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# Simscape Model

## Table of Contents

- 1. Simulink Project - Startup and Shutdown scripts
- 2. Simscape Multibody - Presentation
  - 2.1. Solid bodies
  - 2.2. Frames
  - 2.3. Joints
  - 2.4. Measurements
  - 2.5. Excitation
- 3. Simulink files and signal names
  - 3.1. List of Simscape files
  - 3.2. List of Inputs
    - 3.2.1. Perturbations
    - 3.2.2. Measurement Noise
    - 3.2.3. Control Inputs
  - 3.3. List of Outputs
- 4. Simulink Library
  - 4.1. `inputs`
  - 4.2. `nass_library`
  - 4.3. `pos_error_wrt_nass_base`
  - 4.4. `QuaternionToAngles`
  - 4.5. `RotationMatrixToAngle`
- 5. Functions
  - 5.1. computePsdDispl
  - 5.2. computeSetpoint
  - 5.3. converErrorBasis
  - 5.4. generateDiagPidControl
  - 5.5. identifyPlant
  - 5.6. runSimulation
- 6. Initialize Elements
  - 6.1. Experiment
  - 6.2. Generate Reference Signals
  - 6.3. TODO Inputs
  - 6.4. Ground
  - 6.5. Granite
  - 6.6. Translation Stage
  - 6.7. Tilt Stage
  - 6.8. Spindle
  - 6.9. Initialize Hexapod legs' length
  - 6.10. Micro Hexapod
  - 6.11. Center of gravity compensation
  - 6.12. Mirror
  - 6.13. Nano Hexapod
  - 6.14. Cedrat Actuator
  - 6.15. Sample

This file is used to explain how this Simscape Model works.

- In section 1, the simulink project with the associated scripts are presented
- In section 2, an introduction to Simscape Multibody is done
- In section 3, each simscape files are presented with the associated signal names and joint architectures
- In section 4, the list of the Simulink library elements are described
- In section 5, a list of Matlab function that will be used are defined here
- In section 6, all the functions that are used to initialize the Simscape Multibody elements are defined here. This includes the mass of all solids for instance.

## 1 Simulink Project - Startup and Shutdown scripts

From the Simulink project mathworks page:

> Simulink® and Simulink Projects provide a collaborative, scalable environment that enables teams to manage their files and data in one place.

> With Simulink Projects, you can:

> - Collaborate: Enforce companywide standards such as company tools, libraries, and standard startup and shutdown scripts. Share your work with rich sharing options including MATLAB® toolboxes, email, and archives.
> - Automate: Set up your project environment correctly every time by automating steps such as loading the data, managing the path, and opening the models.
> - Integrate with source control: Enable easy integration with source control and configuration management tools.

The project can be opened using the `simulinkproject` function:

    simulinkproject('./');

When the project opens, a startup script is ran. The startup script is defined below and is exported to the `project_startup.m` script.

    project = simulinkproject;
    projectRoot = project.RootFolder;

    myCacheFolder = fullfile(projectRoot, '.SimulinkCache');
    myCodeFolder = fullfile(projectRoot, '.SimulinkCode');

    Simulink.fileGenControl('set',...
        'CacheFolder', myCacheFolder,...
        'CodeGenFolder', myCodeFolder,...
        'createDir', true);

When the project closes, it runs the `project_shutdown.m` script defined below.

    Simulink.fileGenControl('reset');

The project also permits to automatically add defined folder to the path when the project is opened.

## 2 Simscape Multibody - Presentation

A simscape model permits to model multi-physics systems.

Simscape Multibody permits to model multibody systems using blocks representing bodies, joints, constraints, force elements, and sensors.

### 2.1 Solid bodies

Each solid body is represented by a solid block. The geometry of the solid body can be imported using a `step` file. The properties of the solid body such as its mass, center of mass and moment of inertia can be derived from its density and its geometry as defined by the `step` file. They also can be set by hand.

### 2.2 Frames

Frames are very important in simscape multibody, they defined where the forces are applied, where the joints are located and where the measurements are made.

