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# Identification of the Stewart Platform using Simscape

## Table of Contents

- 1. Identification
  - 1.1. Simscape Model
  - 1.2. Initialize the Stewart Platform
  - 1.3. Identification
- 2. States as the motion of the mobile platform
  - 2.1. Initialize the Stewart Platform
  - 2.2. Identification
  - 2.3. Coordinate transformation
  - 2.4. Analysis
  - 2.5. Visualizing the modes
  - 2.6. Identification
  - 2.7. Change of states
- 3. Simple Model without any sensor
  - 3.1. Simscape Model
  - 3.2. Initialize the Stewart Platform
  - 3.3. Identification
- 4. Cartesian Plot
- 5. From a force to force sensor
- 6. From a force applied in the leg to the displacement of the leg
- 7. Transmissibility
- 8. Compliance
- 9. Inertial

We would like to extract a state space model of the Stewart Platform from the Simscape model.

The inputs are:

Symbol | Meaning
--- | ---
\(\bm{\mathcal{F}}_{d}\) | External forces applied in {B}
\(\bm{\tau}\) | Joint forces
\(\bm{\mathcal{F}}\) | Cartesian forces applied by the Joints
\(\bm{D}_{w}\) | Fixed Based translation and rotations around {A}

The outputs are:

Symbol | Meaning
--- | ---
\(\bm{\mathcal{X}}\) | Relative Motion of {B} with respect to {A}
\(\bm{\mathcal{L}}\) | Joint Displacement
\(\bm{F}_{m}\) | Force Sensors in each strut
\(\bm{v}_{m}\) | Inertial Sensors located at \(b_i\) measuring in the direction of the strut

> An important difference from basic Simulink models is that the states in a physical network are not independent in general, because some states have dependencies on other states through constraints.

## 1 Identification

### 1.1 Simscape Model

### 1.2 Initialize the Stewart Platform

    stewart = initializeFramesPositions();
    stewart = generateGeneralConfiguration(stewart);
    stewart = computeJointsPose(stewart);
    stewart = initializeStrutDynamics(stewart);
    stewart = initializeCylindricalPlatforms(stewart);
    stewart = initializeCylindricalStruts(stewart);
    stewart = computeJacobian(stewart);
    stewart = initializeStewartPose(stewart);

### 1.3 Identification

    %% Options for Linearized
    options = linearizeOptions;
    options.SampleTime = 0;

    %% Name of the Simulink File
    mdl = 'stewart_platform_identification';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/tau'],  1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Fext'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/X'],    1, 'openoutput'); io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Vm'],   1, 'openoutput'); io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Taum'], 1, 'openoutput'); io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Lm'],   1, 'openoutput'); io_i = io_i + 1;

    %% Run the linearization
    G = linearize(mdl, io, options);
    G.InputName  = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6', ...
                    'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

    G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', ...
                    'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6', ...
                    'taum1', 'taum2', 'taum3', 'taum4', 'taum5', 'taum6', ...
                    'Lm1', 'Lm2', 'Lm3', 'Lm4', 'Lm5', 'Lm6'};

## 2 States as the motion of the mobile platform

### 2.1 Initialize the Stewart Platform

    stewart = initializeFramesPositions();
    stewart = generateGeneralConfiguration(stewart);
    stewart = computeJointsPose(stewart);
    stewart = initializeStrutDynamics(stewart);
    stewart = initializeCylindricalPlatforms(stewart);
    stewart = initializeCylindricalStruts(stewart);
    stewart = computeJacobian(stewart);
    stewart = initializeStewartPose(stewart);

### 2.2 Identification

    %% Options for Linearized
    options = linearizeOptions;
    options.SampleTime = 0;

    %% Name of the Simulink File
    mdl = 'stewart_platform_identification_simple';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/tau'],  1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/X'],    1, 'openoutput'); io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Xdot'], 1, 'openoutput'); io_i = io_i + 1;

    %% Run the linearization
    G = linearize(mdl, io);
    % G.InputName  = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
    % G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};

Let’s check the size of `G`:

    size(G)

    size(G)
    State-space model with 12 outputs, 6 inputs, and 18 states.
    'org_babel_eoe'
    ans =
        'org_babel_eoe'

We expect to have only 12 states (corresponding to the 6dof of the mobile platform).

