diff --git a/config_extra.tex b/config_extra.tex index c3581ed..45ff2c2 100644 --- a/config_extra.tex +++ b/config_extra.tex @@ -50,9 +50,9 @@ style=numeric-comp, % bib style hyperref=true, % activate hyperref support backref=true, % activate backrefs - isbn=false, % don't show isbn tags + isbn=true, % don't show isbn tags url=false, % don't show url tags - doi=false, % don't show doi tags + doi=true, % don't show doi tags urldate=long, % display type for dates maxnames=3, % minnames=1, % @@ -72,15 +72,13 @@ {\endlist} {\item} -\AtEveryBibitem{% - \clearfield{urlyear} - \clearfield{urlmonth} - \clearfield{note} - \clearfield{issn} % Remove issn - \clearfield{doi} % Remove doi -\ifentrytype{online}{}{% Remove url except for @online - \clearfield{url} -} +\DeclareSourcemap{ + \maps[datatype=bibtex]{ + \map{ + \step[fieldsource=doi,final] + \step[fieldset=isbn,null] + } + } } \setlength\bibitemsep{1.1\itemsep} @@ -146,9 +144,23 @@ {\savebox\mybox{#1}\begin{quote}\openautoquote\hspace*{-.7ex}} {\unskip\closeautoquote\vspace*{1mm}\signed{\usebox\mybox}\end{quote}} +\usepackage[perpage]{footmisc} + +\let\myRef\ref +\renewcommand\ref{\unskip~\myRef} + +\let\myEqref\eqref +\renewcommand\eqref{\unskip~\myEqref} + +\let\myCite\cite +\renewcommand\cite{\unskip~\myCite} + \usepackage{setspace} \setstretch{1.25} +\renewcommand{\topfraction}{.8} +\renewcommand{\floatpagefraction}{.8} + % \setparsizes{0em}{0.1\baselineskip plus .1\baselineskip}{1em plus 1fil} \numberwithin{equation}{chapter} diff --git a/figs/chapter3_overview.pdf b/figs/chapter3_overview.pdf index e18bcba..09ceb42 100644 Binary files a/figs/chapter3_overview.pdf and b/figs/chapter3_overview.pdf differ diff --git a/figs/chapter3_overview.png b/figs/chapter3_overview.png index f92e953..3b4ae53 100644 Binary files a/figs/chapter3_overview.png and b/figs/chapter3_overview.png differ diff --git a/figs/chapter3_overview.svg b/figs/chapter3_overview.svg index fbaa12a..7767287 100644 --- a/figs/chapter3_overview.svg +++ b/figs/chapter3_overview.svg @@ -2,12 +2,12 @@ C.14.1C.34.3C.54.5C.24.2C.44.4ActuatorsNano Active Stabilization SystemNano Active Stabilization System2\) sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure \ref{fig:detail_control_sensor_fusion_three}. +Fusion can be implemented either ``sequentially,'' utilizing \(n-1\) sets of two complementary filters (Figure \ref{fig:detail_control_sensor_fusion_three_sequential}), or ``in parallel,'' employing a single set of \(n\) complementary filters (Figure \ref{fig:detail_control_sensor_fusion_three_parallel}). + +While conventional sensor fusion synthesis techniques can be applied to the sequential approach, parallel architecture implementation requires a novel synthesis method for multiple complementary filters. +Previous literature has offered only simple analytical formulas for this purpose \cite{stoten01_fusion_kinet_data_using_compos_filter,fonseca15_compl}. +This section presents a generalization of the proposed complementary filter synthesis method to address this gap. + +\begin{figure}[htbp] +\begin{subfigure}{0.58\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.9]{figs/detail_control_sensor_fusion_three_sequential.png} +\end{center} +\subcaption{\label{fig:detail_control_sensor_fusion_three_sequential}Sequential fusion} +\end{subfigure} +\begin{subfigure}{0.38\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.9]{figs/detail_control_sensor_fusion_three_parallel.png} +\end{center} +\subcaption{\label{fig:detail_control_sensor_fusion_three_parallel}Parallel fusion} +\end{subfigure} +\caption{\label{fig:detail_control_sensor_fusion_three}Possible sensor fusion architecture when more than two sensors are to be merged} +\end{figure} + +The synthesis objective is to compute a set of \(n\) stable transfer functions \([H_1(s),\ H_2(s),\ \dots,\ H_n(s)]\) that satisfy conditions \eqref{eq:detail_control_sensor_hinf_cond_compl_gen} and \eqref{eq:detail_control_sensor_hinf_cond_perf_gen}. + +\begin{subequations}\label{eq:detail_control_sensor_hinf_problem_gen} + \begin{align} + & \sum_{i=1}^n H_i(s) = 1 \label{eq:detail_control_sensor_hinf_cond_compl_gen} \\ + & \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:detail_control_sensor_hinf_cond_perf_gen} + \end{align} +\end{subequations} + +The transfer functions \([W_1(s),\ W_2(s),\ \dots,\ W_n(s)]\) are weights selected to specify the maximum complementary filters' norm during synthesis. + +This synthesis objective is closely related to the one described in Section \ref{ssec:detail_control_sensor_hinf_method}, and the proposed synthesis method represents a generalization of the approach previously presented. +A set of \(n\) complementary filters can be shaped by applying standard \(\mathcal{H}_\infty\text{-synthesis}\) to the generalized plant \(P_n(s)\) described by \eqref{eq:detail_control_sensor_generalized_plant_n_filters}. + +\begin{equation}\label{eq:detail_control_sensor_generalized_plant_n_filters} + \begin{bmatrix} z_1 \\ \vdots \\ z_n \\ v \end{bmatrix} = P_n(s) \begin{bmatrix} w \\ u_1 \\ \vdots \\ u_{n-1} \end{bmatrix}; \quad + P_n(s) = \begin{bmatrix} + W_1 & -W_1 & \dots & \dots & -W_1 \\ + 0 & W_2 & 0 & \dots & 0 \\ + \vdots & \ddots & \ddots & \ddots & \vdots \\ + \vdots & & \ddots & \ddots & 0 \\ + 0 & \dots & \dots & 0 & W_n \\ + 1 & 0 & \dots & \dots & 0 + \end{bmatrix} +\end{equation} + +If the synthesis is successful, a set of \(n-1\) filters \([H_2(s),\ H_3(s),\ \dots,\ H_n(s)]\) is obtained such that \eqref{eq:detail_control_sensor_hinf_syn_obj_gen} is satisfied. + +\begin{equation}\label{eq:detail_control_sensor_hinf_syn_obj_gen} + \left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1 +\end{equation} + +\(H_1(s)\) is then defined using \eqref{eq:detail_control_sensor_h1_comp_h2_hn}, which ensures the complementary property for the set of \(n\) filters \eqref{eq:detail_control_sensor_hinf_cond_compl_gen}. +Condition \eqref{eq:detail_control_sensor_hinf_cond_perf_gen} is satisfied through \eqref{eq:detail_control_sensor_hinf_syn_obj_gen}. + +\begin{equation}\label{eq:detail_control_sensor_h1_comp_h2_hn} + H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big] +\end{equation} + +To validate the proposed method for synthesizing a set of three complementary filters, an example is provided. +The sensors to be merged are a displacement sensor (effective from DC up to \(\SI{1}{Hz}\)), a geophone (effective from \(1\) to \(\SI{10}{Hz}\)), and an accelerometer (effective above \(\SI{10}{Hz}\)). +Three weighting functions are designed using formula \eqref{eq:detail_control_sensor_weight_formula}, and their inverse magnitudes are shown in Figure \ref{fig:detail_control_sensor_three_complementary_filters_results} (dashed curves). + +Consider the generalized plant \(P_3(s)\) shown in Figure \ref{fig:detail_control_sensor_comp_filter_three_hinf_fb}, which is also described by \eqref{eq:detail_control_sensor_generalized_plant_three_filters}. + +\begin{equation}\label{eq:detail_control_sensor_generalized_plant_three_filters} + \begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix} +\end{equation} + +\begin{figure}[htbp] +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.9]{figs/detail_control_sensor_comp_filter_three_hinf_fb.png} +\end{center} +\subcaption{\label{fig:detail_control_sensor_comp_filter_three_hinf_fb}Generalized plant with the synthesized filter} +\end{subfigure} +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_sensor_three_complementary_filters_results.png} +\end{center} +\subcaption{\label{fig:detail_control_sensor_three_complementary_filters_results}Weights and obtained filters} +\end{subfigure} +\caption{\label{fig:detail_control_sensor_comp_filter_three_hinf}Architecture for the \(\mathcal{H}_\infty\text{-synthesis}\) of three complementary filters (\subref{fig:detail_control_sensor_comp_filter_three_hinf_fb}). Bode plot of the inverse weighting functions and of the three obtained complementary filters (\subref{fig:detail_control_sensor_three_complementary_filters_results})} +\end{figure} + +Standard \(\mathcal{H}_\infty\text{-synthesis}\) is performed on the generalized plant \(P_3(s)\). +Two filters, \(H_2(s)\) and \(H_3(s)\), are obtained such that the \(\mathcal{H}_\infty\) norm of the closed-loop transfer from \(w\) to \([z_1,\ z_2,\ z_3]\) of the system in Figure \ref{fig:detail_control_sensor_comp_filter_three_hinf_fb} is less than one. +Filter \(H_1(s)\) is defined using \eqref{eq:detail_control_sensor_h1_compl_h2_h3}, thus ensuring the complementary property of the obtained set of filters. + +\begin{equation}\label{eq:detail_control_sensor_h1_compl_h2_h3} + H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big] +\end{equation} + +Figure \ref{fig:detail_control_sensor_three_complementary_filters_results} displays the three synthesized complementary filters (solid lines), confirming the successful synthesis. +\subsubsection{Conclusion} + +A new method for designing complementary filters using the \(\mathcal{H}_\infty\text{-synthesis}\) has been proposed. +This approach allows shaping of the filter magnitudes through the use of weighting functions during synthesis. +This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude. +Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters. + +For the NASS, the HAC-LAC strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune. +Looking forward, it would be interesting to investigate how sensor fusion (particularly between the force sensors and external metrology) compares to the HAC-IFF approach in terms of performance and robustness. +\subsection{Decoupling} +\label{sec:detail_control_decoupling} + +The control of parallel manipulators (and any MIMO system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of SISO control for the decoupled plant (discussed in section \ref{sec:detail_control_cf}). + +When sensors are integrated within the struts, decentralized control may be applied, as the system is already well decoupled at low frequency. +For instance, \cite{furutani04_nanom_cuttin_machin_using_stewar} implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate PI controllers for each strut. +A similar control architecture was proposed in \cite{du14_piezo_actuat_high_precis_flexib} using strain gauge sensors integrated in each strut. + +An alternative strategy involves decoupling the system in the Cartesian frame using Jacobian matrices. +As demonstrated during the study of Stewart platform kinematics, Jacobian matrices can be utilized to map actuator forces to forces and torques applied on the top platform. +This approach enables the implementation of controllers in a defined frame. +It has been applied with various sensor types including force sensors \cite{mcinroy00_desig_contr_flexur_joint_hexap}, relative displacement sensors \cite{kim00_robus_track_contr_desig_dof_paral_manip}, and inertial sensors \cite{li01_simul_vibrat_isolat_point_contr,abbas14_vibrat_stewar_platf}. +The Cartesian frame in which the system is decoupled is typically chosen at the point of interest (i.e., where