16 KiB
16 KiB
Stewart Platform with Simscape
<<matlab-init>>
addpath('src');
addpath('library'); open stewart_identification initializeSample(struct('mass', 50));
initializeHexapod(struct('actuator', 'piezo'));initializeSample(struct('mass', 50));
initializeHexapod(struct('actuator', 'piezo'));
G = identifyPlant();Functions
getMaxPositions
function [X, Y, Z] = getMaxPositions(stewart)
Leg = stewart.Leg;
J = stewart.J;
theta = linspace(0, 2*pi, 100);
phi = linspace(-pi/2 , pi/2, 100);
dmax = zeros(length(theta), length(phi));
for i = 1:length(theta)
for j = 1:length(phi)
L = J*[cos(phi(j))*cos(theta(i)) cos(phi(j))*sin(theta(i)) sin(phi(j)) 0 0 0]';
dmax(i, j) = Leg.stroke/max(abs(L));
end
end
X = dmax.*cos(repmat(phi,length(theta),1)).*cos(repmat(theta,length(phi),1))';
Y = dmax.*cos(repmat(phi,length(theta),1)).*sin(repmat(theta,length(phi),1))';
Z = dmax.*sin(repmat(phi,length(theta),1));
endgetMaxPureDisplacement
function [max_disp] = getMaxPureDisplacement(Leg, J)
max_disp = zeros(6, 1);
max_disp(1) = Leg.stroke/max(abs(J*[1 0 0 0 0 0]'));
max_disp(2) = Leg.stroke/max(abs(J*[0 1 0 0 0 0]'));
max_disp(3) = Leg.stroke/max(abs(J*[0 0 1 0 0 0]'));
max_disp(4) = Leg.stroke/max(abs(J*[0 0 0 1 0 0]'));
max_disp(5) = Leg.stroke/max(abs(J*[0 0 0 0 1 0]'));
max_disp(6) = Leg.stroke/max(abs(J*[0 0 0 0 0 1]'));
endgetStiffnessMatrix
function [K] = getStiffnessMatrix(k, J)
% k - leg stiffness
% J - Jacobian matrix
K = k*(J'*J);
endidentifyPlant
function [sys] = identifyPlant(opts_param)
%% Default values for opts
opts = struct();
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_identification';
%% Input/Output definition
io(1) = linio([mdl, '/F'], 1, 'input'); % Cartesian forces
io(2) = linio([mdl, '/Fl'], 1, 'input'); % Leg forces
io(3) = linio([mdl, '/Fd'], 1, 'input'); % Direct forces
io(4) = linio([mdl, '/Dw'], 1, 'input'); % Base motion
io(5) = linio([mdl, '/Dm'], 1, 'output'); % Relative Motion
io(6) = linio([mdl, '/Dlm'], 1, 'output'); % Displacement of each leg
io(7) = linio([mdl, '/Flm'], 1, 'output'); % Force sensor in each leg
io(8) = linio([mdl, '/Xm'], 1, 'output'); % Absolute motion of platform
%% Run the linearization
G = linearize(mdl, io, 0);
%% Input/Output names
G.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz', ...
'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'};
G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m', ...
'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m', ...
'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'};
%% Cut into sub transfer functions
sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}));
sys.G_forc = minreal(G({'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_legs = G({'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
sys.G_tran = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));
sys.G_comp = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_all = minreal(G);
endinitializeHexapod
The initializeHexapod function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
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
endStewart 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 = 45; % 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 = 45; % 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;
endCompute 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)';
end
endinitializeSample
function [] = initializeSample(opts_param)
%% Default values for opts
sample = struct( ...
'radius', 100, ... % radius of the cylinder [mm]
'height', 300, ... % height of the cylinder [mm]
'mass', 50, ... % mass of the cylinder [kg]
'measheight', 150, ... % measurement point z-offset [mm]
'offset', [0, 0, 0], ... % offset position of the sample [mm]
'color', [0.9 0.1 0.1] ...
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
sample.(opt{1}) = opts_param.(opt{1});
end
end
%% Save
save('./mat/sample.mat', 'sample');
end