% Micro Hexapod % :PROPERTIES: % :header-args:matlab+: :tangle ../src/initializeMicroHexapod.m % :header-args:matlab+: :comments org :mkdirp yes % :header-args:matlab+: :eval no :results none % :END: % <> % This Matlab function is accessible [[file:../src/initializeMicroHexapod.m][here]]. function [micro_hexapod] = initializeMicroHexapod(opts_param) %% Default values for opts opts = struct(... 'rigid', false, ... 'AP', zeros(3, 1), ... % Wanted position in [m] of OB with respect to frame {A} 'ARB', eye(3) ... % Rotation Matrix that represent the wanted orientation of frame {B} with respect to frame {A} ); %% Populate opts with input parameters if exist('opts_param','var') for opt = fieldnames(opts_param)' opts.(opt{1}) = opts_param.(opt{1}); end end %% Stewart Object micro_hexapod = struct(); micro_hexapod.h = 350; % Total height of the platform [mm] micro_hexapod.jacobian = 270; % Distance from the top of the mobile platform to the Jacobian point [mm] %% Bottom Plate - Mechanical Design BP = struct(); BP.rad.int = 110; % Internal Radius [mm] BP.rad.ext = 207.5; % External Radius [mm] BP.thickness = 26; % Thickness [mm] BP.leg.rad = 175.5; % Radius where the legs articulations are positionned [mm] BP.leg.ang = 9.5; % Angle Offset [deg] BP.density = 8000; % Density of the material [kg/m^3] BP.color = [0.6 0.6 0.6]; % Color [rgb] BP.shape = [BP.rad.int BP.thickness; BP.rad.int 0; BP.rad.ext 0; BP.rad.ext BP.thickness]; %% Top Plate - Mechanical Design TP = struct(); TP.rad.int = 82; % Internal Radius [mm] TP.rad.ext = 150; % Internal Radius [mm] TP.thickness = 26; % Thickness [mm] TP.leg.rad = 118; % Radius where the legs articulations are positionned [mm] TP.leg.ang = 12.1; % Angle Offset [deg] TP.density = 8000; % Density of the material [kg/m^3] TP.color = [0.6 0.6 0.6]; % Color [rgb] TP.shape = [TP.rad.int TP.thickness; TP.rad.int 0; TP.rad.ext 0; TP.rad.ext TP.thickness]; %% Struts Leg = struct(); Leg.stroke = 10e-3; % Maximum Stroke of each leg [m] if opts.rigid Leg.k.ax = 1e12; % Stiffness of each leg [N/m] else Leg.k.ax = 2e7; % Stiffness of each leg [N/m] end Leg.ksi.ax = 0.1; % Modal damping ksi = 1/2*c/sqrt(km) [] Leg.rad.bottom = 25; % Radius of the cylinder of the bottom part [mm] Leg.rad.top = 17; % Radius of the cylinder of the top part [mm] Leg.density = 8000; % Density of the material [kg/m^3] 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] 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] Leg = updateDamping(Leg); %% Sphere SP = struct(); SP.height.bottom = 27; % [mm] SP.height.top = 27; % [mm] SP.density.bottom = 8000; % [kg/m^3] SP.density.top = 8000; % [kg/m^3] SP.color.bottom = [0.6 0.6 0.6]; % [rgb] SP.color.top = [0.6 0.6 0.6]; % [rgb] SP.k.ax = 0; % [N*m/deg] SP.ksi.ax = 10; 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] SP.m = SP.density.bottom*2*pi*((SP.rad.bottom*1e-3)^2)*(SP.height.bottom*1e-3); % TODO [kg] SP = updateDamping(SP); %% 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]; %% micro_hexapod.BP = BP; micro_hexapod.TP = TP; micro_hexapod.Leg = Leg; micro_hexapod.SP = SP; %% micro_hexapod = initializeParameters(micro_hexapod); %% Setup equilibrium position of each leg micro_hexapod.L0 = initializeHexapodPosition(micro_hexapod, opts.AP, opts.ARB); %% Save save('./mat/stages.mat', 'micro_hexapod', '-append'); %% function [element] = updateDamping(element) field = fieldnames(element.k); for i = 1:length(field) element.c.(field{i}) = 2*element.ksi.(field{i})*sqrt(element.k.(field{i})*element.m); end end %% 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 = (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 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 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 aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)]; stewart.J = getJacobianMatrix(leg_vectors', aa'); end %% 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 %% function [L] = initializeHexapodPosition(hexapod, AP, ARB) % initializeHexapodPosition - Compute the initial position of each leg to have the wanted Hexapod's position % % Syntax: initializeHexapodPosition(hexapod, AP, ARB) % % Inputs: % - hexapod - Hexapod object containing the geometry of the hexapod % - AP - Position vector of point OB expressed in frame {A} in [m] % - ARB - Rotation Matrix expressed in frame {A} % Wanted Length of the hexapod's legs [m] L = zeros(6, 1); for i = 1:length(L) Bbi = hexapod.pos_top_tranform(i, :)' - 1e-3*[0 ; 0 ; hexapod.TP.thickness+hexapod.Leg.sphere.top+hexapod.SP.thickness.top+hexapod.jacobian]; % [m] Aai = hexapod.pos_base(i, :)' + 1e-3*[0 ; 0 ; hexapod.BP.thickness+hexapod.Leg.sphere.bottom+hexapod.SP.thickness.bottom-micro_hexapod.h-hexapod.jacobian]; % [m] L(i) = sqrt(AP'*AP + Bbi'*Bbi + Aai'*Aai - 2*AP'*Aai + 2*AP'*(ARB*Bbi) - 2*(ARB*Bbi)'*Aai); end end end