They can be defined from the solid body geometry, or using the rigid transform block.

### 2.3 Joints

Solid Bodies are connected with joints (between frames of the two solid bodies).

There are various types of joints that are all described here.

Table 1: Degrees of freedom associated with each joint
Joint Block | Translational DOFs | Rotational DOFs
--- | --- | ---
6-DOF | 3 | 3
Bearing | 1 | 3
Bushing | 3 | 3
Cartesian | 3 | 0
Constant Velocity | 0 | 2
Cylindrical | 1 | 1
Gimbal | 0 | 3
Leadscrew | 1 (coupled) | 1 (coupled)
Pin Slot | 1 | 1
Planar | 2 | 1
Prismatic | 1 | 0
Rectangular | 2 | 0
Revolute | 0 | 1
Spherical | 1 | 3
Telescoping | 1 | 3
Universal | 0 | 2
Weld | 0 | 0

Joint blocks are assortments of joint primitives:

- Prismatic: allows translation along a single standard axis: `Px`, `Py`, `Pz`
- Revolute: allows rotation about a single standard axis: `Rx`, `Ry`, `Rz`
- Spherical: allow rotation about any 3D axis: `S`
- Lead Screw: allows coupled rotation and translation on a standard axis: `LSz`
- Constant Velocity: Allows rotation at constant velocity between intersection through arbitrarily aligned shafts: `CV`

Table 2: Joint primitives for each joint type
Joint Block | Px | Py | Pz | Rx | Ry | Rz | S | CV | LSz
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
6-DOF | x | x | x |  |  |  | x |  |
Bearing |  |  | x | x | x | x |  |  |
Bushing | x | x | x | x | x | x |  |  |
Cartesian | x | x | x |  |  |  |  |  |
Constant Velocity |  |  |  |  |  |  |  | x |
Cylindrical |  |  | x |  |  | x |  |  |
Gimbal |  |  |  | x | x | x |  |  |
Leadscrew |  |  |  |  |  |  |  |  | x
Pin Slot | x |  |  |  |  | x |  |  |
Planar | x | x |  |  |  | x |  |  |
Prismatic |  |  | x |  |  |  |  |  |
Rectangular | x | x |  |  |  |  |  |  |
Revolute |  |  |  |  |  | x |  |  |
Spherical |  |  |  |  |  |  | x |  |
Telescoping |  |  | x |  |  |  | x |  |
Universal |  |  |  | x | x |  |  |  |
Weld |  |  |  |  |  |  |  |  |

For each of the joint primitive, we can specify the dynamical properties:

- The spring stiffness: either linear or rotational one
- The damping coefficient

For the actuation, we can either specify the motion or the force:

- the force applied in the corresponding DOF is provided by the input
- the motion is provided by the input

A sensor can be added to measure either the position, velocity or acceleration of the joint DOF.

Composite Force/Torque sensing:

- Constraint force
- Total force: gives the sum across all joint primitives over all sources: actuation inputs, internal springs and dampers.

### 2.4 Measurements

A transform sensor block measures the spatial relationship between two frames: the base `B` and the follower `F`.

It can give the rotational and translational position, velocity and acceleration.

The measurement frame should be specified: it corresponds to the frame in which to resolve the selected spatial measurements. The default is `world`.

If we want to simulate an inertial sensor, we just have to choose `B` to be the `world` frame.

Force sensors are included in the joints blocks.

### 2.5 Excitation

We can apply external forces to the model by using an external force and torque block.

Internal force, acting reciprocally between base and following origins is implemented using the internal force block even though it is usually included in one joint block.

## 3 Simulink files and signal names

In order to "normalize" things, the names of all the signal are listed here.