the motion is of interest) or at the center of mass. + +Modal decoupling represents another noteworthy decoupling strategy, wherein the ``local'' plant inputs and outputs are mapped to the modal space. +In this approach, multiple SISO plants, each corresponding to a single mode, can be controlled independently. +This decoupling strategy has been implemented for active damping applications \cite{holterman05_activ_dampin_based_decoup_colloc_contr}, which is logical as it is often desirable to dampen specific modes. +The strategy has also been employed in \cite{pu11_six_degree_of_freed_activ} for vibration isolation purposes using geophones, and in \cite{yang19_dynam_model_decoup_contr_flexib} using force sensors. + +Another completely different strategy would be to implement a multivariable control directly on the coupled system. +\(\mathcal{H}_\infty\) and \(\mu\text{-synthesis}\) were applied to a Stewart platform model in \cite{lei08_multi_objec_robus_activ_vibrat}. +In \cite{xie17_model_contr_hybrid_passiv_activ}, decentralized force feedback was first applied, followed by \(\mathcal{H}_2\text{-synthesis}\) for vibration isolation based on accelerometers. +\(\mathcal{H}_\infty\text{-synthesis}\) was also employed in \cite{jiao18_dynam_model_exper_analy_stewar} for active damping based on accelerometers. +A comparative study between \(\mathcal{H}_\infty\text{-synthesis}\) and decentralized control in the frame of the struts was performed in \cite{thayer02_six_axis_vibrat_isolat_system}. +Their experimental closed-loop results indicated that the \(\mathcal{H}_\infty\) controller did not outperform the decentralized controller in the frame of the struts. +These limitations were attributed to the model's poor ability to predict off-diagonal dynamics, which is crucial for \(\mathcal{H}_\infty\text{-synthesis}\). + +The purpose of this section is to compare several methods for the decoupling of parallel manipulators, an analysis that appears to be lacking in the literature. +A simplified parallel manipulator model is introduced in Section \ref{ssec:detail_control_decoupling_model} as a test case for evaluating decoupling strategies. +The decentralized plant (transfer functions from actuators to sensors integrated in the struts) is examined in Section \ref{ssec:detail_control_decoupling_decentralized}. +Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section \ref{ssec:detail_control_decoupling_jacobian}), modal decoupling (Section \ref{ssec:detail_control_decoupling_modal}), and Singular Value Decomposition (SVD) decoupling (Section \ref{ssec:detail_control_decoupling_svd}). +Finally, a comparative analysis with concluding observations is provided in Section \ref{ssec:detail_control_decoupling_comp}. +\subsubsection{Test Model} +\label{ssec:detail_control_decoupling_model} + +Instead of utilizing the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis. +The system illustrated in Figure \ref{fig:detail_control_decoupling_model_test} is used for this purpose. +It possesses three degrees of freedom (DoF) and incorporates three parallel struts. +Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform. + +Two reference frames are defined within this model: frame \(\{M\}\) with origin \(O_M\) at the center of mass of the solid body, and frame \(\{K\}\) with origin \(O_K\) at the center of stiffness of the parallel manipulator. + +\begin{minipage}[b]{0.60\linewidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test.png} +\captionof{figure}{\label{fig:detail_control_decoupling_model_test}Model used to compare decoupling strategies} +\end{center} +\end{minipage} +\hfill +\begin{minipage}[b]{0.36\linewidth} +\begin{scriptsize} +\centering +\begin{tabularx}{\linewidth}{cXc} +\toprule + & \textbf{Description} & \textbf{Value}\\ +\midrule +\(l_a\) & & \(0.5\,m\)\\ +\(h_a\) & & \(0.2\,m\)\\ +\(k\) & Actuator stiffness & \(10\,N/\mu m\)\\ +\(c\) & Actuator damping & \(200\,Ns/m\)\\ +\(m\) & Payload mass & \(40\,\text{kg}\)\\ +\(I\) & Payload \(R_z\) inertia & \(5\,\text{kg}m^2\)\\ +\bottomrule +\end{tabularx} +\captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters} +\end{scriptsize} +\end{minipage} + +The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass \eqref{eq:detail_control_decoupling_model_eom}, where \(\bm{\mathcal{X}}_{\{M\}}\) represents the two translations and one rotation with respect to the center of mass, and \(\bm{\mathcal{F}}_{\{M\}}\) denotes the forces and torque applied at the center of mass. + +\begin{equation}\label{eq:detail_control_decoupling_model_eom} + \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t), \quad + \bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix} + x \\ + y \\ + R_z + \end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix} + F_x \\ + F_y \\ + M_z + \end{bmatrix} +\end{equation} + +The Jacobian matrix \(\bm{J}_{\{M\}}\) is employed to map the spring, damping, and actuator forces to XY forces and Z torque expressed at the center of mass \eqref{eq:detail_control_decoupling_jacobian_CoM}. + +\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM} + \bm{J}_{\{M\}} = \begin{bmatrix} + 1 & 0 & h_a \\ + 0 & 1 & -l_a \\ + 0 & 1 & l_a \\ + \end{bmatrix} +\end{equation} + +Subsequently, the equation of motion relating the actuator forces \(\tau\) to the motion of the mass \(\bm{\mathcal{X}}_{\{M\}}\) is derived \eqref{eq:detail_control_decoupling_plant_cartesian}. + +\begin{equation}\label{eq:detail_control_decoupling_plant_cartesian} + \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t) +\end{equation} + +The matrices representing the payload inertia, actuator stiffness, and damping are shown in \eqref{eq:detail_control_decoupling_system_matrices}. + +\begin{equation}\label{eq:detail_control_decoupling_system_matrices} + \bm{M}_{\{M\}} = \begin{bmatrix} + m & 0 & 0 \\ + 0 & m & 0 \\ + 0 & 0 & I + \end{bmatrix}, \quad + \bm{\mathcal{K}} = \begin{bmatrix} + k & 0 & 0 \\ + 0 & k & 0 \\ + 0 & 0 & k + \end{bmatrix}, \quad + \bm{\mathcal{C}} = \begin{bmatrix} + c & 0 & 0 \\ + 0 & c & 0 \\ + 0 & 0 & c + \end{bmatrix} +\end{equation} + +The parameters employed for the subsequent analysis are summarized in Table \ref{tab:detail_control_decoupling_test_model_params}, which includes values for geometric parameters (\(l_a\), \(h_a\)), mechanical properties (actuator stiffness \(k\) and damping \(c\)), and inertial characteristics (payload mass \(m\) and rotational inertia \(I\)). +\subsubsection{Control in the frame of the struts} +\label{ssec:detail_control_decoupling_decentralized} + +The dynamics in the frame of the struts are first examined. +The equation of motion relating actuator forces \(\bm{\mathcal{\tau}}\) to strut relative motion \(\bm{\mathcal{L}}\) is derived from equation \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 \(\bm{J}_{\{M\}}\) defined in \eqref{eq:detail_control_decoupling_jacobian_CoM}. +The obtained transfer function from \(\bm{\mathcal{\tau}}\) to \(\bm{\mathcal{L}}\) is shown in \eqref{eq:detail_control_decoupling_plant_decentralized}. + +\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized} + \frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1} +\end{equation} + +At low frequencies, the plant converges to a diagonal constant matrix whose diagonal elements are equal to the actuator stiffnesses \eqref{eq:detail_control_decoupling_plant_decentralized_low_freq}. +At high frequencies, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal. + +\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq} + \bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}} +\end{equation} + +The magnitude of the coupled plant \(\bm{G}_{\mathcal{L}}\) is illustrated in Figure \ref{fig:detail_control_decoupling_coupled_plant_bode}. +This representation confirms that at low frequencies (below the first suspension mode), the plant is well decoupled. +Depending on the symmetry present in the system, certain diagonal elements may exhibit identical values, as demonstrated for struts 2 and 3 in this example. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/detail_control_decoupling_coupled_plant_bode.png} +\caption{\label{fig:detail_control_decoupling_coupled_plant_bode}Model dynamics from actuator forces to relative displacement sensor of each strut.} +\end{figure} +\subsubsection{Jacobian Decoupling} +\label{ssec:detail_control_decoupling_jacobian} +\paragraph{Jacobian Matrix} + +The Jacobian matrix \(\bm{J}_{\{O\}}\) serves a dual purpose in the decoupling process: it converts strut velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\), and it transforms actuator forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\), as expressed in equation \eqref{eq:detail_control_decoupling_jacobian}. + +\begin{subequations}\label{eq:detail_control_decoupling_jacobian} + \begin{align} + \dot{\bm{\mathcal{X}}}_{\{O\}} &= \bm{J}_{\{O\}} \dot{\bm{\mathcal{L}}}, \quad \dot{\bm{\mathcal{L}}} = \bm{J}_{\{O\}}^{-1} \dot{\bm{\mathcal{X}}}_{\{O\}} \\ + \bm{\mathcal{F}}_{\{O\}} &= \bm{J}_{\{O\}}^{\intercal} \bm{\tau}, \quad \bm{\tau} = \bm{J}_{\{O\}}^{-\intercal} \bm{\mathcal{F}}_{\{O\}} + \end{align} +\end{subequations} + +The resulting plant (Figure \ref{fig:detail_control_jacobian_decoupling_arch}) have inputs and outputs with clear physical interpretations: +\begin{itemize} +\item \(\bm{\mathcal{F}}_{\{O\}}\) represents forces/torques applied on the payload at the origin of frame \(\{O\}\) +\item \(\bm{\mathcal{X}}_{\{O\}}\) represents translations/rotation of the payload expressed in frame \(\{O\}\) +\end{itemize} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/detail_control_decoupling_control_jacobian.png} +\caption{\label{fig:detail_control_jacobian_decoupling_arch}Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{O\}}\) to \(\bm{\mathcal{X}}_{\{O\}}\)} +\end{figure} + +The transfer function from \(\bm{\mathcal{F}}_{\{O\}\) to \(\bm{\mathcal{X}}_{\{O\}}\), denoted \(\bm{G}_{\{O\}}(s)\) can be computed using \eqref{eq:detail_control_decoupling_plant_jacobian}. + +\begin{equation}\label{eq:detail_control_decoupling_plant_jacobian} + \frac{\bm{\mathcal{X}}_{\{O\}}}{\bm{\mathcal{F}}_{\{O\}}}(s) = \bm{G}_{\{O\}}(s) = \left( \bm{J}_{\{O\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{O\}} s^2 + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{O\}} s + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{O\}} \right)^{-1} +\end{equation} + +The frame \(\{O\}\) can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice. +Two natural reference frames are particularly relevant: the center of mass and the center of stiffness. +\paragraph{Center Of Mass} + +When the decoupling frame is located at the center of mass (frame \(\{M\}\) in Figure \ref{fig:detail_control_decoupling_model_test}), the Jacobian matrix and its inverse are expressed as in \eqref{eq:detail_control_decoupling_jacobian_CoM_inverse}. + +\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse} + \bm{J}_{\{M\}} = \begin{bmatrix} + 1 & 0 & h_a \\ + 0 & 1 & -l_a \\ + 0 & 1 & l_a \\ + \end{bmatrix}, \quad \bm{J}_{\{M\}}^{-1} = \begin{bmatrix} + 1 & \frac{h_a}{2 l_a} & \frac{-h_a}{2 l_a} \\ + 0 & \frac{1}{2} & \frac{1}{2} \\ + 0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} \\ + \end{bmatrix} +\end{equation} + +Analytical formula of the plant \(\bm{G}_{\{M\}}(s)\) is derived \eqref{eq:detail_control_decoupling_plant_CoM}. + +\begin{equation}\label{eq:detail_control_decoupling_plant_CoM} + \frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1} +\end{equation} + +At high frequencies, the plant converges to the inverse of the mass matrix, which is a diagonal matrix \eqref{eq:detail_control_decoupling_plant_CoM_high_freq}. + +\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq} + \bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix} + 1/m & 0 & 0 \\ + 0 & 1/m & 0 \\ + 0 & 0 & 1/I + \end{bmatrix} +\end{equation} + +Consequently, the plant exhibits effective decoupling at frequencies above the highest suspension mode as shown in Figure \ref{fig:detail_control_decoupling_jacobian_plant_CoM}. +This strategy is typically employed in systems with low-frequency suspension modes \cite{butler11_posit_contr_lithog_equip}, where the plant approximates decoupled mass lines. + +The low-frequency coupling observed in this configuration has a clear physical interpretation. +When a static force is applied at the center of mass, the suspended mass rotates around the center of stiffness. +This rotation is due to torque induced by the stiffness of the first actuator (i.e. the one on the left side), which is not aligned with the force application point. +This phenomenon 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_decoupling_jacobian_plant_CoM.png} +\end{center} +\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_decoupling_model_test_CoM.png} +\end{center} +\subcaption{\label{fig:detail_control_decoupling_model_test_CoM}Static force applied at the CoM} +\end{subfigure} +\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} + +When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in \eqref{eq:detail_control_decoupling_jacobian_CoK_inverse}. + +\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoK_inverse} + \bm{J}_{\{K\}} = \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & 1 & -l_a \\ + 0 & 1 & l_a + \end{bmatrix}, \quad \bm{J}_{\{K\}}^{-1} = \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & \frac{1}{2} & \frac{1}{2} \\ + 0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} + \end{bmatrix} +\end{equation} + +The frame \(\{K\}\) was selected based on physical reasoning, positioned in line with the side strut and equidistant between the two vertical struts. +However, it could alternatively be determined through analytical methods to ensure that \(\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}\) forms a diagonal matrix. +It should be noted that the existence of such a center of stiffness (i.e. a frame \(\{K\}\) for which \(\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}\) is diagonal) is not guaranteed for arbitrary systems. +This property is typically achievable only in systems exhibiting specific symmetrical characteristics, as is the case in the present example. + +The analytical expression for the plant in this configuration was then computed \eqref{eq:detail_control_decoupling_plant_CoK}. + +\begin{equation}\label{eq:detail_control_decoupling_plant_CoK} + \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1} +\end{equation} + +Figure \ref{fig:detail_control_decoupling_jacobian_plant_CoK_results} presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the center of stiffness. +The plant is well decoupled below the suspension mode with the lowest frequency \eqref{eq:detail_control_decoupling_plant_CoK_low_freq}, making it particularly suitable for systems with high stiffness. + +\begin{equation}\label{eq:detail_control_decoupling_plant_CoK_low_freq} + \bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-\intercal} +\end{equation} + +The physical reason for high-frequency coupling is illustrated in Figure \ref{fig:detail_control_decoupling_model_test_CoK}. +When a high-frequency force is applied at a point not aligned with the center of mass, it induces 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_decoupling_jacobian_plant_CoK.png} +\end{center} +\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_decoupling_model_test_CoK.png} +\end{center} +\subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK} +\end{subfigure} +\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} +\subsubsection{Modal Decoupling} +\label{ssec:detail_control_decoupling_modal} +Modal decoupling represents an approach based on the principle that a mechanical system's behavior can be understood as a combination of contributions from various modes \cite{rankers98_machin}. +To convert the dynamics in the modal space, the equation of motion are first written with respect to the center of mass \eqref{eq:detail_control_decoupling_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\}}^{\intercal} \bm{\tau}(t) +\end{equation} + +For modal decoupling, a change of variables is introduced \eqref{eq:detail_control_decoupling_modal_coordinates} where \(\bm{\mathcal{X}}_{m}\) represents the modal amplitudes and \(\bm{\Phi}\) is a \(n \times n\)\footnote{\(n\) corresponds to the number of degrees of freedom, here \(n = 3\)} matrix whose columns correspond to the mode shapes of the system, computed from \(\bm{M}_{\{M\}}\) and \(\bm{K}_{\{M\}}\). + +\begin{equation}\label{eq:detail_control_decoupling_modal_coordinates} + \bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m} +\end{equation} + +By pre-multiplying equation \eqref{eq:detail_control_decoupling_equation_motion_CoM} by \(\bm{\Phi}^{\intercal}\) and applying the change of variable \eqref{eq:detail_control_decoupling_modal_coordinates}, a new set of equations of motion is obtained \eqref{eq:detail_control_decoupling_equation_modal_coordinates} where \(\bm{\tau}_m\) represents the modal input, while \(\bm{M}_m\), \(\bm{C}_m\), and \(\bm{K}_m\) denote the modal mass, damping, and stiffness matrices respectively. + +\begin{equation}\label{eq:detail_control_decoupling_equation_modal_coordinates} + \underbrace{\bm{\Phi}^{\intercal} \bm{M} \bm{\Phi}}_{\bm{M}_m} \bm{\ddot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{C} \bm{\Phi}}_{\bm{C}_m} \bm{\dot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{K} \bm{\Phi}}_{\bm{K}_m} \bm{\mathcal{X}}_m(t) = \underbrace{\bm{\Phi}^{\intercal} \bm{J}^{\intercal} \bm{\tau}(t)}_{\bm{\tau}_m(t)} +\end{equation} +The inherent mathematical structure of the mass, damping, and stiffness matrices \cite[, chapt. 8]{lang17_under} ensures that modal matrices are diagonal \cite[, chapt. 2.3]{preumont18_vibrat_contr_activ_struc_fourt_edition}. +This diagonalization transforms equation \eqref{eq:detail_control_decoupling_equation_modal_coordinates} into a set of \(n\) decoupled equations, enabling independent control of each mode without cross-interaction. + +To implement this approach from a decentralized plant, the architecture shown in Figure \ref{fig:detail_control_decoupling_modal} is employed. +Inputs of the decoupling plant are the modal modal inputs \(\bm{\tau}_m\) and the outputs are the modal amplitudes \(\bm{\mathcal{X}}_m\). +This implementation requires knowledge of the system's equations of motion, from which the mode shapes matrix \(\bm{\Phi}\) is derived. +The resulting decoupled system features diagonal elements each representing second-order resonant systems that are straightforward to control individually. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/detail_control_decoupling_modal.png} +\caption{\label{fig:detail_control_decoupling_modal}Modal Decoupling Architecture} +\end{figure} + +Modal decoupling was then applied to the test model. +First, the eigenvectors \(\bm{\Phi}\) of \(\bm{M}_{\{M\}}^{-1}\bm{K}_{\{M\}}\) were computed \eqref{eq:detail_control_decoupling_modal_eigenvectors}. +While analytical derivation of eigenvectors could be obtained for such a simple system, they are typically computed numerically for practical applications. + +\begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors} + \bm{\Phi} = \begin{bmatrix} + \frac{I - h_a^2 m - 2 l_a^2 m - \alpha}{2 h_a m} & 0 & \frac{I - h_a^2 m - 2 l_a^2 m + \alpha}{2 h_a m} \\ + 0 & 1 & 0 \\ + 1 & 0 & 1 + \end{bmatrix},\ \alpha = \sqrt{\left( I + m (h_a^2 - 2 l_a^2) \right)^2 + 8 m^2 h_a^2 l_a^2} +\end{equation} + +The numerical values for the eigenvector matrix and its inverse are shown in \eqref{eq:detail_control_decoupling_modal_eigenvectors_matrices}. + +\begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors_matrices} + \bm{\Phi} = \begin{bmatrix} + -0.905 & 0 & -0.058 \\ + 0 & 1 & 0 \\ + 0.424 & 0 & -0.998 + \end{bmatrix}, \quad + \bm{\Phi}^{-1} = \begin{bmatrix} + -1.075 & 0 & 0.063 \\ + 0 & 1 & 0 \\ + -0.457 & 0 & -0.975 + \end{bmatrix} +\end{equation} + +The two computed matrices were implemented in the control architecture of Figure \ref{fig:detail_control_decoupling_modal}, resulting in three distinct second order plants as depicted in Figure \ref{fig:detail_control_decoupling_modal_plant}. +Each of these diagonal elements corresponds to a specific mode, as shown in Figure \ref{fig:detail_control_decoupling_model_test_modal}, resulting in a perfectly decoupled system. + +\begin{figure}[htbp] +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_modal_plant.png} +\end{center} +\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_decoupling_model_test_modal.png} +\end{center} +\subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes} +\end{subfigure} +\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 invidiually address different modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal})} +\end{figure} +\subsubsection{SVD Decoupling} +\label{ssec:detail_control_decoupling_svd} +\paragraph{Singular Value Decomposition} + +Singular Value Decomposition (SVD) represents a powerful mathematical tool with extensive applications in data analysis \cite[, chapt. 