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52b18b7..afce693 100644 --- a/figs/detail_control_model_test_modal.svg +++ b/figs/detail_control_decoupling_model_test_modal.svg @@ -25,8 +25,8 @@ inkscape:pageopacity="0.0" inkscape:pageshadow="2" inkscape:zoom="5.6568542" - inkscape:cx="152.91184" - inkscape:cy="50.558135" + inkscape:cx="73.185552" + inkscape:cy="54.270446" inkscape:document-units="mm" inkscape:current-layer="layer1" inkscape:document-rotation="0" @@ -2204,7 +2204,7 @@ inkscape:export-ydpi="252" /> + transform="translate(-46.102638,4.9053314)"> - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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experimental ID31 plant ** DONE [#A] Verify why SVD decomposition on the proposed example gives such good performance CLOSED: [2025-04-05 Sat 17:43] @@ -2223,10 +2222,10 @@ Experimental closed-loop control results using the hexapod have shown that contr The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators. It is structured as follow: -- Section ref:ssec:detail_control_decoupling_comp_model: the model used to compare/test decoupling strategies is presented -- Section ref:ssec:detail_control_comp_jacobian: decoupling using Jacobian matrices is presented -- Section ref:ssec:detail_control_comp_modal: modal decoupling is presented -- Section ref:ssec:detail_control_comp_svd: SVD decoupling is presented +- Section ref:ssec:detail_control_decoupling_model: the model used to compare/test decoupling strategies is presented +- Section ref:ssec:detail_control_decoupling_jacobian: decoupling using Jacobian matrices is presented +- Section ref:ssec:detail_control_decoupling_modal: modal decoupling is presented +- Section ref:ssec:detail_control_decoupling_svd: SVD decoupling is presented - Section ref:ssec:detail_control_decoupling_comp: the three decoupling methods are applied on the test model and compared - Conclusions are drawn on the three decoupling methods @@ -2255,11 +2254,11 @@ freqs = logspace(0, 3, 1000); #+end_src ** Test Model -<> +<> - Instead of comparing the decoupling strategies using the Stewart platform, a similar yet much simpler parallel manipulator is used instead -- to render the analysis simpler, the system of Figure ref:fig:detail_control_model_test_decoupling_detail is used -- Fully parallel manipulator: it has 3DoF, and has 3 parallels struts whose model is shown in Figure ref:fig:detail_control_strut_model +- to render the analysis simpler, the system of Figure ref:fig:detail_control_decoupling_model_details is used +- Fully parallel manipulator: it has 3DoF, and has 3 parallels struts whose model is shown in Figure ref:fig:detail_control_decoupling_strut_model As many DoF as actuators and sensors - It is quite similar to the Stewart platform (parallel architecture, as many struts as DoF) @@ -2267,21 +2266,21 @@ Two frames are defined: - $\{M\}$ with origin $O_M$ at the Center of mass of the solid body - $\{K\}$ with origin $O_K$ at the Center of mass of the parallel manipulator -#+name: fig:detail_control_model_test_decoupling_detail +#+name: fig:detail_control_decoupling_model_details #+caption: 3DoF model used to study decoupling strategies #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_decoupling}Geometrical parameters} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test}Geometrical parameters} #+attr_latex: :options {0.58\textwidth} #+begin_subfigure #+attr_latex: :scale 1 -[[file:figs/detail_control_model_test_decoupling.png]] +[[file:figs/detail_control_decoupling_model_test.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_strut_model}Strut model} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_strut_model}Strut model} #+attr_latex: :options {0.38\textwidth} #+begin_subfigure #+attr_latex: :scale 1 -[[file:figs/detail_control_strut_model.png]] +[[file:figs/detail_control_decoupling_strut_model.png]] #+end_subfigure #+end_figure @@ -2358,6 +2357,7 @@ Parameters used for the following analysis are summarized in table ref:tab:detai | $I$ | Payload rotational inertia | $5\,\text{kg}m^2$ | ** Control in the frame of the struts +<> Let's first study the obtained dynamics in the frame of the struts. The equation of motion linking actuator forces $\bm{\mathcal{\tau}}$ to strut relative motion $\bm{\mathcal{L}}$ is obtained from eqref:eq:detail_control_decoupling_plant_cartesian by mapping the cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix $\bm{J}_{\{M\}}$ eqref:eq:detail_control_decoupling_jacobian_CoM . @@ -2433,7 +2433,7 @@ C = J_CoM'*Cr*J_CoM; G_L = J_CoM*inv(M*s^2 + C*s + K)*J_CoM'; #+end_src -The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is