    Gm = minreal(G);

    Gm = minreal(G);
    6 states removed.

And indeed, we obtain 12 states.

### 2.3 Coordinate transformation

We can perform the following transformation using the `ss2ss` command.

    Gt = ss2ss(Gm, Gm.C);

Then, the `C` matrix of `Gt` is the unity matrix which means that the states of the state space model are equal to the measurements \(\bm{Y}\).

The measurements are the 6 displacement and 6 velocities of mobile platform with respect to \(\{B\}\).

We could perform the transformation by hand:

    At = Gm.C*Gm.A*pinv(Gm.C);

    Bt = Gm.C*Gm.B;

    Ct = eye(12);
    Dt = zeros(12, 6);

    Gt = ss(At, Bt, Ct, Dt);

### 2.4 Analysis

    [V,D] = eig(Gt.A);

Mode Number | Resonance Frequency [Hz] | Damping Ratio [%]
--- | --- | ---
1.0 | 174.5 | 0.9
2.0 | 174.5 | 0.7
3.0 | 202.1 | 0.7
4.0 | 237.3 | 0.6
5.0 | 237.3 | 0.5
6.0 | 283.8 | 0.5

### 2.5 Visualizing the modes

To visualize the i’th mode, we may excite the system using the inputs \(U_i\) such that \(B U_i\) is co-linear to \(\xi_i\) (the mode we want to excite).

\[ U(t) = e^{\alpha t} ( ) \]

Let’s first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.

    ws = imag(diag(D));
    [ws,I] = sort(ws)
    ws = ws(7:end); I = I(7:end);

    for i = 1:length(ws)

    i_mode = I(i); % the argument is the i'th mode

    lambda_i = D(i_mode, i_mode);
    xi_i = V(:,i_mode);

    a_i = real(lambda_i);
    b_i = imag(lambda_i);

Let do 10 periods of the mode.

    t = linspace(0, 10/(imag(lambda_i)/2/pi), 1000);
    U_i = pinv(Gt.B) * real(xi_i * lambda_i * (cos(b_i * t) + 1i*sin(b_i * t)));

    U = timeseries(U_i, t);

Simulation:

    load('mat/conf_simscape.mat');
    set_param(conf_simscape, 'StopTime', num2str(t(end)));
    sim(mdl);

Save the movie of the mode shape.

    smwritevideo(mdl, sprintf('figs/mode%i', i), ...
                 'PlaybackSpeedRatio', 1/(b_i/2/pi), ...
                 'FrameRate', 30, ...
                 'FrameSize', [800, 400]);

    end

Figure 1: Identified mode - 1

Figure 2: Identified mode - 3

Figure 3: Identified mode - 5

### 2.6 Identification

    %% Options for Linearized
    options = linearizeOptions;
    options.SampleTime = 0;

    %% Name of the Simulink File
    mdl = 'stewart_platform_identification';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/tau'],  1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Lm'],    1, 'openoutput'); io_i = io_i + 1;

    %% Run the linearization
    G = linearize(mdl, io, options);
    % G.InputName  = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
    % G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};

    size(G)

### 2.7 Change of states

    At = G.C*G.A*pinv(G.C);

    Bt = G.C*G.B;

    Ct = eye(12);
    Dt = zeros(12, 6);

    Gt = ss(At, Bt, Ct, Dt);

    size(Gt)

## 3 Simple Model without any sensor

### 3.1 Simscape Model

    open 'stewart_identification_simple.slx'

### 3.2 Initialize the Stewart Platform

    stewart = initializeFramesPositions();
    stewart = generateGeneralConfiguration(stewart);
    stewart = computeJointsPose(stewart);
    stewart = initializeStrutDynamics(stewart);
    stewart = initializeCylindricalPlatforms(stewart);
    stewart = initializeCylindricalStruts(stewart);
    stewart = computeJacobian(stewart);
    stewart = initializeStewartPose(stewart);

