Update all the initialization of hexapod and simscape of hexapod
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@ -30,17 +30,19 @@
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#+end_src
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#+begin_src matlab :results none
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open stewart_identification
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open stewart
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#+end_src
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#+begin_src matlab :results output
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initializeSample(struct('mass', 50));
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initializeHexapod(struct('actuator', 'piezo'));
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#+begin_src matlab
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hexapod = initializeHexapod();
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#+end_src
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#+RESULTS:
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: initializeSample(struct('mass', 50));
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: initializeHexapod(struct('actuator', 'piezo'));
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: org_babel_eoe
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#+begin_src matlab
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initializeSample();
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#+end_src
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#+begin_src matlab
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G = identifyPlant();
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@ -170,239 +172,450 @@
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:PROPERTIES:
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:HEADER-ARGS:matlab+: :tangle src/initializeHexapod.m
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:END:
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*** Function description and arguments
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The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
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#+begin_src matlab
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function [stewart] = initializeHexapod(opts_param)
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#+end_src
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Default values for opts
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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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'height', 90, ... % Height of the platform [mm]
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'jacobian', 150, ... % Jacobian offset [mm]
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'density', 8000, ... % Density of hexapod [mm]
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'name', 'stewart' ... % Name of the file
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'height', 90, ... % Height of the platform [mm]
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'density', 8000, ... % Density of the material used for the hexapod [kg/m3]
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'k_ax', 1e8, ... % Stiffness of each actuator [N/m]
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'c_ax', 100, ... % Damping of each actuator [N/(m/s)]
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'stroke', 50e-6, ... % Maximum stroke of each actuator [m]
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'name', 'stewart' ... % Name of the file
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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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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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Stewart Object
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*** Initialization of the stewart structure
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We initialize the Stewart structure
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#+begin_src matlab
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stewart = struct();
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stewart.h = opts.height; % Total height of the platform [mm]
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stewart.jacobian = opts.jacobian; % Distance from the center of the top platform
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% where the jacobian is computed [mm]
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#+end_src
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Bottom Plate
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And we defined its total height.
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#+begin_src matlab
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stewart.H = opts.height; % [mm]
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#+end_src
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*** Bottom Plate
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#+name: fig:stewart_bottom_plate
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#+caption: Schematic of the bottom plates with all the parameters
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[[file:./figs/stewart_bottom_plate.png]]
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The bottom plate structure is initialized.
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#+begin_src matlab
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BP = struct();
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BP.rad.int = 0; % Internal Radius [mm]
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BP.rad.ext = 150; % External Radius [mm]
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BP.thickness = 10; % Thickness [mm]
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BP.leg.rad = 100; % Radius where the legs articulations are positionned [mm]
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BP.leg.ang = 45; % Angle Offset [deg]
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BP.density = opts.density; % Density of the material [kg/m3]
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BP.color = [0.7 0.7 0.7]; % Color [rgb]
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BP.shape = [BP.rad.int BP.thickness; BP.rad.int 0; BP.rad.ext 0; BP.rad.ext BP.thickness];
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#+end_src
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Top Plate
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We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
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#+begin_src matlab
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BP.Rint = 0; % Internal Radius [mm]
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BP.Rext = 150; % External Radius [mm]
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#+end_src
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We define its thickness.
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#+begin_src matlab
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BP.H = 10; % Thickness of the Bottom Plate [mm]
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#+end_src
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At which radius legs will be fixed and with that angle offset.
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#+begin_src matlab
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BP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
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BP.alpha = 10; % Angle Offset [deg]
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#+end_src
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We defined the density of the material of the bottom plate.
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#+begin_src matlab
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BP.density = opts.density; % Density of the material [kg/m3]
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#+end_src
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And its color.
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#+begin_src matlab
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BP.color = [0.7 0.7 0.7]; % Color [RGB]
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#+end_src
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Then the profile of the bottom plate is computed and will be used by Simscape
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#+begin_src matlab
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BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
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#+end_src
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The structure is added to the stewart structure
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#+begin_src matlab
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stewart.BP = BP;
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#+end_src
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*** Top Plate
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The top plate structure is initialized.
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#+begin_src matlab
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TP = struct();
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TP.rad.int = 0; % Internal Radius [mm]
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TP.rad.ext = 100; % Internal Radius [mm]
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TP.thickness = 10; % Thickness [mm]
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TP.leg.rad = 90; % Radius where the legs articulations are positionned [mm]
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TP.leg.ang = 45; % Angle Offset [deg]
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TP.density = opts.density; % Density of the material [kg/m3]
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TP.color = [0.7 0.7 0.7]; % Color [rgb]
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TP.shape = [TP.rad.int TP.thickness; TP.rad.int 0; TP.rad.ext 0; TP.rad.ext TP.thickness];
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#+end_src
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Leg
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We defined the internal and external radius of the top plate.
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#+begin_src matlab
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TP.Rint = 0; % [mm]
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TP.Rext = 100; % [mm]
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#+end_src
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The thickness of the top plate.
