21 KiB
21 KiB
Identification of the Stewart Platform using Simscape
- Introduction
- Identification
- States as the motion of the mobile platform
- Simple Model without any sensor
- Cartesian Plot
- From a force to force sensor
- From a force applied in the leg to the displacement of the leg
- Transmissibility
- Compliance
- Inertial
Introduction ignore
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.
Identification
Simscape Model
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);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'};States as the motion of the mobile platform
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);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.
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.
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);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 |
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]); endIdentification
%% 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)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)Simple Model without any sensor
Simscape Model
open 'stewart_identification_simple.slx'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);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.StateNameCartesian 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);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');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');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]');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]');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]');