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# Identification of the Stewart Platform using Simscape

## Table of Contents

- 1. Modal Analysis of the Stewart Platform
  - 1.1. Initialize the Stewart Platform
  - 1.2. Identification
  - 1.3. Coordinate transformation
  - 1.4. Analysis
  - 1.5. Visualizing the modes
- 2. Transmissibility Analysis
  - 2.1. Initialize the Stewart platform
  - 2.2. Transmissibility
- 3. Compliance Analysis
  - 3.1. Initialize the Stewart platform
  - 3.2. Compliance
- 4. Functions
  - 4.1. Compute the Transmissibility
    - Function description
    - Optional Parameters
    - Identification of the Transmissibility Matrix
    - Computation of the Frobenius norm
  - 4.2. Compute the Compliance
    - Function description
    - Optional Parameters
    - Identification of the Compliance Matrix
    - Computation of the Frobenius norm

In this document, we discuss the various methods to identify the behavior of the Stewart platform.

- 1
- 2
- 3

## 1 Modal Analysis of the Stewart Platform

### 1.1 Initialize the Stewart Platform

    stewart = initializeStewartPlatform();
    stewart = initializeFramesPositions(stewart);
    stewart = generateGeneralConfiguration(stewart);
    stewart = computeJointsPose(stewart);
    stewart = initializeStrutDynamics(stewart);
    stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
    stewart = initializeCylindricalPlatforms(stewart);
    stewart = initializeCylindricalStruts(stewart);
    stewart = computeJacobian(stewart);
    stewart = initializeStewartPose(stewart);
    stewart = initializeInertialSensor(stewart);

    ground = initializeGround('type', 'none');
    payload = initializePayload('type', 'none');
    controller = initializeController('type', 'open-loop');

### 1.2 Identification

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

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

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Controller'],              1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
    io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
    io(io_i) = linio([mdl, '/Relative Motion Sensor'],  2, 'openoutput'); io_i = io_i + 1; % Velocity of {B} w.r.t. {A}

    %% 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.

### 1.3 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);

### 1.4 Analysis

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

Mode Number | Resonance Frequency [Hz] | Damping Ratio [%]
--- | --- | ---
1.0 | 780.6 | 0.4
2.0 | 780.6 | 0.3
3.0 | 903.9 | 0.3
4.0 | 1061.4 | 0.3
5.0 | 1061.4 | 0.2
6.0 | 1269.6 | 0.2

### 1.5 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 Transmissibility Analysis

### 2.1 Initialize the Stewart platform

    stewart = initializeStewartPlatform();
    stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
    stewart = generateGeneralConfiguration(stewart);
    stewart = computeJointsPose(stewart);
    stewart = initializeStrutDynamics(stewart);
    stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
    stewart = initializeCylindricalPlatforms(stewart);
    stewart = initializeCylindricalStruts(stewart);
    stewart = computeJacobian(stewart);
    stewart = initializeStewartPose(stewart);
    stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);

We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).

    ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
    payload = initializePayload('type', 'rigid');
    controller = initializeController('type', 'open-loop');

### 2.2 Transmissibility

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

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

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Disturbances/D_w'],        1, 'openinput');  io_i = io_i + 1; % Base Motion [m, rad]
    io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]

    %% Run the linearization
    T = linearize(mdl, io, options);
    T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
    T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

    freqs = logspace(1, 4, 1000);

    figure;
    for ix = 1:6
        for iy = 1:6
            subplot(6, 6, (ix-1)*6 + iy);
            hold on;
            plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
            set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
            ylim([1e-5, 10]);
            xlim([freqs(1), freqs(end)]);
            if ix < 6
                xticklabels({});
            end
            if iy > 1
                yticklabels({});
            end
        end
    end

From (Preumont et al. 2007), one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:

\begin{align*} \| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\ &= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2} \end{align*}

    freqs = logspace(1, 4, 1000);

    T_norm = zeros(length(freqs), 1);

    for i = 1:length(freqs)
        T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
    end

And we normalize by a factor \(\sqrt{6}\) to obtain a performance metric comparable to the transmissibility of a one-axis isolator: \[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]

    Gamma = T_norm/sqrt(6);

    figure;
    plot(freqs, Gamma)
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');

## 3 Compliance Analysis

### 3.1 Initialize the Stewart platform

    stewart = initializeStewartPlatform();
    stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
    stewart = generateGeneralConfiguration(stewart);
    stewart = computeJointsPose(stewart);
    stewart = initializeStrutDynamics(stewart);
    stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
    stewart = initializeCylindricalPlatforms(stewart);
    stewart = initializeCylindricalStruts(stewart);
    stewart = computeJacobian(stewart);
    stewart = initializeStewartPose(stewart);
    stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);

We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).

    ground = initializeGround('type', 'none');
    payload = initializePayload('type', 'rigid');
    controller = initializeController('type', 'open-loop');

### 3.2 Compliance

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

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

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Disturbances/F_ext'],        1, 'openinput');  io_i = io_i + 1; % Base Motion [m, rad]
    io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]