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#+begin_src matlab
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TP.H = 10; % [mm]
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#+end_src
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At which radius and angle are fixed the legs.
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#+begin_src matlab
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TP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
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TP.alpha = 20; % Angle [deg]
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TP.dalpha = 0; % Angle Offset from 0 position [deg]
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#+end_src
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The density of its material.
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#+begin_src matlab
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TP.density = opts.density; % Density of the material [kg/m3]
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#+end_src
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Its color.
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#+begin_src matlab
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TP.color = [0.7 0.7 0.7]; % Color [RGB]
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#+end_src
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Then the shape of the top plate is computed
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#+begin_src matlab
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TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
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#+end_src
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The structure is added to the stewart structure
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#+begin_src matlab
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stewart.TP = TP;
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#+end_src
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*** Legs
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#+name: fig:stewart_legs
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#+caption: Schematic for the legs of the Stewart platform
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[[file:./figs/stewart_legs.png]]
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The leg structure is initialized.
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#+begin_src matlab
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Leg = struct();
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Leg.stroke = 80e-6; % Maximum Stroke of each leg [m]
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if strcmp(opts.actuator, 'piezo')
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Leg.k.ax = 1e7; % Stiffness of each leg [N/m]
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Leg.c.ax = 500; % [N/(m/s)]
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elseif strcmp(opts.actuator, 'lorentz')
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Leg.k.ax = 1e4; % Stiffness of each leg [N/m]
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Leg.c.ax = 200; % [N/(m/s)]
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elseif isnumeric(opts.actuator)
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Leg.k.ax = opts.actuator; % Stiffness of each leg [N/m]
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Leg.c.ax = 100; % [N/(m/s)]
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else
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error('opts.actuator should be piezo or lorentz or numeric value');
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end
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Leg.rad.bottom = 12; % Radius of the cylinder of the bottom part [mm]
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Leg.rad.top = 10; % Radius of the cylinder of the top part [mm]
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Leg.density = opts.density; % Density of the material [kg/m3]
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Leg.color.bottom = [0.5 0.5 0.5]; % Color [rgb]
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Leg.color.top = [0.5 0.5 0.5]; % Color [rgb]
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Leg.sphere.bottom = Leg.rad.bottom; % Size of the sphere at the end of the leg [mm]
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Leg.sphere.top = Leg.rad.top; % Size of the sphere at the end of the leg [mm]
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#+end_src
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Sphere
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The maximum Stroke of each leg is defined.
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#+begin_src matlab
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Leg.stroke = opts.stroke; % [m]
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#+end_src
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The stiffness and damping of each leg are defined
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#+begin_src matlab
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Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
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Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
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#+end_src
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The radius of the legs are defined
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#+begin_src matlab
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Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
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Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
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#+end_src
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The density of its material.
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#+begin_src matlab
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Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
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#+end_src
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Its color.
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#+begin_src matlab
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Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
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#+end_src
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The radius of spheres representing the ball joints are defined.
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#+begin_src matlab
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Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
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#+end_src
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The structure is added to the stewart structure
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#+begin_src matlab
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stewart.Leg = Leg;
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#+end_src
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*** Ball Joints
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#+name: fig:stewart_ball_joints
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#+caption: Schematic of the support for the ball joints
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[[file:./figs/stewart_ball_joints.png]]
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=SP= is the structure representing the support for the ball joints at the extremity of each leg.
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The =SP= structure is initialized.
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#+begin_src matlab
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SP = struct();
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#+end_src
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SP.height.bottom = 15; % [mm]
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SP.height.top = 15; % [mm]
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SP.density.bottom = opts.density; % [kg/m^3]
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SP.density.top = opts.density; % [kg/m^3]
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SP.color.bottom = [0.7 0.7 0.7]; % [rgb]
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SP.color.top = [0.7 0.7 0.7]; % [rgb]
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SP.k.ax = 0; % [N*m/deg]
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SP.c.ax = 0; % [N*m/deg]
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We can define its rotational stiffness and damping. For now, we use perfect joints.
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#+begin_src matlab
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SP.k = 0; % [N*m/deg]
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SP.c = 0; % [N*m/deg]
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#+end_src
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SP.thickness.bottom = SP.height.bottom-Leg.sphere.bottom; % [mm]
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SP.thickness.top = SP.height.top-Leg.sphere.top; % [mm]
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SP.rad.bottom = Leg.sphere.bottom; % [mm]
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SP.rad.top = Leg.sphere.top; % [mm]
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Its height is defined
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#+begin_src matlab
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SP.H = 15; % [mm]
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#+end_src
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Its radius is based on the radius on the sphere at the end of the legs.
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#+begin_src matlab
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SP.R = Leg.R; % [mm]
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#+end_src
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%%
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Leg.support.bottom = [0 SP.thickness.bottom;
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0 0;
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SP.rad.bottom 0;
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SP.rad.bottom SP.height.bottom];
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Leg.support.top = [0 SP.thickness.top;
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0 0;
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SP.rad.top 0;
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SP.rad.top SP.height.top];
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#+begin_src matlab
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SP.section = [0 SP.H-SP.R;
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0 0;
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SP.R 0;
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SP.R SP.H];
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#+end_src
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%%
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stewart.BP = BP;
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stewart.TP = TP;
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stewart.Leg = Leg;
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stewart.SP = SP;
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The density of its material is defined.
