of the Stewart Platform [m]
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stewart.FO_M = [0; 0; stewart.H]; % Position of {M} with respect to {F} [m]
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stewart.MO_B = [0; 0; opts.MO_B]; % Position of {B} with respect to {M} [m]
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stewart.FO_A = stewart.MO_B + stewart.FO_M; % Position of {A} with respect to {F} [m]
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#+end_src
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* =generateCubicConfiguration=: Generate a Cubic Configuration
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:PROPERTIES:
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:header-args:matlab+: :tangle src/generateCubicConfiguration.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:generateCubicConfiguration>>
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This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here]].
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** Function description
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#+begin_src matlab
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function [stewart] = generateCubicConfiguration(stewart, opts_param)
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% generateCubicConfiguration - Generate a Cubic Configuration
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%
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% Syntax: [stewart] = generateCubicConfiguration(stewart, opts_param)
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%
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% Inputs:
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% - stewart - A structure with the following fields
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% - H [1x1] - Total height of the platform [m]
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% - opts_param - Structure with the following fields:
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% - Hc [1x1] - Height of the "useful" part of the cube [m]
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% - FOc [1x1] - Height of the center of the cute with respect to {F} [m]
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% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
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% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
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%
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% Outputs:
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% - stewart - updated Stewart structure with the added fields:
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% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
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% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
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#+end_src
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** Optional Parameters
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Default values for opts.
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#+begin_src matlab
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opts = struct( ...
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'Hc', 60e-3, ... % [m]
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'FOc', 50e-3, ... % [m]
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'FHa', 15e-3, ... % [m]
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'MHb', 15e-3 ... % [m]
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);
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#+end_src
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Populate opts with input parameters
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#+begin_src matlab
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if exist('opts_param','var')
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for opt = fieldnames(opts_param)'
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opts.(opt{1}) = opts_param.(opt{1});
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end
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end
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#+end_src
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** Position of the Cube
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We define the useful points of the cube with respect to the Cube's center.
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${}^{C}C$ 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}.
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#+begin_src matlab
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sx = [ 2; -1; -1];
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sy = [ 0; 1; -1];
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sz = [ 1; 1; 1];
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R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
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L = opts.Hc*sqrt(3);
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Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*opts.Hc];
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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
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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
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#+end_src
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** Compute the pose
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We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).
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#+begin_src matlab
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CSi = (CCm - CCf)./vecnorm(CCm - CCf);
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#+end_src
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We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
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#+begin_src matlab
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stewart.Fa = CCf + [0; 0; opts.FOc] + ((opts.FHa-(opts.FOc-opts.Hc/2))./CSi(3,:)).*CSi;
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stewart.Mb = CCf + [0; 0; opts.FOc-stewart.H] + ((stewart.H-opts.MHb-(opts.FOc-opts.Hc/2))./CSi(3,:)).*CSi;
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#+end_src
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* =computeJointsPose=: Compute the Pose of the Joints
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:PROPERTIES:
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:header-args:matlab+: :tangle src/computeJointsPose.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:computeJointsPose>>
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This Matlab function is accessible [[file:src/computeJointsPose.m][here]].
