doi: 10.3934/jgm.2022013
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Necessary conditions for feedback stabilization and safety

School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA

Received  June 2021 Revised  February 2022 Early access June 2022

Brockett's necessary condition yields a test to determine whether a system can be made to stabilize about some operating point via continuous, purely state-dependent feedback. For many real-world systems, however, one wants to stabilize sets which are more general than a single point. One also wants to control such systems to operate safely by making obstacles and other "dangerous" sets repelling.

We generalize Brockett's necessary condition to the case of stabilizing general compact subsets having a nonzero Euler characteristic in general ambient state spaces (smooth manifolds). Using this generalization, we also formulate a necessary condition for the existence of "safe" control laws. We illustrate the theory in concrete examples and for some general classes of systems including a broad class of nonholonomically constrained Lagrangian systems. We also show that, for the special case of stabilizing a point, the specialization of our general stabilizability test is stronger than Brockett's.

Citation: Matthew D. Kvalheim, Daniel E. Koditschek. Necessary conditions for feedback stabilization and safety. Journal of Geometric Mechanics, doi: 10.3934/jgm.2022013
References:
[1]

A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath and P. Tabuada, Control Barrier Functions: Theory and Applications, 2019 18th European Control Conference (ECC), IEEE, 2019, 3420–3431. doi: 10.1109/LCSYS.2020.2997952.

[2]

A. A. Agrachev and D. Liberzon, Lie-algebraic stability criteria for switched systems, SIAM Journal on Control and Optimization, 40 (2001), no. 1,253–269. doi: 10.1137/S0363012999365704.

[3]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Addison-Wesley, 1987.

[4]

A. Astolfi, Discontinuous control of nonholonomic systems, Systems & Control Letters, 27 (1996), 37-45.  doi: 10.1016/0167-6911(95)00041-0.

[5]

A. D. AmesX. XuJ. W. Grizzle and P. Tabuada, Control barrier function based quadratic programs for safety critical systems, IEEE Trans. Automat. Control, 62 (2017), 3861-3876.  doi: 10.1109/TAC.2016.2638961.

[6]

Y. Baryshnikov, Topological Perplexity in Feedback Stabilization, 2021, preprint available from: http://publish.illinois.edu/ymb/files/2021/08/tp.pdf.

[7]

A. Bloch and S. Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes, Systems & Control Letters, 29 (1996), 91-99.  doi: 10.1016/S0167-6911(96)00049-7.

[8]

A. M. BlochS. V. Drakunov and M. K. Kinyon, Stabilization of nonholonomic systems using isospectral flows, SIAM Journal on Control and Optimization, 38 (2000), 855-874.  doi: 10.1137/S0363012998335607.

[9]

C. I. Byrnes and A. Isidori, On the attitude stabilization of rigid spacecraft, Automatica J. IFAC, 27 (1991), 87-95.  doi: 10.1016/0005-1098(91)90008-P.

[10]

A. M. Bloch, Nonholonomic Mechanics and Control, 2$^{nd}$ edition, 24, Springer-Verlag, 2015. doi: 10.1007/978-1-4939-3017-3.

[11]

A. Bry and N. Roy, Rapidly-exploring random belief trees for motion planning under uncertainty, IEEE, 2011,723–730. doi: 10.1109/ICRA.2011.5980508.

[12]

A. M. BlochM. Reyhanoglu and N. H. McClamroch, Control and stabilization of nonholonomic dynamic systems, IEEE Transactions on Automatic Control, 37 (1992), 1746-1757.  doi: 10.1109/9.173144.

[13]

R. W. Brockett, Control theory and analytical mechanics, Geometric Control Theory, Lie Groups: History, Frontiers and Applications, (1977), 1–46.

[14]

R. W. Brockett, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 27 (1983), 181-191. 

[15]

Y. Baryshnikov and B. Shapiro, How to run a centipede: A topological perspective, Geometric control theory and sub-Riemannian geometry, Springer INdAM Ser., 5 (2014), 37–51. doi: 10.1007/978-3-319-02132-4_3.

[16]

C. I. Byrnes, On Brockett's necessary condition for stabilizability and the topology of Liapunov functions on ${ \mathbb{R}^n }$, Communications in Information and Systems, 8 (2008), 333-352.  doi: 10.4310/CIS.2008.v8.n4.a1.

[17]

J. CulbertsonP. GustafsonD. E. Koditschek and and P. F. Stiller, Formal composition of hybrid systems, Theory and Applications of Categories, 35 (2020), 1634-1682. 

[18]

H. M. Choset, K. M. Lynch, S. Hutchinson, G. Kantor, W. Burgard, L. Kavraki, S. Thrun and R. C. Arkin, Principles of Robot Motion: Theory, Algorithms and Implementation, MIT press, 2005.

[19]

F. H. ClarkeY. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, Journal of Differential Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.

[20]

B. A. Christopherson, B. S. Mordukhovich and F. Jafari, Continuous feedback stabilization of nonlinear control systems by composition operators, ESAIM Control Optim. Calc. Var., 28 (2022), 22 pp.

[21]

C. C. Conley, Isolated Invariant Sets and the Morse Index, 38, American Mathematical Society, 1978.

[22]

J.-M. Coron, A necessary condition for feedback stabilization, Systems & Control Letters, 14 (1990), 227-232.  doi: 10.1016/0167-6911(90)90017-O.

[23]

J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Mathematics of Control, Signals and Systems, 5 (1992), 295-312.  doi: 10.1007/BF01211563.

[24]

J.-M. Coron, Control and Nonlinearity, 136, American Mathematical Soc., 2007. doi: 10.1090/surv/136.