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#+begin_src matlab
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SP.density = opts.density; % [kg/m^3]
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#+end_src
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%%
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Its color is defined.
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#+begin_src matlab
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SP.color = [0.7 0.7 0.7]; % [RGB]
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#+end_src
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The structure is added to the Hexapod structure
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#+begin_src matlab
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stewart.SP = SP;
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#+end_src
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*** More parameters are initialized
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#+begin_src matlab
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stewart = initializeParameters(stewart);
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#+end_src
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%%
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*** Save the Stewart Structure
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#+begin_src matlab
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save('./mat/stewart.mat', 'stewart')
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#+end_src
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Additional Functions
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*** initializeParameters Function
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:PROPERTIES:
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:HEADER-ARGS:matlab+: :tangle no
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:END:
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#+begin_src matlab
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%% Initialize Parameters
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function [stewart] = initializeParameters(stewart)
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%% Connection points on base and top plate w.r.t. World frame at the center of the base plate
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stewart.pos_base = zeros(6, 3);
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stewart.pos_top = zeros(6, 3);
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function [stewart] = initializeParameters(stewart)
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#+end_src
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alpha_b = stewart.BP.leg.ang*pi/180; % angle de décalage par rapport à 120 deg (pour positionner les supports bases)
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alpha_t = stewart.TP.leg.ang*pi/180; % +- offset angle from 120 degree spacing on top
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Computation of the position of the connection points on the base and moving platform
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We first initialize =pos_base= corresponding to $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and =pos_top= corresponding to $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
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#+begin_src matlab
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stewart.pos_base = zeros(6, 3);
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stewart.pos_top = zeros(6, 3);
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#+end_src
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% Height [m] TODO
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height = (stewart.h-stewart.BP.thickness-stewart.TP.thickness-stewart.Leg.sphere.bottom-stewart.Leg.sphere.top-stewart.SP.thickness.bottom-stewart.SP.thickness.top)*0.001;
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We estimate the height between the ball joints of the bottom platform and of the top platform.
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#+begin_src matlab
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height = stewart.H - stewart.BP.H - stewart.TP.H - 2*stewart.SP.H; % [mm]
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#+end_src
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radius_b = stewart.BP.leg.rad*0.001; % rayon emplacement support base
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radius_t = stewart.TP.leg.rad*0.001; % top radius in meters
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#+begin_src matlab
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for i = 1:3
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% base points
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angle_m_b = 120*(i-1) - stewart.BP.alpha;
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angle_p_b = 120*(i-1) + stewart.BP.alpha;
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for i = 1:3
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% base points
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angle_m_b = (2*pi/3)* (i-1) - alpha_b;
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angle_p_b = (2*pi/3)* (i-1) + alpha_b;
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stewart.pos_base(2*i-1,:) = [radius_b*cos(angle_m_b), radius_b*sin(angle_m_b), 0.0];
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stewart.pos_base(2*i,:) = [radius_b*cos(angle_p_b), radius_b*sin(angle_p_b), 0.0];
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stewart.pos_base(2*i-1,:) = [stewart.BP.Rleg*cos(angle_m_b), stewart.BP.Rleg*sin(angle_m_b), 0.0];
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stewart.pos_base(2*i,:) = [stewart.BP.Rleg*cos(angle_p_b), stewart.BP.Rleg*sin(angle_p_b), 0.0];
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% top points
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% Top points are 60 degrees offset
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angle_m_t = (2*pi/3)* (i-1) - alpha_t + 2*pi/6;
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angle_p_t = (2*pi/3)* (i-1) + alpha_t + 2*pi/6;
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stewart.pos_top(2*i-1,:) = [radius_t*cos(angle_m_t), radius_t*sin(angle_m_t), height];
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stewart.pos_top(2*i,:) = [radius_t*cos(angle_p_t), radius_t*sin(angle_p_t), height];
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end
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% top points
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angle_m_t = 120*(i-1) - stewart.TP.alpha + stewart.TP.dalpha;
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angle_p_t = 120*(i-1) + stewart.TP.alpha + stewart.TP.dalpha;
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% permute pos_top points so that legs are end points of base and top points
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stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
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stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
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stewart.pos_top(2*i-1,:) = [stewart.TP.Rleg*cos(angle_m_t), stewart.TP.Rleg*sin(angle_m_t), height];
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stewart.pos_top(2*i,:) = [stewart.TP.Rleg*cos(angle_p_t), stewart.TP.Rleg*sin(angle_p_t), height];
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end
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%% leg vectors
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legs = stewart.pos_top - stewart.pos_base;
|
||||
leg_length = zeros(6, 1);
|
||||
leg_vectors = zeros(6, 3);
|
||||
for i = 1:6
|
||||
leg_length(i) = norm(legs(i,:));
|
||||
leg_vectors(i,:) = legs(i,:) / leg_length(i);
|
||||
end
|
||||
% permute pos_top points so that legs are end points of base and top points
|
||||
stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
|
||||
stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
|
||||
#+end_src
|
||||
|
||||
stewart.Leg.lenght = 1000*leg_length(1)/1.5;
|
||||
stewart.Leg.shape.bot = [0 0; ...