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** Function description
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#+begin_src matlab
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function [stewart] = computeJointsPose(stewart)
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% computeJointsPose -
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%
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% Syntax: [stewart] = computeJointsPose(stewart, opts_param)
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%
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% Inputs:
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% - stewart - A structure with the following fields
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% - FO_A [3x1] - Position of {A} with respect to {F}
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% - MO_B [3x1] - Position of {B} with respect to {M}
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% - FO_M [3x1] - Position of {M} with respect to {F}
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%
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% Outputs:
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% - stewart - A structure with the following added fields
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% - Aa [3x6] - The i'th column is the position of ai with respect to {A}
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% - Ab [3x6] - The i'th column is the position of bi with respect to {A}
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% - Ba [3x6] - The i'th column is the position of ai with respect to {B}
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% - Bb [3x6] - The i'th column is the position of bi with respect to {B}
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% - l [6x1] - The i'th element is the initial length of strut i
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% - As [3x6] - The i'th column is the unit vector of strut i expressed in {A}
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% - Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}
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% - FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
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% - MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
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#+end_src
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** Compute the position of the Joints
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#+begin_src matlab
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stewart.Aa = stewart.Fa - repmat(stewart.FO_A, [1, 6]);
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stewart.Bb = stewart.Mb - repmat(stewart.MO_B, [1, 6]);
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stewart.Ab = stewart.Bb - repmat(-stewart.MO_B-stewart.FO_M+stewart.FO_A, [1, 6]);
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stewart.Ba = stewart.Aa - repmat( stewart.MO_B+stewart.FO_M-stewart.FO_A, [1, 6]);
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#+end_src
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** Compute the strut length and orientation
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#+begin_src matlab
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stewart.As = (stewart.Ab - stewart.Aa)./vecnorm(stewart.Ab - stewart.Aa); % As_i is the i'th vector of As
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stewart.l = vecnorm(stewart.Ab - stewart.Aa)';
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#+end_src
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#+begin_src matlab
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stewart.Bs = (stewart.Bb - stewart.Ba)./vecnorm(stewart.Bb - stewart.Ba);
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#+end_src
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** Compute the orientation of the Joints
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#+begin_src matlab
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stewart.FRa = zeros(3,3,6);
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stewart.MRb = zeros(3,3,6);
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for i = 1:6
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stewart.FRa(:,:,i) = [cross([0;1;0], stewart.As(:,i)) , cross(stewart.As(:,i), cross([0;1;0], stewart.As(:,i))) , stewart.As(:,i)];
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stewart.FRa(:,:,i) = stewart.FRa(:,:,i)./vecnorm(stewart.FRa(:,:,i));
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stewart.MRb(:,:,i) = [cross([0;1;0], stewart.Bs(:,i)) , cross(stewart.Bs(:,i), cross([0;1;0], stewart.Bs(:,i))) , stewart.Bs(:,i)];
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stewart.MRb(:,:,i) = stewart.MRb(:,:,i)./vecnorm(stewart.MRb(:,:,i));
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end
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#+end_src
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* =initializeStrutDynamics=: Add Stiffness and Damping properties of each strut
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:PROPERTIES:
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:header-args:matlab+: :tangle src/initializeStrutDynamics.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:initializeStrutDynamics>>
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This Matlab function is accessible [[file:src/initializeStrutDynamics.m][here]].
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** Function description
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#+begin_src matlab
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function [stewart] = initializeStrutDynamics(stewart, opts_param)
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% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
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%
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% Syntax: [stewart] = initializeStrutDynamics(opts_param)
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%
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% Inputs:
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% - opts_param - Structure with the following fields:
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% - Ki [6x1] - Stiffness of each strut [N/m]
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% - Ci [6x1] - Damping of each strut [N/(m/s)]
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%
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% Outputs:
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% - stewart - updated Stewart structure with the added fields:
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% - Ki [6x1] - Stiffness of each strut [N/m]
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% - Ci [6x1] - Damping of each strut [N/(m/s)]
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#+end_src
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** Optional Parameters
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Default values for opts.
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#+begin_src matlab
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opts = struct( ...