### 3.3 Identification

    stateorder = {...
        'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_1_1_1',...
        'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_2_1_1',...
        'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_3_1_1',...
        'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_4_1_1',...
        'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_5_1_1',...
        'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_6_1_1',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Rz.q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Rz.q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Rz.q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Rz.q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Rz.q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Rz.q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Pz.p',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Pz.p',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Pz.p',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Pz.p',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Pz.p',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Pz.p',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Rz.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Rz.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Rz.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Rz.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Rz.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Rz.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Pz.v',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Pz.v',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Pz.v',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Pz.v',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Pz.v',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Pz.v',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_F.S.Q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.spherical_joint_F.S.Q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.spherical_joint_F.S.Q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.spherical_joint_F.S.Q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.spherical_joint_F.S.Q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.spherical_joint_F.S.Q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.spherical_joint_F.S.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.spherical_joint_F.S.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.spherical_joint_F.S.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.spherical_joint_F.S.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.spherical_joint_F.S.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_F.S.w',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_M.S.Q',...
        'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_M.S.w'};

    %% Options for Linearized
    options = linearizeOptions;
    options.SampleTime = 0;

    %% Name of the Simulink File
    mdl = 'stewart_platform_identification_simple';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/tau'],  1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/X'],     1, 'openoutput'); io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Xdot'],  1, 'openoutput'); io_i = io_i + 1;

    %% Run the linearization
    G = linearize(mdl, io, options);
    G.InputName  = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};

    G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};

    size(G)

    G.StateName

## 4 Cartesian Plot

From a force applied in the Cartesian frame to a displacement in the Cartesian frame.

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_cart(1, 1), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_cart(2, 1), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_cart(3, 1), freqs, 'Hz'))));
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude');

    figure;
    bode(G.G_cart, freqs);

## 5 From a force to force sensor

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
    legend('location', 'southeast');

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
    plot(freqs, abs(squeeze(freqresp(G.G_forc(2, 1), freqs, 'Hz'))), 'k--', 'DisplayName', '$F_{m_j}/F_{i}$');
    plot(freqs, abs(squeeze(freqresp(G.G_forc(3, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    plot(freqs, abs(squeeze(freqresp(G.G_forc(4, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    plot(freqs, abs(squeeze(freqresp(G.G_forc(5, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    plot(freqs, abs(squeeze(freqresp(G.G_forc(6, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
    legend('location', 'southeast');

## 6 From a force applied in the leg to the displacement of the leg

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$D_{i}/F_{i}$');
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$D_{i}/F_{i}$');
    plot(freqs, abs(squeeze(freqresp(G.G_legs(2, 1), freqs, 'Hz'))), 'k--', 'DisplayName', '$D_{j}/F_{i}$');
    plot(freqs, abs(squeeze(freqresp(G.G_legs(3, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    plot(freqs, abs(squeeze(freqresp(G.G_legs(4, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    plot(freqs, abs(squeeze(freqresp(G.G_legs(5, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    plot(freqs, abs(squeeze(freqresp(G.G_legs(6, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
    legend('location', 'northeast');

## 7 Transmissibility

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_tran(2, 2), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_tran(3, 3), freqs, 'Hz'))));
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [m/m]');

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_tran(4, 4), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_tran(5, 5), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_tran(6, 6), freqs, 'Hz'))));
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [$\frac{rad/s}{rad/s}$]');

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 2), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 3), freqs, 'Hz'))));
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [m/m]');

## 8 Compliance

From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_comp(1, 1), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_comp(2, 2), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_comp(3, 3), freqs, 'Hz'))));
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');

## 9 Inertial

From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.

    figure;
    hold on;
    plot(freqs, abs(squeeze(freqresp(G.G_iner(1, 1), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_iner(2, 2), freqs, 'Hz'))));
    plot(freqs, abs(squeeze(freqresp(G.G_iner(3, 3), freqs, 'Hz'))));
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');

Author: Dehaeze Thomas

Created: 2020-01-29 mer. 17:51