1]{brunton22_data} and multivariable control systems where it is particularly valuable for analyzing directional properties in multivariable systems \cite{skogestad07_multiv_feedb_contr}. + +The SVD constitutes a unique matrix decomposition applicable to any complex matrix \(\bm{X} \in \mathbb{C}^{n \times m}\), expressed as: + +\begin{equation}\label{eq:detail_control_svd} + \bm{X} = \bm{U} \bm{\Sigma} \bm{V}^H +\end{equation} + +where \(\bm{U} \in \mathbb{C}^{n \times n}\) and \(\bm{V} \in \mathbb{C}^{m \times m}\) are unitary matrices with orthonormal columns, and \(\bm{\Sigma} \in \mathbb{R}^{n \times n}\) is a diagonal matrix with real, non-negative entries. +For real matrices \(\bm{X}\), the resulting \(\bm{U}\) and \(\bm{V}\) matrices are also real, making them suitable for decoupling applications. +\paragraph{Decoupling using the SVD} + +The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a Frequency Response Function (FRF), which yields a complex matrix \(\bm{G}(\omega_i)\) for multiple frequency points \(\omega_i\). +A specific frequency is then selected for optimal decoupling, with the targeted crossover frequency \(\omega_c\) often serving as an appropriate choice. + +Since real matrices are required for the decoupling transformation, a real approximation of the complex measured response at the selected frequency must be computed. +In this work, the method proposed in \cite{kouvaritakis79_theor_pract_charac_locus_desig_method} was used as it preserves maximal orthogonality in the directional properties of the input complex matrix. + +Following this approximation, a real matrix \(\tilde{\bm{G}}(\omega_c)\) is obtained, and SVD is performed on this matrix. +The resulting (real) unitary matrices \(\bm{U}\) and \(\bm{V}\) are structured such that \(\bm{V}^{-\intercal} \tilde{\bm{G}}(\omega_c) \bm{U}^{-1}\) forms a diagonal matrix. +These singular input and output matrices are then applied to decouple the system as illustrated in Figure \ref{fig:detail_control_decoupling_svd}, and the decoupled plant is described by \eqref{eq:detail_control_decoupling_plant_svd}. + +\begin{equation}\label{eq:detail_control_decoupling_plant_svd} + \bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-\intercal} +\end{equation} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/detail_control_decoupling_svd.png} +\caption{\label{fig:detail_control_decoupling_svd}Decoupled plant \(\bm{G}_{\text{SVD}}\) using the Singular Value Decomposition} +\end{figure} + +Implementation of SVD decoupling requires access to the system's FRF, at least in the vicinity of the desired decoupling frequency. +This information can be obtained either experimentally or derived from a model. +While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency. +Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping. +\paragraph{Example} + +Plant decoupling using the Singular Value Decomposition was then applied on the test model. +A decoupling frequency of \(\SI{100}{Hz}\) was used. +The plant response at that frequency, as well as its real approximation and the obtained \(\bm{U}\) and \(\bm{V}\) matrices are shown in \eqref{eq:detail_control_decoupling_svd_example}. + +\begin{equation}\label{eq:detail_control_decoupling_svd_example} +\begin{align} + & \bm{G}_{\{\mathcal{L}\}}(\omega_c = 2\pi \cdot 100) = 10^{-9} \begin{bmatrix} + -99 - j 2.6 & 74 + j 4.2 & -74 - j 4.2 \\ + 74 + j 4.2 & -247 - j 9.7 & 102 + j 7.0 \\ + -74 - j 4.2 & 102 + j 7.0 & -247 - j 9.7 + \end{bmatrix} \\ + & \xrightarrow[\text{approximation}]{\text{real}} \tilde{\bm{G}}_{\{\mathcal{L}\}}(\omega_c) = 10^{-9} \begin{bmatrix} + -99 & 74 & -74 \\ + 74 & -247 & 102 \\ + -74 & 102 & -247 + \end{bmatrix} \\ + & \xrightarrow[\text{SVD}]{\phantom{\text{approximation}}} \bm{U} = \begin{bmatrix} + 0.34 & 0 & 0.94 \\ + -0.66 & 0.71 & 0.24 \\ + 0.66 & 0.71 & -0.24 + \end{bmatrix}, \ \bm{V} = \begin{bmatrix} + -0.34 & 0 & -0.94 \\ + 0.66 & -0.71 & -0.24 \\ + -0.66 & -0.71 & 0.24 + \end{bmatrix} +\end{align} +\end{equation} + +Using these \(\bm{U}\) and \(\bm{V}\) matrices, the decoupled plant is computed according to equation \eqref{eq:detail_control_decoupling_plant_svd}. +The resulting plant, depicted in Figure \ref{fig:detail_control_decoupling_svd_plant}, exhibits remarkable decoupling across a broad frequency range, extending well beyond the vicinity of \(\omega_c\). +Additionally, the diagonal terms manifest as second-order dynamic systems, facilitating straightforward controller design. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/detail_control_decoupling_svd_plant.png} +\caption{\label{fig:detail_control_decoupling_svd_plant}Plant dynamics \(\bm{G}_{\text{SVD}}(s)\) obtained after decoupling using Singular Value Decomposition} +\end{figure} + +As it was surprising to obtain such a good decoupling at all frequencies, a variant system with identical dynamics but different sensor configurations was examined. +Instead of using relative motion sensors collocated with the struts, three relative motion sensors were positioned as shown in Figure \ref{fig:detail_control_decoupling_model_test_alt}. +Although Jacobian matrices could theoretically be used to map these sensors to the frame of the struts, application of the same SVD decoupling procedure yielded the plant response shown in Figure \ref{fig:detail_control_decoupling_svd_alt_plant}, which exhibits significantly greater coupling. +Notably, the coupling demonstrates local minima near the decoupling frequency, consistent with the fact that the decoupling matrices were derived specifically for that frequency point. + +\begin{figure}[htbp] +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test_alt.png} +\end{center} +\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_decoupling_svd_alt_plant.png} +\end{center} +\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_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}).} +\end{figure} + +The exceptional performance of SVD decoupling on the plant with collocated sensors warrants further investigation. +This effectiveness may be attributed to the symmetrical properties of the plant, as evidenced in the Bode plots of the decentralized plant shown in Figure \ref{fig:detail_control_decoupling_coupled_plant_bode}. +The phenomenon potentially relates to previous research on SVD controllers applied to systems with specific symmetrical characteristics \cite{hovd97_svd_contr_contr}. +\subsubsection{Comparison of decoupling strategies} +\label{ssec:detail_control_decoupling_comp} + +While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table \ref{tab:detail_control_decoupling_strategies_comp}. + +Each method employs a distinct conceptual framework: Jacobian decoupling is ``topology-driven'', relying on the geometric configuration of the system; modal decoupling is ``physics-driven'', based on the system's dynamical equations; and SVD decoupling is ``data-driven'', utilizing measured frequency response functions. + +The physical interpretation of decoupled plant inputs and outputs varies considerably among these methods. +With Jacobian decoupling, inputs and outputs retain clear physical meaning, corresponding to forces/torques and translations/rotations in a specified reference frame. +Modal decoupling arranges inputs to excite individual modes, with outputs combined to measure these modes separately. +For SVD decoupling, inputs and outputs represent special directions ordered by decreasing controllability and observability at the chosen frequency, though physical interpretation becomes challenging for parallel manipulators. + +This difference in interpretation relates directly to the ``control space'' in which the controllers operate. +When these ``control spaces'' meaningfully relate to the control objectives, controllers can be tuned to directly match specific requirements. +For Jacobian decoupling, the controller typically operates in a frame positioned at the point where motion needs to be controlled, for instance where the light is focused in the NASS application. +Modal decoupling provides a natural framework when specific vibrational modes require targeted control. +SVD decoupling generally results in a loss of physical meaning for the ``control space'', potentially complicating the process of relating controller design to practical system requirements. + +The quality of decoupling achieved through these methods also exhibits distinct characteristics. +Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the center of stiffness, or at high frequencies when aligned with the center of mass. +Systems designed with coincident centers of mass and stiffness may achieve excellent decoupling using this approach. +Modal decoupling offers good decoupling across all frequencies, though its effectiveness relies on the model accuracy, with discrepancies potentially resulting in significant off-diagonal elements. +SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point. + +\begin{table}[htbp] +\caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies} +\centering +\scriptsize +\begin{tabularx}{\linewidth}{lXXX} +\toprule + & \textbf{Jacobian} & \textbf{Modal} & \textbf{SVD}\\ +\midrule +\textbf{Philosophy} & Topology Driven & Physics Driven & Data Driven\\ +\midrule +\textbf{Requirements} & Known geometry & Known equations of motion & Identified FRF\\ +\midrule +\textbf{Decoupling Matrices} & Jacobian matrix \(\bm{J}_{\{O\}}\) & Eigenvectors \(\bm{\Phi}\) & SVD matrices \(\bm{U}\) and \(\bm{V}\)\\ +\midrule +\textbf{Decoupled Plant} & \(\bm{G}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-1} \bm{G}_{\mathcal{L}}(s) \bm{J}_{\{O\}}^{-\intercal}\) & \(\bm{G}_m(s) = \bm{\Phi}^{-1} \bm{G}_{\mathcal{X}}(s) \bm{\Phi}^{-\intercal}\) & \(\bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}(s) \bm{V}^{-\intercal}\)\\ +\midrule +\textbf{Controller} & \(\bm{K}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-\intercal} \bm{K}_{d}(s) \bm{J}_{\{O\}}^{-1}\) & \(\bm{K}_m(s) = \bm{\Phi}^{-\intercal} \bm{K}_{d}(s) \bm{\Phi}^{-1}\) & \(\bm{K}_{\text{SVD}}(s) = \bm{V}^{-\intercal} \bm{K}_{d}(s) \bm{U}^{-1}\)\\ +\midrule +\textbf{Interpretation} & Forces/Torques to Displacement/Rotation in chosen frame & Inputs (resp. outputs) to excite (resp. sense) individual modes & Directions of max to min controllability/observability\\ +\midrule +\textbf{Effectiveness} & Decoupling at low or high frequency depending on the chosen frame & Good decoupling at all frequencies & Good decoupling near the chosen frequency\\ +\midrule +\textbf{Pros} & Retain physical meaning of inputs / outputs. Controller acts on a meaningfully ``frame'' & Ability to target specific modes. Simple \(2^{nd}\) order diagonal plants & Good Decoupling near the crossover. Very General and requires no model\\ +\midrule +\textbf{Cons} & Good decoupling at all frequency can only be obtained for specific mechanical architecture & Relies on the accuracy of equation of motions. Robustness to unmodelled dynamics may be poor & Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor\\ +\bottomrule +\end{tabularx} +\end{table} +\subsection{Closed-Loop Shaping using Complementary Filters} +\label{sec:detail_control_cf} + +Once the system is properly decoupled using one of the approaches described in Section \ref{sec:detail_control_decoupling}, SISO controllers can be individually tuned for each decoupled ``directions''. +Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented. + +In some cases ``fixed'' controller structures are utilized, such as PI and PID controllers, whose parameters are manually tuned \cite{furutani04_nanom_cuttin_machin_using_stewar,du14_piezo_actuat_high_precis_flexib,yang19_dynam_model_decoup_contr_flexib}. + +Another popular method is Open-Loop shaping, which was used during the conceptual phase. +Open-loop shaping involves tuning the controller through a series of ``standard'' filters (leads, lags, notches, low-pass filters, \ldots{}) to shape the open-loop transfer function \(G(s)K(s)\) according to desired specifications, including bandwidth, gain and phase margins \cite[, chapt. 