[25]

C. C. De Wit and O. J. Sordalen, Exponential stabilization of mobile robots with nonholonomic constraints, [1991] Proceedings of the 30th IEEE Conference on Decision and Control, IEEE, 1991,692–697. doi: 10.1109/9.173153.

[26]

K. Eda and K. Kawamura, The singular homology of the Hawaiian earring, Journal of the London Mathematical Society, 62 (2000), 305-310.  doi: 10.1112/S0024610700001071.

[27]

R. J. Full and D. E. Koditschek, Templates and anchors: Neuromechanical hypotheses of legged locomotion on land, Journal of Experimental Biology, 202 (1999), 3325-3332.  doi: 10.1242/jeb.202.23.3325.

[28]

A. Fathi and P. Pageault, Smoothing Lyapunov functions, Trans. Amer. Math. Soc., 371 (2019), 1677-1700.  doi: 10.1090/tran/7329.

[29]

R. GuptaF. JafariR. J. Kipka and B. S. Mordukhovich, Linear openness and feedback stabilization of nonlinear control systems, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 1103-1119.  doi: 10.3934/dcdss.2018063.

[30]

J. W. Grizzle and S. I. Marcus, The structure of nonlinear control systems possessing symmetries, IEEE Transactions on Automatic Control, 30 (1985), 248-258.  doi: 10.1109/TAC.1985.1103927.

[31]

M. Gobbino, Topological properties of attractors for dynamical systems, Topology, 40 (2001), 279-298.  doi: 10.1016/S0040-9383(99)00061-0.

[32]

V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, Providence, RI, 2010, Reprint of the 1974 original. doi: 10.1090/chel/370.

[33]

B. Günther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc., 119 (1993), 321-329.  doi: 10.1090/S0002-9939-1993-1170545-4.

[34]

R. GoebelR. G. Sanfelice and A. Teel, Hybrid dynamical systems, Control Systems, IEEE, 29 (2009), 28-93.  doi: 10.1109/MCS.2008.931718.

[35]

P. Hartman, Ordinary Differential Equations, 2$^{nd} $ edition, SIAM, 2002. doi: 10.1137/1.9780898719222.

[36]

H. M. Hastings, Shape Theory and Dynamical Systems, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lecture Notes in Math., 668, Springer, Berlin, 1978,150–159.

[37]

H. M. Hastings, A higher-dimensional Poincaré-Bendixson theorem, Glas. Mat. Ser. III, 14 (1979), 263-268. 

[38]

A. Hatcher, Algebraic topology, 1$^{st}$ edition, Cambridge University Press, 2001.

[39]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994, Corrected reprint of the 1976 original.

[40]

E. HaghverdiP. Tabuada and G. J. Pappas, Bisimulation relations for dynamical, control, and hybrid systems, Theoretical Computer Science, 342 (2005), 229-261.  doi: 10.1016/j.tcs.2005.03.045.

[41]

J. D. Hunter, Matplotlib: A 2D graphics environment, Computing in Science & Engineering, 9 (2007), 90-95.  doi: 10.1109/MCSE.2007.55.

[42]

M. Hurley, Attractors: Persistence, and density of their basins, Transactions of the American Mathematical Society, 269 (1982), 247-271.  doi: 10.1090/S0002-9947-1982-0637037-7.

[43]

M. Ishikawa and M. Sampei, On equilibria set and feedback stabilizability of nonlinear control systems, IFAC Proceedings Volumes, 31 (1998), 609-614.  doi: 10.1016/S1474-6670(17)40404-6.

[44]

W. Jaco, Surfaces embedded in $ {M}^2\times {S}^1$, Canadian Journal of Mathematics 22 (1970), 553–568. doi: 10.4153/CJM-1970-063-x.

[45]

A. M. JohnsonS. A. Burden and D. E. Koditschek, A hybrid systems model for simple manipulation and self-manipulation systems, The International Journal of Robotics Research, 35 (2016), 1354-1392.  doi: 10.1177/0278364916639380.

[46]

E. Kappos, The Role of Morse-Lyapunov Functions in the Design of Nonlinear Global Feedback Dynamics, Variable Structure and Lyapunov Control, Springer, 1994,249–267. doi: 10.1007/BFb0033687.

[47]

E. Kappos, Necessary Conditions for Global Feedback Control, Proc Internat Symp on Nonlinear Theory and its Applications, Las Vegas, LA, Citeseer, 1995.

[48]

H. Khennouf and C. C. De Wit, On the construction of stabilizing discontinuous controllers for nonholonomic systems, IFAC Proceedings Volumes, 28 (1995), 667-672.  doi: 10.1016/S1474-6670(17)46905-9.

[49]

D. E. KoditschekR. J. Full and M. Buehler, Mechanical aspects of legged locomotion control, Arthropod Structure & Development, 33 (2004), 251-272.  doi: 10.1016/j.asd.2004.06.003.

[50]

U. V. KalabićR. GuptaS. Di CairanoA. M. Bloch and I. V. Kolmanovsky, MPC on manifolds with an application to the control of spacecraft attitude on SO(3), Automatica J. IFAC, 76 (2017), 293-300.  doi: 10.1016/j.automatica.2016.10.022.

[51]

M. D. KvalheimP. Gustafson and D. E. Koditschek, Conley's fundamental theorem for a class of hybrid systems, SIAM J. Appl. Dyn. Syst., 20 (2021), 784-825.  doi: 10.1137/20M1336576.

[52]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, New York, 2004. doi: 10.1007/b97315.

[53]

D. E. Koditschek, What is robotics? Why do we need it and how can we get it?, Annual Review of Control, Robotics, and Autonomous Systems, 4 (2021), 1-33.  doi: 10.1146/annurev-control-080320-011601.

