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Identification of the Stewart Platform using Simscape

Table of Contents

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 = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
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 = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
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

mode1.gif

Figure 1: Identified mode - 1

mode3.gif

Figure 2: Identified mode - 3

mode5.gif

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 = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
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: 2020-02-11 mar. 15:50