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'Ki', 1e6*ones(6,1), ... % [N/m]
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'Ci', 1e2*ones(6,1) ... % [N/(m/s)]
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);
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#+end_src
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Populate opts with input parameters
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#+begin_src matlab
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if exist('opts_param','var')
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for opt = fieldnames(opts_param)'
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opts.(opt{1}) = opts_param.(opt{1});
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end
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end
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#+end_src
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** Add Stiffness and Damping properties of each strut
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#+begin_src matlab
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stewart.Ki = opts.Ki;
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stewart.Ci = opts.Ci;
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#+end_src
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* OLD :noexport:
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** Define the Height of the Platform :noexport:
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#+begin_src matlab
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%% 1. Height of the platform. Location of {F} and {M}
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H = 90e-3; % [m]
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FO_M = [0; 0; H];
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#+end_src
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** Define the location of {A} and {B}
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** Define the location of {A} and {B} :noexport:
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#+begin_src matlab
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%% 2. Location of {A} and {B}
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FO_A = [0; 0; 100e-3] + FO_M;% [m,m,m]
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MO_B = [0; 0; 100e-3];% [m,m,m]
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#+end_src
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** Define the position of $a_{i}$ and $b_{i}$
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** Define the position of $a_{i}$ and $b_{i}$ :noexport:
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#+begin_src matlab
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%% 3. Position of ai and bi
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Fa = zeros(3, 6); % Fa_i is the i'th vector of Fa
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@ -130,15 +412,14 @@ The procedure is the following:
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end
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#+end_src
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** Define the dynamical properties of each strut
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** Define the dynamical properties of each strut :noexport:
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#+begin_src matlab
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%% 4. Stiffness and Damping of each strut
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Ki = 1e6*ones(6,1);
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Ci = 1e2*ones(6,1);
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#+end_src
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** Old Introduction :ignore:
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** Old Introduction :noexport:
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First, geometrical parameters are defined:
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- ${}^A\bm{a}_i$ - Position of the joints fixed to the fixed base w.r.t $\{A\}$
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- ${}^A\bm{b}_i$ - Position of the joints fixed to the mobile platform w.r.t $\{A\}$
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@ -170,8 +451,7 @@ Other Parameters are defined for the Simscape simulation:
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- Location of the point for the differential measurements
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- Location of the Jacobian point for velocity/displacement computation
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** Cubic Configuration
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** Cubic Configuration :noexport:
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To define the cubic configuration, we need to define 4 parameters:
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- The size of the cube
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- The location of the cube
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@ -182,7 +462,7 @@ To do so, we specify the following parameters:
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- $H_{C}$ the height of the useful part of the cube
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- ${}^{F}O_{C}$ the position of the center of the cube with respect to $\{F\}$
|
||||
- ${}^{F}H_{A}$: the height of the plane joining the points $a_{i}$ with respect to the frame $\{F\}$
|
||||
- ${}^{M}H_{B}$: the height of the plane joining the points $b_{i}$ with respect to the frame $\{B\}$
|
||||
- ${}^{M}H_{B}$: the height of the plane joining the points $b_{i}$ with respect to the frame $\{M\}$
|
||||
|
||||
We define the parameters
|
||||
#+begin_src matlab
|
||||
@ -225,21 +505,104 @@ We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
|
||||
Mb = CCf + [0; 0; FOc-H] + ((H-MHb-(FOc-Hc/2))./CSi(3,:)).*CSi; % TODO
|
||||
#+end_src
|
||||
|
||||
* initializeCubicConfiguration
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
:HEADER-ARGS:matlab+: :eval no
|
||||
:HEADER-ARGS:matlab+: :tangle src/initializeCubicConfiguration.m
|
||||
:END:
|
||||
<<sec:initializeCubicConfiguration>>
|
||||
** initializeGeneralConfiguration :noexport:
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
:HEADER-ARGS:matlab+: :eval no
|
||||
:HEADER-ARGS:matlab+: :tangle src/initializeGeneralConfiguration.m
|
||||
:END:
|
||||
|
||||
** Function description
|
||||
*** Function description
|
||||
The =initializeGeneralConfiguration= function takes one structure that contains configurations for the hexapod and returns one structure representing the Hexapod.
|
||||
|
||||
#+begin_src matlab
|
||||
function [stewart] = initializeGeneralConfiguration(opts_param)
|
||||
#+end_src
|
||||
|
||||
*** Optional Parameters
|
||||
Default values for opts.
|
||||
#+begin_src matlab
|
||||
opts = struct(...
