Identification of the Stewart Platform using Simscape

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

1 Modal Analysis of the Stewart Platform

1.1 Initialize the Stewart Platform

Copystewart = 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);

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

1.2 Identification

Copy
%% 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:

Copysize(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).

CopyGm = 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.

CopyGt = 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 $Y$ .

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

We could perform the transformation by hand:

Copy
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

Copy[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 $ξ_{i}$ (the mode we want to excite).

$U (t) = e^{α t} ()$

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

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

Copy
for i = 1:length(ws)

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

Copylambda_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.

Copy
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)));

CopyU = timeseries(U_i, t);

Simulation:

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

Save the movie of the mode shape.

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

Copy
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

Copystewart = 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 ${B}$ .

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

2.2 Transmissibility

Copy
%% 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'};

Copyfreqs = 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{aligned} ‖ T (ω) ‖ & = \sqrt{Trace [T (ω) T (ω)^{H}]} \\ = \sqrt{Σ_{i = 1}^{6} Σ_{j = 1}^{6} | T_{i j} |^{2}} \end{aligned}

Copyfreqs = 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: $Γ (ω) = ‖ T (ω) ‖ / \sqrt{6}$

Copy
Gamma = T_norm/sqrt(6);

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

3 Compliance Analysis

3.1 Initialize the Stewart platform

Copystewart = 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 ${B}$ .

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

3.2 Compliance

Copy
%% 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'};

Copyfreqs = 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.

Copyfreqs = 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

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

4 Functions

4.1 Compute the Transmissibility

Function description

Copy
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

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

Copyfreqs = args.freqs;

Identification of the Transmissibility Matrix

Copy
%% 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.

Copy
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

CopyT_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

Copy
T_norm = T_norm/sqrt(6);

Copy
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

Copy
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

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

Copyfreqs = args.freqs;

Identification of the Compliance Matrix

Copy
%% 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.

Copy
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

Copyfreqs = 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

Copy
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.