Rework figures, add colors

journal/journal.org

@@ -254,12 +254,27 @@ Finally, concluding remarks are presented in Section [[*Concluding remarks][5]].
 
 * Complementary Filters Requirements
 <<sec:requirements>>
+** Introduction                                                      :ignore:
+
+Complementary filters provides a framework for fusing signals from different sensors.
+As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
+These requirements are discussed in this section.
+
 ** Sensor Models
 <<sec:sensor_models>>
 
-- Noise + dynamical uncertainty
+- Noise + dynamics
+
+#+name: fig:sensor_model
+#+caption: Basic Sensor Model
+[[file:figs/sensor_model.pdf]]
+
 - Suppose we calibrate the sensors
 
+#+name: fig:sensor_model_calibrated
+#+caption: Calibrated Sensor
+[[file:figs/sensor_model_calibrated.pdf]]
+
 ** Sensor Fusion Architecture
 <<sec:sensor_fusion>>
 
@@ -322,6 +337,15 @@ In practical systems the sensor dynamics is not perfect and eqref:eq:perfect_dyn
 In such case, one can use an inversion filter $\hat{G}_i^{-1}(s)$ to normalize the sensor dynamics, where $\hat{G}_i(s)$ is an estimate of the sensor dynamics $G_i(s)$.
 However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and $\hat{G}_i^{-1}(s) G_i(s) \neq 1$.
 
+#+name: fig:sensor_model_uncertainty
+#+caption: Input Uncertainty
+[[file:figs/sensor_model_uncertainty.png]]
+
+#+name: fig:sensor_model_uncertainty_simplified
+#+caption: Input Uncertainty
+#+RESULTS:
+[[file:figs/sensor_model_uncertainty_simplified.png]]
+
 Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. ref:fig:sensor_fusion_dynamic_uncertainty where $\Delta_i$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $|w_i(s)|$ is a weight representing the magnitude of the uncertainty.
 
 #+name: fig:sensor_fusion_dynamic_uncertainty
@@ -462,7 +486,7 @@ Let's validate the proposed design method of complementary filters with a simple
 - the gain of both filters is equal to $10^{-3}$ away from the merging frequency
 
 The weighting functions $W_1(s)$ and $W_2(s)$ are designed using eqref:eq:weight_formula.
-The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_synthesis_results.
+The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_filters_results.
 
 #+name: tab:weights_params
 #+caption: Parameters used for $W_1(s)$ and $W_2(s)$
@@ -476,16 +500,15 @@ The parameters used are summarized in table ref:tab:weights_params and the magni
 | $G_c$                  | $0.5$    | $0.5$    |
 | $n$                    | $2$      | $3$      |
 
-The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_synthesis_results and their transfer functions in the Laplace domain are given below.
+The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_filters_results and their transfer functions in the Laplace domain are given below.
 \begin{align*}
   H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
   H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
 \end{align*}
 
-#+name: fig:hinf_synthesis_results
+#+name: fig:hinf_filters_results
 #+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
-#+attr_latex: :scale 1
-[[file:figs/hinf_synthesis_results.pdf]]
+[[file:figs/hinf_filters_results.pdf]]
 
 * Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO
 <<sec:application_ligo>>

journal/journal.tex

@@ -1,28 +1,21 @@
-% Created 2021-04-28 mer. 15:56
+% Created 2021-04-30 ven. 11:16
 % Intended LaTeX compiler: pdflatex
 \documentclass[preprint, sort&compress]{elsarticle}
-
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{graphicx}
 \usepackage{grffile}
+\usepackage{longtable}
+\usepackage{wrapfig}
 \usepackage{rotating}
 \usepackage[normalem]{ulem}
+\usepackage{amsmath}
+\usepackage{textcomp}
+\usepackage{amssymb}
 \usepackage{capt-of}
 \usepackage{hyperref}
-\usepackage{bm}
-\usepackage{array}
-\usepackage{amsmath,amssymb,amsfonts}
-\usepackage{algorithmic}
-\usepackage{textcomp}
-\usepackage{cases}
-\usepackage{tabularx,siunitx,booktabs}
-\usepackage{algorithmic}
-\usepackage{import}
-\usepackage{hyperref}
-\usepackage[hyperref]{xcolor}
-\hypersetup{colorlinks=true}
-\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
+\usepackage{amsfonts}
+\usepackage{siunitx}
 \journal{Mechanical Systems and Signal Processing}
 \author[a1,a2]{Thomas Dehaeze\corref{cor1}}
 \author[a3,a4]{Mohit Verma}
@@ -32,6 +25,12 @@
 \address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
 \address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
 \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
+\usepackage{tabularx}
+\usepackage{booktabs}
+\usepackage{array}
+\usepackage[hyperref]{xcolor}
+\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
+\hypersetup{colorlinks=true}
 \date{}
 \title{A new method of designing complementary filters for sensor fusion using \(\mathcal{H}_\infty\) synthesis}
 \begin{document}
@@ -58,7 +57,7 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
 \end{frontmatter}
 
 \section{Introduction}
-\label{sec:orgf244196}
+\label{sec:org341e767}
 \label{sec:introduction}
 \begin{itemize}
 \item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
@@ -78,30 +77,64 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
 \item \cite{robert12_introd_random_signal_applied_kalman} advantage of complementary filters over Kalman filtering
 \end{itemize}
 \begin{itemize}
-\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
-\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
-\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory
-\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
-\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
+\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}
+
 \item Analytical methods:
 \begin{itemize}
 \item first order: \cite{corke04_inert_visual_sensin_system_small_auton_helic}
 \item second order: \cite{baerveldt97_low_cost_low_weigh_attit}, \cite{stoten01_fusion_kinet_data_using_compos_filter}, \cite{jensen13_basic_uas}
 \item higher order: \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, \cite{zimmermann92_high_bandw_orien_measur_contr}, \cite{collette15_sensor_fusion_method_high_perfor}, \cite{matichard15_seism_isolat_advan_ligo}
 \end{itemize}
-\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}
+\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
+\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
+\item Similar to feedback system:
+\begin{itemize}
+\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
+\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory
+\end{itemize}
+
+\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
 \end{itemize}
 \begin{itemize}
 \item Robustness problems: \cite{zimmermann92_high_bandw_orien_measur_contr} change of phase near the merging frequency
 \item Trial and error
 \item Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available.
 \end{itemize}
+Most of the requirements => shape of the complementary filters
+=> propose a way to shape complementary filters.
 
 \section{Complementary Filters Requirements}
-\label{sec:org2279a0f}
+\label{sec:org77471d1}
 \label{sec:requirements}
+Complementary filters provides a framework for fusing signals from different sensors.
+As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
+These requirements are discussed in this section.
+\subsection{Sensor Models}
+\label{sec:org363af04}
+\label{sec:sensor_models}
+
+\begin{itemize}
+\item Noise + dynamics
+\end{itemize}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model.pdf}
+\caption{\label{fig:sensor_model}Basic Sensor Model}
+\end{figure}
+
+\begin{itemize}
+\item Suppose we calibrate the sensors
+\end{itemize}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
+\caption{\label{fig:sensor_model_calibrated}Calibrated Sensor}
+\end{figure}
+
 \subsection{Sensor Fusion Architecture}
-\label{sec:org60aa99d}
+\label{sec:org240da2b}
 \label{sec:sensor_fusion}
 
 Let's consider two sensors measuring the same physical quantity \(x\) with dynamics \(G_1(s)\) and \(G_2(s)\), and with uncorrelated noise characteristics \(n_1\) and \(n_2\).
@@ -125,7 +158,7 @@ The complementary property of \(H_1(s)\) and \(H_2(s)\) implies that their trans
 \end{equation}
 
 \subsection{Noise Sensor Filtering}
-\label{sec:orgf9a3723}
+\label{sec:orgc5064da}
 \label{sec:noise_filtering}
 
 Let's first consider sensors with perfect dynamics
@@ -159,13 +192,25 @@ Usually, the two sensors have high noise levels over distinct frequency regions.
 In order to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\).
 