|
||||
'H_tot', 90, ... % Height of the platform [mm]
|
||||
'H_joint', 15, ... % Height of the joints [mm]
|
||||
'H_plate', 10, ... % Thickness of the fixed and mobile platforms [mm]
|
||||
'R_bot', 100, ... % Radius where the legs articulations are positionned [mm]
|
||||
'R_top', 80, ... % Radius where the legs articulations are positionned [mm]
|
||||
'a_bot', 10, ... % Angle Offset [deg]
|
||||
'a_top', 40, ... % Angle Offset [deg]
|
||||
'da_top', 0 ... % Angle Offset from 0 position [deg]
|
||||
);
|
||||
#+end_src
|
||||
|
||||
Populate opts with input parameters
|
||||
#+begin_src matlab
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
*** Geometry Description
|
||||
#+name: fig:stewart_bottom_plate
|
||||
#+caption: Schematic of the bottom plates with all the parameters
|
||||
[[file:./figs/stewart_bottom_plate.png]]
|
||||
|
||||
*** Compute Aa and Ab
|
||||
We compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
|
||||
|
||||
#+begin_src matlab
|
||||
Aa = zeros(6, 3); % [mm]
|
||||
Ab = zeros(6, 3); % [mm]
|
||||
Bb = zeros(6, 3); % [mm]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:3
|
||||
Aa(2*i-1,:) = [opts.R_bot*cos( pi/180*(120*(i-1) - opts.a_bot) ), ...
|
||||
opts.R_bot*sin( pi/180*(120*(i-1) - opts.a_bot) ), ...
|
||||
opts.H_plate+opts.H_joint];
|
||||
Aa(2*i,:) = [opts.R_bot*cos( pi/180*(120*(i-1) + opts.a_bot) ), ...
|
||||
opts.R_bot*sin( pi/180*(120*(i-1) + opts.a_bot) ), ...
|
||||
opts.H_plate+opts.H_joint];
|
||||
|
||||
Ab(2*i-1,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
|
||||
opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
|
||||
opts.H_tot - opts.H_plate - opts.H_joint];
|
||||
Ab(2*i,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
|
||||
opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
|
||||
opts.H_tot - opts.H_plate - opts.H_joint];
|
||||
end
|
||||
|
||||
Bb = Ab - opts.H_tot*[0,0,1];
|
||||
#+end_src
|
||||
|
||||
*** Returns Stewart Structure
|
||||
#+begin_src matlab :results none
|
||||
stewart = struct();
|
||||
stewart.Aa = Aa;
|
||||
stewart.Ab = Ab;
|
||||
stewart.Bb = Bb;
|
||||
stewart.H_tot = opts.H_tot;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** initializeCubicConfiguration :noexport:
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
:HEADER-ARGS:matlab+: :eval no
|
||||
:HEADER-ARGS:matlab+: :tangle src/initializeCubicConfiguration.m
|
||||
:END:
|
||||
<<sec:initializeCubicConfiguration>>
|
||||
|
||||
*** Function description
|
||||
#+begin_src matlab
|
||||
function [stewart] = initializeCubicConfiguration(opts_param)
|
||||
#+end_src
|
||||
|
||||
** Optional Parameters
|
||||
*** Optional Parameters
|
||||
Default values for opts.
|
||||
#+begin_src matlab
|
||||
opts = struct(...
|
||||
@ -259,7 +622,7 @@ Populate opts with input parameters
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Cube Creation
|
||||
*** Cube Creation
|
||||
#+begin_src matlab :results none
|
||||
points = [0, 0, 0; ...
|
||||
0, 0, 1; ...
|
||||
@ -294,7 +657,7 @@ We use to rotation matrix to rotate the cube
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Vectors of each leg
|
||||
*** Vectors of each leg
|
||||
#+begin_src matlab :results none
|
||||
leg_indices = [3, 4; ...
|
||||
2, 4; ...
|
||||
@ -315,7 +678,7 @@ Vectors are:
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Verification of Height of the Stewart Platform
|
||||
*** Verification of Height of the Stewart Platform
|
||||
If the Stewart platform is not contained in the cube, throw an error.
|
||||
|
||||
#+begin_src matlab :results none
|
||||
@ -328,7 +691,7 @@ If the Stewart platform is not contained in the cube, throw an error.
