From c0ea4268540d4a0aa23c06413c406c2f99cad127 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 6 Nov 2020 18:00:30 +0100 Subject: [PATCH] Change the controller tuning for Stewart platform --- figs/stewart_comp_loop_gain_diagonal.pdf | Bin 0 -> 176194 bytes figs/stewart_comp_loop_gain_diagonal.png | Bin 0 -> 156317 bytes ..._platform_simscape_cl_transmissibility.eps | Bin 568289 -> 544993 bytes ..._platform_simscape_cl_transmissibility.pdf | 2345 +++++++---------- ..._platform_simscape_cl_transmissibility.png | Bin 248491 -> 216963 bytes index.html | 316 +-- index.org | 95 +- 7 files changed, 1143 insertions(+), 1613 deletions(-) create mode 100644 figs/stewart_comp_loop_gain_diagonal.pdf create mode 100644 figs/stewart_comp_loop_gain_diagonal.png diff --git a/figs/stewart_comp_loop_gain_diagonal.pdf b/figs/stewart_comp_loop_gain_diagonal.pdf new file mode 100644 index 0000000000000000000000000000000000000000..eba9b5bfb67f3551c075b5feff4e2959283cb7ea GIT binary patch literal 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    +

    SP_assembly.png

    Figure 5: Stewart Platform CAD View

    @@ -801,22 +801,22 @@ Some notes about the system: The analysis of the SVD control applied to the Stewart platform is performed in the following sections:

      -
    • Section 3.1: The parameters of the Simscape model of the Stewart platform are defined
    • -
    • Section 3.2: The plant is identified from the Simscape model and the system coupling is shown
    • -
    • Section 3.3: The plant is first decoupled using the Jacobian
    • -
    • Section 3.4: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
    • -
    • Section 3.5: The decoupling is performed thanks to the SVD
    • -
    • Section 3.6: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • -
    • Section 3.7: The dynamics of the decoupled plants are shown
    • -
    • Section 3.8: A diagonal controller is defined to control the decoupled plant
    • -
    • Section 3.9: Finally, the closed loop system properties are studied
    • +
    • Section 3.1: The parameters of the Simscape model of the Stewart platform are defined
    • +
    • Section 3.2: The plant is identified from the Simscape model and the system coupling is shown
    • +
    • Section 3.3: The plant is first decoupled using the Jacobian
    • +
    • Section 3.4: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
    • +
    • Section 3.5: The decoupling is performed thanks to the SVD
    • +
    • Section 3.6: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • +
    • Section 3.7: The dynamics of the decoupled plants are shown
    • +
    • Section 3.8: A diagonal controller is defined to control the decoupled plant
    • +
    • Section 3.9: Finally, the closed loop system properties are studied
    -
    -

    3.1 Simscape Model - Parameters

    +
    +

    3.1 Simscape Model - Parameters

    - +

    open('drone_platform.slx');
    @@ -864,14 +864,14 @@ Kc = tf(zeros(6));
     
    -
    +

    stewart_simscape.png

    Figure 6: General view of the Simscape Model

    -
    +

    stewart_platform_details.png

    Figure 7: Simscape model of the Stewart platform

    @@ -879,15 +879,15 @@ Kc = tf(zeros(6));
    -
    -

    3.2 Identification of the plant

    +
    +

    3.2 Identification of the plant

    - +

    -The plant shown in Figure 8 is identified from the Simscape model. +The plant shown in Figure 8 is identified from the Simscape model.

    @@ -903,7 +903,7 @@ The outputs are the 6 accelerations measured by the inertial unit.

    -
    +

    stewart_platform_plant.png

    Figure 8: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform

    @@ -940,7 +940,7 @@ State-space model with 6 outputs, 12 inputs, and 24 states.

    -The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 9. +The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 9.

    @@ -948,7 +948,7 @@ One can easily see that the system is strongly coupled.

    -
    +

    stewart_platform_coupled_plant.png

    Figure 9: Magnitude of all 36 elements of the transfer function matrix \(\bm{G}\)

    @@ -956,17 +956,17 @@ One can easily see that the system is strongly coupled.
    -
    -

    3.3 Physical Decoupling using the Jacobian

    +
    +

    3.3 Physical Decoupling using the Jacobian

    - -Consider the control architecture shown in Figure 10. + +Consider the control architecture shown in Figure 10. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

    -
    +

    plant_decouple_jacobian.png

    Figure 10: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

    @@ -990,11 +990,11 @@ Gx.InputName = {'Dwx',
    -
    -

    3.4 Real Approximation of \(G\) at the decoupling frequency

    +
    +

    3.4 Real Approximation of \(G\) at the decoupling frequency

    - +

    @@ -1174,11 +1174,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown

    -
    -

    3.5 SVD Decoupling

    +
    +

    3.5 SVD Decoupling

    - +

    @@ -1192,11 +1192,11 @@ First, the Singular Value Decomposition of \(H_1\) is performed:

    -The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 11. +The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 11.

    -
    +

    plant_decouple_svd.png

    Figure 11: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

    @@ -1209,11 +1209,11 @@ The decoupled plant is then:
    -
    -

    3.6 Verification of the decoupling using the “Gershgorin Radii”

    +
    +

    3.6 Verification of the decoupling using the “Gershgorin Radii”

    - +

    @@ -1229,7 +1229,7 @@ This is computed over the following frequencies.

    -
    +

    simscape_model_gershgorin_radii.png

    Figure 12: Gershgorin Radii of the Coupled and Decoupled plants

    @@ -1237,30 +1237,30 @@ This is computed over the following frequencies.
    -
    -

    3.7 Obtained Decoupled Plants

    +
    +

    3.7 Obtained Decoupled Plants

    - +

    -The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 13. +The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 13.