### 3.1 List of Simscape files

Few different Simulink files are used:

- kinematics
- identification - micro station
- identification - nano station
- control

Table 3: List of simscape files
Simscape Name | Ty | Ry | Rz | Hexa | NASS
--- | --- | --- | --- | --- | ---
id micro station | F | F | F | F |
id nano station stages | F | F | F | F | F
id nano station config | D | D | D | D | F
control nano station | D | D | D | D | F

### 3.2 List of Inputs

#### 3.2.1 Perturbations

Table 4: List of Disturbances
Variable | Meaning | Size | Unit
--- | --- | --- | ---
`Dw` | Ground motion | 3 | [m]
`Fg` | External force applied on granite | 3 | [N]
`Fs` | External force applied on the Sample | 3 | [N]

#### 3.2.2 Measurement Noise

Table 5: List of Measurement Noise
Variable | Meaning | Size | Unit
--- | --- | --- | ---

#### 3.2.3 Control Inputs

Table 6: List of Control Inputs
Variable | Meaning | Size | Unit
--- | --- | --- | ---
`Fy` | Actuation force for Ty | 1 | [N]
`Dy` | Imposed displacement for Ty | 1 | [m]
`My` | Actuation torque for Ry | 1 | [N.m]
`Ry` | Imposed rotation for Ry | 1 | [rad]
`Mz` | Actuation torque for Rz | 1 | [N.m]
`Rz` | Imposed rotation for Rz | 1 | [rad]
`Fh` | Actuation force/torque for hexapod (cart) | 6 | [N, N.m]
`Fhl` | Actuation force/torque for hexapod (legs) | 6 | [N]
`Dh` | Imposed position for hexapod (cart) | 6 | [m, rad]
`Rm` | Position of the two masses | 2 | [rad]
`Fn` | Actuation force for the NASS (cart) | 6 | [N, N.m]
`Fnl` | Actuation force for the NASS's legs | 6 | [N]
`Dn` | Imposed position for the NASS (cart) | 6 | [m, rad]

### 3.3 List of Outputs

Table 7: List of Outputs
Variable | Meaning | Size | Unit
--- | --- | --- | ---
`Dgm` | Absolute displacement of the granite | 3 | [m]
`Vgm` | Absolute Velocity of the granite | 3 | [m/s]
`Dym` | Measured displacement of Ty | 1 | [m]
`Rym` | Measured rotation of Ry | 1 | [rad]
`Rzm` | Measured rotation of Rz | 1 | [rad]
`Dhm` | Measured position of hexapod (cart) | 6 | [m, rad]
`Fnlm` | Measured force of NASS's legs | 6 | [N]
`Dnlm` | Measured elongation of NASS's legs | 6 | [m]
`Dnm` | Measured position of NASS w.r.t NASS's base | 6 | [m, rad]
`Vnm` | Measured absolute velocity of NASS platform | 6 | [m/s, rad/s]
`Vnlm` | Measured absolute velocity of NASS's legs | 6 | [m/s]
`Dsm` | Position of Sample w.r.t. granite frame | 6 | [m, rad]

## 4 Simulink Library

A simulink library is developed in order to share elements between the different simulink files.

### 4.1 `inputs`

inputs.slx

### 4.2 `nass_library`

nasslibrary.slx

### 4.3 `pos_error_wrt_nass_base`

poserrorwrtnassbase.slx

### 4.4 `QuaternionToAngles`

QuaternionToAngles.slx

### 4.5 `RotationMatrixToAngle`

RotationMatrixToAngle.slx

## 5 Functions

### 5.1 computePsdDispl

This Matlab function is accessible here.

    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

### 5.2 computeSetpoint

This Matlab function is accessible here.

    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

### 5.3 converErrorBasis

This Matlab function is accessible here.

    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

### 5.4 generateDiagPidControl

This Matlab function is accessible here.

    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

### 5.5 identifyPlant

This Matlab function is accessible here.