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Determinate the location of the joints
|
||||
*** Determinate the location of the joints
|
||||
We now determine the location of the joints on the fixed platform w.r.t the fixed frame $\{A\}$.
|
||||
$\{A\}$ is fixed to the bottom of the base.
|
||||
#+begin_src matlab :results none
|
||||
@ -360,7 +723,7 @@ And the location of the joints on the mobile platform with respect to $\{B\}$.
|
||||
Ab = Ab - h*[0, 0, 1];
|
||||
#+end_src
|
||||
|
||||
** Returns Stewart Structure
|
||||
*** Returns Stewart Structure
|
||||
#+begin_src matlab :results none
|
||||
stewart = struct();
|
||||
stewart.Aa = Aa;
|
||||
@ -370,90 +733,7 @@ And the location of the joints on the mobile platform with respect to $\{B\}$.
|
||||
end
|
||||
#+end_src
|
||||
|
||||
* initializeGeneralConfiguration
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
:HEADER-ARGS:matlab+: :eval no
|
||||
:HEADER-ARGS:matlab+: :tangle src/initializeGeneralConfiguration.m
|
||||
:END:
|
||||
|
||||
** Function description
|
||||
The =initializeGeneralConfiguration= function takes one structure that contains configurations for the hexapod and returns one structure representing the Hexapod.
|
||||
|
||||
#+begin_src matlab
|
||||
function [stewart] = initializeGeneralConfiguration(opts_param)
|
||||
#+end_src
|
||||
|
||||
** Optional Parameters
|
||||
Default values for opts.
|
||||
#+begin_src matlab
|
||||
opts = struct(...
|
||||
'H_tot', 90, ... % Height of the platform [mm]
|
||||
'H_joint', 15, ... % Height of the joints [mm]
|
||||
'H_plate', 10, ... % Thickness of the fixed and mobile platforms [mm]
|
||||
'R_bot', 100, ... % Radius where the legs articulations are positionned [mm]
|
||||
'R_top', 80, ... % Radius where the legs articulations are positionned [mm]
|
||||
'a_bot', 10, ... % Angle Offset [deg]
|
||||
'a_top', 40, ... % Angle Offset [deg]
|
||||
'da_top', 0 ... % Angle Offset from 0 position [deg]
|
||||
);
|
||||
#+end_src
|
||||
|
||||
Populate opts with input parameters
|
||||
#+begin_src matlab
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Geometry Description
|
||||
#+name: fig:stewart_bottom_plate
|
||||
#+caption: Schematic of the bottom plates with all the parameters
|
||||
[[file:./figs/stewart_bottom_plate.png]]
|
||||
|
||||
** Compute Aa and Ab
|
||||
We compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
|
||||
|
||||
#+begin_src matlab
|
||||
Aa = zeros(6, 3); % [mm]
|
||||
Ab = zeros(6, 3); % [mm]
|
||||
Bb = zeros(6, 3); % [mm]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:3
|
||||
Aa(2*i-1,:) = [opts.R_bot*cos( pi/180*(120*(i-1) - opts.a_bot) ), ...
|
||||
opts.R_bot*sin( pi/180*(120*(i-1) - opts.a_bot) ), ...
|
||||
opts.H_plate+opts.H_joint];
|
||||
Aa(2*i,:) = [opts.R_bot*cos( pi/180*(120*(i-1) + opts.a_bot) ), ...
|
||||
opts.R_bot*sin( pi/180*(120*(i-1) + opts.a_bot) ), ...
|
||||
opts.H_plate+opts.H_joint];
|
||||
|
||||
Ab(2*i-1,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
|
||||
opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
|
||||
opts.H_tot - opts.H_plate - opts.H_joint];
|
||||
Ab(2*i,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
|
||||
opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
|
||||
opts.H_tot - opts.H_plate - opts.H_joint];
|
||||
end
|
||||
|
||||
Bb = Ab - opts.H_tot*[0,0,1];
|
||||
#+end_src
|
||||
|
||||
** Returns Stewart Structure
|
||||
#+begin_src matlab :results none
|
||||
stewart = struct();
|
||||
stewart.Aa = Aa;
|
||||
stewart.Ab = Ab;
|
||||
stewart.Bb = Bb;
|
||||
stewart.H_tot = opts.H_tot;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
* computeGeometricalProperties
|
||||
** computeGeometricalProperties :noexport:
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
@ -461,12 +741,12 @@ end
|
||||
:HEADER-ARGS:matlab+: :tangle src/computeGeometricalProperties.m
|
||||
:END:
|
||||
|
||||
** Function description
|
||||
*** Function description
|
||||
#+begin_src matlab
|
||||
function [stewart] = computeGeometricalProperties(stewart, opts_param)
|
||||
#+end_src
|
||||
|
||||
** Optional Parameters
|
||||
*** Optional Parameters
|
||||
Default values for opts.
