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@ -81,67 +81,46 @@ Sensor fusion \sep{} Optimal filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{
* Introduction
<<sec:introduction>>
** Introduction to Sensor Fusion :ignore:
** New introduction :ignore:
*** Introduction to Sensor Fusion :ignore:
# Basic explanations of sensor fusion
# "Fusing" several sensor
- cite:anderson53_instr_approac_system_steer_comput earliest application of complementary filters (A simple RC circuit was used to physically realize the complementary filters)
- cite:bendat57_optim_filter_indep_measur_two roots of sensor fusion
** Advantages of Sensor Fusion :ignore:
*** Advantages of Sensor Fusion :ignore:
# Sensor Fusion can have many advantages / can be applied for various purposes
Sensor fusion can have many advantages.
In some situations, it is used to increase the bandwidth of the sensor cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim.
For instance by increasing the high frequency bandwidth of a position sensor using an accelerometer.
- Increase the bandwidth: cite:zimmermann92_high_bandw_orien_measur_contr
- Increased robustness: cite:collette15_sensor_fusion_method_high_perfor
- Decrease the noise:
Decrease the noise: cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur
[[cite:robert12_introd_random_signal_applied_kalman][chapter 8]]
Increased robustness (sensor measuring different quantities): cite:collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage
\par
** Applications :ignore:
*** Applications :ignore:
# The applications of sensor fusion are numerous
The applications of sensor fusion are numerous.
It is widely used for attitude estimation of unmanned aerial vehicle
cite:baerveldt97_low_cost_low_weigh_attit,pascoal99_navig_system_desig_using_time,corke04_inert_visual_sensin_system_small_auton_helic,batista10_optim_posit_veloc_navig_filter_auton_vehic,jensen13_basic_uas,min15_compl_filter_desig_angle_estim
Motion control
cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr
- UAV: cite:pascoal99_navig_system_desig_using_time, cite:jensen13_basic_uas
- Gravitational wave observer: cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,lucia18_low_frequen_optim_perfor_advan,heijningen18_low,akutsu21_vibrat_isolat_system_beam_split
Tjepkema et al. cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip used sensor fusion to isolate precision equipment from the ground motion.
*** Kalman Filtering or Complementary filters :ignore:
Gravitational wave observer cite:heijningen18_low:
LIGO cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system
VIRGO cite:lucia18_low_frequen_optim_perfor_advan
# There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering
** Kalman Filtering / Complementary filters :ignore:
- cite:brown72_integ_navig_system_kalman_filter alternate form of complementary filters => Kalman filtering
- cite:higgins75_compar_compl_kalman_filter Compare Kalman Filtering with sensor fusion using complementary filters
- cite:robert12_introd_random_signal_applied_kalman advantage of complementary filters over Kalman filtering
There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering cite:brown72_integ_navig_system_kalman_filter.
Kalman filtering cite:odry18_kalman_filter_mobil_robot_attit_estim
*** Design Methods of Complementary filters :ignore:
Relations between CF and Kalman: cite:becker15_compl_filter_desig_three_frequen_bands
# In some cases, complementary filters are implemented in an analog way such as in [...], but most of the time it is implemented numerically which allows much more complex
Advantages of complementary filtering over Kalman filtering for sensor fusion:
- Less computation cite:higgins75_compar_compl_kalman_filter
- For Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises cite:robert12_introd_random_signal_applied_kalman
- More intuitive frequency domain technique
- Analog complementary filters: cite:yong16_high_speed_vertic_posit_stage, cite:moore19_capac_instr_sensor_fusion_high_bandw_nanop
** Design Methods of Complementary filters :ignore:
In some cases, complementary filters are implemented in an analog way such as in cite:yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop, but most of the time it is implemented numerically which allows much more complex
Multiple design methods have been used for complementary filters
# Multiple design methods have been used for complementary filters
- Analytical methods:
- first order: cite:corke04_inert_visual_sensin_system_small_auton_helic,yong16_high_speed_vertic_posit_stage
- first order: cite:corke04_inert_visual_sensin_system_small_auton_helic
- second order: cite:baerveldt97_low_cost_low_weigh_attit, cite:stoten01_fusion_kinet_data_using_compos_filter, cite:jensen13_basic_uas
- 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
- cite:pascoal99_navig_system_desig_using_time use LMI to generate complementary filters (convex optimization techniques), specific for navigation systems
@ -152,26 +131,127 @@ Multiple design methods have been used for complementary filters
- 3 complementary filters: cite:becker15_compl_filter_desig_three_frequen_bands
** Problematic / gap in the research :ignore:
*** Problematics / gap in the research :ignore:
- Robustness problems: cite:zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur change of phase near the merging frequency
- Robustness problems: cite:zimmermann92_high_bandw_orien_measur_contr change of phase near the merging frequency
- Trial and error
- 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.
** Describe the paper itself / the problem which is addressed :ignore:
*** Describe the paper itself / the problem which is addressed :ignore:
Most of the requirements => shape of the complementary filters
=> propose a way to shape complementary filters.
** Introduce Each part of the paper :ignore:
*** Introduce Each part of the paper :ignore:
Section ref:sec:requirements
** Old Introduction :ignore:noexport:
*** Establish the importance of the research topic :ignore:
# What are Complementary Filters
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
These filters are used when two or more sensors are measuring the same physical quantity with different noise characteristics.
Unreliable frequencies of each sensor are filtered out by the complementary filters and then combined to form a super sensor giving a better estimate of the physical quantity over a wider bandwidth.
This technique is called sensor fusion and is used in many applications.\par
Section ref:sec:hinf_method
*** Applications of complementary filtering :ignore:
# Improve bandwidth for UAV
In cite:zimmermann92_high_bandw_orien_measur_contr,corke04_inert_visual_sensin_system_small_auton_helic, various sensors (accelerometers, gyroscopes, vision sensors, etc.) are merged using complementary filters for the attitude estimation of Unmanned Aerial Vehicles (UAV).
# Improving the control robustness
In cite:collette15_sensor_fusion_method_high_perfor, several sensor fusion configurations using different types of sensors are discussed in order to increase the control bandwidth of active vibration isolation systems.
# Merging of different sensor types
Furthermore, sensor fusion is used in the isolation systems of the Laser Interferometer Gravitational-Wave Observator (LIGO) to merge inertial sensors with relative sensors
cite:matichard15_seism_isolat_advan_ligo,hua04_polyp_fir_compl_filter_contr_system. \par
Section ref:sec:application_ligo
*** Current design methods for complementary filters :ignore:
# Why Design of Complementary Filter is important
As the super sensor noise characteristics largely depend on the complementary filter norms, their proper design is of primary importance for sensor fusion.
# Discuss the different approach to complementary filter design
In cite:corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas, first and second order analytical formulas of complementary filters have been presented.
# Third Order and Higher orders
Higher order complementary filters have been used in
cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,collette15_sensor_fusion_method_high_perfor.
# Alternate Formulation
In cite:jensen13_basic_uas, the sensitivity and complementary sensitivity transfer functions of a feedback architecture have been proposed to be used as complementary filters. The design of such filters can then benefit from the classical control theory developments.
# LMI / convex Optimization
Linear Matrix Inequalities (LMIs) are used in cite:pascoal99_navig_system_desig_using_time for the synthesis of complementary filters satisfying some frequency-like performance.
# FIR Filters
Finally, a synthesis method of high order Finite Impulse Response (FIR) complementary filters using convex optimization has been developed in cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system. \par
Section ref:sec:discussion
*** Describe a gap in the research :ignore:
# There is a need for easy synthesis methods for complementary filters
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.
*** Describe the paper itself / the problem which is addressed :ignore:
# In this paper, we propose a synthesis method for the shaping of complementary filters using the $\mathcal{H}_\infty$ norm.\par
This paper presents a new design method of complementary filters based on $\mathcal{H}_\infty$ synthesis.
This design method permits to easily shape the norms of the generated filters.\par
*** Introduce Each part of the paper :ignore:
The section ref:sec:requirements gives a brief overview of sensor fusion using complementary filters and explains how the typical requirements for such fusion can be expressed as upper bounds on the filters norms.
In section ref:sec:hinf_method, a new design method for the shaping of complementary filters using $\mathcal{H}_\infty$ synthesis is proposed.
In section ref:sec:application_ligo, the method is used to design complex complementary filters that are used for sensor fusion at the LIGO.
Our conclusions are drawn in the final section.
** Mohit's Introduction :noexport:
The sensors used for measuring physical quantity often works well within a limited frequency range called as the bandwidth of the sensor.
The signals recorded by the sensor beyond its bandwidth are often corrupt with noise and are not reliable.
Many dynamical systems require measurements over a wide frequency range.
Very often a variety of sensors are utilized to sense the same quantity.
These sensors have different operational bandwidth and are reliable only in a particular frequency range.
The signals from the different sensors are fused together in order to get the reliable measurement of the physical quantity over wider frequency band.
The combining of signals from various sensor is called sensor fusion cite:hua04_polyp_fir_compl_filter_contr_system.
The resulting sensor is referred as "super sensor" since it can have better noise characteristics and can operate over a wider frequency band as compared to the individual sensor used for merging cite:shaw90_bandw_enhan_posit_measur_using_measur_accel.
Sensor fusion is most commonly employed in the navigation systems to accurately measure the position of a vehicle.
The GPS sensors, which are accurate in low frequency band, are merged with the high-frequency accelerometers.
Zimmermann and Sulzer cite:zimmermann92_high_bandw_orien_measur_contr used sensor fusion to measure the orientation of a robot.
They merged inclinometer and accelerometers for accurate angular measurements over large frequency band.
Corke cite:corke04_inert_visual_sensin_system_small_auton_helic merged inertial measurement unit with the stereo vision system for measurement of attitude, height and velocity of an unmanned helicopter.
Baerveldt and Klang cite:baerveldt97 used an inclinometer and a gyroscope to measure the orientation of the autonomous helicopter.
The measurement of the 3D orientation using a gyroscope and an accelerometer was demonstrated by Roberts et al. cite:roberts03_low.
Cao et al. cite:cao20_adapt_compl_filter_based_post used sensor fusion to obtain the lateral and longitudinal velocities of the autonomous vehicle.
Sensor fusion is also used for enhancing the working range of the active isolation system.
For example, the active vibration isolation system at the Laser Interferometer Gravitational-Wave Observatory (LIGO) cite:matichard15_seism_isolat_advan_ligo utilizes sensor fusion.
The position sensors, seismometer and geophones are used for measuring the motion of the LIGO platform in different frequency bands cite:hua05_low_ligo.
Tjepkema et al. cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip used sensor fusion to isolate precision equipment from the ground motion.
The feedback from the accelerometer was used for active isolation at low frequency while force sensor was used at high frequency.
Various configurations of sensor fusion for active vibration isolation systems are discussed by Collette and Matichard cite:collette15_sensor_fusion_method_high_perfor.
Ma and Ghasemi-Nejhad cite:ma04_frequen_weigh_adapt_contr_simul used laser sensor and piezoelectric patches for simultaneous tracking and vibration control in smart structures.
Recently, Verma et al. cite:verma21_virtual_sensor_fusion_high_precis_contr presented virtual sensor fusion for high precision control where the signals from a physical sensor are fused with a sensor simulated virtually.
Fusing signals from different sensors can typically be done using Kalman filtering cite:odry18_kalman_filter_mobil_robot_attit_estim, ren19_integ_gnss_hub_motion_estim, faria19_sensor_fusion_rotat_motion_recon, liu18_innov_infor_fusion_method_with, abdel15_const_low_cost_gps_filter, biondi17_attit_recov_from_featur_track or complementary filters cite:brown72_integ_navig_system_kalman_filter.
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
When two filters are complementary, usually one is a low pass filter while the other is an high pass filter.
The complementary filters are designed in such a way that their magnitude is close to one in the bandwidth of the sensor they are combined with.
This enables to measure the physical quantity over larger bandwidth.
There are two different categories of complementary filters --- frequency domain complementary filters and state space complementary filters.
Earliest application of the frequency domain complementary filters was seen in Anderson and Fritze cite:anderson53_instr_approac_system_steer_comput.
A simple RC circuit was used to physically realize the complementary filters.
Frequency domain complementary filters were also used in cite:shaw90_bandw_enhan_posit_measur_using_measur_accel, zimmermann92_high_bandw_orien_measur_contr, baerveldt97, roberts03_low.
State space complementary filter finds application in tracking orientation of the flexible links in a robot cite:bachmann03_desig_marg_dof,salcudean91_global_conver_angul_veloc_obser,mahony08_nonlin_compl_filter_special_orthog_group and are particularly useful for multi-input multi-output systems.
Pascoal et al. cite:pascoal00_navig_system_desig_using_time presented complementary filters which can adapt with time for navigation system capable of estimating position and velocity using GPS and SONAR sensors.
The noise characteristics of the super sensor are governed by the norms of the complementary filters.
Therefore, the proper design of the complementary filters for sensor fusion is of immense importance.
The design of complementary filters is a complex task as they need to tuned as per the specification of the sensor.
In many applications, analytical formulas of first and second order complementary filters are used cite:corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas.
However, these low order complementary filters are not optimal, and high order complementary filters can lead to better fusion cite:jensen13_basic_uas,shaw90_bandw_enhan_posit_measur_using_measur_accel.
Several design techniques have been proposed to design higher order complementary filters.
Pascoal cite:pascoal00_navig_system_desig_using_time used linear matrix inequalities (LMIs) cite:boyd94_linear for the design of time varying complementary filters.
LMIs were also used by Hua et al. cite:hua04_polyp_fir_compl_filter_contr_system to design finite impulse response (FIR) filters for the active vibration isolation system at LIGO.
Plummer cite:plummer06_optim_compl_filter_their_applic_motion_measur proposed an optimal design method using the $\mathcal{H}_{\infty}$ synthesis and weighting functions representing the measurement noise of the sensors.
Although various methods have been presented in the literature for the design of complementary filters, there is a lack of general and simple framework that allows to shape the norm of complementary filters.
Such a method would prove to be very useful as the noise of the "supper sensor" and its dynamical characteristics depend on the norm of the filters.
This paper presents such a framework based on the $\mathcal{H}_\infty$ norm minimization.
The proposed method is quite general and can be easily extended to a case where more than two complementary filters needs to be designed.
The organization of this paper is as follows.
Section 2 presents the design requirements of ideal complementary filters.
It also demonstrates how the noise and robustness characteristics of the "super sensor" can be transformed into upper bounds on the norm of the complementary filters.
The framework for the design of complementary filters is detailed in Section 3.
This is followed by the application of the design method to complementary filter design for the active vibration isolation at LIGO in Section 4.
Finally, concluding remarks are presented in Section 5.
* Sensor Fusion and Complementary Filters Requirements
<<sec:requirements>>
@ -287,7 +367,7 @@ As shown in eqref:eq:noise_filtering_psd, the Power Spectral Density (PSD) of th
\end{equation}
If the two sensors have identical noise characteristics ($\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)$), a simple averaging ($H_1(s) = H_2(s) = 0.5$) is what would minimize the super sensor noise.
This is the simplest form of sensor fusion with complementary filters.
This the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, 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 $\Phi_{n_1}$.
@ -477,7 +557,7 @@ The magnitudes of the weighting functions are shown by dashed lines in Figure re
#+name: tab:weights_params
#+caption: Parameters used for weighting functions $W_1(s)$ and $W_2(s)$ using eqref:eq:weight_formula
#+ATTR_LATEX: :environment tabularx :width 0.29\linewidth :align ccc
#+ATTR_LATEX: :center t :booktabs t
#+ATTR_LATEX: :center t :booktabs t :float t
| Parameters | $W_1(s)$ | $W_2(s)$ |
|------------+---------------+---------------|
| $G_0$ | $0.1$ | $1000$ |
@ -517,7 +597,7 @@ A more complex real life example is taken up in the next section.
<<sec:application_ligo>>
** Introduction :ignore:
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO cite:matichard15_seism_isolat_advan_ligo,hua05_low_ligo, the VIRGO cite:lucia18_low_frequen_optim_perfor_advan,heijningen18_low and the KAGRA [[cite:sekiguchi16_study_low_frequen_vibrat_isolat_system][Chap. 5]].
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO cite:matichard15_seism_isolat_advan_ligo,hua05_low_ligo, the VIRGO cite:lucia18_low_frequen_optim_perfor_advan,heijningen18_low and the KAGRA cite:akutsu21_vibrat_isolat_system_beam_split.
In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop cite:hua04_low_ligo.
A set of complementary filters ($L_2,H_2$) is first used to fuse a seismometer and a geophone.
@ -550,8 +630,6 @@ The specifications for the set of complementary filters ($L_1,H_1$) used at the
These specifications are therefore upper bounds on the complementary filters' magnitudes.
They are physically represented in Figure ref:fig:fir_filter_ligo as well as the obtained magnitude of the FIR filters in cite:hua05_low_ligo.
# Replicated using SeDuMi matlab toolbox cite:sturm99_using_sedum
#+name: fig:fir_filter_ligo
#+caption: Specifications and Bode plot of the obtained FIR filters in cite:hua05_low_ligo
#+attr_latex: :scale 1
@ -688,8 +766,6 @@ Link to literature about doing that with mixed sensitivity
Some applications may require to merge more than two sensors.
For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Figure ref:fig:ligo_super_sensor_architecture). \par
- [ ] cite:becker15_compl_filter_desig_three_frequen_bands
*** Sequential vs Parallel :ignore:
When merging $n>2$ sensors using complementary filters, two architectures can be used as shown in Figure ref:fig:sensor_fusion_three.

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@ -1,4 +1,4 @@
% Created 2021-08-27 ven. 11:24
% Created 2021-06-18 ven. 17:00
% Intended LaTeX compiler: pdflatex
\documentclass[preprint, sort&compress]{elsarticle}
\usepackage[utf8]{inputenc}
@ -58,56 +58,31 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
\end{frontmatter}
\section{Introduction}
\label{sec:org5737795}
\label{sec:org3356a46}
\label{sec:introduction}
\begin{itemize}
\item \cite{anderson53_instr_approac_system_steer_comput} earliest application of complementary filters (A simple RC circuit was used to physically realize the complementary filters)
\item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
\end{itemize}
Sensor fusion can have many advantages.
In some situations, it is used to increase the bandwidth of the sensor \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim}.
For instance by increasing the high frequency bandwidth of a position sensor using an accelerometer.
Decrease the noise: \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur}
\cite[chapter 8]{robert12_introd_random_signal_applied_kalman}
Increased robustness (sensor measuring different quantities): \cite{collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage}
\par
The applications of sensor fusion are numerous.
It is widely used for attitude estimation of unmanned aerial vehicle
\cite{baerveldt97_low_cost_low_weigh_attit,pascoal99_navig_system_desig_using_time,corke04_inert_visual_sensin_system_small_auton_helic,batista10_optim_posit_veloc_navig_filter_auton_vehic,jensen13_basic_uas,min15_compl_filter_desig_angle_estim}
Motion control
\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr}
Tjepkema et al. \cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip} used sensor fusion to isolate precision equipment from the ground motion.
Gravitational wave observer \cite{heijningen18_low}:
LIGO \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}
VIRGO \cite{lucia18_low_frequen_optim_perfor_advan}
There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering \cite{brown72_integ_navig_system_kalman_filter}.