    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, '/Fd'],   1, 'input');  % Disturbance Forces
        io(6)  = linio([mdl, '/Dsm'],  1, 'output'); % Displacement of the sample
        io(7)  = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
        io(8)  = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
        io(9)  = linio([mdl, '/Es'],   1, 'output'); % Position Error w.r.t. NASS base
        io(10) = linio([mdl, '/Vlm'],  1, 'output'); % Measured absolute velocity of the legs

        %% Run the linearization
        G = linearize(mdl, io, options);
        G.InputName  = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz', ...
                        'Dgx', 'Dgy', 'Dgz', ...
                        'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz', ...
                        'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
                        'Frzz', 'Ftyx', 'Ftyz'};
        G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', ...
                        'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6', ...
                        'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ...
                        'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz', ...
                        'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};

        %% Create the sub transfer functions
        minreal_tol = sqrt(eps);
        % 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'}), minreal_tol, false);
        % From ground motion to Sample displacement
        sys.G_gm    = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Dgx', 'Dgy', 'Dgz'}), minreal_tol, false);
        % 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'}), minreal_tol, false);
        % 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'}), minreal_tol, false);
        % 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'}), minreal_tol, false);
        % From forces/torques applied by the NASS to position error
        sys.G_plant = minreal(G({'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'}), minreal_tol, false);
        % From forces/torques applied by the NASS to velocity of the actuator
        sys.G_geoph = minreal(G({'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), minreal_tol, false);
        % From various disturbance forces to position error
        sys.G_dist = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Frzz', 'Ftyx', 'Ftyz'}), minreal_tol, false);

        %% We remove low frequency and high frequency dynamics that are usually unstable
        % using =freqsep= is risky as it may change the shape of the transfer functions
        % f_min = 0.1; % [Hz]
        % f_max = 1e4; % [Hz]

        % [~, sys.G_cart]  = freqsep(freqsep(sys.G_cart,  2*pi*f_max), 2*pi*f_min);
        % [~, sys.G_gm]    = freqsep(freqsep(sys.G_gm,    2*pi*f_max), 2*pi*f_min);
        % [~, sys.G_fs]    = freqsep(freqsep(sys.G_fs,    2*pi*f_max), 2*pi*f_min);
        % [~, sys.G_iff]   = freqsep(freqsep(sys.G_iff,   2*pi*f_max), 2*pi*f_min);
        % [~, sys.G_dleg]  = freqsep(freqsep(sys.G_dleg,  2*pi*f_max), 2*pi*f_min);
        % [~, sys.G_plant] = freqsep(freqsep(sys.G_plant, 2*pi*f_max), 2*pi*f_min);

        %% We finally verify that the system is stable
        if ~isstable(sys.G_cart) || ~isstable(sys.G_gm) || ~isstable(sys.G_fs) || ~isstable(sys.G_iff) || ~isstable(sys.G_dleg) || ~isstable(sys.G_plant)
          warning('One of the identified system is unstable');
        end
    end

### 5.6 runSimulation

This Matlab function is accessible here.

    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

## 6 Initialize Elements

### 6.1 Experiment

This Matlab function is accessible here.

    function [] = initializeExperiment(exp_name, sys_mass)
        if strcmp(exp_name, 'tomography')
            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

### 6.2 Generate Reference Signals

This Matlab function is accessible here.

    function [ref] = initializeReferences(opts_param)
        %% Default values for opts
        opts = struct(   ...
            'Ts',           1e-3, ...       % Sampling Frequency [s]
            'Dy_type',      'constant', ... % Either "constant" / "triangular" / "sinusoidal"
            'Dy_amplitude', 0, ...       % Amplitude of the displacement [m]
            'Dy_period',    1, ...          % Period of the displacement [s]
            'Ry_type',      'constant', ... % Either "constant" / "triangular" / "sinusoidal"
            'Ry_amplitude', 0, ...          % Amplitude [rad]
            'Ry_period',    10, ...         % Period of the displacement [s]
            'Rz_type',      'constant', ... % Either "constant" / "rotating"
            'Rz_amplitude', 0, ...          % Initial angle [rad]
            'Rz_period',    1, ...          % Period of the rotating [s]
            'Dh_type',      'constant', ... % For now, only constant is implemented
            'Dh_pos',       [0; 0; 0; 0; 0; 0], ...  % Initial position [m,m,m,rad,rad,rad] of the top platform
            'Rm_type',      'constant', ... % For now, only constant is implemented
            'Rm_pos',       [0; pi],  ...   % Initial position of the two masses
            'Dn_type',      'constant', ... % For now, only constant is implemented
            'Dn_pos',       [0; 0; 0; 0; 0; 0]  ...  % Initial position [m,m,m,rad,rad,rad] of the top platform
        );