[54]

M. D. Kvalheim and S. Revzen, Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits, Phys. D., 425 (2021), 132959, 20. doi: 10.1016/j.physd.2021.132959.

[55]

L. V. KolmanovksyM Reyhanoglu and N. H. McClamroch, Discontinuous feedback stabilization of nonholonomic systems in extended power form, Proceedings of 1994 33rd IEEE Conference on Decision and Control, 4 (1994), 3469-3474.  doi: 10.1109/CDC.1994.411683.

[56]

W. Lin and C. I. Byrnes, Design of discrete-time nonlinear control systems via smooth feedback, IEEE Trans. Automat. Control, 39 (1994), 2340-2346.  doi: 10.1109/9.333790.

[57]

J. M. Lee, Introduction to Topological Manifolds, 2$^{nd}$ edition, Springer Science & Business Media, 2010. doi: 10.1007/978-1-4419-7940-7.

[58]

J. M. Lee, Introduction to Smooth Manifolds, 2$^{nd}$ edition, Springer-Verlag, 2013.

[59]

E. Lerman, A Category of Hybrid Systems, preprint, arXiv: 1612.01950, (2016).

[60]

E. Lerman, Networks of open systems, Journal of Geometry and Physics, 130 (2018), 81-112.  doi: 10.1016/j.geomphys.2018.03.020.

[61]

G. A. D. Lopes and D. E. Koditschek, Visual servoing for nonholonomically constrained three degree of freedom kinematic systems, The International Journal of Robotics Research, 26 (2007), 715-736.  doi: 10.1177/0278364907080737.

[62]

A.-R. Mansouri, Local asymptotic feedback stabilization to a submanifold: Topological conditions, Systems Control Lett., 56 (2007), 525-528.  doi: 10.1016/j.sysconle.2007.03.001.

[63]

A.-R. Mansouri, Topological obstructions to submanifold stabilization, IEEE Trans. Automat. Control, 55 (2010), 1701-1703.  doi: 10.1109/TAC.2010.2046922.

[64]

A.-R. Mansouri, Topological Obstructions to Distributed Feedback Stabilization, 2013 51st Annual Allerton Conference on Communication, Control, and Computing, IEEE, 2013, 1573–1575.

[65]

A.-R. Mansouri, Topological Obstructions to Distributed Feedback Stabilization to a Submanifold, 2015 Proceedings of the Conference on Control and its Applications, SIAM, 2015, 76–80.

[66]

W. S. Massey, Homology and Cohomology Theory: An Approach Based on Alexander-Spanier Cochains, 46, Marcel Dekker, 1978.

[67]

W. S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991.

[68]

J. Milnor, Topology from the Differentiable Viewpoint, Princeton university press, 1965.

[69]

J. Milnor, On the concept of attractor: Correction and remarks, Communications in Mathematical Physics, 102 (1985), 517-519.  doi: 10.1007/BF01209298.

[70]

J. W. Milnor, Attractor, Scholarpedia, 1 (2006), 1815.  doi: 10.4249/scholarpedia.1815.

[71]

R. T. M'Closkey and R. M. Murray, Convergence Rates for Nonholonomic Systems in Power Form, 1993 American Control Conference, IEEE, 1993, 2967–2972. doi: 10.23919/ACC.1993.4793446.

[72]

J. C. Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Springer, 2004. doi: 10.1007/b84020.

[73]

S. Morita, Geometry of Differential Forms, 201, American Mathematical Soc., 2001. doi: 10.1090/mmono/201.

[74]

P. MorinJ.-B. Pomet and C. Samson, Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of Lie brackets in closed loop, SIAM J. Control Optim., 38 (1999), 22-49.  doi: 10.1137/S0363012997315427.

[75]

P. Morin and C. Samson, Control of nonlinear chained systems: From the Routh-Hurwitz stability criterion to time-varying exponential stabilizers, IEEE Trans. Automat. Control, 45 (2000), 141-146.  doi: 10.1109/9.827372.

[76]

P. W. Michor and C. Vizman, n-transitivity of certain diffeomorphism groups, Acta Math. Univ. Comenianae, 63 (1994), 221-225. 

[77]

R. OrsiL. Praly and I. Mareels, Necessary conditions for stability and attractivity of continuous systems, International Journal of Control, 76 (2003), 1070-1077.  doi: 10.1080/0020717031000122338.

[78]

L. Palmieri and K. O. Arras, A Novel RRT Extend Function for Efficient and Smooth Mobile Robot Motion Planning, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, 2014,205–211. doi: 10.1109/IROS.2014.6942562.

[79]

V. Pacelli, O. Arslan and D. E. Koditschek, Integration of Local Geometry and Metric Information in Sampling-Based motion Planning, 2018 IEEE International Conference on Robotics and Automation (ICRA), IEEE, 2018, 3061–3068. doi: 10.1109/ICRA.2018.8460739.

[80]

J. J. Park and B. Kuipers, Feedback Motion Planning via Non-Holonomic RRT* for Mobile Robots, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2015, 4035–4040. doi: 10.1109/IROS.2015.7353946.

[81]

J.-B. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift, Systems & Control Letters, 18 (1992), 147-158.  doi: 10.1016/0167-6911(92)90019-O.

[82]

A. Papachristodoulou and S. Prajna, A Tutorial on Sum of Squares Techniques for Systems Analysis, Proceedings of the 2005, American Control Conference, 2005., IEEE, 2005, 2686–2700. doi: 10.1007/10997703_2.

