Abstract

In many nonlinear systems, the observability of the system is dependent on its state and control input. Thus, incorporating observability into a control scheme can enhance an observer's ability to recover accurate estimates of unmeasured states, minimize estimation error, and ultimately, allow the original control objective to be achieved. The accommodation of observability, however, may conflict with the original control goal at times. In this paper, we propose the use of control barrier functions (CBFs) to enforce observability and thereby facilitate the convergence of the state estimate to the true state while accommodating the original control objectives. Motivated by practical applications for autonomous robots operating in global positioning system-denied environments, we focus on the problem of target tracking for a unicycle model when only the distance to the target is measured. The proposed approach is compared in simulation with a model predictive control (MPC) approach that treats an observability-related metric as part of the cost function, where several different options for the observability metric are explored. It is found that the CBF-based approach achieves control and estimation performance that is comparable to that of the MPC approach, but with significantly less computational complexity. These findings are further experimentally verified in range-based target tracking with a swimming robotic fish.

References

1.
Ciccarella
,
G.
,
Dalla Mora
,
M.
, and
Germani
,
A.
,
1993
, “
A Luenberger-Like Observer for Nonlinear Systems
,”
Int. J. Control
,
57
(
3
), pp.
537
556
.10.1080/00207179308934406
2.
Khalil
,
H. K.
,
2002
,
Nonlinear Systems
, 3rd ed.,
Prentice Hall
,
Upper Saddle River, NJ
.
3.
Spurgeon
,
S. K.
,
2008
, “
Sliding Mode Observers: A Survey
,”
Int. J. Syst. Sci.
,
39
(
8
), pp.
751
764
.10.1080/00207720701847638
4.
Allgöwer
,
F.
,
Badgwell
,
T. A.
,
Qin
,
J. S.
,
Rawlings
,
J. B.
, and
Wright
,
S. J.
,
1999
, “
Nonlinear Predictive Control and Moving Horizon Estimation-An Introductory Overview
,”
Adv. Control
, pp.
391
449
.10.1007/978-1-4471-0853-5
5.
Welch
,
G.
, and
Bishop
,
G.
,
2001
, “
An Introduction to the Kalman Filter
,” Proc of SIGGRAPH, Course 8.27599-23175, p.
41
.
6.
Gustafsson
,
F.
,
2010
, “
Particle Filter Theory and Practice With Positioning Applications
,”
IEEE Aerosp. Electron. Syst. Mag.
,
25
(
7
), pp.
53
82
.10.1109/MAES.2010.5546308
7.
Bai
,
H.
, and
Taylor
,
C. N.
,
2020
, “
Future Uncertainty-Based Control for Relative Navigation in GPS-Denied Environments
,”
IEEE Trans. Aerosp. Electron. Syst.
,
56
(
5
), pp.
3491
3501
.10.1109/TAES.2020.2974052
8.
Frey
,
K. M.
,
Steiner
,
T. J.
, and
How
,
J. P.
,
2019
, “
Towards Online Observability-Aware Trajectory Optimization for Landmark-Based Estimators
,” arXiv:1908.03790.
9.
Rafieisakhaei
,
M.
,
Chakravorty
,
S.
, and
Kumar
,
P.
,
2017
, “
On the Use of the Observability Gramian for Partially Observed Robotic Path Planning Problems
,” IEEE 56th Annual Conference on Decision and Control (
CDC
),
Melbourne, VIC, Australia, Dec. 12–15,
pp.
1523
1528
.10.1109/CDC.2017.8263868
10.
Antonelli
,
G.
,
Arrichiello
,
F.
,
Chiaverini
,
S.
, and
Sukhatme
,
G. S.
,
2010
, “
Observability Analysis of Relative Localization for AUVs Based on Ranging and Depth Measurements
,”
IEEE International Conference on Robotics and Automation
,
Anchorage, AK, May 3–7,
pp.
4276
4281
.10.1109/ROBOT.2010.5509573
11.
Arrichiello
,
F.
,
Antonelli
,
G.
,
Aguiar
,
A. P.
, and
Pascoal
,
A.
,
2013
, “
An Observability Metric for Underwater Vehicle Localization Using Range Measurements
,”
Sensors
,
13
(
12
), pp.