4.4.7]{schmidt20_desig_high_perfor_mechat_third_revis_edition}. +Open-Loop shaping is very popular because the open-loop transfer function is a linear function of the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics. +Another key advantage is that controllers can be tuned directly from measured frequency response functions of the plant without requiring an explicit model. + +However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions. +Specifications can therefore be expressed in terms of the magnitude of closed-loop transfer functions, such as the sensitivity, plant sensitivity, and complementary sensitivity transfer functions \cite[, chapt. 3]{skogestad07_multiv_feedb_contr}. +With open-loop shaping, closed-loop transfer functions are changed only indirectly, which may make it difficult to directly address the specifications that are in terms of the closed-loop transfer functions. + +In order to synthesize a controller that directly shapes the closed-loop transfer functions (and therefore the performance metric), \(\mathcal{H}_\infty\text{-synthesis}\) may be used \cite{skogestad07_multiv_feedb_contr}. +This approach requires a good model of the plant and expertise in selecting weighting functions that will define the wanted shape of different closed-loop transfer functions \cite{bibel92_guidel_h}. +\(\mathcal{H}_{\infty}\text{-synthesis}\) has been applied for the Stewart platform \cite{jiao18_dynam_model_exper_analy_stewar}, yet when benchmarked against more basic decentralized controllers, the performance gains proved small \cite{thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}. + +In this section, an alternative controller synthesis scheme is proposed in which complementary filters are used for directly shaping the closed-loop transfer functions (i.e., directly addressing the closed-loop performances). +In Section \ref{ssec:detail_control_cf_control_arch}, the proposed control architecture is presented. +In Section \ref{ssec:detail_control_cf_trans_perf}, typical performance requirements are translated into the shape of the complementary filters. +The design of the complementary filters is briefly discussed in Section \ref{ssec:detail_control_cf_analytical_complementary_filters}, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real time. +Finally, in Section \ref{ssec:detail_control_cf_simulations}, a numerical example is used to show how the proposed control architecture can be implemented in practice. +\subsubsection{Control Architecture} +\label{ssec:detail_control_cf_control_arch} +\paragraph{Virtual Sensor Fusion} + +The idea of using complementary filters in the control architecture originates from sensor fusion techniques \cite{collette15_sensor_fusion_method_high_perfor}, where two sensors are combined using complementary filters. +Building upon this concept, ``virtual sensor fusion'' \cite{verma20_virtual_sensor_fusion_high_precis_contr} replaces one physical sensor with a model \(G\) of the plant. +The corresponding control architecture is illustrated in Figure \ref{fig:detail_control_cf_arch}, where \(G^\prime\) represents the physical plant to be controlled, \(G\) is a model of the plant, \(k\) is the controller, and \(H_L\) and \(H_H\) are complementary filters satisfying \(H_L(s) + H_H(s) = 1\). +In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness. + +\begin{figure}[htbp] +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.9]{figs/detail_control_cf_arch.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_arch}Virtual Sensor Fusion} +\end{subfigure} +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.9]{figs/detail_control_cf_arch_eq.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_arch_eq}Equivalent Architecture} +\end{subfigure} +\caption{\label{fig:detail_control_cf_arch_and_eq}Control architecture for virtual sensor fusion (\subref{fig:detail_control_cf_arch}). An equivalent architecture is shown in (\subref{fig:detail_control_cf_arch_eq}). The signals are the reference signal \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\).} +\end{figure} + +Although the control architecture shown in Figure \ref{fig:detail_control_cf_arch} appears to be a multi-loop system, it should be noted that no non-linear saturation-type elements are present in the inner loop (containing \(k\), \(G\), and \(H_H\), all numerically implemented). +Consequently, this structure is mathematically equivalent to the single-loop architecture illustrated in Figure \ref{fig:detail_control_cf_arch_eq}. +\paragraph{Asymptotic behavior} + +When considering the extreme case of very high values for \(k\), the effective controller \(K(s)\) converges to the inverse of the plant model multiplied by the inverse of the high-pass filter, as expressed in \eqref{eq:detail_control_cf_high_k}. + +\begin{equation}\label{eq:detail_control_cf_high_k} + \lim_{k\to\infty} K(s) = \lim_{k\to\infty} \frac{k}{1+H_H(s) G(s) k} = \big( H_H(s) G(s) \big)^{-1} +\end{equation} + +If the resulting \(K\) is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization. +Furthermore, for \(K\) to be stable, both \(G\) and \(H_H\) must be minimum phase transfer functions. + +With these assumptions, the resulting control architecture is illustrated in Figure \ref{fig:detail_control_cf_arch_class}, where the complementary filters \(H_L\) and \(H_H\) remain the only tuning parameters. +The dynamics of this closed-loop system are described by equations \eqref{eq:detail_control_cf_cl_system_y} and \eqref{eq:detail_control_cf_cl_system_y}. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/detail_control_cf_arch_class.png} +\caption{\label{fig:detail_control_cf_arch_class}Equivalent classical feedback control architecture} +\end{figure} + +\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf} + \begin{align} + y &= \frac{ H_H dy + G^{\prime} G^{-1} r - G^{\prime} G^{-1} H_L n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cf_cl_system_y}\\ + u &= \frac{ -G^{-1} H_L dy + G^{-1} r - G^{-1} H_L n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cf_cl_system_u} + \end{align} +\end{subequations} + +At frequencies where the model accurately represents the physical plant (\(G^{-1} G^{\prime} \approx 1\)), the denominator simplifies to \(H_H + G^\prime G^{-1} H_L \approx H_H + H_L = 1\), and the closed-loop transfer functions are then described by equations \eqref{eq:detail_control_cf_cl_performance_y} and \eqref{eq:detail_control_cf_cl_performance_u}. + +\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect} + \begin{alignat}{5} + y &= H_H dy &&+ r &&- H_L n \label{eq:detail_control_cf_cl_performance_y} \\ + u &= -G^{-1} H_L dy &&+ G^{-1} r &&- G^{-1} H_L n \label{eq:detail_control_cf_cl_performance_u} + \end{alignat} +\end{subequations} + +The sensitivity transfer function equals the high-pass filter \(S = \frac{y}{dy} = H_H\), and the complementary sensitivity transfer function equals the low-pass filter \(T = \frac{y}{n} = H_L\). +Hence, when the plant model closely approximates the actual dynamics, the closed-loop transfer functions converge to the designed complementary filters, allowing direct translation of performance requirements into the design of the complementary. +\subsubsection{Translating the performance requirements into the shape of the complementary filters} +\label{ssec:detail_control_cf_trans_perf} +Performance specifications in a feedback system can usually be expressed as upper bounds on the magnitudes of closed-loop transfer functions such as the sensitivity and complementary sensitivity transfer functions \cite{bibel92_guidel_h}. +The design of a controller \(K(s)\) to obtain the desired shape of these closed-loop transfer functions is known as closed-loop shaping. + +In the proposed control architecture, the closed-loop transfer functions \eqref{eq:detail_control_cf_sf_cl_tf_K_inf} are expressed in terms of the complementary filters \(H_L(s)\) and \(H_H(s)\) rather than directly through the controller \(K(s)\). +Therefore, performance requirements must be translated into constraints on the shape of these complementary filters. +\paragraph{Nominal Stability (NS)} +A closed-loop system is stable when all its elements (here \(K\), \(G^\prime\), and \(H_L\)) are stable and the sensitivity function \(S = \frac{1}{1 + G^\prime K H_L}\) is stable. +For the nominal system (\(G^\prime = G\)), the sensitivity transfer function equals the high-pass filter: \(S(s) = H_H(s)\). + +Nominal stability is therefore guaranteed when \(H_L\), \(H_H\), and \(G\) are stable, and both \(G\) and \(H_H\) are minimum phase (ensuring \(K\) is stable). +Consequently, stable and minimum phase complementary filters must be employed. +\paragraph{Nominal Performance (NP)} + +Performance specifications can be formalized using weighting functions \(w_H\) and \(w_L\), where performance is achieved when \eqref{eq:detail_control_cf_weights} is satisfied. +The weighting functions define the maximum magnitude of the closed-loop transfer functions as a function of frequency, effectively determining their ``shape''. + +\begin{subequations}\label{eq:detail_control_cf_weights} + \begin{align} + |w_H(j\omega) S(j\omega)| &\le 1 \quad \forall\omega\\ + |w_L(j\omega) T(j\omega)| &\le 1 \quad \forall\omega + \end{align} +\end{subequations} + +For the nominal system, \(S = H_H\) and \(T = H_L\), hence the performance specifications can be converted on the shape of the complementary filters \eqref{eq:detail_control_cf_nominal_performance}. + +\begin{equation}\label{eq:detail_control_cf_nominal_performance} +\Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*} + |w_H(j\omega) H_H(j\omega)| \le 1 & \forall\omega \\ + |w_L(j\omega) H_L(j\omega)| \le 1 & \forall\omega + \end{cases*}}} +\end{equation} + +For disturbance rejection, the magnitude of the sensitivity function \(|S(j\omega)| = |H_H(j\omega)|\) should be minimized, particularly at low frequencies where disturbances are usually most prominent. +Similarly, for noise attenuation, the magnitude of the complementary sensitivity function \(|T(j\omega)| = |H_L(j\omega)|\) should be minimized, especially at high frequencies where measurement noise typically dominates. +Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function. +Typically, maintaining \(|S|_{\infty} \le 2\) ensures a gain margin of at least 2 and a phase margin of at least \(\SI{29}{\degree}\). + +Therefore, by carefully selecting the shape of the complementary filters, nominal performance specifications can be directly addressed in an intuitive manner. +\paragraph{Robust Stability (RS)} + +Robust stability refers to a control system's ability to maintain stability despite discrepancies between the actual system \(G^\prime\) and the model \(G\) used for controller design. +These discrepancies may arise from unmodeled dynamics or nonlinearities. + +To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure \ref{fig:detail_control_cf_input_uncertainty} is employed. +The set of possible plants \(\Pi_i\) is described by \eqref{eq:detail_control_cf_multiplicative_uncertainty}, with the weighting function \(w_I\) selected such that all possible plants \(G^\prime\) are contained within the set \(\Pi_i\). + +\begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty} + \Pi_i: \quad G^\prime(s) = G(s)\big(1 + w_I(s)\Delta_I(s)\big); \quad |\Delta_I(j\omega)| \le 1 \ \forall\omega +\end{equation} + +\begin{figure}[htbp] +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_cf_input_uncertainty.