Important analysis of the cubic configuration
This commit is contained in:
@@ -52,11 +52,11 @@ In this document, the cubic architecture is analyzed:
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- In section [[sec:cubic_conf_stiffness]], we study the link between the Stiffness matrix and the cubic architecture and we find what are the conditions to obtain a diagonal stiffness matrix
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- In section [[sec:cubic_conf_above_platform]], we study cubic configurations where the cube's center is located above the mobile platform
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- In section [[sec:cubic_conf_size_analysis]], we study the effect of the cube's size on the Stewart platform properties
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- In section [[sec:cubic_conf_coupling]], we study the dynamic coupling of the cubic configuration
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- In section [[sec:cubic_conf_coupling_cartesian]], we study the dynamic coupling of the cubic configuration in the cartesian frame
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- In section [[sec:cubic_conf_coupling_struts]], we study the dynamic coupling of the cubic configuration from actuators to sensors of each strut
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- In section [[sec:functions]], function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function =generateCubicConfiguration= is used (described [[sec:generateCubicConfiguration][here]]).
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To generate and study the Stewart platform with a Cubic configuration, the Matlab function =generateCubicConfiguration= is used (described [[sec:generateCubicConfiguration][here]]).
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* Configuration Analysis - Stiffness Matrix
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* Stiffness Matrix for the Cubic configuration
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<<sec:cubic_conf_stiffness>>
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** Introduction :ignore:
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First, we have to understand what is the physical meaning of the Stiffness matrix $\bm{K}$.
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@@ -552,9 +552,47 @@ We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linear
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In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
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#+end_important
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* Dynamic Coupling
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<<sec:cubic_conf_coupling>>
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* Dynamic Coupling in the Cartesian Frame
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<<sec:cubic_conf_coupling_cartesian>>
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** Introduction :ignore:
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In this section, we study the dynamics of the platform in the cartesian frame.
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We here suppose that there is one relative motion sensor in each strut ($\delta\bm{\mathcal{L}}$ is measured) and we would like to control the position of the top platform pose $\delta \bm{\mathcal{X}}$.
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Thanks to the Jacobian matrix, we can use the "architecture" shown in Figure [[fig:local_to_cartesian_coordinates]] to obtain the dynamics of the system from forces/torques applied by the actuators on the top platform to translations/rotations of the top platform.
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#+begin_src latex :file local_to_cartesian_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports both
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\begin{tikzpicture}
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\node[block] (Jt) at (0, 0) {$\bm{J}^{-T}$};
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\node[block, right= of Jt] (G) {$\bm{G}$};
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\node[block, right= of G] (J) {$\bm{J}^{-1}$};
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\draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}$};
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\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
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\draw[->] (G.east) -- (J.west) node[above left]{$\delta\bm{\mathcal{L}}$};
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\draw[->] (J.east) -- ++(0.8, 0) node[above left]{$\delta\bm{\mathcal{X}}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:local_to_cartesian_coordinates
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#+caption: From Strut coordinate to Cartesian coordinate using the Jacobian matrix
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#+RESULTS:
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[[file:figs/local_to_cartesian_coordinates.png]]
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We here study the dynamics from $\bm{\mathcal{F}}$ to $\delta\bm{\mathcal{X}}$.
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One has to note that when considering the static behavior:
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\[ \bm{G}(s = 0) = \begin{bmatrix}
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1/k_1 & & 0 \\
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& \ddots & 0 \\
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0 & & 1/k_6
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\end{bmatrix}\]
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And thus:
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\[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \]
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We conclude that the *static* behavior of the platform depends on the stiffness matrix.
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For the cubic configuration, we have a diagonal stiffness matrix is the frames $\{A\}$ and $\{B\}$ are coincident with the cube's center.
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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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@@ -569,13 +607,724 @@ We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linear
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simulinkproject('../');
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#+end_src
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** Cube's center at the Center of Mass of the mobile platform
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Let's create a Cubic Stewart Platform where the *Center of Mass of the mobile platform is located at the center of the cube*.
