Remarks on Event-Based Stabilization of Periodic Orbits in Systems with Impulse Effects

This note focuses on control design issues associated with creating and stabilizing periodic orbits in systems with impact events via control decisions that take place only at event-triggered transitions in the dynamics. Many hybrid, underactuated, mechanical systems with impacts arising in legged locomotion can be modeled as systems with impulse effects. Constructive techniques are needed for the design an d analysis of periodic orbits in such systems. I. I NTRODUCTION Mechanical legged locomotion is being studied for its enhanced maneuverability in rough terrain, for its ability to deal with environments with discontinuous supports, such a s the rungs of a ladder, and because of the popular appeal of machines that operate in anthropomorphic ways 1. The dynamic models that arise in the study of legged locomotion are fundamentally hybrid. For example, a bipedal walking motion consists of successive phases of single support (meaning the stance leg is touching the ground and the swing leg is not) and double support (both legs are in contact with the ground), while running consists of successive phases of single support and flight (there is no contact with the ground ). It is common to model the impact that occurs when the swing leg strikes the ground as an instantaneous contact of rigid bodies, which results in an algebraic representation of the impact event; see Figure 1. A canonical problem in legged robots is how to design a controller that generates closed-loop motions, such as walking or running, that are periodic and stable (i.e., limi t cycles). Due to the hybrid nature of the system, this task is far from being solved through existing control methods. New paradigms, concepts, and control analysis techniques are t hus needed to deal with this class of systems [4]. Because the system model is hybrid, it is natural for the controller to be hybrid as well, with control actions taking place during the continuous phase(s) [7], [16] as well as at discrete transit io s [18], [8], [17], [5]. This note focuses on the latter issue. The method of Poincar é sections and return maps is widely used to determine the existence and stability of periodic orbits in a broad range of system models, such as timeinvariant and periodically-time-varying ordinary differ ential equations [14], [9], hybrid systems consisting of several ∗ This work was supported by NSF grant ECS-0322395. J.W. Grizzle is with the Control Systems Laboratory, EECS Department, U iversity of Michigan, Ann Arbor, Michigan 48109-2122, USA. E-mai l: grizzle@umich.edu. 1In this regard, Honda has just announced that Asimo is performi ng the job of a receptionist in an office setting [1]. x = ∆(x) H(x) = 0 ẋ = fol(x) + gol(x)u Fig. 1. A nonlinear system with impulse effects. In the case of a bipedal robot, the state consists of the generalized configu ration and velocity variables, the transition condition corresponds to the hei ght of the swing foot above the ground going to zero, and the re-initialization ru le corresponds to the impact map of two rigid bodies (the swing foot and the grou nd). In this note, it will be assumed that a parameterized family of fe edback laws u(x, a), a ∈ A, has been designed for the continuous phase so that f(x, a) = fol(x) + gol(x)u(x, a). The problem of determining eventbased update laws for the parameter a is investigated for the existence of asymptotically stable periodic orbits. time-invariant ordinary differential equations linked by eventbased switching mechanisms and re-initialization rules [7 ], [13], [15], [12], differential-algebraic equations [10], and relay systems with hysteresis [6]. The conceptual advantag e of the method of Poincar é is that it reduces the study of periodic orbits to the study of equilibrium points of sample ddata systems, with the latter being a more extensively studi ed problem. In particular, this note is based on the fact that parameter values that are held constant within the continuo us phase of the hybrid dynamics and updated at impact events appear as standard sampled-data controls in the Poincar é return map. This observation leads to the formulation of sampled-data feedback design problems for the creation and stabilization of periodic orbits in nonlinear systems with impulse effects. II. BACKGROUND This section reviews the notions of systems with impulse effects, periodic orbits, stability of periodic orbi ts, the method of Poincar é and hybrid invariance. The objective is to set the stage for studying event-based control actions in Section III. A. Systems with impulse effects An autonomous system with impulse effects consists of an autonomous ordinary differential equation, ẋ = f(x), (1) defined on some state space X , a co-dimension one surface S ⊂ X at which solutions of the differential equation undergo a discrete transition that is modeled as an instantaneous re-initialization of the differential equation, a nd a rule ∆ : S → X that specifies the new initial condition as a function of the point at which the solution impacts S [2], [19]. The co-dimension one surface S is called a transition surface or an impact surface (in analogy with biped models) and ∆ is often called an impact map. The system will be denoted by Σ : { ẋ = f(x) x / ∈ S x = ∆(x) x ∈ S, (2) and the following hypotheses will be made: HSH0)X is a smooth embedded submanifold of R; HSH1) f : X → TX is continuous, and a solution of ẋ = f(x) from a given initial condition is unique and depends continuously on the initial condition; HSH2) there exist a differentiable function H : X → R, such thatS = {x ∈ X | H(x) = 0}; moreover, for every s ∈ S, ∂H ∂x (s) 6= 0. HSH3) ∆ : S → X is continuous, whereS is given the subset topology fromX . HSH4) ∆(S) ∩ S = ∅, where∆(S) is the set closure of ∆(S). In simple terms, a solution of (2) is specified by the differential equation (1) until its state “impacts” the hyp er surfaceS at some timetI . At tI , the impulse model∆ compresses the impact event into an instantaneous moment of time, resulting in a discontinuity in the state trajector y. The impact model provides the new initial condition from which the differential equation evolves until the next impact wit h S. In order to avoid the state having to take on two values at the “impact time”tI , the impact event is, roughly speaking, described in terms of the values of the state “just prior to impact” at time “t−I ”, and “just after impact” at time “ t + I ”. These values are represented by x andx, respectively. From this description, a formal definition of a solution is easily written down by piecing together appropriately initialized solutions of (1); see [19], [7], [13], [3]. A cho ice must be made as to whether to take a solution φ(t) of (2) as being a leftor right-continuous function of time at each impact event; here, solutions are assumed to be right continuous [7]. B. Periodic orbits A solution φ(t) of (2) is periodic if there exists a finite T > 0 such thatφ(t + T ) = φ(t) for all t ∈ [t0,∞). A set O ⊂ X is a periodic orbit of (2) if O = {φ(t) | t ≥ t0} for some periodic solutionφ(t). While a system with impulse effects may certainly have periodic solutions that do not involve impact events, they are not of interest here because they could be studied more simply as solutions of (1). If a periodic solution has an impact event, then the correspondi ng periodic orbitO is not closed; see [7] and Fig. 2. Let̄ O denote its set closure. ∆(x) x