Simscape Model of the Nano-Active-Stabilization-System

%% Load all the data used for the simulation
load('sim_conf.mat');

%% Load Controller
load('controllers.mat');

function [psd_object] = computePsdDispl(sys_data, t_init, n_av)
    i_init = find(sys_data.time > t_init, 1);

    han_win = hanning(ceil(length(sys_data.Dx(i_init:end, :))/n_av));
    Fs = 1/sys_data.time(2);

    [pdx, f] = pwelch(sys_data.Dx(i_init:end, :), han_win, [], [], Fs);
    [pdy, ~] = pwelch(sys_data.Dy(i_init:end, :), han_win, [], [], Fs);
    [pdz, ~] = pwelch(sys_data.Dz(i_init:end, :), han_win, [], [], Fs);

    [prx, ~] = pwelch(sys_data.Rx(i_init:end, :), han_win, [], [], Fs);
    [pry, ~] = pwelch(sys_data.Ry(i_init:end, :), han_win, [], [], Fs);
    [prz, ~] = pwelch(sys_data.Rz(i_init:end, :), han_win, [], [], Fs);

    psd_object = struct(...
        'f',  f,   ...
        'dx', pdx, ...
        'dy', pdy, ...
        'dz', pdz, ...
        'rx', prx, ...
        'ry', pry, ...
        'rz', prz);
end

function setpoint = computeSetpoint(ty, ry, rz)
%%
setpoint = zeros(6, 1);

%% Ty
Ty = [1 0 0 0  ;
      0 1 0 ty ;
      0 0 1 0  ;
      0 0 0 1 ];

% Tyinv = [1 0 0  0  ;
%          0 1 0 -ty ;
%          0 0 1  0  ;
%          0 0 0  1 ];

%% Ry
Ry = [ cos(ry) 0 sin(ry) 0 ;
      0       1 0       0 ;
      -sin(ry) 0 cos(ry) 0 ;
      0        0 0       1 ];

% TMry = Ty*Ry*Tyinv;

%% Rz
Rz = [cos(rz) -sin(rz) 0 0 ;
      sin(rz)  cos(rz) 0 0 ;
      0        0       1 0 ;
      0        0       0 1 ];

% TMrz = Ty*TMry*Rz*TMry'*Tyinv;

%% All stages
% TM = TMrz*TMry*Ty;

TM = Ty*Ry*Rz;

[thetax, thetay, thetaz] = RM2angle(TM(1:3, 1:3));

setpoint(1:3) = TM(1:3, 4);
setpoint(4:6) = [thetax, thetay, thetaz];

%% Custom Functions
function [thetax, thetay, thetaz] = RM2angle(R)
    if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1
        thetay = -asin(R(3, 1));
        thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay));
        thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay));
    else
        thetaz = 0;
        if abs(R(3, 1)+1) < 1e-6 % R31 = -1
            thetay = pi/2;
            thetax = thetaz + atan2(R(1, 2), R(1, 3));
        else
            thetay = -pi/2;
            thetax = -thetaz + atan2(-R(1, 2), -R(1, 3));
        end
    end
end
end

function error_nass = convertErrorBasis(pos, setpoint, ty, ry, rz)
% convertErrorBasis -
%
% Syntax: convertErrorBasis(p_error, ty, ry, rz)
%
% Inputs:
%    - p_error - Position error of the sample w.r.t. the granite [m, rad]
%    - ty - Measured translation of the Ty stage [m]
%    - ry - Measured rotation of the Ry stage [rad]
%    - rz - Measured rotation of the Rz stage [rad]
%
% Outputs:
%    - P_nass - Position error of the sample w.r.t. the NASS base [m]
%    - R_nass - Rotation error of the sample w.r.t. the NASS base [rad]
%
% Example:
%

%% If line vector => column vector
if size(pos, 2) == 6
    pos = pos';
end

if size(setpoint, 2) == 6
    setpoint = setpoint';
end

%% Position of the sample in the frame fixed to the Granite
P_granite = [pos(1:3); 1]; % Position [m]
R_granite = [setpoint(1:3); 1]; % Reference [m]

%% Transformation matrices of the stages
% T-y
TMty = [1 0 0 0  ;
        0 1 0 ty ;
        0 0 1 0  ;
        0 0 0 1 ];

% R-y
TMry = [ cos(ry) 0 sin(ry) 0 ;
        0       1 0       0 ;
        -sin(ry) 0 cos(ry) 0 ;
        0        0 0       1 ];