 \subsection{Robustness of the Fusion}
-\label{sec:org227af67}
+\label{sec:orgfc9ea9e}
 \label{sec:fusion_robustness}
 
 In practical systems the sensor dynamics is not perfect and \eqref{eq:perfect_dynamics} is not verified.
 In such case, one can use an inversion filter \(\hat{G}_i^{-1}(s)\) to normalize the sensor dynamics, where \(\hat{G}_i(s)\) is an estimate of the sensor dynamics \(G_i(s)\).
 However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and \(\hat{G}_i^{-1}(s) G_i(s) \neq 1\).
 
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_uncertainty.png}
+\caption{\label{fig:sensor_model_uncertainty}Input Uncertainty}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_uncertainty_simplified.png}
+\caption{\label{fig:sensor_model_uncertainty_simplified}Input Uncertainty}
+\end{figure}
+
 Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. \ref{fig:sensor_fusion_dynamic_uncertainty} where \(\Delta_i\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(|w_i(s)|\) is a weight representing the magnitude of the uncertainty.
 
 \begin{figure}[htbp]
@@ -204,13 +249,13 @@ where \(\Delta \phi_\text{max}\) is the maximum allowed added phase.
 Thus the norm of the complementary filter \(|H_i|\) should be made small at frequencies where \(|w_i|\) is large.
 
 \section{Complementary Filters Shaping using \(\mathcal{H}_\infty\) Synthesis}
-\label{sec:org10fbb17}
+\label{sec:org678f099}
 \label{sec:hinf_method}
 As shown in Sec. \ref{sec:requirements}, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms.
 Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary.
-\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
-\label{sec:org933b14f}
-\label{sec:hinf_synthesis}
+\subsection{Synthesis Objective}
+\label{sec:orgf726b5b}
+\label{sec:synthesis_objective}
 The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:comp_filter}.
 This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:comp_filter_problem_form} are satisfied.
 \begin{subequations}
@@ -223,6 +268,9 @@ This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(
 \end{subequations}
 where \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are chosen to shape the norms of the corresponding filters.
 
+\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
+\label{sec:orga266a36}
+\label{sec:hinf_synthesis}
 In order to express this optimization problem as a standard \(\mathcal{H}_\infty\) problem, the architecture shown in Fig. \ref{fig:h_infinity_robust_fusion} is used where the generalized plant \(P\) is described by \eqref{eq:generalized_plant}.
 \begin{equation}
 \label{eq:generalized_plant}
@@ -257,7 +305,7 @@ The conditions \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} on the filter
 Therefore, all the conditions \eqref{eq:comp_filter_problem_form} are satisfied using this synthesis method based on \(\mathcal{H}_\infty\) synthesis, and thus it permits to shape complementary filters as desired.
 
 \subsection{Weighting Functions Design}
-\label{sec:org3bb6eca}
+\label{sec:org911c399}
 \label{sec:hinf_weighting_func}
 The proper design of the weighting functions is of primary importance for the success of the presented complementary filters \(\mathcal{H}_\infty\) synthesis.
 
@@ -302,7 +350,7 @@ The general shape of a weighting function generated using \eqref{eq:weight_formu
 \end{figure}
 
 \subsection{Validation of the proposed synthesis method}
-\label{sec:org5517901}
+\label{sec:org6867aff}
 \label{sec:hinf_example}
 Let's validate the proposed design method of complementary filters with a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
 \begin{itemize}
@@ -313,7 +361,7 @@ Let's validate the proposed design method of complementary filters with a simple
 \end{itemize}
 
 The weighting functions \(W_1(s)\) and \(W_2(s)\) are designed using \eqref{eq:weight_formula}.
-The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_synthesis_results}.
+The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_filters_results}.
 
 \begin{table}[htbp]
 \caption{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\)}
@@ -331,7 +379,7 @@ Parameter & \(W_1(s)\) & \(W_2(s)\)\\
 \end{tabularx}
 \end{table}
 