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We define the size of the Stewart platform and the position of frames $\{A\}$ and $\{B\}$.
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#+begin_src matlab
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H = 200e-3; % height of the Stewart platform [m]
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MO_B = -10e-3; % Position {B} with respect to {M} [m]
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#+end_src
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** Cube's center at the Center of Mass of the Payload
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Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
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#+begin_src matlab
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Hc = 2.5*H; % Size of the useful part of the cube [m]
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FOc = H + MO_B; % Center of the cube with respect to {F}
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#+end_src
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** Dynamic decoupling between the actuators and sensors
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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#+end_src
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Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with $\{A\}$ and $\{B\}$.
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#+begin_src matlab
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
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'Mpm', 10, ...
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'Mph', 20e-3, ...
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'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
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#+end_src
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And we set small mass for the struts.
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#+begin_src matlab
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stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
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stewart = initializeInertialSensor(stewart);
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#+end_src
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The obtain geometry is shown in figure [[fig:stewart_cubic_conf_decouple_dynamics]].
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#+begin_src matlab :exports none
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displayArchitecture(stewart, 'labels', false, 'view', 'all');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/stewart_cubic_conf_decouple_dynamics.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:stewart_cubic_conf_decouple_dynamics
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#+caption: Geometry used for the simulations - The cube's center, the frames $\{A\}$ and $\{B\}$ and the Center of mass of the mobile platform are coincident ([[./figs/stewart_cubic_conf_decouple_dynamics.png][png]], [[./figs/stewart_cubic_conf_decouple_dynamics.pdf][pdf]])
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[[file:figs/stewart_cubic_conf_decouple_dynamics.png]]
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We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
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#+begin_src matlab
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open('simulink/stewart_active_damping.slx')
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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_active_damping';
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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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
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#+end_src
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Now, thanks to the Jacobian (Figure [[fig:local_to_cartesian_coordinates]]), we compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
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#+begin_src matlab
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Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
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Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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#+end_src
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The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_cubic_decoupled_dynamics_cartesian]].
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#+begin_src matlab :exports none
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freqs = logspace(1, 3, 500);
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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for i = 1:6
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax3 = subplot(2, 2, 3);
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hold on;
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for i = 1:6
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for j = i+1:6
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p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
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p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
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ax2 = subplot(2, 2, 2);
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hold on;
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for i = 1:6
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax4 = subplot(2, 2, 4);
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hold on;
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for i = 1:6
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for j = i+1:6
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p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
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p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
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linkaxes([ax1,ax2,ax3,ax4],'x');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/stewart_cubic_decoupled_dynamics_cartesian.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:stewart_cubic_decoupled_dynamics_cartesian
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#+caption: Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ ([[./figs/stewart_cubic_decoupled_dynamics_cartesian.png][png]], [[./figs/stewart_cubic_decoupled_dynamics_cartesian.pdf][pdf]])
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[[file:figs/stewart_cubic_decoupled_dynamics_cartesian.png]]
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#+begin_important
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The dynamics is well decoupled at all frequencies.
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We have the same dynamics for:
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- $D_x/F_x$, $D_y/F_y$ and $D_z/F_z$
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- $R_x/M_x$ and $D_y/F_y$
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The Dynamics from $F_i$ to $D_i$ is just a 1-dof mass-spring-damper system.
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This is because the Mass, Damping and Stiffness matrices are all diagonal.
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#+end_important
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** Cube's center not coincident with the Mass of the Mobile platform
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Let's create a Stewart platform with a cubic architecture where the cube's center is at the center of the Stewart platform.
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#+begin_src matlab
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H = 200e-3; % height of the Stewart platform [m]
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MO_B = -100e-3; % Position {B} with respect to {M} [m]
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#+end_src
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Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
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#+begin_src matlab
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Hc = 2.5*H; % Size of the useful part of the cube [m]
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FOc = H + MO_B; % Center of the cube with respect to {F}
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#+end_src
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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#+end_src
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However, the Center of Mass of the mobile platform is *not* located at the cube's center.
