nass-simscape/src/initializeMicroHexapod.m

% 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:
% <<sec:initializeMicroHexapod>>

% 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