-
-
- 1. Configuration Analysis - Stiffness Matrix +
- 1. Stiffness Matrix for the Cubic configuration
- 1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
- 1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
- 1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
- 1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center -
- 1.5. Conclusion +
- 1.5. Conclusion
- 2. Configuration with the Cube’s center above the mobile platform
- 3. Cubic size analysis -
- 4. Dynamic Coupling +
- 4. Dynamic Coupling in the Cartesian Frame -
- 5. Functions +
- 5. Dynamic Coupling between actuators and sensors of each strut + +
- 6. Functions
+
-
+
- 6.1.
generateCubicConfiguration: Generate a Cubic Configuration- Function description
- Documentation @@ -337,15 +344,13 @@ In this document, the cubic architecture is analyzed:
- In section 1, 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
- In section 2, we study cubic configurations where the cube’s center is located above the mobile platform
- In section 3, we study the effect of the cube’s size on the Stewart platform properties -
- In section 4, we study the dynamic coupling of the cubic configuration +
- In section 4, we study the dynamic coupling of the cubic configuration in the cartesian frame +
- In section 5, we study the dynamic coupling of the cubic configuration from actuators to sensors of each strut +
- In section 6, function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function
generateCubicConfigurationis used (described here).
-To generate and study the Stewart platform with a Cubic configuration, the Matlab function
- -generateCubicConfigurationis used (described here). --1 Configuration Analysis - Stiffness Matrix
++-1 Stiffness Matrix for the Cubic configuration
-1.5 Conclusion
++1.5 Conclusion
-@@ -1157,8 +1162,8 @@ FOc = H + MO_B; % Cente
-2.2 Conclusion
++2.2 Conclusion
-@@ -1232,8 +1237,8 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi
-3.2 Conclusion
++-3.2 Conclusion
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size. @@ -1249,37 +1254,506 @@ In order to maximize the rotational stiffness of the Stewart platform, the size
-4 Dynamic Coupling
++4 Dynamic Coupling in the Cartesian Frame
+-+In this section, we study the dynamics of the platform in the cartesian frame. +
+ ++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 \(\delta \bm{\mathcal{X}}\). +
+ ++Thanks to the Jacobian matrix, we can use the “architecture” shown in Figure 9 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. +
+ +++ + +\begin{tikzpicture} + \node[block] (Jt) at (0, 0) {$\bm{J}^{-T}$}; + \node[block, right= of Jt] (G) {$\bm{G}$}; + \node[block, right= of G] (J) {$\bm{J}^{-1}$}; + + \draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (J.west) node[above left]{$\delta\bm{\mathcal{L}}$}; + \draw[->] (J.east) -- ++(0.8, 0) node[above left]{$\delta\bm{\mathcal{X}}$}; +\end{tikzpicture} +
+++ +
+
+Figure 9: From Strut coordinate to Cartesian coordinate using the Jacobian matrix
++We here study the dynamics from \(\bm{\mathcal{F}}\) to \(\delta\bm{\mathcal{X}}\). +
+ ++One has to note that when considering the static behavior: +\[ \bm{G}(s = 0) = \begin{bmatrix} + 1/k_1 & & 0 \\ + & \ddots & 0 \\ + 0 & & 1/k_6 + \end{bmatrix}\] +
+ ++And thus: +\[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \] +
+ ++We conclude that the static behavior of the platform depends on the stiffness matrix. +For the cubic configuration, we have a diagonal stiffness matrix is the frames \(\{A\}\) and \(\{B\}\) are coincident with the cube’s center.
-4.1 Cube’s center at the Center of Mass of the Payload
++4.1 Cube’s center at the Center of Mass of the mobile platform
+++ ++Let’s create a Cubic Stewart Platform where the Center of Mass of the mobile platform is located at the center of the cube. +
+ ++We define the size of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\). +
++-H = 200e-3; % height of the Stewart platform [m] +MO_B = -10e-3; % Position {B} with respect to {M} [m] +
-+ +4.2 Dynamic decoupling between the actuators and sensors
++Now, we set the cube’s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\). +
++-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} +
-+4.3 Conclusion
+++ +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); +
++Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with \(\{A\}\) and \(\{B\}\). +
+++ +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))); +
++And we set small mass for the struts. +
+++ +stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3); +stewart = initializeInertialSensor(stewart); +
++The obtain geometry is shown in figure 10. +
+ + +++ +
+
+Figure 10: 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 (png, pdf)
++We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\). +
+++ +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'}; +
++Now, thanks to the Jacobian (Figure 9), we compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\). +
+++ +Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J'); +Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; +Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +
++The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure 11. +
+ + + + ++++The dynamics is well decoupled at all frequencies. +
+ ++We have the same dynamics for: +
+-
+
- \(D_x/F_x\), \(D_y/F_y\) and \(D_z/F_z\) +
- \(R_x/M_x\) and \(D_y/F_y\) +
+The Dynamics from \(F_i\) to \(D_i\) is just a 1-dof mass-spring-damper system. +
+ ++This is because the Mass, Damping and Stiffness matrices are all diagonal. +
+ +++ +4.2 Cube’s center not coincident with the Mass of the Mobile platform
++++Let’s create a Stewart platform with a cubic architecture where the cube’s center is at the center of the Stewart platform. +
+++ +H = 200e-3; % height of the Stewart platform [m] +MO_B = -100e-3; % Position {B} with respect to {M} [m] +
++Now, we set the cube’s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\). +
+++ +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} +
+++ +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); +
++However, the Center of Mass of the mobile platform is not located at the cube’s center. +
+++ +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))); +
++And we set small mass for the struts. +
+++ +stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3); +stewart = initializeInertialSensor(stewart); +
++The obtain geometry is shown in figure 12. +
+ +++ +
+
+Figure 12: 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 (png, pdf)
++We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\). +
