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Safety-Critical Control for Autonomous Systems: Control Barrier Functions via Reduced-Order Models (2403.09865v1)

Published 14 Mar 2024 in cs.SY, cs.RO, and eess.SY

Abstract: Modern autonomous systems, such as flying, legged, and wheeled robots, are generally characterized by high-dimensional nonlinear dynamics, which presents challenges for model-based safety-critical control design. Motivated by the success of reduced-order models in robotics, this paper presents a tutorial on constructive safety-critical control via reduced-order models and control barrier functions (CBFs). To this end, we provide a unified formulation of techniques in the literature that share a common foundation of constructing CBFs for complex systems from CBFs for much simpler systems. Such ideas are illustrated through formal results, simple numerical examples, and case studies of real-world systems to which these techniques have been experimentally applied.

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References (118)
  1. A. D. Ames, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs with application to adaptive cruise control,” in Proc. Conf. Decis. Control, pp. 6271–6278, 2014.
  2. X. Xu, P. Tabuada, J. W. Grizzle, and A. D. Ames, “Robustness of control barrier functions for safety critical control,” in Proc. IFAC Conf. on Analysis and Design of Hybrid Syst., pp. 54–61, 2015.
  3. A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,” IEEE Trans. Autom. Control, vol. 62, no. 8, pp. 3861–3876, 2017.
  4. A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: theory and applications,” in Proc. Eur. Control Conf., pp. 3420–3431, 2019.
  5. T. G. Molnar, R. K. Cosner, A. W. Singletary, W. Ubellacker, and A. D. Ames, “Model-free safety-critical control for robotic systems,” IEEE Robot. Aut. Lett., vol. 7, no. 2, pp. 944–951, 2022.
  6. T. G. Molnar and A. D. Ames, “Safety-critical control with bounded inputs via reduced order models,” in Proc. Amer. Control Conf., pp. 1414–1421, 2023.
  7. A. W. Singletary, K. Klingebiel, J. Bourne, A. Browning, P. Tokumaru, and A. D. Ames, “Comparative analysis of control barrier functions and artificial potential fields for obstacle avoidance,” in Proc. IEEE/RSJ Int. Conf. Int. Robot. and Syst., pp. 8129–8136, 2021.
  8. F. Blanchini and S. Miani, Set-theoretic methods in control. Springer, 2008.
  9. Cambridge University Press, 2017.
  10. L. Hewing, K. P. Wabersich, M. Menner, and M. N. Zeilinger, “Learning-based model predictive control: Toward safe learning in control,” Ann. Rev. Control, Robot. Aut. Syst., vol. 3, pp. 269–296, 2020.
  11. I. M. Mitchell, A. M. Bayen, and C. J. Tomlin, “A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games,” IEEE Trans. Autom. Control, vol. 50, no. 7, pp. 947–957, 2005.
  12. S. Bansal, M. Chen, S. Herbert, and C. J. Tomlin, “Hamilton-Jacobi reachability: A brief overview and recent advances,” in Proc. Conf. Decis. Control, pp. 2242–2253, 2017.
  13. P. Tabuada, Verification and control of hybrid systems: a symbolic approach. Spring Science & Business Media, 2009.
  14. Springer, 2017.
  15. K. P. Wabersich, A. J. Taylor, J. J. Choi, K. Sreenath, C. J. Tomlin, A. D. Ames, and M. N. Zeilinger, “Data-driven safety filters: Hamilton-Jacobi reachability, control barrier functions, and predictive methods for uncertain systems,” IEEE Contr. Syst. Mag., vol. 43, no. 5, pp. 137–177, 2023.
  16. M. Jankovic, “Robust control barrier functions for constrained stabilization of nonlinear systems,” Automatica, vol. 96, pp. 359–367, 2018.
  17. S. Kolathaya and A. D. Ames, “Input-to-state safety with control barrier functions,” IEEE Contr. Syst. Lett., vol. 3, no. 1, pp. 108–113, 2019.
  18. A. Alan, A. J. Taylor, C. R. He, G. Orosz, and A. D. Ames, “Safe controller synthesis with tunable input-to-state safe control barrier functions,” IEEE Contr. Syst. Lett., vol. 6, pp. 908–913, 2022.