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_input_uncertainty}Input multiplicative uncertainty} +\end{subfigure} +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_cf_nyquist_uncertainty.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_nyquist_uncertainty}Nyquist plot - Effect of multiplicative uncertainty} +\end{subfigure} +\caption{\label{fig:detail_control_cf_input_uncertainty_nyquist}Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty})} +\end{figure} + +When considering input multiplicative uncertainty, robust stability can be derived graphically from the Nyquist plot (illustrated in Figure \ref{fig:detail_control_cf_nyquist_uncertainty}), yielding to \eqref{eq:detail_control_cf_robust_stability_graphically}, as demonstrated in \cite[, chapt. 7.5.1]{skogestad07_multiv_feedb_contr}. + +\begin{equation}\label{eq:detail_control_cf_robust_stability_graphically} + \text{RS} \Longleftrightarrow \left|w_I(j\omega) L(j\omega) \right| \le \left| 1 + L(j\omega) \right| \quad \forall\omega +\end{equation} + +After algebraic manipulation, robust stability is guaranteed when the low-pass complementary filter \(H_L\) satisfies \eqref{eq:detail_control_cf_condition_robust_stability}. + +\begin{equation}\label{eq:detail_control_cf_condition_robust_stability} + \boxed{\text{RS} \Longleftrightarrow |w_I(j\omega) H_L(j\omega)| \le 1 \quad \forall \omega} +\end{equation} +\paragraph{Robust Performance (RP)} + +Robust performance ensures that performance specifications \eqref{eq:detail_control_cf_weights} are met even when the plant dynamics fluctuates within specified bounds \eqref{eq:detail_control_cf_robust_perf_S}. + +\begin{equation}\label{eq:detail_control_cf_robust_perf_S} + \text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega +\end{equation} + +Transforming this condition into constraints on the complementary filters yields: + +\begin{equation}\label{eq:detail_control_cf_robust_performance} + \boxed{\text{RP} \Longleftrightarrow | w_H(j\omega) H_H(j\omega) | + | w_I(j\omega) H_L(j\omega) | \le 1, \ \forall\omega} +\end{equation} + +The robust performance condition effectively combines both nominal performance \eqref{eq:detail_control_cf_nominal_performance} and robust stability conditions \eqref{eq:detail_control_cf_condition_robust_stability}. +If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2 \cite[, chapt. 7.6]{skogestad07_multiv_feedb_contr}. +Therefore, for SISO systems, ensuring robust stability and nominal performance is typically sufficient. +\subsubsection{Complementary filter design} +\label{ssec:detail_control_cf_analytical_complementary_filters} + +As proposed in Section \ref{sec:detail_control_sensor}, complementary filters can be shaped using standard \(\mathcal{H}_{\infty}\text{-synthesis}\) techniques. +This approach is particularly well-suited since performance requirements were expressed as upper bounds on the magnitude of the complementary filters. + +Alternatively, analytical formulas for complementary filters may be employed. +For some applications, first-order complementary filters as shown in Equation \eqref{eq:detail_control_cf_1st_order} are sufficient. + +\begin{subequations}\label{eq:detail_control_cf_1st_order} + \begin{align} + H_L(s) &= \frac{1}{1 + s/\omega_0} \\ + H_H(s) &= \frac{s/\omega_0}{1 + s/\omega_0} + \end{align} +\end{subequations} + +These filters can be transformed into the digital domain using the Bilinear transformation, resulting in the digital filter representations shown in Equation \eqref{eq:detail_control_cf_1st_order_z}. + +\begin{subequations}\label{eq:detail_control_cf_1st_order_z} + \begin{align} + H_L(z^{-1}) &= \frac{T_s \omega_0 + T_s \omega_0 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} \\ + H_H(z^{-1}) &= \frac{2 - 2 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} + \end{align} +\end{subequations} + +A significant advantage of using analytical formulas for complementary filters is that key parameters such as \(\omega_0\) can be tuned in real-time, as illustrated in Figure \ref{fig:detail_control_cf_arch_tunable_params}. +This real-time tunability allows rapid testing of different control bandwidths to evaluate performance and robustness characteristics. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/detail_control_cf_arch_tunable_params.png} +\caption{\label{fig:detail_control_cf_arch_tunable_params}Implemented digital complementary filters with parameter \(\omega_0\) that can be changed in real time} +\end{figure} + +For many practical applications, first order complementary filters are not sufficient. +Specifically, a slope of \(+2\) at low frequencies for the sensitivity transfer function (enabling accurate tracking of ramp inputs) and a slope of \(-2\) for the complementary sensitivity transfer function are often desired. +For these cases, the complementary filters analytical formula in Equation \eqref{eq:detail_control_cf_2nd_order} is proposed. + +\begin{subequations}\label{eq:detail_control_cf_2nd_order} + \begin{align} + H_L(s) &= \frac{(1+\alpha) (\frac{s}{\omega_0})+1}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}\\ + H_H(s) &= \frac{(\frac{s}{\omega_0})^2 \left((\frac{s}{\omega_0})+1+\alpha\right)}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)} + \end{align} +\end{subequations} + +The influence of parameters \(\alpha\) and \(\omega_0\) on the frequency response of these complementary filters is illustrated in Figure \ref{fig:detail_control_cf_analytical_effect}. +The parameter \(\alpha\) primarily affects the damping characteristics near the crossover frequency as well as high and low frequency magnitudes, while \(\omega_0\) determines the frequency at which the transition between high-pass and low-pass behavior occurs. +These filters can also be implemented in the digital domain with analytical formulas, preserving the ability to adjust \(\alpha\) and \(\omega_0\) in real-time. + +\begin{figure}[htbp] +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_analytical_effect_alpha.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_analytical_effect_alpha}Effect of $\alpha$} +\end{subfigure} +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_analytical_effect_w0.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_analytical_effect_w0}Effect of $\omega_0$} +\end{subfigure} +\caption{\label{fig:detail_control_cf_analytical_effect}Shape of proposed analytical complementary filters. Effect of \(\alpha\) (\subref{fig:detail_control_cf_analytical_effect_alpha}) and \(\omega_0\) (\subref{fig:detail_control_cf_analytical_effect_w0}) are shown.} +\end{figure} +\subsubsection{Numerical Example} +\label{ssec:detail_control_cf_simulations} +To implement the proposed control architecture in practice, the following procedure is proposed: + +\begin{enumerate} +\item Identify the plant to be controlled to obtain the plant model \(G\). +\item Design the weighting function \(w_I\) such that all possible plants \(G^\prime\) are contained within the uncertainty set \(\Pi_i\). +\item Translate performance requirements into upper bounds on the complementary filters as explained in Section \ref{ssec:detail_control_cf_trans_perf}. +\item Design the weighting functions \(w_H\) and \(w_L\) and generate the complementary filters using \(\mathcal{H}_{\infty}\text{-synthesis}\) as described in Section \ref{ssec:detail_control_sensor_hinf_method}. +If the synthesis fails to produce filters satisfying the defined upper bounds, either revise the requirements or develop a more accurate model \(G\) that will allow for a smaller \(w_I\). +For simpler cases, the analytical formulas for complementary filters presented in Section \ref{ssec:detail_control_cf_analytical_complementary_filters} can be employed. +\item If \(K(s) = H_H^{-1}(s) G^{-1}(s)\) is not proper, add low-pass filters with sufficiently high corner frequencies to ensure realizability. +\end{enumerate} + +To evaluate this control architecture, a simple test model representative of many synchrotron positioning stages is utilized (Figure \ref{fig:detail_control_cf_test_model}). +In this model, a payload with mass \(m\) is positioned on top of a stage. +The objective is to accurately position the sample relative to the X-ray beam. + +The relative position \(y\) between the payload and the X-ray is measured, which typically involves measuring the relative position between the focusing optics and the sample. +Various disturbance forces affect positioning stability, including stage vibrations \(d_w\) and direct forces applied to the sample \(d_F\) (such as cable forces). +The positioning stage itself is characterized by stiffness \(k\), internal damping \(c\), and a controllable force \(F\). + +The model of the plant \(G(s)\) from actuator force \(F\) to displacement \(y\) is described by Equation \eqref{eq:detail_control_cf_test_plant_tf}. + +\begin{equation}\label{eq:detail_control_cf_test_plant_tf} + G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = 1\si{\N/\mu\m},\ c = 10^2\si{\N\per(\m\per\s)} +\end{equation} + +The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics and payload dynamics. +These uncertainties are represented using a multiplicative input uncertainty weight \eqref{eq:detail_control_cf_test_plant_uncertainty}, which specifies the magnitude of uncertainty as a function of frequency. + +\begin{equation}\label{eq:detail_control_cf_test_plant_uncertainty} + w_I(s) = 10 \cdot \frac{(s+100)^2}{(s+1000)^2} +\end{equation} + +Figure \ref{fig:detail_control_cf_bode_plot_mech_sys} illustrates both the nominal plant dynamics and the complete set of possible plants \(\Pi_i\) encompassed by the uncertainty model. + +\begin{figure}[htbp] +\begin{subfigure}{0.3\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_cf_test_model.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_test_model}Test model} +\end{subfigure} +\begin{subfigure}{0.66\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_cf_bode_plot_mech_sys.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_bode_plot_mech_sys}Bode plot of $G(s)$ and associated uncertainty set} +\end{subfigure} +\caption{\label{fig:detail_control_cf_test_model_plant}Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of the transfer function \(G(s)\) from \(F\) to \(y\) and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}).