[83]

C. Prieur and E. Trélat, Robust optimal stabilization of the Brockett integrator via a hybrid feedback, Mathematics of Control, Signals and Systems, 17 (2005), 201-216.  doi: 10.1007/s00498-005-0152-9.

[84]

C. C. Pugh, A generalized Poincaré index formula, Topology, 7 (1968), 217-226.  doi: 10.1016/0040-9383(68)90002-5.

[85]

S. Revzen, D. E. Koditschek and R. J. Full, Towards Testable Neuromechanical Control Architectures for Running, Progress in Motor Control, Springer, 2009, 25–55. doi: 10.1007/978-0-387-77064-2_3.

[86]

S. F. RobertsD. E. Koditschek and L. J. Miracchi, Examples of Gibsonian affordances in legged robotics research using an empirical, generative framework, Frontiers in Neurorobotics, 14 (2020), 12.  doi: 10.3389/fnbot.2020.00012.

[87]

J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynam. Systems, 8 (1988), 375-393.  doi: 10.1017/S0143385700009494.

[88]

E. P. Ryan, On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback, SIAM Journal on Control and Optimization, 32 (1994), 1597-1604.  doi: 10.1137/S0363012992235432.

[89]

O. J. Sordalen and O Egeland, Exponential stabilization of nonholonomic chained systems, IEEE transactions on automatic control, 40 (1995), 35-49.  doi: 10.1109/9.362901.

[90]

M. Shub, Dynamical systems, filtrations and entropy, Bulletin of the American Mathematical Society, 80 (1974), 27-41.  doi: 10.1090/S0002-9904-1974-13344-6.

[91]

S. N. SimićK. H. JohanssonJ. Lygeros and S. Sastry, Towards a geometric theory of hybrid systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 12 (2005), 649-687. 

[92]

J. Seipel, M. Kvalheim, S. Revzen, M. A. Sharbafi and A. Seyfarth, Conceptual Models of Legged Locomotion, Bioinspired Legged Locomotion, Elsevier, 2017, 55–131. doi: 10.1016/B978-0-12-803766-9.00004-X.

[93]

E. D. Sontag, A "universal" construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123.  doi: 10.1016/0167-6911(89)90028-5.

[94]

E. D. Sontag, Control-Lyapunov Functions, Open problems in mathematical systems and control theory, Comm. Control Engrg. Ser., Springer, London, 1999,211–216.

[95]

E. H. Spanier, Algebraic Topology, Springer Science & Business Media, 1966.

[96]

Y.-P. Tian and S. Li, Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control, Automatica, 38 (2002), 1139-1146.  doi: 10.1016/S0005-1098(01)00303-X.

[97]

A. R. Teel, R. M. Murray and G. Walsh, Nonholonomic Control Systems: From Steering to Stabilization with Sinusoids, [1992] Proceedings of the 31st IEEE Conference on Decision and Control, IEEE, 1992, 1603–1609. doi: 10.1109/CDC.1992.371456.

[98]

T. Urakubo, Feedback stabilization of a nonholonomic system with potential fields: Application to a two-wheeled mobile robot among obstacles, Nonlinear Dynamics, 81 (2015), 1475-1487.  doi: 10.1007/s11071-015-2082-5.

[99]

T. Urakubo, Stability analysis and control of nonholonomic systems with potential fields, Journal of Intelligent & Robotic Systems, 89 (2018), 121-137. 

[100]

A. van der Schaft, Symmetries and conservation laws for Hamiltonian systems with inputs and outputs: A generalization of Noether's theorem, Systems Control Lett., 1 (1981/82), 108-115.  doi: 10.1016/S0167-6911(81)80046-1.

[101]

V. VasilopoulosG. PavlakosK. SchmeckpeperK. Daniilidis and D. E. Koditschek, Reactive navigation in partially familiar planar environments using semantic perceptual feedback, The International Journal of Robotics Research, 41 (2022), 85-126. 

[102]

V. Vasilopoulos, T. T. Topping, W. Vega-Brown, N. Roy and D. E. Koditschek, Sensor-Based Reactive Execution of Symbolic Rearrangement Plans by a Legged Mobile Manipulator, 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2018, 3298–3305. doi: 10.1109/IROS.2018.8594342.

[103]

G. C. Walsh and L. G. Bushnell, Stabilization of Multiple Input Chained Form Control Systems, Proceedings of 32nd IEEE Conference on Decision and Control, IEEE, 1993,959–964. doi: 10.1016/0167-6911(94)00061-Y.

[104]

F. W. Wilson Jr, The structure of the level surfaces of a Lyapunov function, J. Differential Equations, 3 (1967), 323-329.  doi: 10.1016/0022-0396(67)90035-6.

[105]

F. W. Wilson Jr, Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc., 139 (1969), 413-428.  doi: 10.1090/S0002-9947-1969-0251747-9.

[106]

J. Zabczyk, Some comments on stabilizability, Applied Mathematics and Optimization, 19 (1989), 1-9.  doi: 10.1007/BF01448189.

show all references

References:
[1]

A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath and P. Tabuada, Control Barrier Functions: Theory and Applications, 2019 18th European Control Conference (ECC), IEEE, 2019, 3420–3431. doi: 10.1109/LCSYS.2020.2997952.

[2]

A. A. Agrachev and D. Liberzon, Lie-algebraic stability criteria for switched systems, SIAM Journal on Control and Optimization, 40 (2001), no. 1,253–269. doi: 10.1137/S0363012999365704.

[3]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Addison-Wesley, 1987.

[4]

A. Astolfi, Discontinuous control of nonholonomic systems, Systems & Control Letters, 27 (1996), 37-45.  doi: 10.1016/0167-6911(95)00041-0.