  19. A. Alan, A. J. Taylor, C. R. He, A. D. Ames, and G. Orosz, “Control barrier functions and input-to-state safety with application to automated vehicles,” IEEE Trans. Contr. Syst. Tech., vol. 31, no. 6, pp. 2744–2759, 2023.
  20. A. J. Taylor and A. D. Ames, “Adaptive safety with control barrier functions,” in Proc. Amer. Control Conf., pp. 1399–1405, 2020.
  21. B. T. Lopez, J. J. Slotine, and J. P. How, “Robust adaptive control barrier functions: An adaptive and data-driven approach to safety,” IEEE Contr. Syst. Lett., vol. 5, no. 3, pp. 1031–1036, 2021.
  22. M. H. Cohen and C. Belta, Adaptive and Learning-based Control of Safety-Critical Systems. Springer Nature, 2023.
  23. A. J. Taylor, A. Singletary, Y. Yue, and A. Ames, “Learning for safety-critical control with control barrier functions,” in Proc. Conf. Learning for Dyn. and Control, vol. 120 of Proceedings of Machine Learning Research, pp. 708–717, 2020.
  24. L. Brunke, S. Zhou, and A. P. Schoellig, “Barrier Bayesian linear regression: Online learning of control barrier conditions for safety-critical control of uncertain systems,” in Proc. Conf. Learning for Dyn. and Control, pp. 881–892, 2022.
  25. V. Dhiman, M. J. Khojasteh, M. Franceschetti, and N. Atanasov, “Control barriers in Bayesian learning of system dynamics,” IEEE Trans. Autom. Control, vol. 68, no. 1, pp. 214–229, 2023.
  26. C. Santoyo, M. Dutreix, and S. Coogan, “A barrier function approach to finite-time stochastic system verification and control,” Automatica, vol. 125, p. 109439, 2021.
  27. R. K. Cosner, P. Culbertson, A. J. Taylor, and A. D. Ames, “Robust safety under stochastic uncertainty with discrete-time control barrier functions,” in Robotics: Science and Syst., 2023.
  28. S. Dean, A. J. Taylor, R. K. Cosner, B. Recht, and A. D. Ames, “Guaranteed safety of learned perception modules via measurement-robust control barrier functions,” in Proc. Conf. Robot Learn., 2020.
  29. D. R. Agrawal and D. Panagou, “Safe and robust observer-controller synthesis using control barrier functions,” IEEE Contr. Syst. Lett., vol. 7, pp. 127–132, 2023.
  30. Y. Wang and X. Xu, “Observer-based control barrier functions for safety critical systems,” in Proc. Amer. Control Conf., pp. 709–714, 2022.
  31. G. Yang, C. Belta, and R. Tron, “Self-triggered control for safety critical systems using control barrier functions,” in Proc. Amer. Control Conf., pp. 4454–4459, 2019.
  32. A. J. Taylor, P. Ong, J. Cortés, and A. D. Ames, “Safety-critical event triggered control via input-to-state safe barrier functions,” IEEE Contr. Syst. Lett., vol. 5, no. 3, pp. 749–754, 2021.
  33. W. Xiao, C. Belta, and C. G. Cassandras, “Event-triggered control for safety-critical systems with unknown dynamics,” IEEE Trans. Autom. Control, vol. 68, no. 7, pp. 4143–4158, 2023.
  34. L. Long and J. Wang, “Safety-critical dynamic event-triggered control of nonlinear systems,” Syst. Control Lett., vol. 162, p. 105176, 2022.
  35. A. Ghaffari, I. Abel, D. Ricketts, S. Lerner, and M. Krstić, “Safety verification using barrier certificates with application to double integrator with input saturation and zero-order hold,” in Proc. Amer. Control Conf., pp. 4664–4669, 2018.
  36. J. Breeden, K. Garg, and D. Panagou, “Control barrier functions in sampled-data systems,” IEEE Contr. Syst. Lett., vol. 6, pp. 367–372, 2021.
  37. A. J. Taylor, V. D. Dorobantu, R. K. Cosner, Y. Yue, and A. D. Ames, “Safety of sampled-data systems with control barrier functions via approximate discrete time models,” in Proc. Conf. Decis. Control, pp. 7127–7134, 2022.
  38. K. Garg and D. Panagou, “Robust control barrier and control Lyapunov functions with fixed-time convergence guarantees,” in Proc. Amer. Control Conf., pp. 2292–2297, 2021.
  39. A. Polyakov and M. Krstic, “Finite-and fixed-time nonovershooting stabilizers and safety filters by homogeneous feedback,” IEEE Trans. Autom. Control, vol. 68, no. 11, pp. 6434–6449, 2023.
  40. I. Abel, D. Steeves, M. Krstić, and M. Janković, “Prescribed-time safety design for strict-feedback nonlinear systems,” IEEE Trans. Autom. Control, 2023.
  41. L. Lindemann and D. V. Dimarogonas, “Control barrier functions for signal temporal logic tasks,” IEEE Contr. Syst. Lett., vol. 3, no. 1, pp. 96–101, 2019.
  42. M. Srinivasan and S. Coogan, “Control of mobile robots using barrier functions under temporal logic specifications,” IEEE Trans. Robot, vol. 37, no. 2, pp. 363–374, 2021.
  43. M. H. Cohen, Z. Serlin, K. Leahy, and C. Belta, “Temporal logic guided safe model-based reinforcement learning: a hybrid systems approach,” Nonlinear Analysis: Hybrid Systems, vol. 47, p. 101295, 2023.
  44. Q. Nguyen and K. Sreenath, “Exponential control barrier functions for enforcing high relative-degree safety-critical constraints,” in Proc. Amer. Control Conf., pp. 322–328, 2016.
  45. W. Xiao and C. Belta, “High order control barrier functions,” IEEE Trans. Autom. Control, vol. 67, no. 7, pp. 3655–3662, 2022.
  46. M. Krstić and M. Bement, “Nonovershooting control of strict-feedback nonlinear systems,” IEEE Trans. Autom. Control, vol. 51, no. 12, pp. 1938–1943, 2006.
  47. Springer Nature, 2023.
  48. X. Tan, W. S. Cortez, and D. V. Dimarogonas, “High-order barrier functions: robustness, safety and performance-critical control,” IEEE Trans. Autom. Control, vol. 67, no. 6, pp. 3021–3028, 2022.
  49. A. Singletary, S. Kolathaya, and A. D. Ames, “Safety-critical kinematic control of robotic systems,” IEEE Contr. Syst. Lett., vol. 6, pp. 139–144, 2022.
  50. W. S. Cortez, C. K. Verginis, and D. V. Dimarogonas, “Safe, passive control for mechanical systems with application to physical human-robot interactions,” in Proc. Int. Conf. Robot. and Autom., pp. 3836–3842, 2021.
  51. W. S. Cortez, D. Oetomo, C. Manzie, and P. Choong, “Control barrier functions for mechanical systems: Theory and application to robotic grasping,” IEEE Trans. Contr. Syst. Tech., vol. 29, no. 2, pp. 530–545, 2021.
  52. W. S. Cortez and D. V. Dimarogonas, “Safe-by-design control for Euler–Lagrange systems,” Automatica, vol. 146, p. 110620, 2022.
  53. Wiley, 1995.
  54. A. J. Taylor, P. Ong, T. G. Molnar, and A. D. Ames, “Safe backstepping with control barrier functions,” in Proc. Conf. Decis. Control, pp. 5775–5782, 2022.
  55. J. Breeden and D. Panagou, “Robust control barrier functions under high relative degree and input constraints for satellite trajectories,” Automatica, vol. 155, p. 111109, 2023.
  56. T. Gurriet, M. Mote, A. Singletary, P. Nilsson, E. Feron, and A. D. Ames, “A scalable safety critical control framework for nonlinear systems,” IEEE Access, vol. 8, pp. 187249–187275, 2020.
  57. Y. Chen, M. Jankovic, M. Santillo, and A. D. Ames, “Backup control barrier functions: Formulation and comparative study,” in Proc. Conf. Decis. Control, pp. 6835–6841, 2021.
  58. J. Breeden and D. Panagou, “Predictive control barrier functions for online safety critical control,” in Proc. Conf. Decis. Control, pp. 924–931, 2022.
  59. K. Wabersich and M. Zeilinger, “Predictive control barrier functions: Enhanced safety mechanisms for learning-based control,” IEEE Trans. Autom. Control, vol. 68, no. 5, 2022.
  60. A. Clark, “Verification and synthesis of control barrier functions,” in Proc. Conf. Decis. Control, pp. 6105–6112, 2021.
  61. A. Clark, “A semi-algebraic framework for verification and synthesis of control barrier functions,” arXiv preprint arXiv:2209.00081, 2022.
  62. H. Dai and F. Permenter, “Convex synthesis and verification of control-Lyapunov and barrier functions with input constraints,” in Proc. Amer. Control Conf., pp. 4116–4123, 2023.
  63. P. Zhao, R. Ghabcheloo, Y. Cheng, H. Abdi, and N. Hovakimyan, “Convex synthesis of control barrier functions under input constraints,” IEEE Contr. Syst. Lett., vol. 7, pp. 3102–3107, 2023.
  64. J. J. Choi, D. Lee, K. Sreenath, C. J. Tomlin, and S. L. Herbert, “Robust control barrier–value functions for safety-critical control,” in Proc. Conf. Decis. Control, pp. 6814–6821, 2021.
  65. S. Tonkens and S. Herbert, “Refining control barrier functions through Hamilton-Jacobi reachability,” in Proc. IEEE/RSJ Int. Conf. Int. Robot. and Syst., pp. 13355–13362, 2022.
  66. J. J. Choi, D. Lee, B. Li, J. P. How, K. Sreenath, S. L. Herbert, and C. J. Tomlin, “A forward reachability perspective on robust control invariance and discount factors in reachability analysis,” arXiv preprint arXiv:2310.17180, 2023.
  67. C. Dawson, Z. Qin, S. Gao, and C. Fan, “Safe nonlinear control using robust neural Lyapunov-barrier functions,” in Proc. Conf. Robot Learn., 2022.
  68. C. Dawson, S. Gao, and C. Fan, “Safe control with learned certificates: A survey of neural Lyapunov, barrier, and contraction methods for robotics and control,” IEEE Trans. Robot, vol. 39, no. 3, pp. 1749–1767, 2023.
  69. O. So, Z. Serlin, M. Mann, J. Gonzales, K. Rutledge, N. Roy, and C. Fan, “How to train your neural control barrier function: Learning safety filters for complex input-constrained systems,” arXiv preprint arXiv:2310.15478, 2023.
  70. A. Robey, H. Hu, L. Lindemann, H. Zhang, D. V. Dimorogonas, S. Tu, and N. Matni, “Learning control barrier functions from expert demonstrations,” in Proc. Conf. Decis. Control, pp. 3717–3724, 2020.
  71. L. Lindemann, H. Hu, A. Robey, H. Zhang, D. V. Dimorogonas, S. Tu, and N. Matni, “Learning hybrid control barrier functions from data,” in Proc. Conf. Robot Learn., 2020.
  72. S. Zhao and Z. Sun, “Defend the practicality of single-integrator models in multi-robot coordination control,” in Proc. Int. Conf Control Autom., pp. 666–671, 2017.
  73. Springer, 2001.
  74. M. H. Raibert, Legged Robots That Balance. MIT Press, 1986.
  75. S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Yokoi, and H. Hirukawa, “A realtime pattern generator for biped walking,” in Proc. Int. Conf. Robot. and Autom., 2002.
  76. X. Xiong and A. D. Ames, “3-D underactuated bipedal walking via H-LIP based gait synthesis and stepping stabilization,” IEEE Trans. Robot, vol. 38, no. 4, pp. 2405–2425, 2022.
  77. A. W. Singletary, W. Guffey, T. Molnar, R. Sinnet, and A. D. Ames, “Safety-critical manipulation for collision-free food preparation,” IEEE Robot. Aut. Lett., vol. 7, no. 4, pp. 10954–10961, 2022.
  78. J. Kim, J. Lee, and A. D. Ames, “Safety-critical coordination for cooperative legged locomotion via control barrier functions,” in Proc. IEEE/RSJ Int. Conf. Int. Robot. and Syst., pp. 2368–2375, 2023.
  79. M. H. Cohen, P. Ong, G. Bahati, and A. D. Ames, “Characterizing smooth safety filters via the implicit function theorem,” IEEE Contr. Syst. Lett., vol. 7, pp. 3890–3895, 2023.
  80. M. W. Spong, “Partial feedback linearization of underactuated mechanical systems,” in Proc. IEEE/RSJ Int. Conf. Int. Robot. and Syst., 1994.
  81. H. K. Khalil, Nonlinear Systems. Prentice Hall, 3 ed., 2002.
  82. G. Bouligand, “Introducion a la geometrie infinitesimale directe,” Paris: Gauthiers-Villars, 1932.
  83. M. Nagumo, “Über die lage der integralkurven gewöhnlicher differentialgleichungen,” Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, vol. 24, pp. 551–559, 1942.
  84. J. M. Bony, “Principe du maximum, inègalite de harnack et unicité du probléme de cauchy pour les opérateurs elliptiques dégénérés,” Annales de l’Institut Fourier, Grenoble, vol. 19, pp. 277–304, 1969.
  85. H. Brezis, “On a characterization of flow‐invariant sets,” Communications on Pure and Applied Mathematics, vol. 23, pp. 261–263, 1970.
  86. R. M. Redheffer, “The theorems of Bony and Brezis on flow-invariant sets,” The American Mathematical Monthly, vol. 79, no. 7, pp. 740–747, 1972.
  87. Addison-Wesley, 1983.
  88. R. Konda, A. D. Ames, and S. Coogan, “Characterizing safety: Minimal control barrier functions from scalar comparison systems,” IEEE Contr. Syst. Lett., vol. 5, no. 2, pp. 523–528, 2021.
  89. S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
  90. T. G. Molnar and A. D. Ames, “Composing control barrier functions for complex safety specifications,” IEEE Contr. Syst. Lett., vol. 7, pp. 3615–3620, 2023.
  91. Springer, 1997.
  92. P. Ong and J. Cortes, “Universal formula for smooth safe stabilization,” in Proc. Conf. Decis. Control, pp. 2373–2378, 2019.
  93. E. D. Sontag, “A ‘universal’ construction of Artstein’s theorem on nonlinear stabilization,” Syst. Control Lett., vol. 13, no. 2, pp. 117–123, 1989.
  94. R. A. Freeman and P. V. Kokotović, “Backstepping design of robust controllers for a class of nonlinear systems,” IFAC Proceedings Volumes, vol. 25, no. 13, pp. 431–436, 1992.
  95. W. Xiao and C. Belta, “Control barrier functions for systems with high relative degree,” in Proc. Conf. Decis. Control, pp. 474–479, 2019.
  96. course notes for MIT 6.832, 2023. https://underactuated.csail.mit.edu.
  97. P. Glotfelter, J. Cortés, and M. Egerstedt, “Nonsmooth barrier functions with applications to multi-robot systems,” IEEE Contr. Syst. Lett., vol. 1, no. 2, pp. 310–315, 2017.
  98. J. Usevitch, K. Garg, and D. Panagou, “Strong invariance using control barrier functions: A Clarke tangent cone approach,” in Proc. Conf. Decis. Control, pp. 2044–2049, 2020.
  99. P. Glotfelter, J. Cortés, and M. Egerstedt, “A nonsmooth approach to controller synthesis for Boolean specifications,” IEEE Trans. Autom. Control, vol. 66, no. 11, pp. 5160–5174, 2020.
  100. T. G. Molnar, S. K. Kannan, J. Cunningham, K. Dunlap, K. L. Hobbs, and A. D. Ames, “Collision avoidance and geofencing for fixed-wing aircraft with control barrier functions,” arXiv preprint arXiv:2403.02508, 2024.