-The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_synthesis_results} and their transfer functions in the Laplace domain are given below.
+The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_filters_results} and their transfer functions in the Laplace domain are given below.
 \begin{align*}
   H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
   H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
@@ -339,50 +387,12 @@ The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,scale=1]{figs/hinf_synthesis_results.pdf}
-\caption{\label{fig:hinf_synthesis_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
-\end{figure}
-
-\subsection{Synthesis of Three Complementary Filters}
-\label{sec:org1fd1484}
-\label{sec:hinf_three_comp_filters}
-Some applications may require to merge more than two sensors.
-In such a case, it is necessary to design as many complementary filters as the number of sensors used.
-The synthesis problem is then to compute \(n\) stable transfer functions \(H_i(s)\) such that \eqref{eq:hinf_problem_gen} is satisfied.
-\begin{subequations}
-\label{eq:hinf_problem_gen}
-  \begin{align}
-  & \sum_{i=0}^n H_i(s) = 1 \label{eq: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:hinf_cond_perf_gen}
-  \end{align}
-\end{subequations}
-The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. \ref{fig:comp_filter_three_hinf}.
-
-The \(\mathcal{H}_\infty\) synthesis objective applied on \(P(s)\) is to design two stable filters \(H_2(s)\) and \(H_3(s)\) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2, \ z_3]\) is less than one \eqref{eq:hinf_syn_obj_three}.
-\begin{equation}
-\label{eq:hinf_syn_obj_three}
-  \left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
-\end{equation}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,scale=1]{figs/comp_filter_three_hinf.pdf}
-\caption{\label{fig:comp_filter_three_hinf}Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters}
-\end{figure}
-
-By choosing \(H_1(s) \triangleq 1 - H_2(s) - H_3(s)\), the proposed \(\mathcal{H}_\infty\) synthesis solves the design problem \eqref{eq:hinf_problem_gen}. \par
-An example is given to validate the method where three sensors are used in different frequency bands (up to \(\SI{1}{Hz}\), from \(1\) to \(\SI{10}{Hz}\) and above \(\SI{10}{Hz}\) respectively).
-Three weighting functions are designed using \eqref{eq:weight_formula} and shown by dashed curves in Fig. \ref{fig:hinf_three_synthesis_results}.
-The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_three_synthesis_results}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,scale=1]{figs/hinf_three_synthesis_results.pdf}
-\caption{\label{fig:hinf_three_synthesis_results}Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
+\includegraphics[scale=1]{figs/hinf_filters_results.pdf}
+\caption{\label{fig:hinf_filters_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
 \end{figure}
 
 \section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
-\label{sec:org925754e}
+\label{sec:org377e66e}
 \label{sec:application_ligo}
 Several complementary filters are used in the active isolation system at the LIGO \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}.
 The requirements on those filters are very tight and thus their design is complex.
@@ -391,7 +401,7 @@ The obtained FIR filters are compliant with the requirements. However they are o
 
 The effectiveness of the proposed method is demonstrated by designing complementary filters with the same requirements as the one described in \cite{hua05_low_ligo}.
 \subsection{Complementary Filters Specifications}
-\label{sec:org4ade5f6}
+\label{sec:org75813ae}
 \label{sec:ligo_specifications}
 The specifications for one pair of complementary filters used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}) and shown in Fig. \ref{fig:ligo_weights}:
 \begin{itemize}
@@ -402,7 +412,7 @@ The specifications for one pair of complementary filters used at the LIGO are su
 \end{itemize}
 
 \subsection{Weighting Functions Design}
-\label{sec:org8c85120}
+\label{sec:org7015511}
 \label{sec:ligo_weights}
 The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem.
 However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.
@@ -418,7 +428,7 @@ The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weight
 \end{figure}
 
 \subsection{\(\mathcal{H}_\infty\) Synthesis}
-\label{sec:org89d27e1}
+\label{sec:orge5b6fe6}
 \label{sec:ligo_results}
 \(\mathcal{H}_\infty\) synthesis is performed using the architecture shown in Fig. \ref{eq:generalized_plant}.
 The complementary filters obtained are of order \(27\).
@@ -432,7 +442,7 @@ They are found to be very close to each other and this shows the effectiveness o
 \end{figure}
 
 \section{Conclusion}
-\label{sec:org153df0f}
+\label{sec:org39b90d9}
 \label{sec:conclusion}
 This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.
 Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters.
@@ -440,7 +450,7 @@ Therefore, a synthesis method that permits the shaping of the complementary filt
 Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion.
 
 \section*{Acknowledgment}
-\label{sec:org6c8e8b0}
+\label{sec:org1ece332}
 This research benefited from a FRIA grant from the French Community of Belgium.
 
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