Update - New Simscape Model
This commit is contained in:
@@ -39,32 +39,9 @@
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:END:
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* Introduction :ignore:
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We would like to extract a state space model of the Stewart Platform from the Simscape model.
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The inputs are:
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| Symbol | Meaning |
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|------------------------+--------------------------------------------------|
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| $\bm{\mathcal{F}}_{d}$ | External forces applied in {B} |
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| $\bm{\tau}$ | Joint forces |
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| $\bm{\mathcal{F}}$ | Cartesian forces applied by the Joints |
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| $\bm{D}_{w}$ | Fixed Based translation and rotations around {A} |
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The outputs are:
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| Symbol | Meaning |
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|--------------------+---------------------------------------------------------------------------|
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| $\bm{\mathcal{X}}$ | Relative Motion of {B} with respect to {A} |
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| $\bm{\mathcal{L}}$ | Joint Displacement |
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| $\bm{F}_{m}$ | Force Sensors in each strut |
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| $\bm{v}_{m}$ | Inertial Sensors located at $b_i$ measuring in the direction of the strut |
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#+begin_quote
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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.
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#+end_quote
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* Identification
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* Modal Analysis of the Stewart Platform
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** Introduction :ignore:
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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@@ -78,62 +55,8 @@ An important difference from basic Simulink models is that the states in a physi
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simulinkproject('../');
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#+end_src
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** Simscape Model
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** Initialize the Stewart Platform
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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#+end_src
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** Identification
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_platform_identification';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/tau'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Fext'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Vm'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Taum'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Lm'], 1, 'openoutput'); io_i = io_i + 1;
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6', ...
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'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', ...
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'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6', ...
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'taum1', 'taum2', 'taum3', 'taum4', 'taum5', 'taum6', ...
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'Lm1', 'Lm2', 'Lm3', 'Lm4', 'Lm5', 'Lm6'};
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#+end_src
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* States as the motion of the mobile platform
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :results none :exports none
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simulinkproject('../');
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open('stewart_platform_model.slx')
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#+end_src
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** Initialize the Stewart Platform
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@@ -143,10 +66,17 @@ An important difference from basic Simulink models is that the states in a physi
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart);
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#+end_src
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'none');
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#+end_src
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** Identification
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@@ -156,13 +86,13 @@ An important difference from basic Simulink models is that the states in a physi
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_platform_identification_simple';
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mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/tau'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Xdot'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 2, 'openoutput'); io_i = io_i + 1; % Velocity of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io);
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@@ -233,12 +163,12 @@ We could perform the transformation by hand:
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#+RESULTS:
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| Mode Number | Resonance Frequency [Hz] | Damping Ratio [%] |
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|-------------+--------------------------+-------------------|
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| 1.0 | 174.5 | 0.9 |
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| 2.0 | 174.5 | 0.7 |
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| 3.0 | 202.1 | 0.7 |
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| 4.0 | 237.3 | 0.6 |
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| 5.0 | 237.3 | 0.5 |
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| 6.0 | 283.8 | 0.5 |
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| 1.0 | 780.6 | 0.4 |
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| 2.0 | 780.6 | 0.3 |
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| 3.0 | 903.9 | 0.3 |
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| 4.0 | 1061.4 | 0.3 |
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| 5.0 | 1061.4 | 0.2 |
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| 6.0 | 1269.6 | 0.2 |
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** Visualizing the modes
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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).
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@@ -309,288 +239,3 @@ Save the movie of the mode shape.
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#+caption: Identified mode - 5
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[[file:figs/mode5.gif]]
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** Identification
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_platform_identification';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/tau'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Lm'], 1, 'openoutput'); io_i = io_i + 1;
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%% Run the linearization
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G = linearize(mdl, io, options);
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% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
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% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
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#+end_src
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#+begin_src matlab
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size(G)
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#+end_src
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** Change of states
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#+begin_src matlab
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At = G.C*G.A*pinv(G.C);
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Bt = G.C*G.B;
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Ct = eye(12);
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Dt = zeros(12, 6);
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#+end_src
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#+begin_src matlab
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Gt = ss(At, Bt, Ct, Dt);
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#+end_src
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#+begin_src matlab
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size(Gt)
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#+end_src
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* Simple Model without any sensor
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
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** Simscape Model
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#+begin_src matlab
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open 'stewart_identification_simple.slx'
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#+end_src
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** Initialize the Stewart Platform
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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#+end_src
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** Identification
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#+begin_src matlab
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stateorder = {...
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'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_1_1_1',...
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'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_2_1_1',...
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'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_3_1_1',...
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'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_4_1_1',...
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'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_5_1_1',...
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'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_6_1_1',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Rz.q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Rz.q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Rz.q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Rz.q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Rz.q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Rz.q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Pz.p',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Pz.p',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Pz.p',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Pz.p',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Pz.p',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Pz.p',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Rz.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Rz.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Rz.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Rz.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Rz.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Rz.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Pz.v',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Pz.v',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Pz.v',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Pz.v',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Pz.v',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Pz.v',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_F.S.Q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.spherical_joint_F.S.Q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.spherical_joint_F.S.Q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.spherical_joint_F.S.Q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.spherical_joint_F.S.Q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.spherical_joint_F.S.Q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.spherical_joint_F.S.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.spherical_joint_F.S.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.spherical_joint_F.S.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.spherical_joint_F.S.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.spherical_joint_F.S.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_F.S.w',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_M.S.Q',...
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'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_M.S.w'};
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#+end_src
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_platform_identification_simple';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/tau'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Xdot'], 1, 'openoutput'); io_i = io_i + 1;
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
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G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
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#+end_src
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#+begin_src matlab
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size(G)
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#+end_src
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#+begin_src matlab
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G.StateName
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#+end_src
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* Cartesian Plot
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From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
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#+begin_src matlab :results none
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_cart(1, 1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart(2, 1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart(3, 1), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Amplitude');
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#+end_src
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#+begin_src matlab :results none
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figure;
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bode(G.G_cart, freqs);
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#+end_src
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* From a force to force sensor
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#+begin_src matlab :results none
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
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legend('location', 'southeast');
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#+end_src
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#+begin_src matlab :results none
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
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plot(freqs, abs(squeeze(freqresp(G.G_forc(2, 1), freqs, 'Hz'))), 'k--', 'DisplayName', '$F_{m_j}/F_{i}$');
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plot(freqs, abs(squeeze(freqresp(G.G_forc(3, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G.G_forc(4, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G.G_forc(5, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G.G_forc(6, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
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hold off;
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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
|
||||
|
||||
|
||||
Reference in New Issue
Block a user