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

        %% Set Sampling Time
        Ts = opts.Ts;

        %% Translation stage - Dy
        t = 0:Ts:opts.Dy_period-Ts; % Time Vector [s]
        Dy = zeros(length(t), 1);
        switch opts.Dy_type
          case 'constant'
            Dy(:) = opts.Dy_amplitude;
          case 'triangular'
            Dy(:)                     = -4*opts.Dy_amplitude + 4*opts.Dy_amplitude/opts.Dy_period*t;
            Dy(t<0.75*opts.Dy_period) =  2*opts.Dy_amplitude - 4*opts.Dy_amplitude/opts.Dy_period*t(t<0.75*opts.Dy_period);
            Dy(t<0.25*opts.Dy_period) =  4*opts.Dy_amplitude/opts.Dy_period*t(t<0.25*opts.Dy_period);
          case 'sinusoidal'
            Dy(:) = opts.Dy_amplitude*sin(2*pi/opts.Dy_period*t);
          otherwise
            warning('Dy_type is not set correctly');
        end
        Dy = struct('time', t, 'signals', struct('values', Dy));

        %% Tilt Stage - Ry
        t = 0:Ts:opts.Ry_period-Ts;
        Ry = zeros(length(t), 1);

        switch opts.Ry_type
          case 'constant'
            Ry(:) = opts.Ry_amplitude;
          case 'triangular'
            Ry(:)                     = -4*opts.Ry_amplitude + 4*opts.Ry_amplitude/opts.Ry_period*t;
            Ry(t<0.75*opts.Ry_period) =  2*opts.Ry_amplitude - 4*opts.Ry_amplitude/opts.Ry_period*t(t<0.75*opts.Ry_period);
            Ry(t<0.25*opts.Ry_period) =  4*opts.Ry_amplitude/opts.Ry_period*t(t<0.25*opts.Ry_period);
          case 'sinusoidal'

          otherwise
            warning('Ry_type is not set correctly');
        end
        Ry = struct('time', t, 'signals', struct('values', Ry));

        %% Spindle - Rz
        t = 0:Ts:opts.Rz_period-Ts;
        Rz = zeros(length(t), 1);

        switch opts.Rz_type
          case 'constant'
            Rz(:) = opts.Rz_amplitude;
          case 'rotating'
            Rz(:) = opts.Rz_amplitude+2*pi/opts.Rz_period*t;
          otherwise
            warning('Rz_type is not set correctly');
        end
        Rz = struct('time', t, 'signals', struct('values', Rz));

        %% Micro-Hexapod
        t = [0, Ts];
        Dh = zeros(length(t), 6);

        switch opts.Dh_type
          case 'constant'
            Dh = [opts.Dh_pos, opts.Dh_pos];
          otherwise
            warning('Dh_type is not set correctly');
        end
        Dh = struct('time', t, 'signals', struct('values', Dh));

        %% Axis Compensation - Rm
        t = [0, Ts];
        Rm = [opts.Rm_pos, opts.Rm_pos];
        Rm = struct('time', t, 'signals', struct('values', Rm));

        %% Nano-Hexapod
        t = [0, Ts];
        Dn = zeros(length(t), 6);

        switch opts.Dn_type
          case 'constant'
            Dn = [opts.Dn_pos, opts.Dn_pos];
          otherwise
            warning('Dn_type is not set correctly');
        end
        Dn = struct('time', t, 'signals', struct('values', Dn));

        %% Save
        save('./mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Rm', 'Dn', 'Ts');
    end

### 6.3 TODO Inputs

- `[ ]` This function should not be used anymore. Now there are two functions to initialize disturbances and references.