[5]

A. D. AmesX. XuJ. W. Grizzle and P. Tabuada, Control barrier function based quadratic programs for safety critical systems, IEEE Trans. Automat. Control, 62 (2017), 3861-3876.  doi: 10.1109/TAC.2016.2638961.

[6]

Y. Baryshnikov, Topological Perplexity in Feedback Stabilization, 2021, preprint available from: http://publish.illinois.edu/ymb/files/2021/08/tp.pdf.

[7]

A. Bloch and S. Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes, Systems & Control Letters, 29 (1996), 91-99.  doi: 10.1016/S0167-6911(96)00049-7.

[8]

A. M. BlochS. V. Drakunov and M. K. Kinyon, Stabilization of nonholonomic systems using isospectral flows, SIAM Journal on Control and Optimization, 38 (2000), 855-874.  doi: 10.1137/S0363012998335607.

[9]

C. I. Byrnes and A. Isidori, On the attitude stabilization of rigid spacecraft, Automatica J. IFAC, 27 (1991), 87-95.  doi: 10.1016/0005-1098(91)90008-P.

[10]

A. M. Bloch, Nonholonomic Mechanics and Control, 2$^{nd}$ edition, 24, Springer-Verlag, 2015. doi: 10.1007/978-1-4939-3017-3.

[11]

A. Bry and N. Roy, Rapidly-exploring random belief trees for motion planning under uncertainty, IEEE, 2011,723–730. doi: 10.1109/ICRA.2011.5980508.

[12]

A. M. BlochM. Reyhanoglu and N. H. McClamroch, Control and stabilization of nonholonomic dynamic systems, IEEE Transactions on Automatic Control, 37 (1992), 1746-1757.  doi: 10.1109/9.173144.

[13]

R. W. Brockett, Control theory and analytical mechanics, Geometric Control Theory, Lie Groups: History, Frontiers and Applications, (1977), 1–46.

[14]

R. W. Brockett, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 27 (1983), 181-191. 

[15]

Y. Baryshnikov and B. Shapiro, How to run a centipede: A topological perspective, Geometric control theory and sub-Riemannian geometry, Springer INdAM Ser., 5 (2014), 37–51. doi: 10.1007/978-3-319-02132-4_3.

[16]

C. I. Byrnes, On Brockett's necessary condition for stabilizability and the topology of Liapunov functions on ${ \mathbb{R}^n }$, Communications in Information and Systems, 8 (2008), 333-352.  doi: 10.4310/CIS.2008.v8.n4.a1.

[17]

J. CulbertsonP. GustafsonD. E. Koditschek and and P. F. Stiller, Formal composition of hybrid systems, Theory and Applications of Categories, 35 (2020), 1634-1682. 

[18]

H. M. Choset, K. M. Lynch, S. Hutchinson, G. Kantor, W. Burgard, L. Kavraki, S. Thrun and R. C. Arkin, Principles of Robot Motion: Theory, Algorithms and Implementation, MIT press, 2005.

[19]

F. H. ClarkeY. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, Journal of Differential Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.

[20]

B. A. Christopherson, B. S. Mordukhovich and F. Jafari, Continuous feedback stabilization of nonlinear control systems by composition operators, ESAIM Control Optim. Calc. Var., 28 (2022), 22 pp.

[21]

C. C. Conley, Isolated Invariant Sets and the Morse Index, 38, American Mathematical Society, 1978.

[22]

J.-M. Coron, A necessary condition for feedback stabilization, Systems & Control Letters, 14 (1990), 227-232.  doi: 10.1016/0167-6911(90)90017-O.

[23]

J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Mathematics of Control, Signals and Systems, 5 (1992), 295-312.  doi: 10.1007/BF01211563.

[24]

J.-M. Coron, Control and Nonlinearity, 136, American Mathematical Soc., 2007. doi: 10.1090/surv/136.

[25]

C. C. De Wit and O. J. Sordalen, Exponential stabilization of mobile robots with nonholonomic constraints, [1991] Proceedings of the 30th IEEE Conference on Decision and Control, IEEE, 1991,692–697. doi: 10.1109/9.173153.

[26]

K. Eda and K. Kawamura, The singular homology of the Hawaiian earring, Journal of the London Mathematical Society, 62 (2000), 305-310.  doi: 10.1112/S0024610700001071.

[27]

R. J. Full and D. E. Koditschek, Templates and anchors: Neuromechanical hypotheses of legged locomotion on land, Journal of Experimental Biology, 202 (1999), 3325-3332.  doi: 10.1242/jeb.202.23.3325.

[28]

A. Fathi and P. Pageault, Smoothing Lyapunov functions, Trans. Amer. Math. Soc., 371 (2019), 1677-1700.  doi: 10.1090/tran/7329.

[29]

R. GuptaF. JafariR. J. Kipka and B. S. Mordukhovich, Linear openness and feedback stabilization of nonlinear control systems, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 1103-1119.  doi: 10.3934/dcdss.2018063.

[30]

J. W. Grizzle and S. I. Marcus, The structure of nonlinear control systems possessing symmetries, IEEE Transactions on Automatic Control, 30 (1985), 248-258.  doi: 10.1109/TAC.1985.1103927.

[31]

M. Gobbino, Topological properties of attractors for dynamical systems, Topology, 40 (2001), 279-298.  doi: 10.1016/S0040-9383(99)00061-0.

[32]

V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, Providence, RI, 2010, Reprint of the 1974 original. doi: 10.1090/chel/370.

[33]

B. Günther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc., 119 (1993), 321-329.  doi: 10.1090/S0002-9939-1993-1170545-4.