% R-z
TMrz = [cos(rz) -sin(rz) 0 0 ;
        sin(rz)  cos(rz) 0 0 ;
        0        0       1 0 ;
        0        0       0 1 ];

%% Compute Point coordinates in the new reference fixed to the NASS base
% P_nass = TMrz*TMry*TMty*P_granite;
P_nass = TMrz\TMry\TMty\P_granite;
R_nass = TMrz\TMry\TMty\R_granite;

dx = R_nass(1)-P_nass(1);
dy = R_nass(2)-P_nass(2);
dz = R_nass(3)-P_nass(3);

%% Compute new basis vectors linked to the NASS base
% ux_nass = TMrz*TMry*TMty*[1; 0; 0; 0];
% ux_nass = ux_nass(1:3);
% uy_nass = TMrz*TMry*TMty*[0; 1; 0; 0];
% uy_nass = uy_nass(1:3);
% uz_nass = TMrz*TMry*TMty*[0; 0; 1; 0];
% uz_nass = uz_nass(1:3);

ux_nass = TMrz\TMry\TMty\[1; 0; 0; 0];
ux_nass = ux_nass(1:3);
uy_nass = TMrz\TMry\TMty\[0; 1; 0; 0];
uy_nass = uy_nass(1:3);
uz_nass = TMrz\TMry\TMty\[0; 0; 1; 0];
uz_nass = uz_nass(1:3);

%% Rotations error w.r.t. granite Frame
% Rotations error w.r.t. granite Frame
rx_nass = pos(4);
ry_nass = pos(5);
rz_nass = pos(6);

% Rotation matrices of the Sample w.r.t. the Granite
Mrx_error = [1 0              0 ;
            0 cos(-rx_nass) -sin(-rx_nass) ;
            0 sin(-rx_nass)  cos(-rx_nass)];

Mry_error = [ cos(-ry_nass) 0 sin(-ry_nass) ;
              0             1 0 ;
            -sin(-ry_nass) 0 cos(-ry_nass)];

Mrz_error = [cos(-rz_nass) -sin(-rz_nass) 0 ;
            sin(-rz_nass)  cos(-rz_nass) 0 ;
            0              0             1];

% Rotation matrix of the Sample w.r.t. the Granite
Mr_error = Mrz_error*Mry_error*Mrx_error;

%% Use matrix to solve
R = Mr_error/[ux_nass, uy_nass, uz_nass]; % Rotation matrix from NASS base to Sample

[thetax, thetay, thetaz] = RM2angle(R);

error_nass = [dx; dy; dz; thetax; thetay; thetaz];

%% Custom Functions
function [thetax, thetay, thetaz] = RM2angle(R)
    if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1
        thetay = -asin(R(3, 1));
        % thetaybis = pi-thetay;
        thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay));
        % thetaxbis = atan2(R(3, 2)/cos(thetaybis), R(3, 3)/cos(thetaybis));
        thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay));
        % thetazbis = atan2(R(2, 1)/cos(thetaybis), R(1, 1)/cos(thetaybis));
    else
        thetaz = 0;
        if abs(R(3, 1)+1) < 1e-6 % R31 = -1
            thetay = pi/2;
            thetax = thetaz + atan2(R(1, 2), R(1, 3));
        else
            thetay = -pi/2;
            thetax = -thetaz + atan2(-R(1, 2), -R(1, 3));
        end
    end
end

end

function [K] = generateDiagPidControl(G, fs)
    %%
    pid_opts = pidtuneOptions(...
        'PhaseMargin', 50, ...
        'DesignFocus', 'disturbance-rejection');

    %%
    K = tf(zeros(6));

    for i = 1:6
        input_name = G.InputName(i);
        output_name = G.OutputName(i);
        K(i, i) = tf(pidtune(minreal(G(output_name, input_name)), 'PIDF', 2*pi*fs, pid_opts));
    end

    K.InputName = G.OutputName;
    K.OutputName = G.InputName;
end

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 = 'sim_nano_station_id';

    %% Input/Output definition
    io(1) = linio([mdl, '/Fn'],   1, 'input'); % Cartesian forces applied by NASS
    io(2) = linio([mdl, '/Dw'],   1, 'input'); % Ground Motion
    io(3) = linio([mdl, '/Fs'],   1, 'input'); % External forces on the sample
    io(4) = linio([mdl, '/Fnl'],  1, 'input'); % Forces applied on the NASS's legs
    io(5) = linio([mdl, '/Dsm'],  1, 'output'); % Displacement of the sample
    io(6) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
    io(7) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
    io(8) = linio([mdl, '/Es'],   1, 'output'); % Position Error w.r.t. NASS base