    -
    +

    simscape_model_decoupled_plant_svd.png

    Figure 13: Decoupled Plant using SVD

    -Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 14. +Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 14.

    -
    +

    simscape_model_decoupled_plant_jacobian.png

    Figure 14: Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

    @@ -1268,27 +1268,15 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
    -
    -

    3.8 Diagonal Controller

    +
    +

    3.8 Diagonal Controller

    - +

    -The controller \(K_c\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\). -

    - -
    -
    wc = 2*pi*0.1; % Crossover Frequency [rad/s]
    -C_g = 50; % DC Gain
    -
    -Kc = eye(6)*C_g/(s+wc);
    -
    -
    - -

    -The control diagram for the centralized control is shown in Figure 15. +The control diagram for the centralized control is shown in Figure 15.

    @@ -1297,55 +1285,71 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied

    -
    +

    centralized_control.png

    Figure 15: Control Diagram for the Centralized control

    -The feedback system is computed as shown below. -

    -
    -
    G_cen = feedback(G, inv(J')*Kc, [7:12], [1:6]);
    -
    -
    - -

    -The SVD control architecture is shown in Figure 16. +The SVD control architecture is shown in Figure 16. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

    -
    +

    svd_control.png

    Figure 16: Control Diagram for the SVD control

    +

    -The feedback system is computed as shown below. +We choose the controller to be a low pass filter: +\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]

    + +

    +\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\) +

    +
    -
    G_svd = feedback(G, pinv(V')*Kc*pinv(U), [7:12], [1:6]);
    +
    wc = 2*pi*80;
    +w0 = 2*pi*0.1;
    +
    +
    + +
    +
    K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
    +L_cen = K_cen*Gx;
    +G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
    +
    +
    + +
    +
    K_svd = diag(1./diag(abs(evalfr(Gd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
    +L_svd = K_svd*Gd;
    +G_svd = feedback(G, pinv(V')*K_svd*pinv(U), [7:12], [1:6]);
     

    -Let’s look at the Loop Gains. +The obtained diagonal elements of the loop gains are shown in Figure 17.

    -
    -
    L_svd =
    -
    + +
    +

    stewart_comp_loop_gain_diagonal.png +

    +

    Figure 17: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

    -
    -

    3.9 Closed-Loop system Performances

    +
    +

    3.9 Closed-Loop system Performances

    - +

    @@ -1376,14 +1380,14 @@ ans =

    -The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 18.

    -
    +

    stewart_platform_simscape_cl_transmissibility.png

    -

    Figure 17: Obtained Transmissibility

    +

    Figure 18: Obtained Transmissibility

    @@ -1391,7 +1395,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well

    Author: Dehaeze Thomas

    -

    Created: 2020-11-06 ven. 17:02

    +

    Created: 2020-11-06 ven. 18:00

    diff --git a/index.org b/index.org index dec1674..2d0d0a6 100644 --- a/index.org +++ b/index.org @@ -1229,9 +1229,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of ** Diagonal Controller <> -The controller $K_c$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$. - -#+begin_src matlab +#+begin_src matlab :exports none :tangle no wc = 2*pi*0.1; % Crossover Frequency [rad/s] C_g = 50; % DC Gain @@ -1268,11 +1266,6 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied #+RESULTS: [[file:figs/centralized_control.png]] -The feedback system is computed as shown below. -#+begin_src matlab - G_cen = feedback(G, inv(J')*Kc, [7:12], [1:6]); -#+end_src - The SVD control architecture is shown in Figure [[fig:svd_control]]. The matrices $U$ and $V$ are used to decoupled the plant $G$. #+begin_src latex :file svd_control.pdf :tangle no :exports results @@ -1301,17 +1294,89 @@ The matrices $U$ and $V$ are used to decoupled the plant $G$. #+RESULTS: [[file:figs/svd_control.png]] -The feedback system is computed as shown below. -#+begin_src matlab - G_svd = feedback(G, pinv(V')*Kc*pinv(U), [7:12], [1:6]); -#+end_src -Let's look at the Loop Gains. +We choose the controller to be a low pass filter: +\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \] + +$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$ #+begin_src matlab - L_svd = + wc = 2*pi*80; + w0 = 2*pi*0.1; #+end_src +#+begin_src matlab + K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); + L_cen = K_cen*Gx; + G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]); +#+end_src + +#+begin_src matlab + K_svd = diag(1./diag(abs(evalfr(Gd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); + L_svd = K_svd*Gd; + G_svd = feedback(G, pinv(V')*K_svd*pinv(U), [7:12], [1:6]); +#+end_src + +The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]]. + +#+begin_src matlab :exports none + freqs = logspace(-1, 2, 1000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile([2, 1]); + hold on; + plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$'); + for i_in_out = 2:6 + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off'); + end + + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ... + 'DisplayName', '$L_{J}(i,i)$'); + for i_in_out = 2:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + legend('location', 'northwest'); + ylim([5e-2, 2e3]) + + % Phase + ax2 = nexttile; + hold on; + for i_in_out = 1:6 + set(gca,'ColorOrderIndex',1) + plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz')))); + end + set(gca,'ColorOrderIndex',2) + for i_in_out = 1:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180:90:360]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/stewart_comp_loop_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:stewart_comp_loop_gain_diagonal +#+caption: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one +#+RESULTS: +[[file:figs/stewart_comp_loop_gain_diagonal.png]] + ** Closed-Loop system Performances <> @@ -1389,7 +1454,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well linkaxes([ax1,ax2,ax3,ax4],'xy'); xlim([freqs(1), freqs(end)]); - ylim([1e-5, 1e2]); + ylim([1e-3, 1e2]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace