564 lines
21 KiB
Org Mode
564 lines
21 KiB
Org Mode
#+TITLE: Identification of the Stewart Platform using Simscape
|
|
:DRAWER:
|
|
#+HTML_LINK_HOME: ./index.html
|
|
#+HTML_LINK_UP: ./index.html
|
|
|
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
|
#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
|
|
#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
|
|
#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
|
|
#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
|
|
|
|
#+PROPERTY: header-args:matlab :session *MATLAB*
|
|
#+PROPERTY: header-args:matlab+ :tangle matlab/identification.m
|
|
#+PROPERTY: header-args:matlab+ :comments org
|
|
#+PROPERTY: header-args:matlab+ :exports both
|
|
#+PROPERTY: header-args:matlab+ :results none
|
|
#+PROPERTY: header-args:matlab+ :eval no-export
|
|
#+PROPERTY: header-args:matlab+ :noweb yes
|
|
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
|
#+PROPERTY: header-args:matlab+ :output-dir figs
|
|
:END:
|
|
|
|
* 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 |
|
|
|
|
|
|
#+begin_quote
|
|
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.
|
|
#+end_quote
|
|
|
|
|
|
|
|
* Identification
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :results none :exports none
|
|
simulinkproject('./');
|
|
#+end_src
|
|
|
|
** Simscape Model
|
|
|
|
** Initialize the Stewart Platform
|
|
#+begin_src matlab
|
|
stewart = initializeFramesPositions();
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
#+end_src
|
|
|
|
** Identification
|
|
#+begin_src matlab
|
|
%% 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'};
|
|
#+end_src
|
|
|
|
* States as the motion of the mobile platform
|
|
** Initialize the Stewart Platform
|
|
#+begin_src matlab
|
|
stewart = initializeFramesPositions();
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
#+end_src
|
|
|
|
** Identification
|
|
#+begin_src matlab
|
|
%% 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'};
|
|
#+end_src
|
|
|
|
Let's check the size of =G=:
|
|
#+begin_src matlab :results replace output
|
|
size(G)
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
: 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).
|
|
#+begin_src matlab :results replace output
|
|
Gm = minreal(G);
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
: Gm = minreal(G);
|
|
: 6 states removed.
|
|
|
|
And indeed, we obtain 12 states.
|
|
|
|
** Coordinate transformation
|
|
We can perform the following transformation using the =ss2ss= command.
|
|
#+begin_src matlab
|
|
Gt = ss2ss(Gm, Gm.C);
|
|
#+end_src
|
|
|
|
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:
|
|
#+begin_src matlab
|
|
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);
|
|
#+end_src
|
|
|
|
** Analysis
|
|
#+begin_src matlab
|
|
[V,D] = eig(Gt.A);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
ws = imag(diag(D))/2/pi;
|
|
[ws,I] = sort(ws)
|
|
|
|
xi = 100*real(diag(D))./imag(diag(D));
|
|
xi = xi(I);
|
|
|
|
data2orgtable([[1:length(ws(ws>0))]', ws(ws>0), xi(xi>0)], {}, {'Mode Number', 'Resonance Frequency [Hz]', 'Damping Ratio [%]'}, ' %.1f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| 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.
|
|
#+begin_src matlab
|
|
ws = imag(diag(D));
|
|
[ws,I] = sort(ws)
|
|
ws = ws(7:end); I = I(7:end);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
for i = 1:length(ws)
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
i_mode = I(i); % the argument is the i'th mode
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
lambda_i = D(i_mode, i_mode);
|
|
xi_i = V(:,i_mode);
|
|
|
|
a_i = real(lambda_i);
|
|
b_i = imag(lambda_i);
|
|
#+end_src
|
|
|
|
Let do 10 periods of the mode.
|
|
#+begin_src matlab
|
|
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)));
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
U = timeseries(U_i, t);
|
|
#+end_src
|
|
|
|
Simulation:
|
|
#+begin_src matlab
|
|
load('mat/conf_simscape.mat');
|
|
set_param(conf_simscape, 'StopTime', num2str(t(end)));
|
|
sim(mdl);
|
|
#+end_src
|
|
|
|
Save the movie of the mode shape.
|
|
#+begin_src matlab
|
|
smwritevideo(mdl, sprintf('figs/mode%i', i), ...
|
|
'PlaybackSpeedRatio', 1/(b_i/2/pi), ...
|
|
'FrameRate', 30, ...
|
|
'FrameSize', [800, 400]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
end
|
|
#+end_src
|
|
|
|
#+name: fig:mode1
|
|
#+caption: Identified mode - 1
|
|
[[file:figs/mode1.gif]]
|
|
|
|
#+name: fig:mode3
|
|
#+caption: Identified mode - 3
|
|
[[file:figs/mode3.gif]]
|
|
|
|
#+name: fig:mode5
|
|
#+caption: Identified mode - 5
|
|
[[file:figs/mode5.gif]]
|
|
|
|
** Identification
|
|
#+begin_src matlab
|
|
%% 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'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
size(G)
|
|
#+end_src
|
|
|
|
** Change of states
|
|
#+begin_src matlab
|
|
At = G.C*G.A*pinv(G.C);
|
|
|
|
Bt = G.C*G.B;
|
|
|
|
Ct = eye(12);
|
|
Dt = zeros(12, 6);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Gt = ss(At, Bt, Ct, Dt);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
size(Gt)
|
|
#+end_src
|
|
|
|
* Simple Model without any sensor
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :results none :exports none
|
|
simulinkproject('./');
|
|
#+end_src
|
|
|
|
** Simscape Model
|
|
#+begin_src matlab
|
|
open 'stewart_identification_simple.slx'
|
|
#+end_src
|
|
|
|
|
|
** Initialize the Stewart Platform
|
|
#+begin_src matlab
|
|
stewart = initializeFramesPositions();
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
#+end_src
|
|
|
|
** Identification
|
|
#+begin_src matlab
|
|
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'};
|
|
#+end_src
|
|
|
|
|
|
#+begin_src matlab
|
|
%% 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'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
size(G)
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
G.StateName
|
|
#+end_src
|
|
|
|
* Cartesian Plot
|
|
From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
|
|
#+begin_src matlab :results none
|
|
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');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :results none
|
|
figure;
|
|
bode(G.G_cart, freqs);
|
|
#+end_src
|
|
|
|
* From a force to force sensor
|
|
#+begin_src matlab :results none
|
|
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');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :results none
|
|
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');
|
|
#+end_src
|
|
|
|
* From a force applied in the leg to the displacement of the leg
|
|
#+begin_src matlab :results none
|
|
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]');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :results none
|
|
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');
|
|
#+end_src
|
|
|
|
* Transmissibility
|
|
#+begin_src matlab :results none
|
|
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]');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :results none
|
|
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 [$\frac{rad/s}{rad/s}$]');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :results none
|
|
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]');
|
|
#+end_src
|
|
|
|
* Compliance
|
|
From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.
|
|
|
|
#+begin_src matlab :results none
|
|
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]');
|
|
#+end_src
|
|
|
|
* Inertial
|
|
From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.
|
|
|
|
#+begin_src matlab :results none
|
|
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]');
|
|
#+end_src
|
|
|