Kalman filtering \cite{odry18_kalman_filter_mobil_robot_attit_estim}
Relations between CF and Kalman: \cite{becker15_compl_filter_desig_three_frequen_bands}
Advantages of complementary filtering over Kalman filtering for sensor fusion:
\begin{itemize}
\item Less computation \cite{higgins75_compar_compl_kalman_filter}
\item For Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises \cite{robert12_introd_random_signal_applied_kalman}
\item More intuitive frequency domain technique
\item Increase the bandwidth: \cite{zimmermann92_high_bandw_orien_measur_contr}
\item Increased robustness: \cite{collette15_sensor_fusion_method_high_perfor}
\item Decrease the noise:
\end{itemize}
In some cases, complementary filters are implemented in an analog way such as in \cite{yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop}, but most of the time it is implemented numerically which allows much more complex
Multiple design methods have been used for complementary filters
\begin{itemize}
\item UAV: \cite{pascoal99_navig_system_desig_using_time}, \cite{jensen13_basic_uas}
\item Gravitational wave observer: \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,lucia18_low_frequen_optim_perfor_advan,heijningen18_low,akutsu21_vibrat_isolat_system_beam_split}
\end{itemize}
\begin{itemize}
\item \cite{brown72_integ_navig_system_kalman_filter} alternate form of complementary filters => Kalman filtering
\item \cite{higgins75_compar_compl_kalman_filter} Compare Kalman Filtering with sensor fusion using complementary filters
\item \cite{robert12_introd_random_signal_applied_kalman} advantage of complementary filters over Kalman filtering
\end{itemize}
\begin{itemize}
\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,yong16_high_speed_vertic_posit_stage}
\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}
@ -122,28 +97,21 @@ Multiple design methods have been used for complementary filters
\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,plummer06_optim_compl_filter_their_applic_motion_measur} change of phase near the merging frequency
\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 \ref{sec:requirements}
Section \ref{sec:hinf_method}
Section \ref{sec:application_ligo}
Section \ref{sec:discussion}
\section{Sensor Fusion and Complementary Filters Requirements}
\label{sec:orgbd86d49}
\label{sec:org32c05cb}
\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 Fusion Architecture}
\label{sec:org56b9e47}
\label{sec:orgcfc6167}
\label{sec:sensor_fusion}
A general sensor fusion architecture using complementary filters is shown in Figure \ref{fig:sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
@ -170,7 +138,7 @@ Therefore, a pair of strict complementary filter needs to satisfy the following
It will soon become clear why the complementary property is important.
\subsection{Sensor Models and Sensor Normalization}
\label{sec:org684f136}
\label{sec:orga2c7e39}
\label{sec:sensor_models}
In order to study such sensor fusion architecture, a model of the sensors is required.
@ -219,7 +187,7 @@ The super sensor output is therefore equal to:
\end{figure}
\subsection{Noise Sensor Filtering}
\label{sec:org99631b9}
\label{sec:org5397108}
\label{sec:noise_filtering}
In this section, it is supposed that all the sensors are perfectly calibrated, such that:
@ -252,14 +220,14 @@ As shown in \eqref{eq:noise_filtering_psd}, the Power Spectral Density (PSD) of
\end{equation}
If the two sensors have identical noise characteristics (\(\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)\)), a simple averaging (\(H_1(s) = H_2(s) = 0.5\)) is what would minimize the super sensor noise.
This is the simplest form of sensor fusion with complementary filters.
This the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, 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 \(\Phi_{n_1}\).
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
\subsection{Sensor Fusion Robustness}
\label{sec:org3f9e403}
\label{sec:org6cbe7ea}
\label{sec:fusion_robustness}
In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified.
@ -321,14 +289,14 @@ As it is generally desired to limit the maximum phase added by the super sensor,
Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
\section{Complementary Filters Shaping}
\label{sec:org82bc276}
\label{sec:org3fcce50}
\label{sec:hinf_method}
As shown in Section \ref{sec:requirements}, the noise and robustness of the ``super sensor'' are determined by the complementary filters norms.
Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use.
In this section, such synthesis is proposed by expressing this problem as a \(\mathcal{H}_\infty\) norm optimization.
\subsection{Synthesis Objective}
\label{sec:orgceb5825}
\label{sec:org006154f}
\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}.
@ -345,7 +313,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
where \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
\label{sec:org79feac5}
\label{sec:orgd8cba14}
\label{sec:hinf_synthesis}
In this section, it is shown that the synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimal control problem and therefore solved using convenient tools readily available.
@ -386,7 +354,7 @@ Therefore, applying the \(\mathcal{H}_\infty\) synthesis on the standard plant \
The above optimization problem can be efficiently solved in Matlab \cite{matlab20} using the Robust Control Toolbox.
\subsection{Weighting Functions Design}
\label{sec:orgd27beed}
\label{sec:org7aa4ffb}
\label{sec:hinf_weighting_func}
Weighting functions are used during the synthesis to specify what is the maximum allowed norms of the complementary filters.
