Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics

By Aaron D. Ames, Kevin S. Galloway, Koushil Sreenath, Jessy W. Grizzle. IEEE Transactions on Automatic Control, 2014

452 citations recorded by the source corpus.

This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models-systems with impulse effects-through control Lyapunov functions. The periodic orbit is assumed to lie in a C1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the full-order dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions.

Subfields: Biomedical Engineering, Genetics

View Paper: https://www.semanticscholar.org/paper/5d5e595181a487dbd603ad9643e3ae2c51550354

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Citations in the corpus

Safe Control With Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction Methods for Robotics and Control (2022, background): “2]) and has a closed-form solution so long as U is convex [18].”
RMA: Rapid Motor Adaptation for Legged Robots (2021, methodology): “Great progress has been made in legged robotics over the last forty years through the modeling of physical dynamics and the tools of control theory [36, 43, 46, 16, 56, 63, 50, 26, 28, 2, 24].”
Control barrier function based quadratic programs with application to adaptive cruise control (2014, background): “Recently, these results were extended to achieve rapid exponential convergence in order to stabilize periodic orbits in hybrid systems, with experimental realization on bipedal robots [2], [4], [9].”
Control Barrier Functions: Theory and Applications (2019, methodology): “To motivate these considerations, we will begin by reviewing control Lyapunov functions (CLFs) and discuss how they can be used to synthesize controllers that enforce stability.”
Modular Hopping and Running via Parallel Composition (2017, methodology): “(2003), intuitively a two-step anchoring process Ames et al. (2014). In contrast, in our experience, these conventional notions of anchoring are too rigid to allow for the kinds of compositions we would like to describe.”
Models, feedback control, and open problems of 3D bipedal robotic walking (2014, methodology): “The point-wise minimum norm of [142, 42] is extended to RES-CLFs and validated experimentally on MABEL in [14].”
Exponential Control Barrier Functions for enforcing high relative-degree safety-critical constraints (2016, methodology): “Control Lyapunov Function based Quadratic Programs A control approach based on control Lyapunov functions, introduced in [2], provides guarantees of exponential stability for the traverse variables η.”
Legged Locomotion in Challenging Terrains using Egocentric Vision (2022, unknown): “233 Several classical works use model based techniques, or define heuristic reactive controllers to 234 achieve the task of walking [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].”