[34]

R. GoebelR. G. Sanfelice and A. Teel, Hybrid dynamical systems, Control Systems, IEEE, 29 (2009), 28-93.  doi: 10.1109/MCS.2008.931718.

[35]

P. Hartman, Ordinary Differential Equations, 2$^{nd} $ edition, SIAM, 2002. doi: 10.1137/1.9780898719222.

[36]

H. M. Hastings, Shape Theory and Dynamical Systems, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lecture Notes in Math., 668, Springer, Berlin, 1978,150–159.

[37]

H. M. Hastings, A higher-dimensional Poincaré-Bendixson theorem, Glas. Mat. Ser. III, 14 (1979), 263-268. 

[38]

A. Hatcher, Algebraic topology, 1$^{st}$ edition, Cambridge University Press, 2001.

[39]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994, Corrected reprint of the 1976 original.

[40]

E. HaghverdiP. Tabuada and G. J. Pappas, Bisimulation relations for dynamical, control, and hybrid systems, Theoretical Computer Science, 342 (2005), 229-261.  doi: 10.1016/j.tcs.2005.03.045.

[41]

J. D. Hunter, Matplotlib: A 2D graphics environment, Computing in Science & Engineering, 9 (2007), 90-95.  doi: 10.1109/MCSE.2007.55.

[42]

M. Hurley, Attractors: Persistence, and density of their basins, Transactions of the American Mathematical Society, 269 (1982), 247-271.  doi: 10.1090/S0002-9947-1982-0637037-7.

[43]

M. Ishikawa and M. Sampei, On equilibria set and feedback stabilizability of nonlinear control systems, IFAC Proceedings Volumes, 31 (1998), 609-614.  doi: 10.1016/S1474-6670(17)40404-6.

[44]

W. Jaco, Surfaces embedded in $ {M}^2\times {S}^1$, Canadian Journal of Mathematics 22 (1970), 553–568. doi: 10.4153/CJM-1970-063-x.

[45]

A. M. JohnsonS. A. Burden and D. E. Koditschek, A hybrid systems model for simple manipulation and self-manipulation systems, The International Journal of Robotics Research, 35 (2016), 1354-1392.  doi: 10.1177/0278364916639380.

[46]

E. Kappos, The Role of Morse-Lyapunov Functions in the Design of Nonlinear Global Feedback Dynamics, Variable Structure and Lyapunov Control, Springer, 1994,249–267. doi: 10.1007/BFb0033687.

[47]

E. Kappos, Necessary Conditions for Global Feedback Control, Proc Internat Symp on Nonlinear Theory and its Applications, Las Vegas, LA, Citeseer, 1995.

[48]

H. Khennouf and C. C. De Wit, On the construction of stabilizing discontinuous controllers for nonholonomic systems, IFAC Proceedings Volumes, 28 (1995), 667-672.  doi: 10.1016/S1474-6670(17)46905-9.

[49]

D. E. KoditschekR. J. Full and M. Buehler, Mechanical aspects of legged locomotion control, Arthropod Structure & Development, 33 (2004), 251-272.  doi: 10.1016/j.asd.2004.06.003.

[50]

U. V. KalabićR. GuptaS. Di CairanoA. M. Bloch and I. V. Kolmanovsky, MPC on manifolds with an application to the control of spacecraft attitude on SO(3), Automatica J. IFAC, 76 (2017), 293-300.  doi: 10.1016/j.automatica.2016.10.022.

[51]

M. D. KvalheimP. Gustafson and D. E. Koditschek, Conley's fundamental theorem for a class of hybrid systems, SIAM J. Appl. Dyn. Syst., 20 (2021), 784-825.  doi: 10.1137/20M1336576.

[52]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, New York, 2004. doi: 10.1007/b97315.

[53]

D. E. Koditschek, What is robotics? Why do we need it and how can we get it?, Annual Review of Control, Robotics, and Autonomous Systems, 4 (2021), 1-33.  doi: 10.1146/annurev-control-080320-011601.

[54]

M. D. Kvalheim and S. Revzen, Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits, Phys. D., 425 (2021), 132959, 20. doi: 10.1016/j.physd.2021.132959.

[55]

L. V. KolmanovksyM Reyhanoglu and N. H. McClamroch, Discontinuous feedback stabilization of nonholonomic systems in extended power form, Proceedings of 1994 33rd IEEE Conference on Decision and Control, 4 (1994), 3469-3474.  doi: 10.1109/CDC.1994.411683.

[56]

W. Lin and C. I. Byrnes, Design of discrete-time nonlinear control systems via smooth feedback, IEEE Trans. Automat. Control, 39 (1994), 2340-2346.  doi: 10.1109/9.333790.

[57]

J. M. Lee, Introduction to Topological Manifolds, 2$^{nd}$ edition, Springer Science & Business Media, 2010. doi: 10.1007/978-1-4419-7940-7.

[58]

J. M. Lee, Introduction to Smooth Manifolds, 2$^{nd}$ edition, Springer-Verlag, 2013.

[59]

E. Lerman, A Category of Hybrid Systems, preprint, arXiv: 1612.01950, (2016).

[60]

E. Lerman, Networks of open systems, Journal of Geometry and Physics, 130 (2018), 81-112.  doi: 10.1016/j.geomphys.2018.03.020.

[61]

G. A. D. Lopes and D. E. Koditschek, Visual servoing for nonholonomically constrained three degree of freedom kinematic systems, The International Journal of Robotics Research, 26 (2007), 715-736.  doi: 10.1177/0278364907080737.

[62]

A.-R. Mansouri, Local asymptotic feedback stabilization to a submanifold: Topological conditions, Systems Control Lett., 56 (2007), 525-528.  doi: 10.1016/j.sysconle.2007.03.001.