shown in Figure ref:fig:detail_control_coupled_plant_bode. +The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is shown in Figure ref:fig:detail_control_decoupling_coupled_plant_bode. This confirms that at low frequency (below the first suspension mode), the plant is well decoupled. Depending on the symmetry in the system, some diagonal elements may be equal (such as for struts 2 and 3 in this example). @@ -2467,17 +2467,16 @@ end #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_coupled_plant_bode.pdf', 'width', 'full', 'height', 'tall'); +exportFig('figs/detail_control_decoupling_coupled_plant_bode.pdf', 'width', 'full', 'height', 'tall'); #+end_src -#+name: fig:detail_control_coupled_plant_bode +#+name: fig:detail_control_decoupling_coupled_plant_bode #+caption: Magnitude of the coupled plant. #+RESULTS: -[[file:figs/detail_control_coupled_plant_bode.png]] - +[[file:figs/detail_control_decoupling_coupled_plant_bode.png]] ** Jacobian Decoupling -<> +<> **** Jacobian Matrix As already explained, the Jacobian matrix can be used to both convert strut velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$ and Convert actuators forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$ eqref:eq:detail_control_decoupling_jacobian. @@ -2587,7 +2586,7 @@ Such strategy is usually applied on systems with low frequency suspension modes, The coupling at low frequency can easily be understood physically. When a static (or with frequency lower than the suspension modes) force is applied at the center of mass, rotation is induced by the stiffness of the first actuator, not in line with the force application point. -this is illustrated in Figure ref:fig:detail_control_model_test_CoM. +this is illustrated in Figure ref:fig:detail_control_decoupling_model_test_CoM. #+begin_src matlab %% Jacobian Decoupling - Center of Mass @@ -2615,24 +2614,24 @@ leg.ItemTokenSize(1) = 18; #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_jacobian_plant_CoM.pdf', 'width', 'half', 'height', 'normal'); +exportFig('figs/detail_control_decoupling_jacobian_plant_CoM.pdf', 'width', 'half', 'height', 'normal'); #+end_src -#+name: fig:detail_control_jacobian_plant_CoM_results -#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoM}). +#+name: fig:detail_control_jacobian_decoupling_plant_CoM_results +#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_decoupling_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoM}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_jacobian_plant_CoM}Dynamics at the CoM} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoM}Dynamics at the CoM} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth -[[file:figs/detail_control_jacobian_plant_CoM.png]] +[[file:figs/detail_control_decoupling_jacobian_plant_CoM.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_CoM}Static force applied at the CoM} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoM}Static force applied at the CoM} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 1 -[[file:figs/detail_control_model_test_CoM.png]] +[[file:figs/detail_control_model_decoupling_test_CoM.png]] #+end_subfigure #+end_figure @@ -2687,7 +2686,7 @@ This is usually suited for systems which high stiffness. \end{equation} -The physical reason for high frequency coupling is schematically shown in Figure ref:fig:detail_control_model_test_CoK. +The physical reason for high frequency coupling is schematically shown in Figure ref:fig:detail_control_decoupling_model_test_CoK. At high frequency, a force applied on a point which is not aligned with the center of mass. Therefore, it will induce some rotation around the center of mass. @@ -2727,29 +2726,29 @@ leg.ItemTokenSize(1) = 18; #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_jacobian_plant_CoK.pdf', 'width', 'half', 'height', 'normal'); +exportFig('figs/detail_control_decoupling_jacobian_plant_CoK.pdf', 'width', 'half', 'height', 'normal'); #+end_src -#+name: fig:detail_control_jacobian_plant_CoK_results -#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoK}). +#+name: fig:detail_control_decoupling_jacobian_plant_CoK_results +#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_jacobian_plant_CoK}Dynamics at the CoK} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth -[[file:figs/detail_control_jacobian_plant_CoK.png]] +[[file:figs/detail_control_decoupling_jacobian_plant_CoK.