    %% Run the linearization
    G = linearize(mdl, io, 0);
    G.InputName  = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz', ...
                    'Dgx', 'Dgy', 'Dgz', ...
                    'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz', ...
                    'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', ...
                    'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6', ...
                    'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ...
                    'Edx', 'Rdy', 'Edz', 'Erx', 'Ery', 'Erz'};

    %% Create the sub transfer functions
    % From forces applied in the cartesian frame to displacement of the sample in the cartesian frame
    sys.G_cart = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'}));
    % From ground motion to Sample displacement
    sys.G_gm   = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Dgx', 'Dgy', 'Dgz'}));
    % From direct forces applied on the sample to displacement of the sample
    sys.G_fs   = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz'}));
    % From forces applied on NASS's legs to force sensor in each leg
    sys.G_iff  = minreal(G({'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
    % From forces applied on NASS's legs to displacement of each leg
    sys.G_dleg = minreal(G({'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
    % From forces applied on NASS's legs to displacement of each leg
    sys.G_plant = minreal(G({'Edx', 'Rdy', 'Edz', 'Erx', 'Ery', 'Erz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'}));
end

function [] = runSimulation(sys_name, sys_mass, ctrl_type, act_damp)
    %% Load the controller and save it for the simulation
    if strcmp(ctrl_type, 'cl') && strcmp(act_damp, 'none')
      K_obj = load('./mat/K_fb.mat');
      K = K_obj.(sprintf('K_%s_%s', sys_mass, sys_name)); %#ok
      save('./mat/controllers.mat', 'K');
    elseif strcmp(ctrl_type, 'cl') && strcmp(act_damp, 'iff')
      K_obj = load('./mat/K_fb_iff.mat');
      K = K_obj.(sprintf('K_%s_%s_iff', sys_mass, sys_name)); %#ok
      save('./mat/controllers.mat', 'K');
    elseif strcmp(ctrl_type, 'ol')
      K = tf(zeros(6)); %#ok
      save('./mat/controllers.mat', 'K');
    else
      error('ctrl_type should be cl or ol');
    end

    %% Active Damping
    if strcmp(act_damp, 'iff')
      K_iff_crit = load('./mat/K_iff_crit.mat');
      K_iff = K_iff_crit.(sprintf('K_iff_%s_%s', sys_mass, sys_name)); %#ok
      save('./mat/controllers.mat', 'K_iff', '-append');
    elseif strcmp(act_damp, 'none')
      K_iff = tf(zeros(6)); %#ok
      save('./mat/controllers.mat', 'K_iff', '-append');
    end

    %%
    if strcmp(sys_name, 'pz')
      initializeNanoHexapod(struct('actuator', 'piezo'));
    elseif strcmp(sys_name, 'vc')
      initializeNanoHexapod(struct('actuator', 'lorentz'));
    else
      error('sys_name should be pz or vc');
    end

    if strcmp(sys_mass, 'light')
      initializeSample(struct('mass', 1));
    elseif strcmp(sys_mass, 'heavy')
      initializeSample(struct('mass', 50));
    else
      error('sys_mass should be light or heavy');
    end

    %% Run the simulation
    sim('sim_nano_station_ctrl.slx');

    %% Split the Dsample matrix into vectors
    [Dx, Dy, Dz, Rx, Ry, Rz] = matSplit(Es.Data, 1); %#ok
    time = Dsample.Time; %#ok

    %% Save the result
    filename = sprintf('sim_%s_%s_%s_%s', sys_mass, sys_name, ctrl_type, act_damp);
    save(sprintf('./mat/%s.mat', filename), ...
        'time', 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', 'K');
end

function [] = initializeSimConf(opts_param)
    %% Default values for opts
    opts = struct('Ts',      1e-4,  ... % Sampling time [s]
                  'Tsim',    10,    ... % Simulation time [s]
                  'cl_time', 0,     ... % Close Loop time [s]
                  'gravity', false  ... % Gravity along the z axis
    );

    %% Populate opts with input parameters
    if exist('opts_param','var')
        for opt = fieldnames(opts_param)'
            opts.(opt{1}) = opts_param.(opt{1});
        end
    end

    %%
    sim_conf = struct();

    %%
    sim_conf.Ts = opts.Ts;
    sim_conf.Tsim = opts.Tsim;
    sim_conf.cl_time = opts.cl_time;

    %% Gravity
    if opts.gravity
        sim_conf.g = -9.8; %#ok
    else
        sim_conf.g = 0; %#ok
    end

    %% Save
    save('./mat/sim_conf.mat', 'sim_conf');
end

function [] = initializeExperiment(exp_name, sys_mass)
    if strcmp(exp_name, 'tomography')
        opts_sim = struct(...
            'Tsim',    5, ...
            'cl_time', 5  ...
        );
        initializeSimConf(opts_sim);

        if strcmp(sys_mass, 'light')
            opts_inputs = struct(...
                'Dw', true,  ...
                'Rz', 60     ... % rpm
            );
        elseif strcpm(sys_mass, 'heavy')
            opts_inputs = struct(...
                'Dw', true,  ...
                'Rz', 1     ... % rpm
            );
        else
            error('sys_mass should be light or heavy');
        end

        initializeInputs(opts_inputs);
    else
        error('exp_name is only configured for tomography');
    end
end

function [inputs] = initializeInputs(opts_param)
    %% Default values for opts
    opts = struct(   ...
        'Dw', false, ...
        'Dy', false, ...
        'Ry', false, ...
        'Rz', false, ...
        'Dh', false, ...
        'Rm', false, ...
        'Dn', false  ...
    );

    %% Populate opts with input parameters
    if exist('opts_param','var')
        for opt = fieldnames(opts_param)'
            opts.(opt{1}) = opts_param.(opt{1});
        end
    end

    %% Load Sampling Time and Simulation Time
    load('./mat/sim_conf.mat', 'sim_conf');

    %% Define the time vector
    t = 0:sim_conf.Ts:sim_conf.Tsim;

    %% Ground motion - Dw
    if islogical(opts.Dw) && opts.Dw == true
        load('./mat/perturbations.mat', 'Wxg');
        Dw = 1/sqrt(2)*100*random('norm', 0, 1, length(t), 3);
        Dw(:, 1) = lsim(Wxg, Dw(:, 1), t);
        Dw(:, 2) = lsim(Wxg, Dw(:, 2), t);
        Dw(:, 3) = lsim(Wxg, Dw(:, 3), t);
    elseif islogical(opts.Dw) && opts.Dw == false
        Dw = zeros(length(t), 3);
    else
        Dw = opts.Dw;
    end

    %% Translation stage - Dy
    if islogical(opts.Dy) && opts.Dy == true
        Dy = zeros(length(t), 1);
    elseif islogical(opts.Dy) && opts.Dy == false
        Dy = zeros(length(t), 1);
    else
        Dy = opts.Dy;
    end

    %% Tilt Stage - Ry
    if islogical(opts.Ry) && opts.Ry == true
        Ry = 3*(2*pi/360)*sin(2*pi*0.2*t);
    elseif islogical(opts.Ry) && opts.Ry == false
        Ry = zeros(length(t), 1);
    elseif isnumeric(opts.Ry) && length(opts.Ry) == 1
        Ry = opts.Ry*(2*pi/360)*ones(length(t), 1);
    else
        Ry = opts.Ry;
    end

    %% Spindle - Rz
    if islogical(opts.Rz) && opts.Rz == true
        Rz = 2*pi*0.5*t;
    elseif islogical(opts.Rz) && opts.Rz == false
        Rz = zeros(length(t), 1);
    elseif isnumeric(opts.Rz) && length(opts.Rz) == 1
        Rz = opts.Rz*(2*pi/60)*t;
    else
        Rz = opts.Rz;
    end

    %% Micro Hexapod - Dh
    if islogical(opts.Dh) && opts.Dh == true
        Dh = zeros(length(t), 6);
    elseif islogical(opts.Dh) && opts.Dh == false
        Dh = zeros(length(t), 6);
    else
        Dh = opts.Dh;
    end

    %% Axis Compensation - Rm
    if islogical(opts.Rm)
        Rm = zeros(length(t), 2);
        Rm(:, 2) = pi*ones(length(t), 1);
    else
        Rm = opts.Rm;
    end

    %% Nano Hexapod - Dn
    if islogical(opts.Dn) && opts.Dn == true
        Dn = zeros(length(t), 6);
    elseif islogical(opts.Dn) && opts.Dn == false
        Dn = zeros(length(t), 6);
    else
        Dn = opts.Dn;
    end

    %% Setpoint - Ds
    Ds = zeros(length(t), 6);
    for i = 1:length(t)
        Ds(i, :) = computeSetpoint(Dy(i), Ry(i), Rz(i));
    end

    %% External Forces applied on the Granite
    Fg = zeros(length(t), 3);

    %% External Forces applied on the Sample
    Fs = zeros(length(t), 6);

    %% Create the input Structure that will contain all the inputs
    inputs = struct( ...
        'Ts', sim_conf.Ts,       ...
        'Dw', timeseries(Dw, t), ...
        'Dy', timeseries(Dy, t), ...
        'Ry', timeseries(Ry, t), ...
        'Rz', timeseries(Rz, t), ...
        'Dh', timeseries(Dh, t), ...
        'Rm', timeseries(Rm, t), ...
        'Dn', timeseries(Dn, t), ...
        'Ds', timeseries(Ds, t), ...
        'Fg', timeseries(Fg, t), ...
        'Fs', timeseries(Fs, t)  ...
    );

    %% Save
    save('./mat/inputs.mat', 'inputs');

    %% Custom Functions
    function setpoint = computeSetpoint(ty, ry, rz)
    %%
    setpoint = zeros(6, 1);

    %% Ty
    TMTy = [1 0 0 0  ;
          0 1 0 ty ;
          0 0 1 0  ;
          0 0 0 1 ];

    %% Ry
    TMRy = [ cos(ry) 0 sin(ry) 0 ;
          0       1 0       0 ;
          -sin(ry) 0 cos(ry) 0 ;
          0        0 0       1 ];

    %% Rz
    TMRz = [cos(rz) -sin(rz) 0 0 ;
          sin(rz)  cos(rz) 0 0 ;
          0        0       1 0 ;
          0        0       0 1 ];

    %% All stages
    TM = TMTy*TMRy*TMRz;

    [thetax, thetay, thetaz] = RM2angle(TM(1:3, 1:3));

    setpoint(1:3) = TM(1:3, 4);
    setpoint(4:6) = [thetax, thetay, thetaz];

    %% Custom Functions
    function [thetax, thetay, thetaz] = RM2angle(R)
        if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1
            thetay = -asin(R(3, 1));
            thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay));
            thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay));
        else
            thetaz = 0;
            if abs(R(3, 1)+1) < 1e-6 % R31 = -1
                thetay = pi/2;
                thetax = thetaz + atan2(R(1, 2), R(1, 3));
            else
                thetay = -pi/2;
                thetax = -thetaz + atan2(-R(1, 2), -R(1, 3));
            end
        end
    end
    end
end

function [ground] = initializeGround()
    %%
    ground = struct();

    ground.shape = [2, 2, 0.5]; % [m]
    ground.density = 2800; % [kg/m3]
    ground.color = [0.5, 0.5, 0.5];

    %% Save
    save('./mat/stages.mat', 'ground', '-append');
end

function [granite] = initializeGranite()
    %%
    granite = struct();

    %% Static Properties
    granite.density = 2800; % [kg/m3]
    granite.volume  = 0.72; % [m3] TODO - should
    granite.mass    = granite.density*granite.volume; % [kg]
    granite.color   = [1 1 1];
    granite.STEP    = './STEPS/granite/granite.STEP';

    %% Dynamical Properties
    granite.k.x = 1e8; % [N/m]
    granite.c.x = 1e4; % [N/(m/s)]

    granite.k.y = 1e8; % [N/m]
    granite.c.y = 1e4; % [N/(m/s)]

    granite.k.z = 1e8; % [N/m]
    granite.c.z = 1e4; % [N/(m/s)]

    %% Positioning parameters
    granite.sample_pos = 0.8; % Z-offset for the initial position of the sample [m]

    %% Save
    save('./mat/stages.mat', 'granite', '-append');
end

function [ty] = initializeTy(opts_param)
    %% Default values for opts
    opts = struct('rigid', false);

    %% Populate opts with input parameters
    if exist('opts_param','var')
        for opt = fieldnames(opts_param)'
            opts.(opt{1}) = opts_param.(opt{1});
        end
    end

    %%
    ty = struct();

    %% Y-Translation - Static Properties
    % Ty Granite frame
    ty.granite_frame.density = 7800; % [kg/m3]
    ty.granite_frame.color   = [0.753 1 0.753];
    ty.granite_frame.STEP    = './STEPS/Ty/Ty_Granite_Frame.STEP';
    % Guide Translation Ty
    ty.guide.density         = 7800; % [kg/m3]
    ty.guide.color           = [0.792 0.820 0.933];
    ty.guide.STEP            = './STEPS/ty/Ty_Guide.STEP';
    % Ty - Guide_Translation12
    ty.guide12.density       = 7800; % [kg/m3]
    ty.guide12.color         = [0.792 0.820 0.933];
    ty.guide12.STEP          = './STEPS/Ty/Ty_Guide_12.STEP';
    % Ty - Guide_Translation11
    ty.guide11.density       = 7800; % [kg/m3]
    ty.guide11.color         = [0.792 0.820 0.933];
    ty.guide11.STEP          = './STEPS/ty/Ty_Guide_11.STEP';
    % Ty - Guide_Translation22
    ty.guide22.density       = 7800; % [kg/m3]
    ty.guide22.color         = [0.792 0.820 0.933];
    ty.guide22.STEP          = './STEPS/ty/Ty_Guide_22.STEP';
    % Ty - Guide_Translation21
    ty.guide21.density       = 7800; % [kg/m3]
    ty.guide21.color         = [0.792 0.820 0.933];
    ty.guide21.STEP          = './STEPS/Ty/Ty_Guide_21.STEP';
    % Ty - Plateau translation
    ty.frame.density         = 7800; % [kg/m3]
    ty.frame.color           = [0.792 0.820 0.933];
    ty.frame.STEP            = './STEPS/ty/Ty_Stage.STEP';
    % Ty Stator Part
    ty.stator.density        = 5400; % [kg/m3]
    ty.stator.color          = [0.792 0.820 0.933];
    ty.stator.STEP           = './STEPS/ty/Ty_Motor_Stator.STEP';
    % Ty Rotor Part
    ty.rotor.density         = 5400; % [kg/m3]
    ty.rotor.color           = [0.792 0.820 0.933];
    ty.rotor.STEP            = './STEPS/ty/Ty_Motor_Rotor.STEP';

    ty.m = 250; % TODO [kg]

    %% Y-Translation - Dynamicals Properties
    if opts.rigid
        ty.k.ax  = 1e10; % Axial Stiffness for each of the 4 guidance (y) [N/m]
    else
        ty.k.ax  = 1e7/4; % Axial Stiffness for each of the 4 guidance (y) [N/m]
    end
    ty.k.rad = 9e9/4; % Radial Stiffness for each of the 4 guidance (x-z) [N/m]

    ty.c.ax  = 100*(1/5)*sqrt(ty.k.ax/ty.m);
    ty.c.rad = 100*(1/5)*sqrt(ty.k.rad/ty.m);

    %% Save
    save('./mat/stages.mat', 'ty', '-append');
end

function [ry] = initializeRy(opts_param)
    %% Default values for opts
    opts = struct('rigid', false);

    %% Populate opts with input parameters
    if exist('opts_param','var')
        for opt = fieldnames(opts_param)'
            opts.(opt{1}) = opts_param.(opt{1});
        end
    end

    %%
    ry = struct();

    %% Tilt Stage - Static Properties
    % Ry - Guide for the tilt stage
    ry.guide.density = 7800; % [kg/m3]
    ry.guide.color   = [0.792 0.820 0.933];
    ry.guide.STEP    = './STEPS/ry/Tilt_Guide.STEP';
    % Ry - Rotor of the motor
    ry.rotor.density = 2400; % [kg/m3]
    ry.rotor.color   = [0.792 0.820 0.933];
    ry.rotor.STEP    = './STEPS/ry/Tilt_Motor_Axis.STEP';
    % Ry - Motor
    ry.motor.density = 3200; % [kg/m3]
    ry.motor.color   = [0.792 0.820 0.933];
    ry.motor.STEP    = './STEPS/ry/Tilt_Motor.STEP';
    % Ry - Plateau Tilt
    ry.stage.density = 7800; % [kg/m3]
    ry.stage.color   = [0.792 0.820 0.933];
    ry.stage.STEP    = './STEPS/ry/Tilt_Stage.STEP';

    ry.m = 200; % TODO [kg]

    %% Tilt Stage - Dynamical Properties
    if opts.rigid
        ry.k.tilt = 1e10; % Rotation stiffness around y [N*m/deg]
    else
        ry.k.tilt = 1e4; % Rotation stiffness around y [N*m/deg]
    end

    ry.k.h    = 357e6/4; % Stiffness in the direction of the guidance [N/m]
    ry.k.rad  = 555e6/4; % Stiffness in the top direction [N/m]
    ry.k.rrad = 238e6/4; % Stiffness in the side direction [N/m]

    ry.c.h    = 10*(1/5)*sqrt(ry.k.h/ry.m);
    ry.c.rad  = 10*(1/5)*sqrt(ry.k.rad/ry.m);
    ry.c.rrad = 10*(1/5)*sqrt(ry.k.rrad/ry.m);
    ry.c.tilt = 10*(1/1)*sqrt(ry.k.tilt/ry.m);

    %% Positioning parameters
    ry.z_offset = 0.58178; % Z-Offset so that the center of rotation matches the sample center [m]

    %% Save
    save('./mat/stages.mat', 'ry', '-append');
end

function [rz] = initializeRz(opts_param)
    %% Default values for opts
    opts = struct('rigid', false);

    %% Populate opts with input parameters
    if exist('opts_param','var')
        for opt = fieldnames(opts_param)'
            opts.(opt{1}) = opts_param.(opt{1});
        end
    end

    %%
    rz = struct();

    %% Spindle - Static Properties
    % Spindle - Slip Ring
    rz.slipring.density = 7800; % [kg/m3]
    rz.slipring.color   = [0.792 0.820 0.933];
    rz.slipring.STEP    = './STEPS/rz/Spindle_Slip_Ring.STEP';
    % Spindle - Rotor
    rz.rotor.density    = 7800; % [kg/m3]
    rz.rotor.color      = [0.792 0.820 0.933];
    rz.rotor.STEP       = './STEPS/rz/Spindle_Rotor.STEP';
    % Spindle - Stator
    rz.stator.density   = 7800; % [kg/m3]
    rz.stator.color     = [0.792 0.820 0.933];
    rz.stator.STEP      = './STEPS/rz/Spindle_Stator.STEP';

    % Estimated mass of the mooving part
    rz.m = 250; % [kg]

    %% Spindle - Dynamical Properties
    % Estimated stiffnesses
    rz.k.ax   = 2e9; % Axial Stiffness [N/m]
    rz.k.rad  = 7e8; % Radial Stiffness [N/m]
    rz.k.rot  = 100e6*2*pi/360; % Rotational Stiffness [N*m/deg]

    if opts.rigid
        rz.k.tilt = 1e10; % Vertical Rotational Stiffness [N*m/deg]
    else
        rz.k.tilt = 1e2; % TODO what value should I put? [N*m/deg]
    end

    % TODO
    rz.c.ax   = 2*sqrt(rz.k.ax/rz.m);
    rz.c.rad  = 2*sqrt(rz.k.rad/rz.m);
    rz.c.tilt = 100*sqrt(rz.k.tilt/rz.m);
    rz.c.rot  = 100*sqrt(rz.k.rot/rz.m);

    %% Save
    save('./mat/stages.mat', 'rz', '-append');
end

function [micro_hexapod] = initializeMicroHexapod(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

    %% Stewart Object
    micro_hexapod = struct();
    micro_hexapod.h        = 350; % Total height of the platform [mm]
%     micro_hexapod.jacobian = 269.26; % Distance from the top platform to the Jacobian point [mm]
    micro_hexapod.jacobian = 270; % Distance from the top 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]
    Leg.k.ax       = 5e7;   % Stiffness of each leg [N/m]
    Leg.ksi.ax     = 3;     % Maximum amplification at resonance []
    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);

    %% 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}) = 1/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
end

function [axisc] = initializeAxisc()
    %%
    axisc = struct();

    %% Axis Compensator - Static Properties
    % Structure
    axisc.structure.density = 3400; % [kg/m3]
    axisc.structure.color   = [0.792 0.820 0.933];
    axisc.structure.STEP    = './STEPS/axisc/axisc_structure.STEP';
    % Wheel
    axisc.wheel.density     = 2700; % [kg/m3]
    axisc.wheel.color       = [0.753 0.753 0.753];
    axisc.wheel.STEP        = './STEPS/axisc/axisc_wheel.STEP';
    % Mass
    axisc.mass.density      = 7800; % [kg/m3]
    axisc.mass.color        = [0.792 0.820 0.933];
    axisc.mass.STEP         = './STEPS/axisc/axisc_mass.STEP';
    % Gear
    axisc.gear.density      = 7800; % [kg/m3]
    axisc.gear.color        = [0.792 0.820 0.933];
    axisc.gear.STEP         = './STEPS/axisc/axisc_gear.STEP';

    axisc.m      = 40; % TODO [kg]

    %% Axis Compensator - Dynamical Properties
    axisc.k.ax   = 1; % TODO [N*m/deg)]
    axisc.c.ax   = (1/1)*sqrt(axisc.k.ax/axisc.m);

    %% Save
    save('./mat/stages.mat', 'axisc', '-append');
end

function [] = initializeMirror(opts_param)
    %% Default values for opts
    opts = struct(...
        'shape', 'spherical', ... % spherical or conical
        'angle', 45 ...
    );