This Matlab function is accessible here.

    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

### 6.4 Ground

This Matlab function is accessible here.

    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

### 6.5 Granite

This Matlab function is accessible here.

    function [granite] = initializeGranite(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

        %%
        granite = struct();

        %% Static Properties
        granite.density = 2800; % [kg/m3]
        granite.volume  = 0.749; % [m3] TODO - should
        granite.mass    = granite.density*granite.volume; % [kg]
        granite.color   = [1 1 1];
        granite.STEP    = './STEPS/granite/granite.STEP';

        granite.mass_top = 4000; % [kg] TODO

        %% Dynamical Properties
        if opts.rigid
          granite.k.x = 1e12; % [N/m]
          granite.k.y = 1e12; % [N/m]
          granite.k.z = 1e12; % [N/m]
          granite.k.rx = 1e10; % [N*m/deg]
          granite.k.ry = 1e10; % [N*m/deg]
          granite.k.rz = 1e10; % [N*m/deg]
        else
          granite.k.x = 4e9; % [N/m]
          granite.k.y = 3e8; % [N/m]
          granite.k.z = 8e8; % [N/m]
          granite.k.rx = 1e4; % [N*m/deg]
          granite.k.ry = 1e4; % [N*m/deg]
          granite.k.rz = 1e6; % [N*m/deg]
        end

        granite.c.x = 0.1*sqrt(granite.mass_top*granite.k.x); % [N/(m/s)]
        granite.c.y = 0.1*sqrt(granite.mass_top*granite.k.y); % [N/(m/s)]
        granite.c.z = 0.5*sqrt(granite.mass_top*granite.k.z); % [N/(m/s)]
        granite.c.rx = 0.1*sqrt(granite.mass_top*granite.k.rx); % [N*m/(deg/s)]
        granite.c.ry = 0.1*sqrt(granite.mass_top*granite.k.ry); % [N*m/(deg/s)]
        granite.c.rz = 0.1*sqrt(granite.mass_top*granite.k.rz); % [N*m/(deg/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

### 6.6 Translation Stage

This Matlab function is accessible here.

    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 = 1000; % TODO [kg]

        %% Y-Translation - Dynamicals Properties
        if opts.rigid
          ty.k.ax  = 1e12; % Axial Stiffness for each of the 4 guidance (y) [N/m]
          ty.k.rad = 1e12; % Radial Stiffness for each of the 4 guidance (x-z) [N/m]
        else
          ty.k.ax  = 5e8; % Axial Stiffness for each of the 4 guidance (y) [N/m]
          ty.k.rad = 5e7; % Radial Stiffness for each of the 4 guidance (x-z) [N/m]
        end

        ty.c.ax  = 0.1*sqrt(ty.k.ax*ty.m);
        ty.c.rad = 0.1*sqrt(ty.k.rad*ty.m);

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

### 6.7 Tilt Stage

This Matlab function is accessible here.

    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 = 800; % TODO [kg]

        %% Tilt Stage - Dynamical Properties
        if opts.rigid
          ry.k.tilt = 1e10; % Rotation stiffness around y [N*m/deg]
          ry.k.h    = 1e12; % Stiffness in the direction of the guidance [N/m]
          ry.k.rad  = 1e12; % Stiffness in the top direction [N/m]
          ry.k.rrad = 1e12; % Stiffness in the side direction [N/m]
        else
          ry.k.tilt = 1e4; % Rotation stiffness around y [N*m/deg]
          ry.k.h    = 1e8; % Stiffness in the direction of the guidance [N/m]
          ry.k.rad  = 1e8; % Stiffness in the top direction [N/m]
          ry.k.rrad = 1e8; % Stiffness in the side direction [N/m]
        end

        ry.c.h    = 0.1*sqrt(ry.k.h*ry.m);
        ry.c.rad  = 0.1*sqrt(ry.k.rad*ry.m);
        ry.c.rrad = 0.1*sqrt(ry.k.rrad*ry.m);
        ry.c.tilt = 0.1*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

### 6.8 Spindle

This Matlab function is accessible here.