} +\end{figure} +\paragraph{Requirements and choice of complementary filters} + +As discussed in Section \ref{ssec:detail_control_cf_trans_perf}, nominal performance requirements can be expressed as upper bounds on the shape of the complementary filters. +For this example, the requirements are: +\begin{itemize} +\item track ramp inputs (i.e. constant velocity scans) with zero steady-state error: a \(+2\) slope at low frequencies for the magnitude of the sensitivity function \(|S(j\omega)|\) is required +\item filtering of measurement noise above \(\SI{300}{Hz}\), where sensor noise is significant (requiring a filtering factor of approximately 100 above this frequency) +\item maximizing disturbance rejection +\end{itemize} + +Additionally, robust stability must be ensured, requiring the closed-loop system to remain stable despite the dynamic uncertainties modeled by \(w_I\). +This condition is satisfied when the magnitude of the low-pass complementary filter \(|H_L(j\omega)|\) remains below the inverse of the uncertainty weight magnitude \(|w_I(j\omega)|\), as expressed in Equation \eqref{eq:detail_control_cf_condition_robust_stability}. + +Robust performance is achieved when both nominal performance and robust stability conditions are simultaneously satisfied. + +All requirements imposed on \(H_L\) and \(H_H\) are visualized in Figure \ref{fig:detail_control_cf_specs_S_T}. +While \(\mathcal{H}_\infty\text{-synthesis}\) could be employed to design the complementary filters, analytical formulas were used for this relatively simple example. +The second-order complementary filters from Equation \eqref{eq:detail_control_cf_2nd_order} were selected with parameters \(\alpha = 1\) and \(\omega_0 = 2\pi \cdot 20\,\text{Hz}\). +There magnitudes are displayed in Figure \ref{fig:detail_control_cf_specs_S_T}, confirming that these complementary filters are fulfilling the specifications. + +\begin{figure}[htbp] +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_specs_S_T.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_specs_S_T}Specifications and complementary filters} +\end{subfigure} +\begin{subfigure}{0.48\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_bode_Kfb.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_bode_Kfb}Bode plot of $K(s) \cdot H_L(s)$} +\end{subfigure} +\caption{\label{fig:detail_control_cf_specs_S_T_obtained_filters}Performance requirement and complementary filters used (\subref{fig:detail_control_cf_specs_S_T}). Obtained controller from the complementary filters and the plant inverse is shown in (\subref{fig:detail_control_cf_bode_Kfb}).} +\end{figure} +\paragraph{Controller analysis} + +The controller to be implemented takes the form \(K(s) = \tilde{G}^{-1}(s) H_H^{-1}(s)\), where \(\tilde{G}^{-1}(s)\) represents the plant inverse, which must be both stable and proper. +To ensure properness, low-pass filters with high corner frequencies are added as shown in Equation \eqref{eq:detail_control_cf_test_plant_inverse}. + +\begin{equation}\label{eq:detail_control_cf_test_plant_inverse} + \tilde{G}^{-1}(s) = \frac{m s^2 + c s + k}{1 + \frac{s}{2\pi \cdot 1000} + \left( \frac{s}{2\pi \cdot 1000} \right)^2} +\end{equation} + +The Bode plot of the controller multiplied by the complementary low-pass filter, \(K(s) \cdot H_L(s)\), is presented in Figure \ref{fig:detail_control_cf_bode_Kfb}. +The frequency response reveals several important characteristics: +\begin{itemize} +\item The presence of two integrators at low frequencies, enabling accurate tracking of ramp inputs +\item A notch at the plant resonance frequency (arising from the plant inverse) +\item A lead component near the control bandwidth of approximately 20 Hz, enhancing stability margins +\end{itemize} +\paragraph{Robustness and Performance analysis} + +Robust stability is assessed using the Nyquist plot shown in Figure \ref{fig:detail_control_cf_nyquist_robustness}. +Even when considering all possible plants within the uncertainty set, the Nyquist plot remains sufficiently distant from the critical point \((-1,0)\), indicating robust stability with adequate margins. + +Performance is evaluated by examining the closed-loop sensitivity and complementary sensitivity transfer functions, as illustrated in Figure \ref{fig:detail_control_cf_robust_perf}. +It is shown that the sensitivity transfer function achieves the desired \(+2\) slope at low frequencies and that the complementary sensitivity transfer function nominally provides the wanted noise filtering. + +\begin{figure}[htbp] +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_nyquist_robustness.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust Stability} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_robust_perf.png} +\end{center} +\subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and Robust performance} +\end{subfigure} +\caption{\label{fig:detail_control_cf_simulation_results}Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf})} +\end{figure} +\subsubsection{Conclusion} + +In this section, a control architecture in which complementary filters are used for closed-loop shaping has been presented. +This approach differs from traditional open-loop shaping in that no controller is manually designed; rather, appropriate complementary filters are selected to achieve the desired closed-loop behavior. +The method shares conceptual similarities with mixed-sensitivity \(\mathcal{H}_{\infty}\text{-synthesis}\), as both approaches aim to shape closed-loop transfer functions, but with notable distinctions in implementation and complexity. + +While \(\mathcal{H}_{\infty}\text{-synthesis}\) offers greater flexibility and can be readily generalized to MIMO plants, the presented approach provides a simpler alternative that requires minimal design effort. +Implementation only necessitates extracting a model of the plant and selecting appropriate analytical complementary filters, making it particularly interesting for applications where simplicity and intuitive parameter tuning are valued. + +Due to time constraints, an extensive literature review comparing this approach with similar existing architectures, such as Internal Model Control \cite{saxena12_advan_inter_model_contr_techn}, was not conducted. +Consequently, it remains unclear whether the proposed architecture offers significant advantages over existing methods in the literature. + +The control architecture has been presented for SISO systems, but can be applied to MIMO systems when sufficient decoupling is achieved. +It will be experimentally validated with the NASS during the experimental phase. +\subsection{Conclusion} +\label{sec:detail_control_conclusion} + +In order to optimize the control of the Nano Active Stabilization System, several aspects of control theory were studied. +Different approaches to combine sensors were compared in Section \ref{sec:detail_control_sensor}. +While High Authority Control-Low Authority Control (HAC-LAC) was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters. +It was demonstrated that the performance of such fusion depends significantly on the magnitude of the complementary filters. +To address this challenge, a synthesis method based on \(\mathcal{H}_\infty\text{-synthesis}\) was proposed, allowing for intuitive shaping of the complementary filters through weighting functions. +For the NASS, while HAC-LAC remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation. + +Various decoupling strategies for parallel manipulators were examined in Section \ref{sec:detail_control_decoupling}, including decentralized control, Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. +The main characteristics of each approach were highlighted, providing valuable insights into their respective strengths and limitations. +Among the examined methods, Jacobian decoupling was determined to be most appropriate for the NASS, as it provides straightforward implementation while preserving the physical meaning of inputs and outputs. + +With the system successfully decoupled, attention shifted to designing appropriate SISO controllers for each decoupled direction. +A control architecture for directly shaping closed-loop transfer functions was proposed. +It is based on complementary filters that can be designed using either the proposed \(\mathcal{H}_\infty\text{-synthesis}\) approach described earlier or through analytical formulas. +Experimental validation of this method on the NASS will be conducted during the experimental tests on ID31. \section{Choice of Instrumentation} \label{sec:detail_instrumentation} This chapter presents an approach to select and validate appropriate instrumentation for the Nano Active Stabilization System (NASS), ensuring each component meets specific performance requirements. @@ -6788,7 +8353,7 @@ Small Signal Bandwidth \(> 5\,kHz\) & \(6.4\,kHz\) & \(300\,Hz\) & \(30\,kHz\) Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\) & n/a & n/a\\ \bottomrule \end{tabularx} -\end{table}\footnotetext[25]{\label{orgb6f9e1e}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)} +\end{table}\footnotetext[27]{\label{orgb6be1d7}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)} \subsubsection{ADC and DAC} Analog-to-digital converters and digital-to-analog converters play key roles in the system, serving as the interface between the digital RT controller and the analog physical plant. The proper selection of these components is critical for system performance. @@ -6951,7 +8516,7 @@ All ADC channels demonstrated similar performance, so only one channel's noise p If necessary, oversampling can be applied to further reduce the noise \cite{lab13_improv_adc}. To gain \(w\) additional bits of resolution, the oversampling frequency \(f_{os}\) should be set to \(f_{os} = 4^w \cdot F_s\). Given that the ADC can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4). -This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV) \cite{hauser91_princ_overs_conver}. +This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV) \cite{hauser91_princ_overs_d_conver}. \begin{figure}[htbp] \centering @@ -7207,14 +8772,62 @@ This rigorous methodology spanning requirement formulation, component selection, \end{itemize} \section*{Detailed Design - Conclusion} \label{sec:detail_conclusion} + +In this chapter, a comprehensive approach to the detailed design of the nano-hexapod for the Nano Active Stabilization System has been presented. +The design process was structured around four key aspects: geometry optimization, component design, control strategy refinement, and instrumentation selection. + +The geometry optimization began with a review of existing Stewart platform designs, followed by analytical modeling of the relationship between geometric parameters and performance characteristics. +While cubic architectures are prevalent in the literature due to their purported advantages in decoupling and uniform stiffness, the analysis revealed that these benefits are more nuanced than commonly described. +For the nano-hexapod application, struts were oriented more vertically than in a cubic configuration to address the stringent vertical performance requirements and to better match the micro-station's modal characteristics. + +For component optimization, a hybrid modeling methodology was used that combined finite element