16191
16215
.10.3390/s131216191
12.
Grebe
,
C.
,
Wise
,
E.
, and
Kelly
,
J.
,
2021
, “
Observability-Aware Trajectory Optimization: Theory, Viability, and State of the Art
,” IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (
MFI
),
Karlsruhe, Germany, Sept. 23–25,
pp.
1
8
.10.1109/MFI52462.2021.9591177
13.
Krener
,
A. J.
, and
Ide
,
K.
,
2009
, “
Measures of Unobservability
,”
Proceedings of the 48th IEEE Conference on Decision and Control (CDC) Held Jointly With 2009 28th Chinese Control Conference
,
Shanghai, China, Dec. 15–18,
pp.
6401
6406
.10.1109/CDC.2009.5400067
14.
Wu
,
J.
,
Bingham
,
R. C.
,
Ting
,
S.
,
Yager
,
K.
,
Wood
,
Z. J.
,
Gambin
,
T.
, and
Clark
,
C. M.
,
2019
, “
Multi-AUV Motion Planning for Archeological Site Mapping and Photogrammetric Reconstruction
,”
J. Field Rob.
,
36
(
7
), pp.
1250
1269
.10.1002/rob.21905
15.
Williams
,
S. B.
,
Pizarro
,
O.
,
Jakuba
,
M.
, and
Barrett
,
N.
,
2010
, “
AUV Benthic Habitat Mapping in South Eastern Tasmania
,”
Field and Service Robotics
,
Springer
,
Berlin
, pp.
275
284
.
16.
Woods Hole Oceanographic Institution
,
2021
, “
Acoustic Micromodem
,” accessed Feb. 23, 2024, https://acomms.whoi.edu/micro-modem/
17.
Paull
,
L.
,
Saeedi
,
S.
,
Seto
,
M.
, and
Li
,
H.
,
2014
, “
AUV Navigation and Localization: A Review
,”
IEEE J. Oceanic Eng.
,
39
(
1
), pp.
131
149
.10.1109/JOE.2013.2278891
18.
Gadre
,
A. S.
, and
Stilwell
,
D. J.
,
2005
, “
A Complete Solution to Underwater Navigation in the Presence of Unknown Currents Based on Range Measurements From a Single Location
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
,
Edmonton, AB, Canada, Aug. 2–6,
pp.
1420
1425
.10.1109/IROS.2005.1545230
19.
Chen
,
Q.
,
You
,
K.
, and
Song
,
S.
,
2017
, “
Cooperative Localization for Autonomous Underwater Vehicles Using Parallel Projection
,” 13th IEEE International Conference on Control & Automation (
ICCA
),
Ohrid, Macedonia, July 3–6,
pp.
788
793
.10.1109/ICCA.2017.8003160
20.
Bahr
,
A.
,
Leonard
,
J. J.
, and
Fallon
,
M. F.
,
2009
, “
Cooperative Localization for Autonomous Underwater Vehicles
,”
Int. J. Rob. Res.
,
28
(
6
), pp.
714
728
.10.1177/0278364908100561
21.
Huang
,
Y.
,
Zhang
,
Y.
,
Xu
,
B.
,
Wu
,
Z.
, and
Chambers
,
J. A.
,
2018
, “
A New Adaptive Extended Kalman Filter for Cooperative Localization
,”
IEEE Trans. Aerosp. Electron. Syst.
,
54
(
1
), pp.
353
368
.10.1109/TAES.2017.2756763
22.
Fallon
,
M. F.
,
Papadopoulos
,
G.
,
Leonard
,
J. J.
, and
Patrikalakis
,
N. M.
,
2010
, “
Cooperative AUV Navigation Using a Single Maneuvering Surface Craft
,”
Int. J. Rob. Res.
,
29
(
12
), pp.
1461
1474
.10.1177/0278364910380760
23.
Baccou
,
P.
, and
Jouvencel
,
B.
,
2002
, “
Homing and Navigation Using One Transponder for AUV, Postprocessing Comparisons Results With Long Base-Line Navigation
,”
Proceedings 2002 IEEE International Conference on Robotics and Automation
, Washington, DC, May 11–15,
Vol.