|
||||
stewart.Leg.rad.bottom 0; ...
|
||||
stewart.Leg.rad.bottom stewart.Leg.lenght; ...
|
||||
stewart.Leg.rad.top stewart.Leg.lenght; ...
|
||||
stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
|
||||
0 0.2*stewart.Leg.lenght];
|
||||
leg vectors
|
||||
#+begin_src matlab
|
||||
legs = stewart.pos_top - stewart.pos_base;
|
||||
leg_length = zeros(6, 1);
|
||||
leg_vectors = zeros(6, 3);
|
||||
for i = 1:6
|
||||
leg_length(i) = norm(legs(i,:));
|
||||
leg_vectors(i,:) = legs(i,:) / leg_length(i);
|
||||
end
|
||||
|
||||
%% Calculate revolute and cylindrical axes
|
||||
rev1 = zeros(6, 3);
|
||||
rev2 = zeros(6, 3);
|
||||
cyl1 = zeros(6, 3);
|
||||
for i = 1:6
|
||||
rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
|
||||
rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
|
||||
stewart.Leg.lenght = 1000*leg_length(1)/1.5;
|
||||
stewart.Leg.shape.bot = [0 0; ...
|
||||
stewart.Leg.rad.bottom 0; ...
|
||||
stewart.Leg.rad.bottom stewart.Leg.lenght; ...
|
||||
stewart.Leg.rad.top stewart.Leg.lenght; ...
|
||||
stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
|
||||
0 0.2*stewart.Leg.lenght];
|
||||
#+end_src
|
||||
|
||||
rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
|
||||
rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
|
||||
Calculate revolute and cylindrical axes
|
||||
#+begin_src matlab
|
||||
rev1 = zeros(6, 3);
|
||||
rev2 = zeros(6, 3);
|
||||
cyl1 = zeros(6, 3);
|
||||
for i = 1:6
|
||||
rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
|
||||
rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
|
||||
|
||||
cyl1(i,:) = leg_vectors(i,:);
|
||||
end
|
||||
rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
|
||||
rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
|
||||
|
||||
|
||||
%% Coordinate systems
|
||||
stewart.lower_leg = struct('rotation', eye(3));
|
||||
stewart.upper_leg = struct('rotation', eye(3));
|
||||
|
||||
for i = 1:6
|
||||
stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
|
||||
stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
|
||||
end
|
||||
|
||||
%% Position Matrix
|
||||
% TODO
|
||||
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.thickness+stewart.Leg.sphere.top+stewart.SP.thickness.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
|
||||
|
||||
%% Compute Jacobian Matrix
|
||||
% TODO
|
||||
% aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
bb = stewart.pos_top_tranform - (stewart.TP.thickness + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
stewart.J = getJacobianMatrix(leg_vectors', bb');
|
||||
|
||||
stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
|
||||
cyl1(i,:) = leg_vectors(i,:);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Compute the Jacobian Matrix
|
||||
Coordinate systems
|
||||
#+begin_src matlab
|
||||
function J = getJacobianMatrix(RM, M_pos_base)
|
||||
% RM - [3x6] unit vector of each leg in the fixed frame
|
||||
% M_pos_base - [3x6] vector of the leg connection at the top platform location in the fixed frame
|
||||
J = zeros(6);
|
||||
stewart.lower_leg = struct('rotation', eye(3));
|
||||
stewart.upper_leg = struct('rotation', eye(3));
|
||||
|
||||
J(:, 1:3) = RM';
|
||||
J(:, 4:6) = cross(M_pos_base, RM)';
|
||||
for i = 1:6
|
||||
stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
|
||||
stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Position Matrix
|
||||
#+begin_src matlab
|
||||
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.h+stewart.Leg.sphere.top+stewart.SP.h.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
|
||||
#+end_src
|
||||
|
||||
Compute Jacobian Matrix
|
||||
#+begin_src matlab
|
||||
% aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.h - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