|
||||
#+begin_src matlab
|
||||
opts = struct(...
|
||||
@ -484,7 +764,7 @@ Populate opts with input parameters
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Rotation matrices
|
||||
*** Rotation matrices
|
||||
We initialize $l_i$ and $\hat{s}_i$
|
||||
#+begin_src matlab
|
||||
leg_length = zeros(6, 1); % [mm]
|
||||
@ -525,7 +805,7 @@ The rotation matrix transforms the $z$ axis to the axis of the leg. The other ax
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Jacobian matrices
|
||||
*** Jacobian matrices
|
||||
Compute Jacobian Matrix
|
||||
#+begin_src matlab
|
||||
Jd = zeros(6);
|
||||
@ -555,7 +835,7 @@ Compute Jacobian Matrix
|
||||
end
|
||||
#+end_src
|
||||
|
||||
* initializeMechanicalElements
|
||||
** initializeMechanicalElements :noexport:
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
@ -563,12 +843,12 @@ Compute Jacobian Matrix
|
||||
:HEADER-ARGS:matlab+: :tangle src/initializeMechanicalElements.m
|
||||
:END:
|
||||
|
||||
** Function description
|
||||
*** Function description
|
||||
#+begin_src matlab
|
||||
function [stewart] = initializeMechanicalElements(stewart, opts_param)
|
||||
#+end_src
|
||||
|
||||
** Optional Parameters
|
||||
*** Optional Parameters
|
||||
Default values for opts.
|
||||
#+begin_src matlab
|
||||
opts = struct(...
|
||||
@ -589,7 +869,7 @@ Populate opts with input parameters
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Bottom Plate
|
||||
*** Bottom Plate
|
||||
#+name: fig:stewart_bottom_plate
|
||||
#+caption: Schematic of the bottom plates with all the parameters
|
||||
[[file:./figs/stewart_bottom_plate.png]]
|
||||
@ -630,7 +910,7 @@ The structure is added to the stewart structure
|
||||
stewart.BP = BP;
|
||||
#+end_src
|
||||
|
||||
** Top Plate
|
||||
*** Top Plate
|
||||
The top plate structure is initialized.
|
||||
#+begin_src matlab
|
||||
TP = struct();
|
||||
@ -667,7 +947,7 @@ The structure is added to the stewart structure
|
||||
stewart.TP = TP;
|
||||
#+end_src
|
||||
|
||||
** Legs
|
||||
*** Legs
|
||||
#+name: fig:stewart_legs
|
||||
#+caption: Schematic for the legs of the Stewart platform
|
||||
[[file:./figs/stewart_legs.png]]
|
||||
@ -731,7 +1011,7 @@ The structure is added to the stewart structure
|
||||
stewart.Leg = Leg;
|
||||
#+end_src
|
||||
|
||||
** Ball Joints
|
||||
*** Ball Joints
|
||||
#+name: fig:stewart_ball_joints
|
||||
#+caption: Schematic of the support for the ball joints
|
||||
[[file:./figs/stewart_ball_joints.png]]
|
||||
@ -781,7 +1061,7 @@ The structure is added to the Hexapod structure
|
||||
stewart.SP = SP;
|
||||
#+end_src
|
||||
|
||||
* initializeSample
|
||||
** initializeSample :noexport:
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
@ -789,12 +1069,12 @@ The structure is added to the Hexapod structure
|
||||
:HEADER-ARGS:matlab+: :tangle src/initializeSample.m
|
||||
:END:
|
||||
|
||||
** Function description
|
||||
*** Function description
|
||||
#+begin_src matlab
|
||||
function [] = initializeSample(opts_param)
|
||||
#+end_src
|
||||
|
||||
** Optional Parameters
|
||||
*** Optional Parameters
|
||||
Default values for opts.