[63]

A.-R. Mansouri, Topological obstructions to submanifold stabilization, IEEE Trans. Automat. Control, 55 (2010), 1701-1703.  doi: 10.1109/TAC.2010.2046922.

[64]

A.-R. Mansouri, Topological Obstructions to Distributed Feedback Stabilization, 2013 51st Annual Allerton Conference on Communication, Control, and Computing, IEEE, 2013, 1573–1575.

[65]

A.-R. Mansouri, Topological Obstructions to Distributed Feedback Stabilization to a Submanifold, 2015 Proceedings of the Conference on Control and its Applications, SIAM, 2015, 76–80.

[66]

W. S. Massey, Homology and Cohomology Theory: An Approach Based on Alexander-Spanier Cochains, 46, Marcel Dekker, 1978.

[67]

W. S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991.

[68]

J. Milnor, Topology from the Differentiable Viewpoint, Princeton university press, 1965.

[69]

J. Milnor, On the concept of attractor: Correction and remarks, Communications in Mathematical Physics, 102 (1985), 517-519.  doi: 10.1007/BF01209298.

[70]

J. W. Milnor, Attractor, Scholarpedia, 1 (2006), 1815.  doi: 10.4249/scholarpedia.1815.

[71]

R. T. M'Closkey and R. M. Murray, Convergence Rates for Nonholonomic Systems in Power Form, 1993 American Control Conference, IEEE, 1993, 2967–2972. doi: 10.23919/ACC.1993.4793446.

[72]

J. C. Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Springer, 2004. doi: 10.1007/b84020.

[73]

S. Morita, Geometry of Differential Forms, 201, American Mathematical Soc., 2001. doi: 10.1090/mmono/201.

[74]

P. MorinJ.-B. Pomet and C. Samson, Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of Lie brackets in closed loop, SIAM J. Control Optim., 38 (1999), 22-49.  doi: 10.1137/S0363012997315427.

[75]

P. Morin and C. Samson, Control of nonlinear chained systems: From the Routh-Hurwitz stability criterion to time-varying exponential stabilizers, IEEE Trans. Automat. Control, 45 (2000), 141-146.  doi: 10.1109/9.827372.

[76]

P. W. Michor and C. Vizman, n-transitivity of certain diffeomorphism groups, Acta Math. Univ. Comenianae, 63 (1994), 221-225. 

[77]

R. OrsiL. Praly and I. Mareels, Necessary conditions for stability and attractivity of continuous systems, International Journal of Control, 76 (2003), 1070-1077.  doi: 10.1080/0020717031000122338.

[78]

L. Palmieri and K. O. Arras, A Novel RRT Extend Function for Efficient and Smooth Mobile Robot Motion Planning, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, 2014,205–211. doi: 10.1109/IROS.2014.6942562.

[79]

V. Pacelli, O. Arslan and D. E. Koditschek, Integration of Local Geometry and Metric Information in Sampling-Based motion Planning, 2018 IEEE International Conference on Robotics and Automation (ICRA), IEEE, 2018, 3061–3068. doi: 10.1109/ICRA.2018.8460739.

[80]

J. J. Park and B. Kuipers, Feedback Motion Planning via Non-Holonomic RRT* for Mobile Robots, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2015, 4035–4040. doi: 10.1109/IROS.2015.7353946.

[81]

J.-B. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift, Systems & Control Letters, 18 (1992), 147-158.  doi: 10.1016/0167-6911(92)90019-O.

[82]

A. Papachristodoulou and S. Prajna, A Tutorial on Sum of Squares Techniques for Systems Analysis, Proceedings of the 2005, American Control Conference, 2005., IEEE, 2005, 2686–2700. doi: 10.1007/10997703_2.

[83]

C. Prieur and E. Trélat, Robust optimal stabilization of the Brockett integrator via a hybrid feedback, Mathematics of Control, Signals and Systems, 17 (2005), 201-216.  doi: 10.1007/s00498-005-0152-9.

[84]

C. C. Pugh, A generalized Poincaré index formula, Topology, 7 (1968), 217-226.  doi: 10.1016/0040-9383(68)90002-5.

[85]

S. Revzen, D. E. Koditschek and R. J. Full, Towards Testable Neuromechanical Control Architectures for Running, Progress in Motor Control, Springer, 2009, 25–55. doi: 10.1007/978-0-387-77064-2_3.

[86]

S. F. RobertsD. E. Koditschek and L. J. Miracchi, Examples of Gibsonian affordances in legged robotics research using an empirical, generative framework, Frontiers in Neurorobotics, 14 (2020), 12.  doi: 10.3389/fnbot.2020.00012.

[87]

J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynam. Systems, 8 (1988), 375-393.  doi: 10.1017/S0143385700009494.

[88]

E. P. Ryan, On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback, SIAM Journal on Control and Optimization, 32 (1994), 1597-1604.  doi: 10.1137/S0363012992235432.

[89]

O. J. Sordalen and O Egeland, Exponential stabilization of nonholonomic chained systems, IEEE transactions on automatic control, 40 (1995), 35-49.  doi: 10.1109/9.362901.

[90]

M. Shub, Dynamical systems, filtrations and entropy, Bulletin of the American Mathematical Society, 80 (1974), 27-41.  doi: 10.1090/S0002-9904-1974-13344-6.

[91]

S. N. SimićK. H. JohanssonJ. Lygeros and S. Sastry, Towards a geometric theory of hybrid systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 12 (2005), 649-687. 