bb = stewart.pos_top_tranform - (stewart.TP.h + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
stewart.J = getJacobianMatrix(leg_vectors', bb');
|
||||
|
||||
stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
*** initializeParameters Function - BIS
|
||||
#+begin_src matlab
|
||||
function [stewart] = initializeParameters(stewart)
|
||||
#+end_src
|
||||
|
||||
We first 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
|
||||
stewart.Aa = zeros(6, 3); % [mm]
|
||||
stewart.Ab = zeros(6, 3); % [mm]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:3
|
||||
stewart.Aa(2*i-1,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
|
||||
stewart.BP.Rleg*sin( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
|
||||
stewart.BP.H+stewart.SP.H];
|
||||
stewart.Aa(2*i,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
|
||||
stewart.BP.Rleg*sin( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
|
||||
stewart.BP.H+stewart.SP.H];
|
||||
|
||||
stewart.Ab(2*i-1,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
|
||||
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
|
||||
stewart.H - stewart.TP.H - stewart.SP.H];
|
||||
stewart.Ab(2*i,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
|
||||
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
|
||||
stewart.H - stewart.TP.H - stewart.SP.H];
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
|
||||
\[ b_i - a_i = l_i \hat{s}_i \]
|
||||
|
||||
We initialize $l_i$ and $\hat{s}_i$
|
||||
#+begin_src matlab
|
||||
leg_length = zeros(6, 1); % [mm]
|
||||
leg_vectors = zeros(6, 3);
|
||||
#+end_src
|
||||
|
||||
We compute $b_i - a_i$, and then:
|
||||
\begin{align*}
|
||||
l_i &= \left|b_i - a_i\right| \\
|
||||
\hat{s}_i &= \frac{b_i - a_i}{l_i}
|
||||
\end{align*}
|
||||
|
||||
#+begin_src matlab
|
||||
legs = stewart.Ab - stewart.Aa;
|
||||
|
||||
for i = 1:6
|
||||
leg_length(i) = norm(legs(i,:));
|
||||
leg_vectors(i,:) = legs(i,:) / leg_length(i);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Then the shape of the bottom leg is estimated
|
||||
#+begin_src matlab
|
||||
stewart.Leg.lenght = leg_length(1)/1.5;
|
||||
stewart.Leg.shape.bot = ...
|
||||
[0 0; ...
|
||||
stewart.Leg.Rbot 0; ...
|
||||
stewart.Leg.Rbot stewart.Leg.lenght; ...
|
||||
stewart.Leg.Rtop stewart.Leg.lenght; ...
|
||||
stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
|
||||
0 0.2*stewart.Leg.lenght];
|
||||
#+end_src
|
||||
|
||||
We compute rotation matrices to have the orientation of the legs.
|
||||
The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
|
||||
#+begin_src matlab
|
||||
stewart.Rm = struct('R', eye(3));
|
||||
|
||||
for i = 1:6
|
||||
sx = cross(leg_vectors(i,:), [1 0 0]);
|
||||
sx = sx/norm(sx);
|
||||
|
||||
sy = -cross(sx, leg_vectors(i,:));
|
||||
sy = sy/norm(sy);
|
||||
|
||||
sz = leg_vectors(i,:);
|
||||
sz = sz/norm(sz);
|
||||
|
||||
stewart.Rm(i).R = [sx', sy', sz'];
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Compute Jacobian Matrix
|
||||
#+begin_src matlab
|
||||
J = zeros(6);
|
||||
|
||||
for i = 1:6
|
||||
J(i, 1:3) = leg_vectors(i, :);
|
||||
J(i, 4:6) = cross(0.001*(stewart.Ab - stewart.H*[0,0,1]), leg_vectors(i, :));
|
||||
end
|
||||
|
||||
stewart.J = J;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
|
||||
@ -1,208 +1,171 @@
|
||||
function [stewart] = initializeHexapod(opts_param)
|
||||
%% Default values for opts
|
||||
opts = struct(...
|
||||
'height', 90, ... % Height of the platform [mm]
|
||||
'jacobian', 150, ... % Jacobian offset [mm]
|
||||
'density', 8000, ... % Density of hexapod [mm]
|
||||
'name', 'stewart' ... % Name of the file
|
||||
);
|
||||
|
||||
%% Populate opts with input parameters
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
opts = struct(...