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_CoK}High frequency force applied at the CoK} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 1 -[[file:figs/detail_control_model_test_CoK.png]] +[[file:figs/detail_control_decoupling_model_test_CoK.png]] #+end_subfigure #+end_figure ** Modal Decoupling -<> +<> **** Theory :ignore: - A mechanical system consists of several modes: @@ -2763,21 +2762,21 @@ The physical interpretation of the above two equations is that any motion of the IFF in modal space [[cite:&holterman05_activ_dampin_based_decoup_colloc_contr]] very interesting paper [[cite:&pu11_six_degree_of_freed_activ]] -\begin{equation}\label{eq:detail_control_equation_motion_CoM} +\begin{equation}\label{eq:detail_control_decoupling_equation_motion_CoM} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^t \bm{\tau}(t) \end{equation} Let's make a change of variables: -\begin{equation}\label{eq:detail_control_modal_coordinates} +\begin{equation}\label{eq:detail_control_decoupling_modal_coordinates} \bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m} \end{equation} with: - $\bm{\mathcal{X}}_{m}$ the modal amplitudes - $\bm{\Phi}$ a matrix whose columns are the modes shapes of the system which can be computed from $\bm{M}_{\{M\}}$ and $\bm{K}_{\{M\}}$. -By pre-multiplying the equation of motion eqref:eq:detail_control_equation_motion_CoM by $\bm{\Phi}^t$ and using the change of variable eqref:eq:detail_control_modal_coordinates, a new set of equation of motion are obtained +By pre-multiplying the equation of motion eqref:eq:detail_control_decoupling_equation_motion_CoM by $\bm{\Phi}^t$ and using the change of variable eqref:eq:detail_control_decoupling_modal_coordinates, a new set of equation of motion are obtained -\begin{equation}\label{eq:detail_control_equation_modal_coordinates} +\begin{equation}\label{eq:detail_control_decoupling_equation_modal_coordinates} \underbrace{\bm{\Phi}^t \bm{M} \bm{\Phi}}_{\bm{M}_m} \bm{\ddot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^t \bm{C} \bm{\Phi}}_{\bm{C}_m} \bm{\dot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^t \bm{K} \bm{\Phi}}_{\bm{K}_m} \bm{\mathcal{X}}_m(t) = \underbrace{\bm{\Phi}^t \bm{J}^t \bm{\tau}(t)}_{\bm{\tau}_m(t)} \end{equation} @@ -2866,7 +2865,7 @@ For the present test system, obtained eigen vectors are Eigenvectors are arranged for increasing eigenvalues (i.e. resonance frequencies). -\begin{equation}\label{eq:} +\begin{equation} \bm{\phi} = \begin{bmatrix} -0.905 & 0 & -0.058 \\ 0 & 1 & 0 \\ @@ -2909,29 +2908,29 @@ leg.ItemTokenSize(1) = 18; #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_modal_plant.pdf', 'width', 'half', 'height', 'normal'); +exportFig('figs/detail_control_decoupling_modal_plant.pdf', 'width', 'half', 'height', 'normal'); #+end_src -#+name: fig:detail_control_modal_plant_decoupling -#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_modal_plant}) which can be used to control separately different modes (\subref{fig:detail_control_model_test_modal}) +#+name: fig:detail_control_decoupling_modal_plant_modes +#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to control separately different modes (\subref{fig:detail_control_decoupling_model_test_modal}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_modal_plant}Decoupled plant in modal space} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth -[[file:figs/detail_control_modal_plant.png]] +[[file:figs/detail_control_decoupling_modal_plant.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_modal}Individually controlled modes} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth -[[file:figs/detail_control_model_test_modal.png]] +[[file:figs/detail_control_decoupling_model_test_modal.png]] #+end_subfigure #+end_figure ** SVD Decoupling -<> +<> **** Singular Value Decomposition Singular Value Decomposition (SVD) @@ -3047,7 +3046,7 @@ Once the $\bm{U}$ and $\bm{V}$ matrices are obtained, the decoupled plant can be \bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-t} \end{equation} -The obtained plant shown in Figure ref:fig:detail_control_svd_plant is very well decoupled. and not only around $\omega_c$. +The obtained plant shown in Figure ref:fig:detail_control_decoupling_svd_plant is very well decoupled. and not only around $\omega_c$. On top of that, the diagonal terms are second order plants. #+begin_src matlab @@ -3089,18 +3088,18 @@ leg.ItemTokenSize(1) = 18; #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_svd_plant.pdf', 'width', 'wide', 'height', 'normal'); +exportFig('figs/detail_control_decoupling_svd_plant.pdf', 'width', 'wide', 'height', 'normal'); #+end_src -#+name: fig:detail_control_svd_plant +#+name: fig:detail_control_decoupling_svd_plant #+caption: Svd plant $G_m(s)$ #+RESULTS: -[[file:figs/detail_control_svd_plant.png]] +[[file:figs/detail_control_decoupling_svd_plant.png]] - [ ] Do we have something special when applying