    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

        if opts.rigid
          rz.k.rot  = 1e10; % Rotational Stiffness (Rz) [N*m/deg]
          rz.k.tilt = 1e10; % Rotational Stiffness (Rx, Ry) [N*m/deg]
          rz.k.ax   = 1e12; % Axial Stiffness (Z) [N/m]
          rz.k.rad  = 1e12; % Radial Stiffness (X, Y) [N/m]
        else
          rz.k.rot  = 1e6; % TODO - Rotational Stiffness (Rz) [N*m/deg]
          rz.k.tilt = 1e6; % Rotational Stiffness (Rx, Ry) [N*m/deg]
          rz.k.ax   = 2e9; % Axial Stiffness (Z) [N/m]
          rz.k.rad  = 7e8; % Radial Stiffness (X, Y) [N/m]
        end

        % Damping
        rz.c.ax   = 0.1*sqrt(rz.k.ax*rz.m);
        rz.c.rad  = 0.1*sqrt(rz.k.rad*rz.m);
        rz.c.tilt = 0.1*sqrt(rz.k.tilt*rz.m);
        rz.c.rot  = 0.1*sqrt(rz.k.rot*rz.m);

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

### 6.9 Initialize Hexapod legs' length

This Matlab function is accessible here.

    function [hexapod] = initializeHexapodPosition(hexapod, AP, ARB)
    % initializeHexapodPosition -
    %
    % 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}
    %    - ARB - Rotation Matrix expressed in frame {A}

      L0 = zeros(6, 1);

      for i = 1:length(L)
        Bbi = hexapod.pos_top_tranform(i, :)';
        Aai = hexapod.pos_base(i, :)';

        L0(i) = sqrt(AP'*AP + Bbi'*Bbi + Aai'*Aai - 2*AP'*Aai + 2*AP'*(ARB*Bbi) - 2*(ARB*Bbi)'*Aai);
      end

      hexapod.L0 = L0;
    end

### 6.10 Micro Hexapod

This Matlab function is accessible 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

### 6.11 Center of gravity compensation

This Matlab function is accessible here.

    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

### 6.12 Mirror

This Matlab function is accessible here.

    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

### 6.13 Nano Hexapod

This Matlab function is accessible here.

    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         = 0;

        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)
                if element.ksi.(field{i}) > 0
                  element.c.(field{i}) = 1/element.ksi.(field{i})*sqrt(element.k.(field{i})/element.m);
                else
                  element.c.(field{i}) = 0;
                end
            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

### 6.14 Cedrat Actuator

This Matlab function is accessible here.

    function [cedrat] = initializeCedratPiezo(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
      cedrat = struct();
      cedrat.k = 10e7; % Linear Stiffness of each "blade" [N/m]
      cedrat.ka = 10e7; % Linear Stiffness of the stack [N/m]

      cedrat.c = 0.1*sqrt(1*cedrat.k); % [N/(m/s)]
      cedrat.ca = 0.1*sqrt(1*cedrat.ka); % [N/(m/s)]

      cedrat.L = 80; % Total Width of the Actuator[mm]
      cedrat.H = 45; % Total Height of the Actuator [mm]
      cedrat.L2 = sqrt((cedrat.L/2)^2 + (cedrat.H/2)^2); % Length of the elipsoidal sections [mm]
      cedrat.alpha = 180/pi*atan2(cedrat.L/2, cedrat.H/2); % [deg]

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

### 6.15 Sample

This Matlab function is accessible here.

    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 = 0.1*sqrt(sample.k.x*sample.mass);

        sample.k.y = 1e8;
        sample.c.y = 0.1*sqrt(sample.k.y*sample.mass);

        sample.k.z = 1e8;
        sample.c.z = 0.1*sqrt(sample.k.z*sample.mass);

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

Author: Dehaeze Thomas

Created: 2019-12-10 mar. 18:05

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