4
, pp.
4004
4009
.10.1109/ROBOT.2002.1014361
24.
Hinson
,
B. T.
,
Binder
,
M. K.
, and
Morgansen
,
K. A.
,
2013
, “
Path Planning to Optimize Observability in a Planar Uniform Flow Field
,”
American Control Conference
,
Washington, DC, June 17–19,
pp.
1392
1399
.10.1109/ACC.2013.6580031
25.
Ross
,
A.
, and
Jouffroy
,
J.
,
2005
, “
Remarks on the Observability of Single Beacon Underwater Navigation
,”
Proceedings of International Symposium on Unmanned Untethered Submersible Technology
, Durham, NH.https://www.researchgate.net/publication/281436020_Remarks_on_the_Observability_of_Single_Beacon_Underwater_Navigation
26.
Ferreira
,
B.
,
Matos
,
A.
, and
Cruz
,
N.
,
2010
, “
Single Beacon Navigation: Localization and Control of the MARES AUV
,”
Oceans 2010 MTS/IEEE Seattle
,
Seattle, WA, Sept. 20–23,
pp.
1
9
.10.1109/OCEANS.2010.5664518
27.
Quenzer
,
J. D.
, and
Morgansen
,
K. A.
,
2014
, “
Observability Based Control in Range-Only Underwater Vehicle Localization
,”
2014 American Control Conference
,
Portland, OR, June 4–6,
pp.
4702
4707
.10.1109/ACC.2014.6859032
28.
Ames
,
A. D.
,
Grizzle
,
J. W.
, and
Tabuada
,
P.
,
2014
, “
Control Barrier Function Based Quadratic Programs With Application to Adaptive Cruise Control
,”
53rd IEEE Conference on Decision and Control
,
Los Angeles, CA, Dec. 15–17,
pp.
6271
6278
.10.1109/CDC.2014.7040372
29.
Borrmann
,
U.
,
Wang
,
L.
,
Ames
,
A. D.
, and
Egerstedt
,
M.
,
2015
, “
Control Barrier Certificates for Safe Swarm Behavior
,”
IFAC-PapersOnLine
,
48
(
27
), pp.
68
73
.10.1016/j.ifacol.2015.11.154
30.
Ames
,
A. D.
,
Xu
,
X.
,
Grizzle
,
J. W.
, and
Tabuada
,
P.
,
2017
, “
Control Barrier Function Based Quadratic Programs for Safety Critical Systems
,”
IEEE Trans. Autom. Control
,
62
(
8
), pp.
3861
3876
.10.1109/TAC.2016.2638961
31.
Ames
,
A. D.
,
Coogan
,
S.
,
Egerstedt
,
M.
,
Notomista
,
G.
,
Sreenath
,
K.
, and
Tabuada
,
P.
,
2019
, “
Control Barrier Functions: Theory and Applications
,” 18th European Control Conference (
ECC
),
Naples, Italy, June 25–28,
pp.
3420
3431
.10.23919/ECC.2019.8796030
32.
Ames
,
A. D.
,
Notomista
,
G.
,
Wardi
,
Y.
, and
Egerstedt
,
M.
,
2021
, “
Integral Control Barrier Functions for Dynamically Defined Control Laws
,”
IEEE Control Syst. Lett.
,
5
(
3
), pp.
887
892
.10.1109/LCSYS.2020.3006764
33.
Huang
,
Y.
,
Yong
,
S. Z.
, and
Chen
,
Y.
,
2021
, “
Stability Control of Autonomous Ground Vehicles Using Control-Dependent Barrier Functions
,”
IEEE Trans. Intell. Veh.
,
6
(
4
), pp.
699
710
.10.1109/TIV.2021.3058064
34.
Nguyen
,
Q.
, and
Sreenath
,
K.
,
2022
, “
L1 Adaptive Control Barrier Functions for Nonlinear Underactuated Systems
,” 2022 American Control Conference (
ACC
),
Atlanta, GA, June 8–10,
pp.
721
728
.10.23919/ACC53348.2022.9867596
35.
Isaly
,
A.