SVD to a collocated MIMO system? - As shown in Figure ref:fig:detail_control_coupled_plant_bode, the plant is symmetrical. + As shown in Figure ref:fig:detail_control_decoupling_coupled_plant_bode, the plant is symmetrical. Paper by Skogestad mention that. "symmetric circular plants" [[cite:&hovd97_svd_contr_contr]] @@ -3109,11 +3108,11 @@ exportFig('figs/detail_control_svd_plant.pdf', 'width', 'wide', 'height', 'norma A second system, identical to the first in terms of dynamics. Just the sensor are changed. -Instead of having relative motion sensors in the frame of the struts, three relative motion sensors are used as shown in Figure ref:fig:detail_control_model_test_decoupling_alt. +Instead of having relative motion sensors in the frame of the struts, three relative motion sensors are used as shown in Figure ref:fig:detail_control_decoupling_model_test_alt. Using Jacobian matrices, it is possible to compute the relative motion of each struts. So theoretically, it should be possible to control both systems the same way. -However, when applying the same SVD decoupling, plant of Figure ref:fig:detail_control_svd_alt_plant is obtained. +However, when applying the same SVD decoupling, plant of Figure ref:fig:detail_control_decoupling_svd_alt_plant is obtained. It has much more coupling. It is interesting to note that the coupling have local minimum near the chosen decoupling frequency. This is very logical as the decoupling matrices were computed from the plant response at that particular frequency. @@ -3177,24 +3176,24 @@ leg.ItemTokenSize(1) = 18; #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_svd_alt_plant.pdf', 'width', 'half', 'height', 'normal'); +exportFig('figs/detail_control_decoupling_svd_alt_plant.pdf', 'width', 'half', 'height', 'normal'); #+end_src #+name: fig:detail_control_svd_decoupling_not_symmetrical -#+caption: Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_model_test_decoupling_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_svd_alt_plant}). +#+caption: Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_decoupling_alt}Alternative location of sensors} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_alt}Alternative location of sensors} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 1 -[[file:figs/detail_control_model_test_decoupling_alt.png]] +[[file:figs/detail_control_decoupling_model_test_alt.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_svd_alt_plant}Obtained decoupled plant} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth -[[file:figs/detail_control_svd_alt_plant.png]] +[[file:figs/detail_control_decoupling_svd_alt_plant.png]] #+end_subfigure #+end_figure diff --git a/nass-control.pdf b/nass-control.pdf index 627433c..c068570 100644 Binary files a/nass-control.pdf and b/nass-control.pdf differ diff --git a/nass-control.tex b/nass-control.tex index 68d0089..d633c0b 100644 --- a/nass-control.tex +++ b/nass-control.tex @@ -1,4 +1,4 @@ -% Created 2025-04-05 Sat 21:56 +% Created 2025-04-05 Sat 22:14 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -658,20 +658,20 @@ The goal of this section is to compare the use of several methods for the decoup It is structured as follow: \begin{itemize} -\item Section \ref{ssec:detail_control_decoupling_comp_model}: the model used to compare/test decoupling strategies is presented -\item Section \ref{ssec:detail_control_comp_jacobian}: decoupling using Jacobian matrices is presented -\item Section \ref{ssec:detail_control_comp_modal}: modal decoupling is presented -\item Section \ref{ssec:detail_control_comp_svd}: SVD decoupling is presented +\item Section \ref{ssec:detail_control_decoupling_model}: the model used to compare/test decoupling strategies is presented +\item Section \ref{ssec:detail_control_decoupling_jacobian}: decoupling using Jacobian matrices is presented +\item Section \ref{ssec:detail_control_decoupling_modal}: modal decoupling is presented +\item Section \ref{ssec:detail_control_decoupling_svd}: SVD decoupling is presented \item Section \ref{ssec:detail_control_decoupling_comp}: the three decoupling methods are applied on the test model and compared \item Conclusions are drawn on the three decoupling methods \end{itemize} \section{Test Model} -\label{ssec:detail_control_decoupling_comp_model} +\label{ssec:detail_control_decoupling_model} \begin{itemize} \item Instead of comparing the decoupling strategies using the Stewart platform, a similar yet much simpler parallel manipulator is used instead -\item to render the analysis simpler, the system of Figure \ref{fig:detail_control_model_test_decoupling_detail} is used -\item Fully parallel manipulator: it has 3DoF, and has 3 parallels struts whose model is shown in Figure \ref{fig:detail_control_strut_model} +\item to render the analysis simpler, the system of Figure \ref{fig:detail_control_decoupling_model_details} is used +\item Fully parallel manipulator: it has 3DoF, and has 3 parallels struts whose model is shown in Figure \ref{fig:detail_control_decoupling_strut_model} As many DoF as actuators and sensors \item It is quite similar to the Stewart platform (parallel architecture, as many struts as DoF) \end{itemize} @@ -685,17 +685,17 @@ Two frames are defined: \begin{figure}[htbp] \begin{subfigure}{0.58\textwidth} \begin{center} -\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_decoupling.png} +\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test.png} \end{center} -\subcaption{\label{fig:detail_control_model_test_decoupling}Geometrical parameters} +\subcaption{\label{fig:detail_control_decoupling_model_test}Geometrical parameters} \end{subfigure} \begin{subfigure}{0.38\textwidth} \begin{center} -\includegraphics[scale=1,scale=1]{figs/detail_control_strut_model.png} +\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_strut_model.png} \end{center} -\subcaption{\label{fig:detail_control_strut_model}Strut model} +\subcaption{\label{fig:detail_control_decoupling_strut_model}Strut model} \end{subfigure} -\caption{\label{fig:detail_control_model_test_decoupling_detail}3DoF model used to study decoupling strategies} +\caption{\label{fig:detail_control_decoupling_model_details}3DoF model used to study decoupling strategies} \end{figure} First, the equation of motion are derived. @@ -775,6 +775,7 @@ Parameters used for the following analysis are summarized in table \ref{tab:deta \end{tabularx} \end{table} \section{Control in the frame of the struts} +\label{ssec:detail_control_decoupling_decentralized} Let's first study the obtained dynamics in the frame of the struts. The equation of motion linking actuator forces \(\bm{\mathcal{\tau}}\) to strut relative motion \(\bm{\mathcal{L}}\) is obtained from \eqref{eq:detail_control_decoupling_plant_cartesian} by mapping the cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix \(\bm{J}_{\{M\}}\) \eqref{eq:detail_control_decoupling_jacobian_CoM} . @@ -798,17 +799,17 @@ At low frequency the plant converges to a diagonal constant matrix whose diagona At high frequency, the plant converges to the mass matrix mapped in the frame of the struts, which is in general highly non-diagonal. -The magnitude of the coupled plant \(\bm{G}_{\mathcal{L}}\) is shown in Figure \ref{fig:detail_control_coupled_plant_bode}. +The magnitude of the coupled plant \(\bm{G}_{\mathcal{L}}\) is shown in Figure \ref{fig:detail_control_decoupling_coupled_plant_bode}. This confirms that at low frequency (below the first suspension mode), the plant is well decoupled. Depending on the symmetry in the system, some diagonal elements may be equal (such as for struts 2 and 3 in this example). \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/detail_control_coupled_plant_bode.png} -\caption{\label{fig:detail_control_coupled_plant_bode}Magnitude of the coupled plant.} +\includegraphics[scale=1]{figs/detail_control_decoupling_coupled_plant_bode.png} +\caption{\label{fig:detail_control_decoupling_coupled_plant_bode}Magnitude of the coupled plant.} \end{figure} \section{Jacobian Decoupling} -\label{ssec:detail_control_comp_jacobian} +\label{ssec:detail_control_decoupling_jacobian} \paragraph{Jacobian Matrix} As already explained, the Jacobian matrix can be used to both convert strut velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\) and Convert actuators forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\) \eqref{eq:detail_control_decoupling_jacobian}. @@ -881,22 +882,22 @@ Such strategy is usually applied on systems with low frequency suspension modes, The coupling at low frequency can easily be understood physically. When a static (or with frequency lower than the suspension modes) force is applied at the center of mass, rotation is induced by the stiffness of the first actuator, not in line with the force application point. -this is illustrated in Figure \ref{fig:detail_control_model_test_CoM}. +this is illustrated in Figure \ref{fig:detail_control_decoupling_model_test_CoM}. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_jacobian_plant_CoM.png} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_jacobian_plant_CoM.png} \end{center} -\subcaption{\label{fig:detail_control_jacobian_plant_CoM}Dynamics at the CoM} +\subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoM}Dynamics at the CoM} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_CoM.png} +\includegraphics[scale=1,scale=1]{figs/detail_control_model_decoupling_test_CoM.png} \end{center} -\subcaption{\label{fig:detail_control_model_test_CoM}Static force applied at the CoM} +\subcaption{\label{fig:detail_control_decoupling_model_test_CoM}Static force applied at the CoM} \end{subfigure} -\caption{\label{fig:detail_control_jacobian_plant_CoM_results}Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoM}).} +\caption{\label{fig:detail_control_jacobian_decoupling_plant_CoM_results}Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_decoupling_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoM}).} \end{figure} \paragraph{Center Of Stiffness} @@ -927,27 +928,27 @@ This is usually suited for systems which high stiffness. \end{equation} -The physical reason for high frequency coupling is schematically shown in Figure \ref{fig:detail_control_model_test_CoK}. +The physical reason for high frequency coupling is schematically shown in Figure \ref{fig:detail_control_decoupling_model_test_CoK}. At high frequency, a force applied on a point which is not aligned with the center of mass. Therefore, it will induce some rotation around the center of mass. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_jacobian_plant_CoK.png} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_jacobian_plant_CoK.png} \end{center} -\subcaption{\label{fig:detail_control_jacobian_plant_CoK}Dynamics at the CoK} +\subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_CoK.png} +\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test_CoK.png} \end{center} -\subcaption{\label{fig:detail_control_model_test_CoK}High frequency force applied at the CoK} +\subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK} \end{subfigure} -\caption{\label{fig:detail_control_jacobian_plant_CoK_results}Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_model_test_CoK}).} +\caption{\label{fig:detail_control_decoupling_jacobian_plant_CoK_results}Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).} \end{figure} \section{Modal Decoupling} -\label{ssec:detail_control_comp_modal} +\label{ssec:detail_control_decoupling_modal} \begin{itemize} \item A mechanical system consists of several modes: \begin{itemize} @@ -965,12 +966,12 @@ IFF in modal space \cite{holterman05_activ_dampin_based_decoup_colloc_contr} ver \cite{pu11_six_degree_of_freed_activ} \end{itemize} -\begin{equation}\label{eq:detail_control_equation_motion_CoM} +\begin{equation}\label{eq:detail_control_decoupling_equation_motion_CoM} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^t \bm{\tau}(t) \end{equation} Let's make a change of variables: -\begin{equation}\label{eq:detail_control_modal_coordinates} +\begin{equation}\label{eq:detail_control_decoupling_modal_coordinates} \bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m} \end{equation} with: @@ -979,9 +980,9 @@ with: \item \(\bm{\Phi}\) a matrix whose columns are the modes shapes of the system which can be computed from \(\bm{M}_{\{M\}}\) and \(\bm{K}_{\{M\}}\). \end{itemize} -By pre-multiplying the equation of motion \eqref{eq:detail_control_equation_motion_CoM} by \(\bm{\Phi}^t\) and using the change of variable \eqref{eq:detail_control_modal_coordinates}, a new set of equation of motion are obtained +By pre-multiplying the equation of motion \eqref{eq:detail_control_decoupling_equation_motion_CoM} by \(\bm{\Phi}^t\) and using the change of variable \eqref{eq:detail_control_decoupling_modal_coordinates}, a new set of equation of motion are obtained -\begin{equation}\label{eq:detail_control_equation_modal_coordinates} +\begin{equation}\label{eq:detail_control_decoupling_equation_modal_coordinates} \underbrace{\bm{\Phi}^t \bm{M} \bm{\Phi}}_{\bm{M}_m} \bm{\ddot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^t \bm{C} \bm{\Phi}}_{\bm{C}_m} \bm{\dot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^t \bm{K} \bm{\Phi}}_{\bm{K}_m} \bm{\mathcal{X}}_m(t) = \underbrace{\bm{\Phi}^t \bm{J}^t \bm{\tau}(t)}_{\bm{\tau}_m(t)} \end{equation} @@ -1051,7 +1052,7 @@ For the present test system, obtained eigen vectors are Eigenvectors are arranged for increasing eigenvalues (i.e. resonance frequencies). -\begin{equation}\label{eq:} +\begin{equation} \bm{\phi} = \begin{bmatrix} -0.905 & 0 & -0.058 \\ 0 & 1 & 0 \\ @@ -1071,20 +1072,20 @@ Eigenvectors are arranged for increasing eigenvalues (i.e. resonance frequencies \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_modal_plant.png} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_modal_plant.png} \end{center} -\subcaption{\label{fig:detail_control_modal_plant}Decoupled plant in modal