|
||||
#+begin_src matlab
|
||||
sample = struct( ...
|
||||
@ -816,7 +1096,7 @@ Populate opts with input parameters
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Save the Sample structure
|
||||
*** Save the Sample structure
|
||||
#+begin_src matlab
|
||||
save('./mat/sample.mat', 'sample');
|
||||
#+end_src
|
||||
|
||||
45
src/computeJointsPose.m
Normal file
45
src/computeJointsPose.m
Normal file
@ -0,0 +1,45 @@
|
||||
function [stewart] = computeJointsPose(stewart)
|
||||
% computeJointsPose -
|
||||
%
|
||||
% Syntax: [stewart] = computeJointsPose(stewart, opts_param)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - A structure with the following fields
|
||||
% - FO_A [3x1] - Position of {A} with respect to {F}
|
||||
% - MO_B [3x1] - Position of {B} with respect to {M}
|
||||
% - FO_M [3x1] - Position of {M} with respect to {F}
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - A structure with the following added fields
|
||||
% - Aa [3x6] - The i'th column is the position of ai with respect to {A}
|
||||
% - Ab [3x6] - The i'th column is the position of bi with respect to {A}
|
||||
% - Ba [3x6] - The i'th column is the position of ai with respect to {B}
|
||||
% - Bb [3x6] - The i'th column is the position of bi with respect to {B}
|
||||
% - l [6x1] - The i'th element is the initial length of strut i
|
||||
% - As [3x6] - The i'th column is the unit vector of strut i expressed in {A}
|
||||
% - Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}
|
||||
% - FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
|
||||
% - MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
|
||||
|
||||
stewart.Aa = stewart.Fa - repmat(stewart.FO_A, [1, 6]);
|
||||
stewart.Bb = stewart.Mb - repmat(stewart.MO_B, [1, 6]);
|
||||
|
||||
stewart.Ab = stewart.Bb - repmat(-stewart.MO_B-stewart.FO_M+stewart.FO_A, [1, 6]);
|
||||
stewart.Ba = stewart.Aa - repmat( stewart.MO_B+stewart.FO_M-stewart.FO_A, [1, 6]);
|
||||
|
||||
stewart.As = (stewart.Ab - stewart.Aa)./vecnorm(stewart.Ab - stewart.Aa); % As_i is the i'th vector of As
|
||||
|
||||
stewart.l = vecnorm(stewart.Ab - stewart.Aa)';
|
||||
|
||||
stewart.Bs = (stewart.Bb - stewart.Ba)./vecnorm(stewart.Bb - stewart.Ba);
|
||||
|
||||
stewart.FRa = zeros(3,3,6);
|
||||
stewart.MRb = zeros(3,3,6);
|
||||
|
||||
for i = 1:6
|
||||
stewart.FRa(:,:,i) = [cross([0;1;0], stewart.As(:,i)) , cross(stewart.As(:,i), cross([0;1;0], stewart.As(:,i))) , stewart.As(:,i)];
|
||||
stewart.FRa(:,:,i) = stewart.FRa(:,:,i)./vecnorm(stewart.FRa(:,:,i));
|
||||
|
||||
stewart.MRb(:,:,i) = [cross([0;1;0], stewart.Bs(:,i)) , cross(stewart.Bs(:,i), cross([0;1;0], stewart.Bs(:,i))) , stewart.Bs(:,i)];
|
||||
stewart.MRb(:,:,i) = stewart.MRb(:,:,i)./vecnorm(stewart.MRb(:,:,i));
|
||||
end
|
||||
48
src/generateCubicConfiguration.m
Normal file
48
src/generateCubicConfiguration.m
Normal file
@ -0,0 +1,48 @@
|
||||
function [stewart] = generateCubicConfiguration(stewart, opts_param)
|
||||
% generateCubicConfiguration -
|
||||
%
|
||||
% Syntax: [stewart] = generateCubicConfiguration(stewart, opts_param)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - the Stewart struct should have a parameter "H" corresponding to the total height of the platform
|
||||
% - opts_param - Structure with the following parameters
|
||||
% - Hc [1x1] - Height of the "useful" part of the cube [m]
|
||||
% - FOc [1x1] - Height of the center of the cute 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 parameters:
|
||||
% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
|
||||
% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
|
||||
|
||||
opts = struct( ...