|
||||
'height', 90, ... % Height of the platform [mm]
|
||||
'density', 8000, ... % Density of the material used for the hexapod [kg/m3]
|
||||
'k_ax', 1e8, ... % Stiffness of each actuator [N/m]
|
||||
'c_ax', 100, ... % Damping of each actuator [N/(m/s)]
|
||||
'stroke', 50e-6, ... % Maximum stroke of each actuator [m]
|
||||
'name', 'stewart' ... % Name of the file
|
||||
);
|
||||
|
||||
%% Stewart Object
|
||||
stewart = struct();
|
||||
stewart.h = opts.height; % Total height of the platform [mm]
|
||||
stewart.jacobian = opts.jacobian; % distance from the center of the top platform
|
||||
% where the jacobian is computed [mm]
|
||||
|
||||
%% Bottom Plate
|
||||
BP = struct();
|
||||
|
||||
BP.rad.int = 0; % Internal Radius [mm]
|
||||
BP.rad.ext = 150; % External Radius [mm]
|
||||
BP.thickness = 10; % Thickness [mm]
|
||||
BP.leg.rad = 100; % Radius where the legs articulations are positionned [mm]
|
||||
BP.leg.ang = 5; % Angle Offset [deg]
|
||||
BP.density = opts.density; % Density of the material [kg/m3]
|
||||
BP.color = [0.7 0.7 0.7]; % Color [rgb]
|
||||
BP.shape = [BP.rad.int BP.thickness; BP.rad.int 0; BP.rad.ext 0; BP.rad.ext BP.thickness];
|
||||
|
||||
%% Top Plate
|
||||
TP = struct();
|
||||
|
||||
TP.rad.int = 0; % Internal Radius [mm]
|
||||
TP.rad.ext = 100; % Internal Radius [mm]
|
||||
TP.thickness = 10; % Thickness [mm]
|
||||
TP.leg.rad = 90; % Radius where the legs articulations are positionned [mm]
|
||||
TP.leg.ang = 5; % Angle Offset [deg]
|
||||
TP.density = opts.density; % Density of the material [kg/m3]
|
||||
TP.color = [0.7 0.7 0.7]; % Color [rgb]
|
||||
TP.shape = [TP.rad.int TP.thickness; TP.rad.int 0; TP.rad.ext 0; TP.rad.ext TP.thickness];
|
||||
|
||||
%% Leg
|
||||
Leg = struct();
|
||||
|
||||
Leg.stroke = 80e-6; % Maximum Stroke of each leg [m]
|
||||
if strcmp(opts.actuator, 'piezo')
|
||||
Leg.k.ax = 1e7; % Stiffness of each leg [N/m]
|
||||
Leg.c.ax = 500; % [N/(m/s)]
|
||||
elseif strcmp(opts.actuator, 'lorentz')
|
||||
Leg.k.ax = 1e4; % Stiffness of each leg [N/m]
|
||||
Leg.c.ax = 200; % [N/(m/s)]
|
||||
elseif isnumeric(opts.actuator)
|
||||
Leg.k.ax = opts.actuator; % Stiffness of each leg [N/m]
|
||||
Leg.c.ax = 100; % [N/(m/s)]
|
||||
else
|
||||
error('opts.actuator should be piezo or lorentz or numeric value');
|
||||
end
|
||||
Leg.rad.bottom = 12; % Radius of the cylinder of the bottom part [mm]
|
||||
Leg.rad.top = 10; % Radius of the cylinder of the top part [mm]
|
||||
Leg.density = opts.density; % Density of the material [kg/m3]
|
||||
Leg.color.bottom = [0.5 0.5 0.5]; % Color [rgb]
|
||||
Leg.color.top = [0.5 0.5 0.5]; % Color [rgb]
|
||||
|
||||
Leg.sphere.bottom = Leg.rad.bottom; % Size of the sphere at the end of the leg [mm]
|
||||
Leg.sphere.top = Leg.rad.top; % Size of the sphere at the end of the leg [mm]
|
||||
|
||||
%% Sphere
|
||||
SP = struct();
|
||||
|
||||
SP.height.bottom = 15; % [mm]
|
||||
SP.height.top = 15; % [mm]
|
||||
SP.density.bottom = opts.density; % [kg/m^3]
|
||||
SP.density.top = opts.density; % [kg/m^3]
|
||||
SP.color.bottom = [0.7 0.7 0.7]; % [rgb]
|
||||
SP.color.top = [0.7 0.7 0.7]; % [rgb]
|
||||
SP.k.ax = 0; % [N*m/deg]
|
||||
SP.c.ax = 0; % [N*m/deg]
|
||||
|
||||
SP.thickness.bottom = SP.height.bottom-Leg.sphere.bottom; % [mm]
|
||||
SP.thickness.top = SP.height.top-Leg.sphere.top; % [mm]
|
||||
SP.rad.bottom = Leg.sphere.bottom; % [mm]
|
||||
SP.rad.top = Leg.sphere.top; % [mm]
|
||||
|
||||
|
||||
%%
|
||||
Leg.support.bottom = [0 SP.thickness.bottom; 0 0; SP.rad.bottom 0; SP.rad.bottom SP.height.bottom];
|
||||
Leg.support.top = [0 SP.thickness.top; 0 0; SP.rad.top 0; SP.rad.top SP.height.top];
|
||||
|
||||
%%
|
||||
stewart.BP = BP;
|
||||
stewart.TP = TP;
|
||||
stewart.Leg = Leg;
|
||||
stewart.SP = SP;
|
||||
|
||||
%%
|
||||
stewart = initializeParameters(stewart);
|
||||
|
||||
%%
|
||||
save('./mat/stewart.mat', 'stewart')
|
||||
|
||||
%% ==============================================================
|
||||
% Additional Functions
|
||||
% ===============================================================
|
||||
|
||||
%% Initialize Parameters
|
||||
function [stewart] = initializeParameters(stewart)
|
||||
%% Connection points on base and top plate w.r.t. World frame at the center of the base plate
|