space} +\subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_model_test_modal.png} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_model_test_modal.png} \end{center} -\subcaption{\label{fig:detail_control_model_test_modal}Individually controlled modes} +\subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes} \end{subfigure} -\caption{\label{fig:detail_control_modal_plant_decoupling}Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_modal_plant}) which can be used to control separately different modes (\subref{fig:detail_control_model_test_modal})} +\caption{\label{fig:detail_control_decoupling_modal_plant_modes}Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to control separately different modes (\subref{fig:detail_control_decoupling_model_test_modal})} \end{figure} \section{SVD Decoupling} -\label{ssec:detail_control_comp_svd} +\label{ssec:detail_control_decoupling_svd} \paragraph{Singular Value Decomposition} Singular Value Decomposition (SVD) @@ -1184,20 +1185,20 @@ Once the \(\bm{U}\) and \(\bm{V}\) matrices are obtained, the decoupled plant ca \bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-t} \end{equation} -The obtained plant shown in Figure \ref{fig:detail_control_svd_plant} is very well decoupled. and not only around \(\omega_c\). +The obtained plant shown in Figure \ref{fig:detail_control_decoupling_svd_plant} is very well decoupled. and not only around \(\omega_c\). On top of that, the diagonal terms are second order plants. \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/detail_control_svd_plant.png} -\caption{\label{fig:detail_control_svd_plant}Svd plant \(G_m(s)\)} +\includegraphics[scale=1]{figs/detail_control_decoupling_svd_plant.png} +\caption{\label{fig:detail_control_decoupling_svd_plant}Svd plant \(G_m(s)\)} \end{figure} \begin{itemize} \item[{$\square$}] Do we have something special when applying SVD to a collocated MIMO system? -As shown in Figure \ref{fig:detail_control_coupled_plant_bode}, the plant is symmetrical. +As shown in Figure \ref{fig:detail_control_decoupling_coupled_plant_bode}, the plant is symmetrical. Paper by Skogestad mention that. ``symmetric circular plants'' \cite{hovd97_svd_contr_contr} \end{itemize} @@ -1207,11 +1208,11 @@ Paper by Skogestad mention that. A second system, identical to the first in terms of dynamics. Just the sensor are changed. -Instead of having relative motion sensors in the frame of the struts, three relative motion sensors are used as shown in Figure \ref{fig:detail_control_model_test_decoupling_alt}. +Instead of having relative motion sensors in the frame of the struts, three relative motion sensors are used as shown in Figure \ref{fig:detail_control_decoupling_model_test_alt}. Using Jacobian matrices, it is possible to compute the relative motion of each struts. So theoretically, it should be possible to control both systems the same way. -However, when applying the same SVD decoupling, plant of Figure \ref{fig:detail_control_svd_alt_plant} is obtained. +However, when applying the same SVD decoupling, plant of Figure \ref{fig:detail_control_decoupling_svd_alt_plant} is obtained. It has much more coupling. It is interesting to note that the coupling have local minimum near the chosen decoupling frequency. This is very logical as the decoupling matrices were computed from the plant response at that particular frequency. @@ -1219,17 +1220,17 @@ This is very logical as the decoupling matrices were computed from the plant res \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_decoupling_alt.png} +\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test_alt.png} \end{center} -\subcaption{\label{fig:detail_control_model_test_decoupling_alt}Alternative location of sensors} +\subcaption{\label{fig:detail_control_decoupling_model_test_alt}Alternative location of sensors} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center} -\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_svd_alt_plant.png} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_svd_alt_plant.png} \end{center} -\subcaption{\label{fig:detail_control_svd_alt_plant}Obtained decoupled plant} +\subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant} \end{subfigure} -\caption{\label{fig:detail_control_svd_decoupling_not_symmetrical}Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_model_test_decoupling_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_svd_alt_plant}).} +\caption{\label{fig:detail_control_svd_decoupling_not_symmetrical}Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}).} \end{figure} \section{Comparison of decoupling strategies} \label{ssec:detail_control_decoupling_comp}