197 lines
8.0 KiB
Mathematica
197 lines
8.0 KiB
Mathematica
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function [micro_hexapod] = initializeMicroHexapod(args)
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arguments
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args.rigid logical {mustBeNumericOrLogical} = false
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args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
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args.ARB (3,3) double {mustBeNumeric} = eye(3)
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end
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%% Stewart Object
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micro_hexapod = struct();
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micro_hexapod.h = 350; % Total height of the platform [mm]
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micro_hexapod.jacobian = 270; % Distance from the top of the mobile platform to the Jacobian point [mm]
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%% Bottom Plate - Mechanical Design
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BP = struct();
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BP.rad.int = 110; % Internal Radius [mm]
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BP.rad.ext = 207.5; % External Radius [mm]
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BP.thickness = 26; % Thickness [mm]
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BP.leg.rad = 175.5; % Radius where the legs articulations are positionned [mm]
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BP.leg.ang = 9.5; % Angle Offset [deg]
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BP.density = 8000; % Density of the material [kg/m^3]
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BP.color = [0.6 0.6 0.6]; % 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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%% Top Plate - Mechanical Design
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TP = struct();
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TP.rad.int = 82; % Internal Radius [mm]
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TP.rad.ext = 150; % Internal Radius [mm]
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TP.thickness = 26; % Thickness [mm]
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TP.leg.rad = 118; % Radius where the legs articulations are positionned [mm]
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TP.leg.ang = 12.1; % Angle Offset [deg]
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TP.density = 8000; % Density of the material [kg/m^3]
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TP.color = [0.6 0.6 0.6]; % 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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%% Struts
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Leg = struct();
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Leg.stroke = 10e-3; % Maximum Stroke of each leg [m]
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if args.rigid
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Leg.k.ax = 1e12; % Stiffness of each leg [N/m]
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else
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Leg.k.ax = 2e7; % Stiffness of each leg [N/m]
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end
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Leg.ksi.ax = 0.1; % Modal damping ksi = 1/2*c/sqrt(km) []
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Leg.rad.bottom = 25; % Radius of the cylinder of the bottom part [mm]
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Leg.rad.top = 17; % Radius of the cylinder of the top part [mm]
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Leg.density = 8000; % Density of the material [kg/m^3]
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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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Leg.m = TP.density*((pi*(TP.rad.ext/1000)^2)*(TP.thickness/1000)-(pi*(TP.rad.int/1000^2))*(TP.thickness/1000))/6; % TODO [kg]
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Leg = updateDamping(Leg);
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%% Sphere
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SP = struct();
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SP.height.bottom = 27; % [mm]
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SP.height.top = 27; % [mm]
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SP.density.bottom = 8000; % [kg/m^3]
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SP.density.top = 8000; % [kg/m^3]
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SP.color.bottom = [0.6 0.6 0.6]; % [rgb]
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SP.color.top = [0.6 0.6 0.6]; % [rgb]
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SP.k.ax = 0; % [N*m/deg]
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SP.ksi.ax = 10;
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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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SP.m = SP.density.bottom*2*pi*((SP.rad.bottom*1e-3)^2)*(SP.height.bottom*1e-3); % TODO [kg]
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SP = updateDamping(SP);
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%%
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Leg.support.bottom = [0 SP.thickness.bottom; 0 0; SP.rad.bottom 0; SP.rad.bottom SP.height.bottom];
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Leg.support.top = [0 SP.thickness.top; 0 0; SP.rad.top 0; SP.rad.top SP.height.top];
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%%
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micro_hexapod.BP = BP;
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micro_hexapod.TP = TP;
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micro_hexapod.Leg = Leg;
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micro_hexapod.SP = SP;
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%%
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micro_hexapod = initializeParameters(micro_hexapod);
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%% Setup equilibrium position of each leg
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micro_hexapod.L0 = inverseKinematicsHexapod(micro_hexapod, args.AP, args.ARB);
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%% Save
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save('./mat/stages.mat', 'micro_hexapod', '-append');
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%%
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function [element] = updateDamping(element)
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field = fieldnames(element.k);
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for i = 1:length(field)
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element.c.(field{i}) = 2*element.ksi.(field{i})*sqrt(element.k.(field{i})*element.m);
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end
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end
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%%
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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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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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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; % TODO
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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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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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% 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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% 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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%% leg vectors
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legs = stewart.pos_top - stewart.pos_base;
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leg_length = zeros(6, 1);
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leg_vectors = zeros(6, 3);
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for i = 1:6
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leg_length(i) = norm(legs(i,:));
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leg_vectors(i,:) = legs(i,:) / leg_length(i);
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end
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stewart.Leg.lenght = 1000*leg_length(1)/1.5;
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stewart.Leg.shape.bot = [0 0; ...
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stewart.Leg.rad.bottom 0; ...
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stewart.Leg.rad.bottom stewart.Leg.lenght; ...
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stewart.Leg.rad.top stewart.Leg.lenght; ...
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stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
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0 0.2*stewart.Leg.lenght];
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%% Calculate revolute and cylindrical axes
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rev1 = zeros(6, 3);
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rev2 = zeros(6, 3);
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cyl1 = zeros(6, 3);
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for i = 1:6
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rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
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rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
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rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
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rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
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cyl1(i,:) = leg_vectors(i,:);
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end
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%% Coordinate systems
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stewart.lower_leg = struct('rotation', eye(3));
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stewart.upper_leg = struct('rotation', eye(3));
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for i = 1:6
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stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
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stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
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end
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%% Position Matrix
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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)];
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%% Compute Jacobian Matrix
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aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
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stewart.J = getJacobianMatrix(leg_vectors', aa');
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end
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%%
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function J = getJacobianMatrix(RM, M_pos_base)
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% RM: [3x6] unit vector of each leg in the fixed frame
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% M_pos_base: [3x6] vector of the leg connection at the top platform location in the fixed frame
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J = zeros(6);
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J(:, 1:3) = RM';
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J(:, 4:6) = cross(M_pos_base, RM)';
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end
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end
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