||||
stewart.pos_base = zeros(6, 3);
|
||||
stewart.pos_top = zeros(6, 3);
|
||||
|
||||
alpha_b = stewart.BP.leg.ang*pi/180; % angle de décalage par rapport à 120 deg (pour positionner les supports bases)
|
||||
alpha_t = stewart.TP.leg.ang*pi/180; % +- offset angle from 120 degree spacing on top
|
||||
|
||||
% Height [m] TODO
|
||||
height = (stewart.h-stewart.BP.thickness-stewart.TP.thickness-stewart.Leg.sphere.bottom-stewart.Leg.sphere.top-stewart.SP.thickness.bottom-stewart.SP.thickness.top)*0.001;
|
||||
|
||||
radius_b = stewart.BP.leg.rad*0.001; % rayon emplacement support base
|
||||
radius_t = stewart.TP.leg.rad*0.001; % top radius in meters
|
||||
|
||||
for i = 1:3
|
||||
% base points
|
||||
angle_m_b = (2*pi/3)* (i-1) - alpha_b;
|
||||
angle_p_b = (2*pi/3)* (i-1) + alpha_b;
|
||||
stewart.pos_base(2*i-1,:) = [radius_b*cos(angle_m_b), radius_b*sin(angle_m_b), 0.0];
|
||||
stewart.pos_base(2*i,:) = [radius_b*cos(angle_p_b), radius_b*sin(angle_p_b), 0.0];
|
||||
|
||||
% top points
|
||||
% Top points are 60 degrees offset
|
||||
angle_m_t = (2*pi/3)* (i-1) - alpha_t + 2*pi/6;
|
||||
angle_p_t = (2*pi/3)* (i-1) + alpha_t + 2*pi/6;
|
||||
stewart.pos_top(2*i-1,:) = [radius_t*cos(angle_m_t), radius_t*sin(angle_m_t), height];
|
||||
stewart.pos_top(2*i,:) = [radius_t*cos(angle_p_t), radius_t*sin(angle_p_t), height];
|
||||
end
|
||||
|
||||
% permute pos_top points so that legs are end points of base and top points
|
||||
stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
|
||||
stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
|
||||
|
||||
%% leg vectors
|
||||
legs = stewart.pos_top - stewart.pos_base;
|
||||
leg_length = zeros(6, 1);
|
||||
leg_vectors = zeros(6, 3);
|
||||
for i = 1:6
|
||||
leg_length(i) = norm(legs(i,:));
|
||||
leg_vectors(i,:) = legs(i,:) / leg_length(i);
|
||||
end
|
||||
|
||||
stewart.Leg.lenght = 1000*leg_length(1)/1.5;
|
||||
stewart.Leg.shape.bot = [0 0; ...
|
||||
stewart.Leg.rad.bottom 0; ...
|
||||
stewart.Leg.rad.bottom stewart.Leg.lenght; ...
|
||||
stewart.Leg.rad.top stewart.Leg.lenght; ...
|
||||
stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
|
||||
0 0.2*stewart.Leg.lenght];
|
||||
|
||||
%% Calculate revolute and cylindrical axes
|
||||
rev1 = zeros(6, 3);
|
||||
rev2 = zeros(6, 3);
|
||||
cyl1 = zeros(6, 3);
|
||||
for i = 1:6
|
||||
rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
|
||||
rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
|
||||
|
||||
rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
|
||||
rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
|
||||
|
||||
cyl1(i,:) = leg_vectors(i,:);
|
||||
end
|
||||
|
||||
|
||||
%% Coordinate systems
|
||||
stewart.lower_leg = struct('rotation', eye(3));
|
||||
stewart.upper_leg = struct('rotation', eye(3));
|
||||
|
||||
for i = 1:6
|
||||
stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
|
||||
stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
|
||||
end
|
||||
|
||||
%% Position Matrix
|
||||
% TODO
|
||||
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.thickness+stewart.Leg.sphere.top+stewart.SP.thickness.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
|
||||
|
||||
%% Compute Jacobian Matrix
|
||||
% TODO
|
||||
% aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
bb = stewart.pos_top_tranform - (stewart.TP.thickness + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
|
||||
stewart.J = getJacobianMatrix(leg_vectors', bb');
|
||||
|
||||
stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
|
||||
end
|
||||
|
||||
%% Compute the Jacobian Matrix
|
||||
function J = getJacobianMatrix(RM, M_pos_base)
|
||||
% RM - [3x6] unit vector of each leg in the fixed frame
|
||||
% M_pos_base - [3x6] vector of the leg connection at the top platform location in the fixed frame
|
||||
J = zeros(6);
|
||||
|
||||
J(:, 1:3) = RM';
|
||||
J(:, 4:6) = cross(M_pos_base, RM)';
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
|
||||
stewart = struct();
|
||||
|
||||
stewart.H = opts.height; % [mm]
|
||||
|
||||
BP = struct();
|
||||
|
||||
BP.Rint = 0; % Internal Radius [mm]
|
||||
BP.Rext = 150; % External Radius [mm]
|
||||
|
||||
BP.H = 10; % Thickness of the Bottom Plate [mm]
|
||||
|
||||
BP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