  101. A. Singletary, A. Swann, Y. Chen, and A. D. Ames, “Onboard safety guarantees for racing drones: High-speed geofencing with control barrier functions,” IEEE Robot. Aut. Lett., vol. 7, no. 2, pp. 2897–2904, 2022.
  102. W. Ubellacker, N. Csomay-Shanklin, T. G. Molnar, and A. D. Ames, “Verifying safe transitions between dynamic motion primitives on legged robots,” in Proc. IEEE/RSJ Int. Conf. Int. Robot. and Syst., pp. 8477–8484, 2021.
  103. K. A. Hamed, J. Kim, and A. Pandala, “Quadrupedal locomotion via event-based predictive control and QP-based virtual constraints,” IEEE Robot. Aut. Lett., vol. 5, no. 3, pp. 4463–4470, 2020.
  104. J. Kim, R. T. Fawcett, V. R. Ramidi, A. D. Ames, and K. A. Hamed, “Layered control for cooperative locomotion of two quadrupedal robots: Centralized and distributed approaches,” IEEE Trans. Robot, vol. 39, no. 6, pp. 4728–4748, 2023.
  105. C. R. He, A. Alan, T. G. Molnár, S. S. Avedisov, A. H. Bell, R. Zukouski, M. Hunkler, J. Yan, and G. Orosz, “Improving fuel economy of heavy-duty vehicles in daily driving,” in Proc. Amer. Control Conf., pp. 2306–2311, 2020.
  106. L. Zhang and G. Orosz, “Motif-based design for connected vehicle systems in presence of heterogeneous connectivity structures and time delays,” IEEE Trans. Intell. Trans. Syst., vol. 17, no. 6, pp. 1638–1651, 2016.
  107. T. Gurriet, A. Singletary, J. Reher, L. Ciarletta, E. Feron, and A. Ames, “Towards a framework for realizable safety critical control through active set invariance,” in Proc. ACM/IEEE Int. Conf. Cyber-Physical Syst., pp. 98–106, 2018.
  108. D. Agrawal and D. Panagou, “Safe control synthesis via input constrained control barrier functions,” in Proc. Conf. Decis. Control, pp. 6113–6118, 2021.
  109. A. D. Ames, G. Notomista, Y. Wardi, and M. Egerstedt, “Integral control barrier functions for dynamically defined control laws,” IEEE Contr. Syst. Lett., vol. 5, no. 3, pp. 887–892, 2021.
  110. N. Csomay-Shanklin, A. J. Taylor, U. Rosolia, and A. D. Ames, “Multi-rate planning and control of uncertain nonlinear systems: Model predictive control and control Lyapunov functions,” in Proc. Conf. Decis. Control, pp. 3732–3739, 2022.
  111. A. Isidori, Nonlinear Control Systems. Springer, third ed., 1995.
  112. A. Isidori, “The zero dynamics of a nonlinear system: From the origin to the latest progresses of a long successful story,” European Journal of Control, vol. 19, pp. 369–378, 2013.
  113. CRC Press, 2007.
  114. M. Maggiore and L. Consolini, “Virtual holonomic constraints for euler–lagrange systems,” IEEE Trans. Autom. Control, vol. 58, no. 4, pp. 1001–1008, 2013.
  115. K. A. Hamed and A. D. Ames, “Nonholonomic hybrid zero dynamics for the stabilization of periodic orbits: Application to underactuated robotic walking,” IEEE Trans. Contr. Syst. Tech., vol. 28, no. 6, pp. 2689–2696, 2020.
  116. A. Isidori and C. Byrnes, “Output regulation of nonlinear systems,” IEEE Trans. Autom. Control, vol. 35, no. 2, pp. 131–140, 1990.
  117. M. D. D. Benedetto and J. W. Grizzle, “Asymptotic model matching for nonlinear systems,” IEEE Trans. Autom. Control, vol. 39, no. 8, pp. 1539–1550, 1994.
  118. J. W. Grizzle, M. D. D. Benedetto, and F. Lamnabhi-Lagarrigue, “Necessary conditions for asymptotic tracking in nonlinear systems,” IEEE Trans. Autom. Control, vol. 39, no. 9, pp. 1782–1794, 1994.
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