|
||||
'Hc', 60e-3, ... % [m]
|
||||
'FOc', 50e-3, ... % [m]
|
||||
'FHa', 15e-3, ... % [m]
|
||||
'MHb', 15e-3 ... % [m]
|
||||
);
|
||||
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
|
||||
sx = [ 2; -1; -1];
|
||||
sy = [ 0; 1; -1];
|
||||
sz = [ 1; 1; 1];
|
||||
|
||||
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
|
||||
|
||||
L = opts.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*opts.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
|
||||
|
||||
CSi = (CCm - CCf)./vecnorm(CCm - CCf);
|
||||
|
||||
stewart.Fa = CCf + [0; 0; opts.FOc] + ((opts.FHa-(opts.FOc-opts.Hc/2))./CSi(3,:)).*CSi;
|
||||
stewart.Mb = CCf + [0; 0; opts.FOc-stewart.H] + ((stewart.H-opts.MHb-(opts.FOc-opts.Hc/2))./CSi(3,:)).*CSi;
|
||||
37
src/initializeFramesPositions.m
Normal file
37
src/initializeFramesPositions.m
Normal file
@ -0,0 +1,37 @@
|
||||
function [stewart] = initializeFramesPositions(opts_param)
|
||||
% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
|
||||
%
|
||||
% Syntax: [stewart] = initializeFramesPositions(H, MO_B)
|
||||
%
|
||||
% Inputs:
|
||||
% - opts_param - Structure with the following fields:
|
||||
% - H [1x1] - Total Height of the Stewart Platform [m]
|
||||
% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - A structure with the following fields:
|
||||
% - H [1x1] - Total Height of the Stewart Platform [m]
|
||||
% - FO_M [3x1] - Position of {M} with respect to {F} [m]
|
||||
% - MO_B [3x1] - Position of {B} with respect to {M} [m]
|
||||
% - FO_A [3x1] - Position of {A} with respect to {F} [m]
|
||||
|
||||
opts = struct( ...
|
||||
'H', 90e-3, ... % [m]
|
||||
'MO_B', 50e-3 ... % [m]
|
||||
);
|
||||
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
|
||||
stewart = struct();
|
||||
|
||||
stewart.H = opts.H; % Total Height of the Stewart Platform [m]
|
||||
|
||||
stewart.FO_M = [0; 0; stewart.H]; % Position of {M} with respect to {F} [m]
|
||||
|
||||
stewart.MO_B = [0; 0; opts.MO_B]; % Position of {B} with respect to {M} [m]
|
||||
|
||||
stewart.FO_A = stewart.MO_B + stewart.FO_M; % Position of {A} with respect to {F} [m]
|
||||
28
src/initializeStrutDynamics.m
Normal file
28
src/initializeStrutDynamics.m
Normal file
@ -0,0 +1,28 @@
|
||||
function [stewart] = initializeStrutDynamics(stewart, opts_param)
|
||||
% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
|
||||
%
|
||||
% Syntax: [stewart] = initializeStrutDynamics(opts_param)
|
||||
%
|
||||
% Inputs:
|
||||
% - opts_param - Structure with the following fields:
|
||||
% - Ki [6x1] - Stiffness of each strut [N/m]
|
||||
% - Ci [6x1] - Damping of each strut [N/(m/s)]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - Ki [6x1] - Stiffness of each strut [N/m]
|
||||
% - Ci [6x1] - Damping of each strut [N/(m/s)]
|
||||
|
||||
opts = struct( ...
|
||||
'Ki', 1e6*ones(6,1), ... % [N/m]
|
||||
'Ci', 1e2*ones(6,1) ... % [N/(m/s)]
|
||||
);
|
||||
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
|
||||
stewart.Ki = opts.Ki;
|
||||
stewart.Ci = opts.Ci;
|
||||
BIN
stewart_strut.slx
Normal file
BIN
stewart_strut.slx
Normal file
Binary file not shown.
Loading…
Reference in New Issue
Block a user