|
||||
BP.alpha = 10; % Angle Offset [deg]
|
||||
|
||||
BP.density = opts.density; % Density of the material [kg/m3]
|
||||
|
||||
BP.color = [0.7 0.7 0.7]; % Color [RGB]
|
||||
|
||||
BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
|
||||
|
||||
stewart.BP = BP;
|
||||
|
||||
TP = struct();
|
||||
|
||||
TP.Rint = 0; % [mm]
|
||||
TP.Rext = 100; % [mm]
|
||||
|
||||
TP.H = 10; % [mm]
|
||||
|
||||
TP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
|
||||
TP.alpha = 20; % Angle [deg]
|
||||
TP.dalpha = 0; % Angle Offset from 0 position [deg]
|
||||
|
||||
TP.density = opts.density; % Density of the material [kg/m3]
|
||||
|
||||
TP.color = [0.7 0.7 0.7]; % Color [RGB]
|
||||
|
||||
TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
|
||||
|
||||
stewart.TP = TP;
|
||||
|
||||
Leg = struct();
|
||||
|
||||
Leg.stroke = opts.stroke; % [m]
|
||||
|
||||
Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
|
||||
Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
|
||||
|
||||
Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
|
||||
Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
|
||||
|
||||
Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
|
||||
|
||||
Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
|
||||
|
||||
Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
|
||||
|
||||
stewart.Leg = Leg;
|
||||
|
||||
SP = struct();
|
||||
|
||||
SP.k = 0; % [N*m/deg]
|
||||
SP.c = 0; % [N*m/deg]
|
||||
|
||||
SP.H = 15; % [mm]
|
||||
|
||||
SP.R = Leg.R; % [mm]
|
||||
|
||||
SP.section = [0 SP.H-SP.R;
|
||||
0 0;
|
||||
SP.R 0;
|
||||
SP.R SP.H];
|
||||
|
||||
SP.density = opts.density; % [kg/m^3]
|
||||
|
||||
SP.color = [0.7 0.7 0.7]; % [RGB]
|
||||
|
||||
stewart.SP = SP;
|
||||
|
||||
stewart = initializeParameters(stewart);
|
||||
|
||||
save('./mat/stewart.mat', 'stewart')
|
||||
|
||||
function [stewart] = initializeParameters(stewart)
|
||||
|
||||
stewart.Aa = zeros(6, 3); % [mm]
|
||||
stewart.Ab = zeros(6, 3); % [mm]
|
||||
stewart.Bb = zeros(6, 3); % [mm]
|
||||
|
||||
for i = 1:3
|
||||
stewart.Aa(2*i-1,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
|
||||
stewart.BP.Rleg*sin( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
|
||||
stewart.BP.H+stewart.SP.H];
|
||||
stewart.Aa(2*i,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
|
||||
stewart.BP.Rleg*sin( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
|
||||
stewart.BP.H+stewart.SP.H];
|
||||
|
||||
stewart.Ab(2*i-1,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
|
||||
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
|
||||
stewart.H - stewart.TP.H - stewart.SP.H];
|
||||
stewart.Ab(2*i,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
|
||||
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
|
||||
stewart.H - stewart.TP.H - stewart.SP.H];
|
||||
end
|
||||
|
||||
stewart.Bb = stewart.Ab - stewart.H*[0,0,1];
|
||||
|
||||
leg_length = zeros(6, 1); % [mm]
|
||||
leg_vectors = zeros(6, 3);
|
||||
|
||||
legs = stewart.Ab - stewart.Aa;
|
||||
|
||||
for i = 1:6
|
||||
leg_length(i) = norm(legs(i,:));
|
||||
leg_vectors(i,:) = legs(i,:) / leg_length(i);
|
||||
end
|
||||
|
||||
stewart.Leg.lenght = leg_length(1)/1.5;
|
||||
stewart.Leg.shape.bot = ...
|
||||
[0 0; ...
|
||||
stewart.Leg.Rbot 0; ...
|
||||
stewart.Leg.Rbot stewart.Leg.lenght; ...
|
||||
stewart.Leg.Rtop stewart.Leg.lenght; ...
|
||||
stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
|
||||
0 0.2*stewart.Leg.lenght];
|
||||
|
||||
stewart.Rm = struct('R', eye(3));
|
||||
|
||||
for i = 1:6
|
||||
sx = cross(leg_vectors(i,:), [1 0 0]);
|
||||
sx = sx/norm(sx);
|
||||
|
||||
sy = -cross(sx, leg_vectors(i,:));
|
||||
sy = sy/norm(sy);
|
||||
|
||||
sz = leg_vectors(i,:);
|
||||
sz = sz/norm(sz);
|
||||
|
||||
stewart.Rm(i).R = [sx', sy', sz'];
|
||||
end
|
||||
|
||||
J = zeros(6);
|
||||
|
||||
for i = 1:6
|
||||
J(i, 1:3) = leg_vectors(i, :);
|
||||
J(i, 4:6) = cross(0.001*stewart.Bb(i, :), leg_vectors(i, :));
|
||||
end
|
||||
|
||||
stewart.J = J;
|
||||
|
||||
stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;
|
||||
|
||||
end
|
||||
end
|
||||
|
||||
BIN
stewart.slx
Normal file
BIN
stewart.slx
Normal file
Binary file not shown.
Binary file not shown.
Loading…
Reference in New Issue
Block a user