Rework introduction and cleaned BIB file
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@ -81,46 +81,67 @@ Sensor fusion \sep{} Optimal filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{
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* Introduction
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<<sec:introduction>>
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** New introduction :ignore:
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*** Introduction to Sensor Fusion :ignore:
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** Introduction to Sensor Fusion :ignore:
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# Basic explanations of sensor fusion
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# "Fusing" several sensor
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- cite:anderson53_instr_approac_system_steer_comput earliest application of complementary filters (A simple RC circuit was used to physically realize the complementary filters)
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- cite:bendat57_optim_filter_indep_measur_two roots of sensor fusion
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*** Advantages of Sensor Fusion :ignore:
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** Advantages of Sensor Fusion :ignore:
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# Sensor Fusion can have many advantages / can be applied for various purposes
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Sensor fusion can have many advantages.
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- Increase the bandwidth: cite:zimmermann92_high_bandw_orien_measur_contr
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- Increased robustness: cite:collette15_sensor_fusion_method_high_perfor
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- Decrease the noise:
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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.
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For instance by increasing the high frequency bandwidth of a position sensor using an accelerometer.
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*** Applications :ignore:
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Decrease the noise: cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur
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[[cite:robert12_introd_random_signal_applied_kalman][chapter 8]]
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Increased robustness (sensor measuring different quantities): cite:collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage
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\par
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** Applications :ignore:
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# The applications of sensor fusion are numerous
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The applications of sensor fusion are numerous.
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It is widely used for attitude estimation of unmanned aerial vehicle
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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
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- UAV: cite:pascoal99_navig_system_desig_using_time, cite:jensen13_basic_uas
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- 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
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Motion control
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cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr
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*** Kalman Filtering or Complementary filters :ignore:
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Tjepkema et al. cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip used sensor fusion to isolate precision equipment from the ground motion.
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# There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering
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Gravitational wave observer cite:heijningen18_low:
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LIGO cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system
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VIRGO cite:lucia18_low_frequen_optim_perfor_advan
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- cite:brown72_integ_navig_system_kalman_filter alternate form of complementary filters => Kalman filtering
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- cite:higgins75_compar_compl_kalman_filter Compare Kalman Filtering with sensor fusion using complementary filters
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- cite:robert12_introd_random_signal_applied_kalman advantage of complementary filters over Kalman filtering
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** Kalman Filtering / Complementary filters :ignore:
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*** Design Methods of Complementary filters :ignore:
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There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering cite:brown72_integ_navig_system_kalman_filter.
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Kalman filtering cite:odry18_kalman_filter_mobil_robot_attit_estim
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# 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
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Relations between CF and Kalman: cite:becker15_compl_filter_desig_three_frequen_bands
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- Analog complementary filters: cite:yong16_high_speed_vertic_posit_stage, cite:moore19_capac_instr_sensor_fusion_high_bandw_nanop
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Advantages of complementary filtering over Kalman filtering for sensor fusion:
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- Less computation cite:higgins75_compar_compl_kalman_filter
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- For Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises cite:robert12_introd_random_signal_applied_kalman
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- More intuitive frequency domain technique
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# Multiple design methods have been used for complementary filters
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** Design Methods of Complementary filters :ignore:
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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
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Multiple design methods have been used for complementary filters
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- Analytical methods:
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- first order: cite:corke04_inert_visual_sensin_system_small_auton_helic
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- first order: cite:corke04_inert_visual_sensin_system_small_auton_helic,yong16_high_speed_vertic_posit_stage
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- second order: cite:baerveldt97_low_cost_low_weigh_attit, cite:stoten01_fusion_kinet_data_using_compos_filter, cite:jensen13_basic_uas
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- 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
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- cite:pascoal99_navig_system_desig_using_time use LMI to generate complementary filters (convex optimization techniques), specific for navigation systems
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@ -131,127 +152,26 @@ Sensor fusion \sep{} Optimal filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{
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- 3 complementary filters: cite:becker15_compl_filter_desig_three_frequen_bands
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*** Problematics / gap in the research :ignore:
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** Problematic / gap in the research :ignore:
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- Robustness problems: cite:zimmermann92_high_bandw_orien_measur_contr change of phase near the merging frequency
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- Robustness problems: cite:zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur change of phase near the merging frequency
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- Trial and error
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- 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.
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*** Describe the paper itself / the problem which is addressed :ignore:
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** Describe the paper itself / the problem which is addressed :ignore:
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Most of the requirements => shape of the complementary filters
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=> propose a way to shape complementary filters.
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*** Introduce Each part of the paper :ignore:
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** Introduce Each part of the paper :ignore:
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** Old Introduction :ignore:noexport:
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*** Establish the importance of the research topic :ignore:
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# What are Complementary Filters
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A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
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These filters are used when two or more sensors are measuring the same physical quantity with different noise characteristics.
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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.
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This technique is called sensor fusion and is used in many applications.\par
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Section ref:sec:requirements
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*** Applications of complementary filtering :ignore:
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# Improve bandwidth for UAV
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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).
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# Improving the control robustness
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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.
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# Merging of different sensor types
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Furthermore, sensor fusion is used in the isolation systems of the Laser Interferometer Gravitational-Wave Observator (LIGO) to merge inertial sensors with relative sensors
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cite:matichard15_seism_isolat_advan_ligo,hua04_polyp_fir_compl_filter_contr_system. \par
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Section ref:sec:hinf_method
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*** Current design methods for complementary filters :ignore:
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# Why Design of Complementary Filter is important
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As the super sensor noise characteristics largely depend on the complementary filter norms, their proper design is of primary importance for sensor fusion.
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# Discuss the different approach to complementary filter design
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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.
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# Third Order and Higher orders
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Higher order complementary filters have been used in
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cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,collette15_sensor_fusion_method_high_perfor.
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# Alternate Formulation
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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.
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# LMI / convex Optimization
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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.
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# FIR Filters
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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
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Section ref:sec:application_ligo
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*** Describe a gap in the research :ignore:
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# There is a need for easy synthesis methods for complementary filters
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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.
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*** Describe the paper itself / the problem which is addressed :ignore:
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# In this paper, we propose a synthesis method for the shaping of complementary filters using the $\mathcal{H}_\infty$ norm.\par
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This paper presents a new design method of complementary filters based on $\mathcal{H}_\infty$ synthesis.
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This design method permits to easily shape the norms of the generated filters.\par
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*** Introduce Each part of the paper :ignore:
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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.
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In section ref:sec:hinf_method, a new design method for the shaping of complementary filters using $\mathcal{H}_\infty$ synthesis is proposed.
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In section ref:sec:application_ligo, the method is used to design complex complementary filters that are used for sensor fusion at the LIGO.
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Our conclusions are drawn in the final section.
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** Mohit's Introduction :noexport:
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The sensors used for measuring physical quantity often works well within a limited frequency range called as the bandwidth of the sensor.
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The signals recorded by the sensor beyond its bandwidth are often corrupt with noise and are not reliable.
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Many dynamical systems require measurements over a wide frequency range.
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Very often a variety of sensors are utilized to sense the same quantity.
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These sensors have different operational bandwidth and are reliable only in a particular frequency range.
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The signals from the different sensors are fused together in order to get the reliable measurement of the physical quantity over wider frequency band.
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The combining of signals from various sensor is called sensor fusion cite:hua04_polyp_fir_compl_filter_contr_system.
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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.
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Sensor fusion is most commonly employed in the navigation systems to accurately measure the position of a vehicle.
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The GPS sensors, which are accurate in low frequency band, are merged with the high-frequency accelerometers.
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Zimmermann and Sulzer cite:zimmermann92_high_bandw_orien_measur_contr used sensor fusion to measure the orientation of a robot.
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They merged inclinometer and accelerometers for accurate angular measurements over large frequency band.
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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.
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Baerveldt and Klang cite:baerveldt97 used an inclinometer and a gyroscope to measure the orientation of the autonomous helicopter.
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The measurement of the 3D orientation using a gyroscope and an accelerometer was demonstrated by Roberts et al. cite:roberts03_low.
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Cao et al. cite:cao20_adapt_compl_filter_based_post used sensor fusion to obtain the lateral and longitudinal velocities of the autonomous vehicle.
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Sensor fusion is also used for enhancing the working range of the active isolation system.
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For example, the active vibration isolation system at the Laser Interferometer Gravitational-Wave Observatory (LIGO) cite:matichard15_seism_isolat_advan_ligo utilizes sensor fusion.
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The position sensors, seismometer and geophones are used for measuring the motion of the LIGO platform in different frequency bands cite:hua05_low_ligo.
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Tjepkema et al. cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip used sensor fusion to isolate precision equipment from the ground motion.
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The feedback from the accelerometer was used for active isolation at low frequency while force sensor was used at high frequency.
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Various configurations of sensor fusion for active vibration isolation systems are discussed by Collette and Matichard cite:collette15_sensor_fusion_method_high_perfor.
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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.
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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.
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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.
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A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
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When two filters are complementary, usually one is a low pass filter while the other is an high pass filter.
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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.
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This enables to measure the physical quantity over larger bandwidth.
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There are two different categories of complementary filters --- frequency domain complementary filters and state space complementary filters.
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Earliest application of the frequency domain complementary filters was seen in Anderson and Fritze cite:anderson53_instr_approac_system_steer_comput.
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A simple RC circuit was used to physically realize the complementary filters.
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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.
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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.
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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.
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The noise characteristics of the super sensor are governed by the norms of the complementary filters.
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Therefore, the proper design of the complementary filters for sensor fusion is of immense importance.
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The design of complementary filters is a complex task as they need to tuned as per the specification of the sensor.
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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.
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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.
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Several design techniques have been proposed to design higher order complementary filters.
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Pascoal cite:pascoal00_navig_system_desig_using_time used linear matrix inequalities (LMIs) cite:boyd94_linear for the design of time varying complementary filters.
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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.
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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.
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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.
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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.
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This paper presents such a framework based on the $\mathcal{H}_\infty$ norm minimization.
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The proposed method is quite general and can be easily extended to a case where more than two complementary filters needs to be designed.
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The organization of this paper is as follows.
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Section 2 presents the design requirements of ideal complementary filters.
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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.
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The framework for the design of complementary filters is detailed in Section 3.
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This is followed by the application of the design method to complementary filter design for the active vibration isolation at LIGO in Section 4.
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Finally, concluding remarks are presented in Section 5.
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Section ref:sec:discussion
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* Sensor Fusion and Complementary Filters Requirements
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<<sec:requirements>>
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@ -367,7 +287,7 @@ As shown in eqref:eq:noise_filtering_psd, the Power Spectral Density (PSD) of th
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\end{equation}
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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.
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This the simplest form of sensor fusion with complementary filters.
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This is the simplest form of sensor fusion with complementary filters.
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However, the two sensors have usually high noise levels over distinct frequency regions.
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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 $\Phi_{n_1}$.
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@ -557,7 +477,7 @@ The magnitudes of the weighting functions are shown by dashed lines in Figure re
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#+name: tab:weights_params
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#+caption: Parameters used for weighting functions $W_1(s)$ and $W_2(s)$ using eqref:eq:weight_formula
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#+ATTR_LATEX: :environment tabularx :width 0.29\linewidth :align ccc
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#+ATTR_LATEX: :center t :booktabs t :float t
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#+ATTR_LATEX: :center t :booktabs t
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| Parameters | $W_1(s)$ | $W_2(s)$ |
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|------------+---------------+---------------|
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| $G_0$ | $0.1$ | $1000$ |
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@ -597,7 +517,7 @@ A more complex real life example is taken up in the next section.
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<<sec:application_ligo>>
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** Introduction :ignore:
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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.
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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]].
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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.
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A set of complementary filters ($L_2,H_2$) is first used to fuse a seismometer and a geophone.
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@ -630,6 +550,8 @@ The specifications for the set of complementary filters ($L_1,H_1$) used at the
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These specifications are therefore upper bounds on the complementary filters' magnitudes.
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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.
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# Replicated using SeDuMi matlab toolbox cite:sturm99_using_sedum
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#+name: fig:fir_filter_ligo
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#+caption: Specifications and Bode plot of the obtained FIR filters in cite:hua05_low_ligo
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#+attr_latex: :scale 1
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@ -766,6 +688,8 @@ Link to literature about doing that with mixed sensitivity
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Some applications may require to merge more than two sensors.
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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
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- [ ] cite:becker15_compl_filter_desig_three_frequen_bands
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*** Sequential vs Parallel :ignore:
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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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313
journal/ref.bib
313
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@ -7,7 +7,6 @@
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pages = {1-21},
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year = 2015,
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doi = {10.1016/j.jsv.2015.01.006},
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url = {https://doi.org/10.1016/j.jsv.2015.01.006},
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}
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@phdthesis{hua05_low_ligo,
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@ -15,15 +14,14 @@
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doi = {10.1117/12.552518},
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school = {stanford university},
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title = {Low frequency vibration isolation and alignment system for
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advanced LIGO},
|
||||
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keywords = {sensor fusion, complementary filters},
|
||||
publisher = {SAGE Publications Sage UK: London, England},
|
||||
}
|
||||
|
||||
@ -524,7 +279,6 @@
|
||||
journal = {IRE Transactions on Circuit Theory},
|
||||
year = 1957,
|
||||
doi = {10.1109/tct.1957.1086345},
|
||||
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|
||||
page = {--},
|
||||
}
|
||||
|
||||
@ -533,7 +287,7 @@
|
||||
Giaime, Joseph A and Hammond, Giles Dominic and Hardham, C and
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
@ -541,21 +295,6 @@
|
||||
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|
||||
}
|
||||
|
||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
|
||||
@phdthesis{heijningen18_low,
|
||||
author = {van Heijningen, JV},
|
||||
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|
||||
@ -582,6 +321,36 @@
|
||||
year = 2005,
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
|
||||
@article{min15_compl_filter_desig_angle_estim,
|
||||
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|
||||
title = {Complementary Filter Design for Angle Estimation Using Mems
|
||||
Accelerometer and Gyroscope},
|
||||
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|
||||
National University, Changwon, Korea},
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
title = {Optimal Position and Velocity Navigation Filters for
|
||||
Autonomous Vehicles},
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
|
||||
@phdthesis{sekiguchi16_study_low_frequen_vibrat_isolat_system,
|
||||
author = {Sekiguchi, Takanori},
|
||||
school = {University of Tokyo},
|
||||
title = {A Study of Low Frequency Vibration Isolation System for
|
||||
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|
||||
year = 2016,
|
||||
}
|
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|
||||
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Reference in New Issue
Block a user