[92]

J. Seipel, M. Kvalheim, S. Revzen, M. A. Sharbafi and A. Seyfarth, Conceptual Models of Legged Locomotion, Bioinspired Legged Locomotion, Elsevier, 2017, 55–131. doi: 10.1016/B978-0-12-803766-9.00004-X.

[93]

E. D. Sontag, A "universal" construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123.  doi: 10.1016/0167-6911(89)90028-5.

[94]

E. D. Sontag, Control-Lyapunov Functions, Open problems in mathematical systems and control theory, Comm. Control Engrg. Ser., Springer, London, 1999,211–216.

[95]

E. H. Spanier, Algebraic Topology, Springer Science & Business Media, 1966.

[96]

Y.-P. Tian and S. Li, Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control, Automatica, 38 (2002), 1139-1146.  doi: 10.1016/S0005-1098(01)00303-X.

[97]

A. R. Teel, R. M. Murray and G. Walsh, Nonholonomic Control Systems: From Steering to Stabilization with Sinusoids, [1992] Proceedings of the 31st IEEE Conference on Decision and Control, IEEE, 1992, 1603–1609. doi: 10.1109/CDC.1992.371456.

[98]

T. Urakubo, Feedback stabilization of a nonholonomic system with potential fields: Application to a two-wheeled mobile robot among obstacles, Nonlinear Dynamics, 81 (2015), 1475-1487.  doi: 10.1007/s11071-015-2082-5.

[99]

T. Urakubo, Stability analysis and control of nonholonomic systems with potential fields, Journal of Intelligent & Robotic Systems, 89 (2018), 121-137. 

[100]

A. van der Schaft, Symmetries and conservation laws for Hamiltonian systems with inputs and outputs: A generalization of Noether's theorem, Systems Control Lett., 1 (1981/82), 108-115.  doi: 10.1016/S0167-6911(81)80046-1.

[101]

V. VasilopoulosG. PavlakosK. SchmeckpeperK. Daniilidis and D. E. Koditschek, Reactive navigation in partially familiar planar environments using semantic perceptual feedback, The International Journal of Robotics Research, 41 (2022), 85-126. 

[102]

V. Vasilopoulos, T. T. Topping, W. Vega-Brown, N. Roy and D. E. Koditschek, Sensor-Based Reactive Execution of Symbolic Rearrangement Plans by a Legged Mobile Manipulator, 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2018, 3298–3305. doi: 10.1109/IROS.2018.8594342.

[103]

G. C. Walsh and L. G. Bushnell, Stabilization of Multiple Input Chained Form Control Systems, Proceedings of 32nd IEEE Conference on Decision and Control, IEEE, 1993,959–964. doi: 10.1016/0167-6911(94)00061-Y.

[104]

F. W. Wilson Jr, The structure of the level surfaces of a Lyapunov function, J. Differential Equations, 3 (1967), 323-329.  doi: 10.1016/0022-0396(67)90035-6.

[105]

F. W. Wilson Jr, Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc., 139 (1969), 413-428.  doi: 10.1090/S0002-9947-1969-0251747-9.

[106]

J. Zabczyk, Some comments on stabilizability, Applied Mathematics and Optimization, 19 (1989), 1-9.  doi: 10.1007/BF01448189.

Figure 1.  Depiction of the objects from Theorem 3.2 with a general continuous adversary $ X $ and with a constant (with respect to some local trivialization of $ T M $) adversary (see Rem. 6)
Figure 2.  An illustration of the proof of Theorem 3.2. Here $ A $ is homeomorphic to the wedge product of two circles, a "figure eight", which has Euler characteristic $ -1\neq 0 $
Figure 3.  An illustration of the proof of Theorem 3.6. The idea is that, if $ S $ is savable (Def. 3.4), then $ S $ can be made strictly positively invariant (Def. 3.3) for some closed-loop vector field $ F $, which implies (Lem. 3.5) that there exists some asymptotically stable invariant set $ A \subset \text{int}(S) $ satisfying $ \chi( A) = \chi(S) $. It follows that the situation in the proof of Theorem 3.2 becomes embedded within $ \text{int}(S) $, as illustrated by Fig. 2 and the right side of the present figure. This yields an immediate proof of Theorem 3.6. In the present figure we are also emphasizing the fact that $ \partial S $ does not need to be smooth
Figure 4.  This figure illustrates Ex. 2. It is desired for a differential-drive robot to simultaneously (i) aim the line of sight of a mounted camera to point within 90 degrees of the origin, (ii) avoid $ n\geq 1 $ obstacles, and (iii) remain inside a big disk and outside a small disk centered at the origin. For simplicity, in Ex. 2 we assume that the size of the robot itself is negligible. (In the figure, the camera is aimed away from the origin.)
Figure 5.  This figure illustrates Ex. 5. Shown are numerical approximations of trajectory segments of the vector fields: $ f $ defined by (34) (left); $ -X_{\epsilon} $, with $ \epsilon = 0.01 $, defined by the left side of (36) (middle); and $ f-X_{0.01} $ (right), illustrating the fact that $ f-X_{\epsilon} $ has no equilibria for any $ \epsilon > 0 $. For visualization purposes, the width of each trajectory is proportional to the norm of its instantaneous velocity, and the same proportionality constant was used in all three plots. Each plot was generated using the function $\mathtt{streamplot} $ from the Python library matplotlib [41]
Figure 6.  This figure illustrates Ex. 6. Shown are numerical approximations of trajectory segments of the vector field $ g $ defined by the right side of (46). For visualization purposes, the width of each trajectory is proportional to the norm of its instantaneous velocity. This plot was generated using the function $\mathtt{streamplot}$ from the Python library matplotlib [41]
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