Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems

Abstract

We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constantdimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincaré map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high-dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior. I. INTRODUCTION Rhythmic phenomena are pervasive, appearing in physical situations as diverse as legged locomotion [1], dexterous manipulation [2], gene regulation [3], and electrical power generation [4]. The most natural dynamical models for these systems are piecewise-defined or discontinuous owing to intermittent changes in the mechanical contact state of a locomotor or manipulator, or to rapid switches in protein synthesis or constraint activation in a gene or power network. Such hybrid systems generally exhibit dynamical behaviors that are distinct from those of smooth systems [5]. Restricting our attention to the dynamics near periodic orbits in hybrid dynamical systems, we demonstrate that a class of hybrid models for rhythmic phenomena reduce to classical (smooth) dynamical systems.Although the results of this paper do not depend on the phenomenology of the physical system under investigation, a principal application domain for this work is terrestrial locomotion. Numerous architectures have been proposed to explain how animals control their limbs; for steady-state locomotion, most posit a principle of coordination, synergy, symmetry or synchronization, and there is a surfeit of neurophysiological data to support these hypotheses [6]- [10]. Taken together, the empirical evidence suggests that the large number of degrees-of-freedom (DOF) available to a locomotor can collapse during regular motion to a low-dimensional dynamical attractor (a template) embedded within a higher-dimensional model (an anchor) that respects the locomotor's physiology [1], [11]. We provide a mathematical framework to model this empirically observed dimensionality reduction in the deterministic setting.A stable hybrid periodic orbit provides a natural abstraction for the dynamics of steady-state legged locomotion. This widely-adopted approach has generated models of bipedal [12] . In certain cases, it has been possibl...

Data driven models of legged locomotion

“…23,24 To these naturally correspond their "hybrid tangent space" TM and "hybrid boundary" ∂M . As a consequence of the topological structure, a continuous map R : M → N between smooth hybrid manifolds M = j∈J M j and N := i∈I N i must map each M j into a unique N i , allowing the concept of a smooth map to be naturally extended to smooth hybrid manifolds.…”

System identification of rhythmic hybrid dynamical systems via discrete time harmonic transfer functions

“…Each element of this union, i.e. V α , is called a chart The dimension of the charts can typically depend on α [5,6,20]. A state of the overall hybrid dynamical system consists of an index α together with a point in the chart V α .…”

“…Similar to the previous work on rhythmic hybrid dynamical systems [6], we limit our attention to hybrid systems undergoing a finite number of isolated hybrid transitions near the limit cycle. We also assume that in a local neighborhood of the limit cycle that the number and order of hybrid transitions is fixed (although the transition times can and typically will vary) and both the threshold functions and transition maps associated with these transitions are locally smooth.…”

“…A Poincaré section, Σ i ⊂ V , is an embedded submanifold that intersects the periodic orbit at exactly one point [6]: Let Σ j be another Poincaré section with associated fixed point ( j , α j ). We define a mapping, P i→j : W i × R lmax → Σ j by assumingû = u i is constant (i.e.…”

“…Using (6) we can compute the linearized Poincaré return map at any section. Let i ∈ {0, · · · , N − 1} be the section that we are interested.…”