,
Patil
,
O. S.
,
Sanfelice
,
R. G.
, and
Dixon
,
W. E.
,
2021
, “
Adaptive Safety With Multiple Barrier Functions Using Integral Concurrent Learning
,”
2021 American Control Conference (ACC)
,
New Orleans, LA, May 25–28,
pp.
3719
3724
.
36.
Breeden
,
J.
, and
Panagou
,
D.
,
2021
, “
High Relative Degree Control Barrier Functions Under Input Constraints
,” 60th IEEE Conference on Decision and Control (
CDC
),
Austin, TX, Dec. 14–17,
pp.
6119
6124
.10.1109/CDC45484.2021.9683705
37.
Panagou
,
D.
,
Stipanovič
,
D. M.
, and
Voulgaris
,
P. G.
,
2013
, “
Multi-Objective Control for Multi-Agent Systems Using Lyapunov-Like Barrier Functions
,”
52nd IEEE Conference on Decision and Control
,
Firenze, Italy, Dec. 10–13,
pp.
1478
1483
.10.1109/CDC.2013.6760091
38.
Panagou
,
D.
,
Stipanović
,
D. M.
, and
Voulgaris
,
P. G.
,
2016
, “
Distributed Coordination Control for Multi-Robot Networks Using Lyapunov-Like Barrier Functions
,”
IEEE Trans. Autom. Control
,
61
(
3
), pp.
617
632
.10.1109/TAC.2015.2444131
39.
Wang
,
L.
,
Ames
,
A. D.
, and
Egerstedt
,
M.
,
2016
, “
Multi-Objective Compositions for Collision-Free Connectivity Maintenance in Teams of Mobile Robots
,” IEEE 55th Conference on Decision and Control (
CDC
),
Las Vegas, NV, Dec. 12–14,
pp.
2659
2664
.10.1109/CDC.2016.7798663
40.
Glotfelter
,
P.
,
Cortés
,
J.
, and
Egerstedt
,
M.
,
2017
, “
Nonsmooth Barrier Functions With Applications to Multi-Robot Systems
,”
IEEE Control Syst. Lett.
,
1
(
2
), pp.
310
315
.10.1109/LCSYS.2017.2710943
41.
Coleman
,
D.
,
Bopardikar
,
S. D.
, and
Tan
,
X.
,
2021
, “
Incorporating Observability Via Control Barrier Functions With Application to Range-Based Target Tracking
,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics (
AIM
), Delft, The Netherlands, July 12–16
pp.
713
719
.10.1109/AIM46487.2021.9517467
42.
Coleman
,
D.
,
Bopardikar
,
S. D.
, and
Tan
,
X.
,
2021
, “
Observability-Aware Target Tracking With Range Only Measurement
,” 2021 American Control Conference (
ACC
), New Orleans, LA,
pp.
4217
4224
.10.23919/ACC50511.2021.9483280
43.
Kou
,
S. R.
,
Elliott
,
D. L.
, and
Tarn
,
T. J.
,
1973
, “
Observability of Nonlinear Systems
,”
Inf. Control
,
22
(
1
), pp.
89
99
.10.1016/S0019-9958(73)90508-1
44.
Hermann
,
R.
, and
Krener
,
A.
,
1977
, “
Nonlinear Controllability and Observability
,”
IEEE Trans. Autom. Control
,
22
(
5
), pp.
728
740
.10.1109/TAC.1977.1101601
45.
Powel
,
N. D.
, and
Morgansen
,
K. A.
,
2015
, “
Empirical Observability Gramian Rank Condition for Weak Observability of Nonlinear Systems With Control
,” 54th IEEE Conference on Decision and Control (
CDC
),
Osaka, Japan, Dec. 15–18,
pp.
6342
6348
.10.1109/CDC.2015.7403218
46.
Powel
,
N.
,
2016
, “
Noise-Enabled Observability of Nonlinear Dynamic Systems Using the Empirical Observability Gramian
,”
Ph.D. dissertation
, University of Washington Seattle, WA.https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/35976/Powel_washington_0250E_15547.pdf?sequence=1
47.
MathWorks
,
2020
,
Matlab Optimization Toolbox
,
The MathWorks
,
Natick, MA
.
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