analysis with multi-body dynamics. +This approach, validated experimentally using an Amplified Piezoelectric Actuator, enabled both detailed component-level optimization and efficient system-level simulation. +Through this methodology, the APA300ML was selected as the optimal actuator, offering the necessary combination of stroke, stiffness, and force sensing capabilities required for the application. +Similarly, the flexible joints were designed with careful consideration of bending and axial stiffness requirements, resulting in a design that balances competing mechanical demands. + +For control optimization, three critical challenges were addressed. +First, the problem of optimally combining multiple sensors was investigated and was focused on the design of complementary filters for sensor fusion. +A \(\mathcal{H}_{\infty}\text{-synthesis}\) technique was formulated for designing complementary filters with precisely shaped magnitude responses. +Second, various decoupling strategies for parallel manipulators were compared, filling a notable gap in current literature. +Among the evaluated techniques (decentralized control, Jacobian decoupling, modal decoupling, and SVD decoupling), Jacobian decoupling was identified as the most suitable for the NASS due to its simplicity and ability to maintain physical interpretation of the decoupled plant's inputs and outputs. +Third, a novel control architecture was developed that leverages complementary filters for direct shaping of closed-loop transfer functions. +This framework, which will be validated during the experimental phase, offers an intuitive alternative to traditional methods by allowing designers to directly specify desired closed-loop characteristics in a simple and intuitive way. + +The instrumentation selection was guided by dynamic error budgeting, which established maximum acceptable noise specifications for each component. +The selected components—including the IO131 ADC/DAC board, PD200 voltage amplifiers, and Vionic linear encoders—were then experimentally characterized to verify their performance. +All components were found to meet or exceed their specifications, with the combined effect of all noise sources estimated to induce vertical sample vibrations of only \(1.5\,\text{nm RMS}\), well below the \(15\,\text{nm RMS}\) requirement. + +The outcome of this detailed design process is a nano-hexapod and associated instrumentation specifically tailored to the NASS applications. +Following the completion of this design phase and the subsequent procurement of all specified components, the project progressed to the experimental validation stage, which forms the focus of the next chapter. \chapter{Experimental Validation} \label{chap:test} \minitoc \subsubsection*{Abstract} +The experimental validation follows a systematic approach, beginning with the characterization of individual components before advancing to evaluate the assembled system's performance (illustrated in Figure \ref{fig:chapter3_overview}). +Section \ref{sec:test_apa} focuses on the Amplified Piezoelectric Actuator (APA300ML), examining its electrical properties, and dynamical behavior. +Two models are developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from finite element analysis. +The implementation of Integral Force Feedback is also experimentally evaluated to assess its effectiveness in adding damping to the system. + +In Section \ref{sec:test_joints}, the flexible joints are characterized to ensure they meet the required specifications for stiffness and stroke. +A dedicated test bench is developed to measure the bending stiffness, with error analysis performed to validate the measurement accuracy. + +Section \ref{sec:test_struts} examines the assembly and testing of the struts, which integrate the APAs and flexible joints. +The mounting procedure is detailed, with particular attention to ensure consistent performance across multiple struts. +Dynamical measurements are performed to verify whether the dynamics of the struts are corresponding to the multi-body model. + +The assembly and testing of the complete nano-hexapod is presented in Section \ref{sec:test_nhexa}. +A suspended table is developed to isolate the hexapod's dynamics from support dynamics, enabling accurate identification of its dynamical properties. +The experimental frequency response functions are compared with the multi-body model predictions to validate the modeling approach. +The effects of various payload masses are also investigated. + +Finally, Section \ref{sec:test_id31} presents the validation of the NASS on the ID31 beamline. +A short-stroke metrology system is developed to measure the sample position relative to the granite base. +The HAC-LAC control architecture is implemented and tested under various experimental conditions, including payload masses up to \(39\,\text{kg}\) and for typical experiments, including tomography scans, reflectivity measurements, and diffraction tomography. + \begin{figure}[htbp] \centering \includegraphics[scale=1,width=\linewidth]{figs/chapter3_overview.png} -\caption{\label{fig:chapter3_overview}Figure caption} +\caption{\label{fig:chapter3_overview}Overview of the Experimental validation phase. The actuators and flexible joints and individual tested and then integrated into the struts. The Nano-hexapod is then mounted and the complete system is validated on the ID31 beamline.} \end{figure} \section{Amplified Piezoelectric Actuator} \label{sec:test_apa} @@ -10535,10 +12148,34 @@ The system's ability to maintain precise sample positioning across a wide range Moreover, the systematic approach to system development and validation, along with a detailed understanding of performance limitations, provides valuable insights for future improvements and potential applications in similar high-precision positioning systems. \section*{Experimental Validation - Conclusion} \label{sec:test_conclusion} + +The experimental validation detailed in this chapter confirms that the Nano Active Stabilization System successfully augments the positioning capabilities of the micro-station, thereby enabling full utilization of the ESRF's new light source potential. +A methodical approach was employed—first characterizing individual components and subsequently testing the integrated system—to comprehensively evaluate the NASS performance. + +Initially, the Amplified Piezoelectric Actuators (APA300ML) were characterized, revealing consistent mechanical and electrical properties across multiple units. +The implementation of Integral Force Feedback was shown to add significant damping to the system. +Two models of the APA300ML were developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from finite element analysis. +Both models accurately represented the axial dynamics of the actuators, with the super-element model additionally capturing flexible modes. + +The flexible joints were examined for geometric accuracy and bending stiffness, with measurements confirming compliance with design specifications. +The experimental values demonstrated good agreement with finite element predictions, validating the joint design and enabling accurate modeling of their behavior within the system. + +The strut assembly process was optimized to ensure consistent performance across the six struts. +Dynamic testing revealed complex behavior in the transfer function from piezoelectric voltage to encoder displacement, attributed to misalignment between flexible joints and actuators. +This finding led to the strategic decision to mount encoders on the nano-hexapod plates rather than on the struts. + +The nano-hexapod was then assembled and mounted on a suspended table to characterize its dynamic behavior. +The measurement setup enabled isolation of the hexapod's dynamics from potential influence of complex support dynamics. +The experimental frequency response functions exhibited good correlation with the multi-body model, confirming that the model can be used for control system design. + +Finally, the complete NASS was validated on the ID31 beamline using a short-stroke metrology system. +The HAC-LAC control architecture successfully provided robust active damping of suspension modes and rejection of low-frequency disturbances across various payload conditions. +Comprehensive testing under typical experimental scenarios—including tomography scans, reflectivity scans, and diffraction tomography—demonstrated the NASS ability to maintain the positioning errors within specifications (\(30\,\text{nm RMS}\) in lateral direction, \(15\,\text{nm RMS}\) in vertical direction, and \(250\,\text{nrad RMS}\) in tilt direction). +The system performed exceptionally well during vertical scans, though some limitations were identified during rapid lateral scanning and with heavier payloads. + +With the implementation of an accurate online metrology system, the NASS will be ready for integration into the beamline environment, significantly enhancing the capabilities of high-precision X-ray experimentation on the ID31 beamline. \chapter{Conclusion and Future Work} \label{chap:conclusion} -\section{Alternative Architecture} -\url{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/alternative-micro-station-architecture.org} \printbibliography[heading=bibintoc,title={Bibliography}] \chapter*{List of Publications} \begin{refsection}[ref.bib] diff --git a/setup.org b/setup.org index cdcdb06..662c1bb 100644 --- a/setup.org +++ b/setup.org @@ -162,9 +162,9 @@ See more information [[https://tex.stackexchange.com/questions/68516/avoid-that- style=numeric-comp, % bib style hyperref=true, % activate hyperref support backref=true, % activate backrefs - isbn=false, % don't show isbn tags + isbn=true, % don't show isbn tags url=false, % don't show url tags - doi=false, % don't show doi tags + doi=true, % don't show doi tags urldate=long, % display type for dates maxnames=3, % minnames=1, % @@ -187,17 +187,15 @@ See more information [[https://tex.stackexchange.com/questions/68516/avoid-that- {\item} #+end_src -Remove ISSN, DOI and URL to shorten the bibliography. +Remove ISBN if DOI is available. #+begin_src latex -\AtEveryBibitem{% - \clearfield{urlyear} - \clearfield{urlmonth} - \clearfield{note} - \clearfield{issn} % Remove issn - \clearfield{doi} % Remove doi -\ifentrytype{online}{}{% Remove url except for @online - \clearfield{url} -} +\DeclareSourcemap{ + \maps[datatype=bibtex]{ + \map{ + \step[fieldsource=doi,final] + \step[fieldset=isbn,null] + } + } } #+end_src @@ -301,6 +299,23 @@ And then use quotes as: Note that org-ref links won't work here because the attr latex will be pasted as-is in the .tex file. +** Footnotes +#+begin_src latex +\usepackage[perpage]{footmisc} +#+end_src + +** Automatic non-breaking space for citations +#+begin_src latex +\let\myRef\ref +\renewcommand\ref{\unskip~\myRef} + +\let\myEqref\eqref +\renewcommand\eqref{\unskip~\myEqref} + +\let\myCite\cite +\renewcommand\cite{\unskip~\myCite} +#+end_src + ** Setspace for controlling line spacing #+begin_src latex \usepackage{setspace}