    %% Run the linearization
    C = linearize(mdl, io, options);
    C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
    C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

    freqs = logspace(1, 4, 1000);

    figure;
    for ix = 1:6
        for iy = 1:6
            subplot(6, 6, (ix-1)*6 + iy);
            hold on;
            plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
            set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
            ylim([1e-10, 1e-3]);
            xlim([freqs(1), freqs(end)]);
            if ix < 6
                xticklabels({});
            end
            if iy > 1
                yticklabels({});
            end
        end
    end

We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.

    freqs = logspace(1, 4, 1000);

    C_norm = zeros(length(freqs), 1);

    for i = 1:length(freqs)
        C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
    end

    figure;
    plot(freqs, C_norm)
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');

## 4 Functions

### 4.1 Compute the Transmissibility

#### Function description

    function [T, T_norm, freqs] = computeTransmissibility(args)
    % computeTransmissibility -
    %
    % Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
    %
    % Inputs:
    %    - args - Structure with the following fields:
    %        - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
    %        - freqs [] - Frequency vector to estimate the Frobenius norm
    %
    % Outputs:
    %    - T      [6x6 ss] - Transmissibility matrix
    %    - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
    %    - freqs  [length(freqs)x1] - Frequency vector in [Hz]

#### Optional Parameters

    arguments
        args.plots logical {mustBeNumericOrLogical} = false
        args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
    end

    freqs = args.freqs;

#### Identification of the Transmissibility Matrix

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

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

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Disturbances/D_w'],        1, 'openinput');  io_i = io_i + 1; % Base Motion [m, rad]
    io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]

    %% Run the linearization
    T = linearize(mdl, io, options);
    T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
    T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

If wanted, the 6x6 transmissibility matrix is plotted.

    p_handle = zeros(6*6,1);

    if args.plots
        fig = figure;
        for ix = 1:6
            for iy = 1:6
                p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
                hold on;
                plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
                set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
                if ix < 6
                    xticklabels({});
                end
                if iy > 1
                    yticklabels({});
                end
            end
        end

        linkaxes(p_handle, 'xy')
        xlim([freqs(1), freqs(end)]);
        ylim([1e-5, 1e2]);

        han = axes(fig, 'visible', 'off');
        han.XLabel.Visible = 'on';
        han.YLabel.Visible = 'on';
        xlabel(han, 'Frequency [Hz]');
        ylabel(han, 'Transmissibility [m/m]');
    end

#### Computation of the Frobenius norm

    T_norm = zeros(length(freqs), 1);

    for i = 1:length(freqs)
        T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
    end

    T_norm = T_norm/sqrt(6);

    if args.plots
        figure;
        plot(freqs, T_norm)
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
        xlabel('Frequency [Hz]');
        ylabel('Transmissibility - Frobenius Norm');
    end

### 4.2 Compute the Compliance

#### Function description

    function [C, C_norm, freqs] = computeCompliance(args)
    % computeCompliance -
    %
    % Syntax: [C, C_norm, freqs] = computeCompliance(args)
    %
    % Inputs:
    %    - args - Structure with the following fields:
    %        - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
    %        - freqs [] - Frequency vector to estimate the Frobenius norm
    %
    % Outputs:
    %    - C      [6x6 ss] - Compliance matrix
    %    - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
    %    - freqs  [length(freqs)x1] - Frequency vector in [Hz]

#### Optional Parameters

    arguments
        args.plots logical {mustBeNumericOrLogical} = false
        args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
    end

    freqs = args.freqs;

#### Identification of the Compliance Matrix

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

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

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Disturbances/F_ext'],      1, 'openinput');  io_i = io_i + 1; % External forces [N, N*m]
    io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]

    %% Run the linearization
    C = linearize(mdl, io, options);
    C.InputName  = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
    C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

If wanted, the 6x6 transmissibility matrix is plotted.

    p_handle = zeros(6*6,1);

    if args.plots
        fig = figure;
        for ix = 1:6
            for iy = 1:6
                p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
                hold on;
                plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
                set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
                if ix < 6
                    xticklabels({});
                end
                if iy > 1
                    yticklabels({});
                end
            end
        end

        linkaxes(p_handle, 'xy')
        xlim([freqs(1), freqs(end)]);

        han = axes(fig, 'visible', 'off');
        han.XLabel.Visible = 'on';
        han.YLabel.Visible = 'on';
        xlabel(han, 'Frequency [Hz]');
        ylabel(han, 'Compliance [m/N, rad/(N*m)]');
    end

#### Computation of the Frobenius norm

    freqs = args.freqs;

    C_norm = zeros(length(freqs), 1);

    for i = 1:length(freqs)
        C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
    end

    if args.plots
        figure;
        plot(freqs, C_norm)
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
        xlabel('Frequency [Hz]');
        ylabel('Compliance - Frobenius Norm');
    end

## Bibliography

Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” Journal of Sound and Vibration 300 (3-5):644–61. https://doi.org/10.1016/j.jsv.2006.07.050.

Author: Dehaeze Thomas

Created: 2021-01-08 ven. 15:52