    %% Populate opts with input parameters
    if exist('opts_param','var')
        for opt = fieldnames(opts_param)'
            opts.(opt{1}) = opts_param.(opt{1});
        end
    end

    %%
    mirror = struct();
    mirror.h = 50; % height of the mirror [mm]
    mirror.thickness = 25; % Thickness of the plate supporting the sample [mm]
    mirror.hole_rad = 120; % radius of the hole in the mirror [mm]
    mirror.support_rad = 100; % radius of the support plate [mm]
    mirror.jacobian = 150; % point of interest offset in z (above the top surfave) [mm]
    mirror.rad = 180; % radius of the mirror (at the bottom surface) [mm]

    mirror.density = 2400; % Density of the mirror [kg/m3]
    mirror.color = [0.4 1.0 1.0]; % Color of the mirror

    mirror.cone_length = mirror.rad*tand(opts.angle)+mirror.h+mirror.jacobian; % Distance from Apex point of the cone to jacobian point

    %% Shape
    mirror.shape = [...
        0 mirror.h-mirror.thickness
        mirror.hole_rad mirror.h-mirror.thickness; ...
        mirror.hole_rad 0; ...
        mirror.rad 0 ...
    ];

    if strcmp(opts.shape, 'spherical')
        mirror.sphere_radius = sqrt((mirror.jacobian+mirror.h)^2+mirror.rad^2); % Radius of the sphere [mm]

        for z = linspace(0, mirror.h, 101)
            mirror.shape = [mirror.shape; sqrt(mirror.sphere_radius^2-(z-mirror.jacobian-mirror.h)^2) z];
        end
    elseif strcmp(opts.shape, 'conical')
        mirror.shape = [mirror.shape; mirror.rad+mirror.h/tand(opts.angle) mirror.h];
    else
        error('Shape should be either conical or spherical');
    end

    mirror.shape = [mirror.shape; 0 mirror.h];

    %% Save
    save('./mat/stages.mat', 'mirror', '-append');
end

function [nano_hexapod] = initializeNanoHexapod(opts_param)
    %% Default values for opts
    opts = struct('actuator', 'piezo');

    %% 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
    nano_hexapod = struct();
    nano_hexapod.h        = 90;  % Total height of the platform [mm]
    nano_hexapod.jacobian = 175; % Point where the Jacobian is computed => Center of rotation [mm]
%     nano_hexapod.jacobian = 174.26; % Point where the Jacobian is computed => Center of rotation [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   = 8000;% Density of the material [kg/m^3]
    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   = 8000;% Density of the material [kg/m^3]
    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]
    elseif strcmp(opts.actuator, 'lorentz')
        Leg.k.ax   = 1e4;   % Stiffness of each leg [N/m]
    else
        error('opts.actuator should be piezo or lorentz');
    end
    Leg.ksi.ax     = 10;    % Maximum amplification at resonance []
    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    = 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  = 15; % [mm]
    SP.height.top     = 15; % [mm]
    SP.density.bottom = 8000; % [kg/m^3]
    SP.density.top    = 8000; % [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.ksi.ax         = 3;

    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];

    %%
    nano_hexapod.BP = BP;
    nano_hexapod.TP = TP;
    nano_hexapod.Leg = Leg;
    nano_hexapod.SP = SP;

    %%
    nano_hexapod = initializeParameters(nano_hexapod);

    %% Save
    save('./mat/stages.mat', 'nano_hexapod', '-append');

    %%
    function [element] = updateDamping(element)
        field = fieldnames(element.k);
        for i = 1:length(field)
            element.c.(field{i}) = 1/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
end

function [sample] = initializeSample(opts_param)
    %% Default values for opts
    sample = struct('radius', 100, ...
                    'height', 300, ...
                    'mass',   50,  ...
                    'offset', 0,   ...
                    'color',  [0.45, 0.45, 0.45] ...
    );

    %% Populate opts with input parameters
    if exist('opts_param','var')
        for opt = fieldnames(opts_param)'
            sample.(opt{1}) = opts_param.(opt{1});
        end
    end

    %%
    sample.k.x = 1e8;
    sample.c.x = sqrt(sample.k.x*sample.mass)/10;

    sample.k.y = 1e8;
    sample.c.y = sqrt(sample.k.y*sample.mass)/10;

    sample.k.z = 1e8;
    sample.c.z = sqrt(sample.k.y*sample.mass)/10;

    %% Save
    save('./mat/stages.mat', 'sample', '-append');
end