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#+begin_src matlab
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
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'Mpm', 10, ...
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'Mph', 20e-3, ...
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'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
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#+end_src
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And we set small mass for the struts.
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#+begin_src matlab
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stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
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stewart = initializeInertialSensor(stewart);
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#+end_src
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The obtain geometry is shown in figure [[fig:stewart_cubic_conf_mass_above]].
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#+begin_src matlab :exports none
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displayArchitecture(stewart, 'labels', false, 'view', 'all');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/stewart_cubic_conf_mass_above.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:stewart_cubic_conf_mass_above
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#+caption: Geometry used for the simulations - The cube's center is coincident with the frames $\{A\}$ and $\{B\}$ but not with the Center of mass of the mobile platform ([[./figs/stewart_cubic_conf_mass_above.png][png]], [[./figs/stewart_cubic_conf_mass_above.pdf][pdf]])
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[[file:figs/stewart_cubic_conf_mass_above.png]]
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We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
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#+begin_src matlab
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open('simulink/stewart_active_damping.slx')
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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_active_damping';
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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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
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#+end_src
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And we use the Jacobian to compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
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#+begin_src matlab
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Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
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Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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#+end_src
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|
||||
The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_conf_coupling_mass_matrix]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 500);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax3 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
|
||||
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
|
||||
|
||||
ax2 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax4 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
||||
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/stewart_conf_coupling_mass_matrix.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_conf_coupling_mass_matrix
|
||||
#+caption: Obtained Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ ([[./figs/stewart_conf_coupling_mass_matrix.png][png]], [[./figs/stewart_conf_coupling_mass_matrix.pdf][pdf]])
|
||||
[[file:figs/stewart_conf_coupling_mass_matrix.png]]
|
||||
|
||||
#+begin_important
|
||||
The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is *not* decoupled at all frequencies.
|
||||
|
||||
This was expected as the mass matrix is not diagonal (the Center of Mass of the mobile platform not being coincident with the frame $\{B\}$).
|
||||
#+end_important
|
||||
|
||||
** Conclusion
|
||||
#+begin_important
|
||||
Some conclusions can be drawn from the above analysis:
|
||||
- Static Decoupling <=> Diagonal Stiffness matrix <=> {A} and {B} at the cube's center
|
||||
- Dynamic Decoupling <=> Static Decoupling + CoM of mobile platform coincident with {A} and {B}.
|
||||
#+end_important
|
||||
|
||||
* Dynamic Coupling between actuators and sensors of each strut
|
||||
<<sec:cubic_conf_coupling_struts>>
|
||||
** Introduction :ignore:
|
||||
From cite:preumont07_six_axis_singl_stage_activ, the cubic configuration "/minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other)/".
|
||||
|
||||
In this section, we wish to study such properties of the cubic architecture.
|
||||
|
||||
We will compare the transfer function from sensors to actuators in each strut for a cubic architecture and for a non-cubic architecture (where the struts are not orthogonal with each other).
|
||||
|
||||
** 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
|
||||
|
||||
** Coupling between the actuators and sensors - Cubic Architecture
|
||||
Let's generate a Cubic architecture where the cube's center and the frames $\{A\}$ and $\{B\}$ are coincident.
|
||||
|
||||
#+begin_src matlab
|
||||
H = 200e-3; % height of the Stewart platform [m]
|
||||
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
||||
Hc = 2.5*H; % Size of the useful part of the cube [m]
|
||||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
'Mpm', 10, ...
|
||||
'Mph', 20e-3, ...
|
||||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
||||
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
displayArchitecture(stewart, 'labels', false, 'view', 'all');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/stewart_architecture_coupling_struts_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_architecture_coupling_struts_cubic
|
||||
#+caption: Geometry of the generated Stewart platform ([[./figs/stewart_architecture_coupling_struts_cubic.png][png]], [[./figs/stewart_architecture_coupling_struts_cubic.pdf][pdf]])
|
||||
[[file:figs/stewart_architecture_coupling_struts_cubic.png]]
|
||||
|
||||
And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure [[fig:coupling_struts_relative_sensor_cubic]]) and to the force sensors $\tau_{m,i}$ (Figure [[fig:coupling_struts_force_sensor_cubic]]).
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
open('simulink/stewart_active_damping.slx')
|
||||
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_active_damping';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax3 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$L_i/\tau_i$', '$L_i/\tau_j$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/coupling_struts_relative_sensor_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:coupling_struts_relative_sensor_cubic
|
||||
#+caption: Dynamics from the force actuators to the relative motion sensors ([[./figs/coupling_struts_relative_sensor_cubic.png][png]], [[./figs/coupling_struts_relative_sensor_cubic.pdf][pdf]])
|
||||
[[file:figs/coupling_struts_relative_sensor_cubic.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 500);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax3 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$F_{m,i}/\tau_i$', '$F_{m,i}/\tau_j$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/coupling_struts_force_sensor_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:coupling_struts_force_sensor_cubic
|
||||
#+caption: Dynamics from the force actuators to the force sensors ([[./figs/coupling_struts_force_sensor_cubic.png][png]], [[./figs/coupling_struts_force_sensor_cubic.pdf][pdf]])
|
||||
[[file:figs/coupling_struts_force_sensor_cubic.png]]
|
||||
|
||||
** Coupling between the actuators and sensors - Non-Cubic Architecture
|
||||
Now we generate a Stewart platform which is not cubic but with approximately the same size as the previous cubic architecture.
|
||||
|
||||
#+begin_src matlab
|
||||
H = 200e-3; % height of the Stewart platform [m]
|
||||
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateGeneralConfiguration(stewart, 'FR', 250e-3, 'MR', 150e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
'Mpm', 10, ...
|
||||
'Mph', 20e-3, ...
|
||||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
||||
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
displayArchitecture(stewart, 'labels', false, 'view', 'all');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/stewart_architecture_coupling_struts_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_architecture_coupling_struts_non_cubic
|
||||
#+caption: Geometry of the generated Stewart platform ([[./figs/stewart_architecture_coupling_struts_non_cubic.png][png]], [[./figs/stewart_architecture_coupling_struts_non_cubic.pdf][pdf]])
|
||||
[[file:figs/stewart_architecture_coupling_struts_non_cubic.png]]
|
||||
|
||||
And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure [[fig:coupling_struts_relative_sensor_non_cubic]]) and to the force sensors $\tau_{m,i}$ (Figure [[fig:coupling_struts_force_sensor_non_cubic]]).
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
open('simulink/stewart_active_damping.slx')
|
||||
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_active_damping';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax3 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$L_i/\tau_i$', '$L_i/\tau_j$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/coupling_struts_relative_sensor_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:coupling_struts_relative_sensor_non_cubic
|
||||
#+caption: Dynamics from the force actuators to the relative motion sensors ([[./figs/coupling_struts_relative_sensor_non_cubic.png][png]], [[./figs/coupling_struts_relative_sensor_non_cubic.pdf][pdf]])
|
||||
[[file:figs/coupling_struts_relative_sensor_non_cubic.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 500);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax3 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
for j = i+1:6
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$F_{m,i}/\tau_i$', '$F_{m,i}/\tau_j$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/coupling_struts_force_sensor_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:coupling_struts_force_sensor_non_cubic
|
||||
#+caption: Dynamics from the force actuators to the force sensors ([[./figs/coupling_struts_force_sensor_non_cubic.png][png]], [[./figs/coupling_struts_force_sensor_non_cubic.pdf][pdf]])
|
||||
[[file:figs/coupling_struts_force_sensor_non_cubic.png]]
|
||||
|
||||
** Conclusion
|
||||
#+begin_important
|
||||
The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut.
|
||||
#+end_important
|
||||
|
||||
* Functions
|
||||
<<sec:functions>>
|
||||
|
||||
Reference in New Issue
Block a user