@ -436,7 +404,7 @@ The typical shape of a weighting function generated using \eqref{eq:weight_formu
\end{figure}
\subsection{Validation of the proposed synthesis method}
\label{sec:orgc8f3eb3}
\label{sec:orgb562cf2}
\label{sec:hinf_example}
The proposed methodology for the design of complementary filters is now applied on a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
@ -497,9 +465,9 @@ This simple example illustrates the fact that the proposed methodology for compl
A more complex real life example is taken up in the next section.
\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
\label{sec:org8cb3b2e}
\label{sec:org60805ba}
\label{sec:application_ligo}
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO \cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO \cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite[Chap. 5]{sekiguchi16_study_low_frequen_vibrat_isolat_system}.
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO \cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO \cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite{akutsu21_vibrat_isolat_system_beam_split}.
In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop \cite{hua04_low_ligo}.
A set of complementary filters (\(L_2,H_2\)) is first used to fuse a seismometer and a geophone.
@ -520,7 +488,7 @@ After synthesis, the obtained FIR filters were found to be compliant with the re
However they are of very high order so their implementation is quite complex.
In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.
\subsection{Complementary Filters Specifications}
\label{sec:orgb603be6}
\label{sec:orgfdd63d0}
\label{sec:ligo_specifications}
The specifications for the set of complementary filters (\(L_1,H_1\)) used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}):
\begin{itemize}
@ -540,7 +508,7 @@ They are physically represented in Figure \ref{fig:fir_filter_ligo} as well as t
\end{figure}
\subsection{Weighting Functions Design}
\label{sec:orgd94a6e5}
\label{sec:org916b9d5}
\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.
@ -556,7 +524,7 @@ The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weight
\end{figure}
\subsection{\(\mathcal{H}_\infty\) Synthesis}
\label{sec:org1f03af8}
\label{sec:orgab74bf1}
\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 \(27\).
@ -570,10 +538,10 @@ They are found to be very close to each other and this shows the effectiveness o
\end{figure}
\section{Discussion}
\label{sec:org013b9e6}
\label{sec:org5bc126e}
\label{sec:discussion}
\subsection{``Closed-Loop'' complementary filters}
\label{sec:orga1ea439}
\label{sec:org8731218}
\label{sec:closed_loop_complementary_filters}
It is possible to use the fundamental properties of a feedback architecture to generate complementary filters.
@ -658,7 +626,7 @@ L = H_H^{-1} - 1
(provided \(H_H\) is invertible, therefore bi-proper)
\subsection{Imposing zero at origin / roll-off}
\label{sec:org293cf77}
\label{sec:orgdea775a}
\label{sec:add_features_in_filters}
3 methods:
@ -666,14 +634,10 @@ L = H_H^{-1} - 1
Link to literature about doing that with mixed sensitivity
\subsection{Synthesis of Three Complementary Filters}
\label{sec:orgd44eb72}
\label{sec:org6446998}
\label{sec:hinf_three_comp_filters}
Some applications may require to merge more than two sensors.
For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Figure \ref{fig:ligo_super_sensor_architecture}). \par
\begin{itemize}
\item[{$\square$}] \cite{becker15_compl_filter_desig_three_frequen_bands}
\end{itemize}
When merging \(n>2\) sensors using complementary filters, two architectures can be used as shown in Figure \ref{fig:sensor_fusion_three}.
The fusion can either be done in a ``sequential'' way where \(n-1\) sets of two complementary filters are used (Figure \ref{fig:sensor_fusion_three_sequential}), or in a ``parallel'' way where one set of \(n\) complementary filters is used (Figure \ref{fig:sensor_fusion_three_parallel}).
@ -763,7 +727,7 @@ Such synthesis method can be generalized to a set of \(n\) complementary filters
\end{equation}
\section{Conclusion}
\label{sec:orgc6071ad}
\label{sec:orgcba6c13}
\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.
@ -771,7 +735,7 @@ 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:org4efce57}
\label{sec:orgf175dee}
This research benefited from a FRIA grant from the French Community of Belgium.
\bibliographystyle{elsarticle-num}

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