+++ +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'}; +
++And we use the Jacobian to compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\). +
+++ +Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J'); +Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; +Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +
++The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure 13. +
+ + + + ++++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 \(\{B\}\)). +
+ +++4.3 Conclusion
++++++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}. +
+5 Dynamic Coupling between actuators and sensors of each strut
++ +++From 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). +
+++ +5.1 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. +
+ +++ +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} +
+++ + + + +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); +
++And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure 15) and to the force sensors \(\tau_{m,i}\) (Figure 16). +
+ + + + + + +++ +5.2 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. +
+ +++ +H = 200e-3; % height of the Stewart platform [m] +MO_B = -10e-3; % Position {B} with respect to {M} [m] +
+++ + + + +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); +
++And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure 18) and to the force sensors \(\tau_{m,i}\) (Figure 19). +
+ + + + + + ++5.3 Conclusion
+++++The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut. +
+ +-5 Functions
-+6 Functions
+-@@ -1428,7 +1902,7 @@ stewart.platform_M.Mb = Mb;5.1
-generateCubicConfiguration: Generate a Cubic Configuration+6.1
+generateCubicConfiguration: Generate a Cubic Configuration@@ -1323,7 +1797,7 @@ This Matlab function is accessible
-
Figure 9: Cubic Configuration
+Figure 20: Cubic Configuration
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diagonal stiffness matrix - In section [[sec:cubic_conf_above_platform]], we study cubic configurations where the cube's center is located above the mobile platform - In section [[sec:cubic_conf_size_analysis]], we study the effect of the cube's size on the Stewart platform properties -- In section [[sec:cubic_conf_coupling]], we study the dynamic coupling of the cubic configuration +- In section [[sec:cubic_conf_coupling_cartesian]], we study the dynamic coupling of the cubic configuration in the cartesian frame +- In section [[sec:cubic_conf_coupling_struts]], we study the dynamic coupling of the cubic configuration from actuators to sensors of each strut +- 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]]). -To generate and study the Stewart platform with a Cubic configuration, the Matlab function =generateCubicConfiguration= is used (described [[sec:generateCubicConfiguration][here]]). - -* Configuration Analysis - Stiffness Matrix +* Stiffness Matrix for the Cubic configuration <Created: 2020-02-12 mer. 11:23
+Created: 2020-02-12 mer. 18:26
> +#+end_src + +#+name: fig:stewart_cubic_conf_decouple_dynamics +#+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]]) +[[file:figs/stewart_cubic_conf_decouple_dynamics.png]] + +We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$. +#+begin_src matlab + 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 + +Now, thanks to the Jacobian (Figure [[fig:local_to_cartesian_coordinates]]), we compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$. +#+begin_src matlab + Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J'); + Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +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]]. + +#+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_cubic_decoupled_dynamics_cartesian.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +< > +#+end_src + +#+name: fig:stewart_cubic_decoupled_dynamics_cartesian +#+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]]) +[[file:figs/stewart_cubic_decoupled_dynamics_cartesian.png]] + +#+begin_important +The dynamics is well decoupled at all frequencies. + +We have the same dynamics for: +- $D_x/F_x$, $D_y/F_y$ and $D_z/F_z$ +- $R_x/M_x$ and $D_y/F_y$ + +The Dynamics from $F_i$ to $D_i$ is just a 1-dof mass-spring-damper system. + +This is because the Mass, Damping and Stiffness matrices are all diagonal. +#+end_important + +** Cube's center not coincident with the Mass of the Mobile platform +Let's create a Stewart platform with a cubic architecture where the cube's center is at the center of the Stewart platform. +#+begin_src matlab + H = 200e-3; % height of the Stewart platform [m] + MO_B = -100e-3; % Position {B} with respect to {M} [m] +#+end_src + +Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$. +#+begin_src matlab + 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); +#+end_src + +However, the Center of Mass of the mobile platform is *not* located at the cube's center. +#+begin_src matlab + 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))); +#+end_src + +And we set small mass for the struts. +#+begin_src matlab + stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3); + stewart = initializeInertialSensor(stewart); +#+end_src + +The obtain geometry is shown in figure [[fig:stewart_cubic_conf_mass_above]]. +#+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_cubic_conf_mass_above.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +< > +#+end_src + +#+name: fig:stewart_cubic_conf_mass_above +#+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]]) +[[file:figs/stewart_cubic_conf_mass_above.png]] + +We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$. +#+begin_src matlab + 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 + +And we use the Jacobian to compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$. +#+begin_src matlab + Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J'); + Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +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") +< > +#+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 +< > +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + < > +#+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") +< > +#+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") +< > +#+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") +< > +#+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") +< > +#+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") +< > +#+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") +< > +#+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 <
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