Asymptotically stable walking for biped robots: analysis via systems with impulse effects

Abstract

Biped robots form a subclass of legged or walking robots. The study of mechanical legged motion has been motivated by its potential use as a means of locomotion in rough terrain, as well as its potential benefits to prothesis development and testing. This paper concentrates on issues related to the automatic control of biped robots. More precisely, its primary goal is to contribute a means to prove asymptotically-stable walking in planar, under actuated biped robot models. Since normal walking can be viewed as a periodic solution of the robot model, the method of Poincaré sections is the natural means to study asymptotic stability of a walking cycle. However, due to the complexity of the associated dynamic models, this approach has had limited success. The principal contribution of the present work is to show that the control strategy can be designed in a way that greatly simplifies the application of the method of Poincaré to a class of biped models, and, in fact, to reduce the stability assessment problem to the calculation of a continuous map from a subinterval of IR to itself. The mapping in question is directly computable from a simulation model. The stability analysis is based on a careful formulation of the robot model as a system with impulse effects and the extension of the method of Poincaré sections to this class of models.

Mathematical Analysis of the Transmission Dynamics of HIV/AIDS: Role of Female Sex Workers

“…In the present time, mathematical models are extensively used to study epidemics to figure out the analytical conclusions based upon the epidemiological assumptions [17,18,19,20,21,22,23,24,25]. Several modelling studies have been published recently that are intended to provide information to policy makers to support the development of policies and strategies for reducing the overall transmission rate of HIV infection.…”

Achieving Bipedal Running with RABBIT: Six Steps Toward Infinity

“…Using the idea of a restricted Poincaré return map-the Poincaré return map associated with the HZD-explicit criteria are given for determining the existence and stability of periodic orbits. The Poincaré return map is a well-known tool for determining the existence of periodic orbits and their stability properties; for its use in hybrid systems, see for example [7,9,11,16]. Its application to periodic orbits of (20) can be carried out using the results in [9] and a construction presented in [6, Thm.…”

“…To perform reduced-order stability analysis similar to that outlined in Section 3.8, a finite-time converging controller must be used in either the stance or the flight phase. A finite-time controller is used in the stance phase to render the stance phase constraint surface finite-time attractive so that the analysis of running will be similar to that of walking [9]. For further discussion of this point, refer to Section 3.8.…”

“…Perfect enforcement of virtual constraints results in low-dimensional surfaces that are invariant under the differential equations of the closed-loop model 8 and are also invariant under the transition maps. 9 To achieve the invariance at landing, a deadbeat action is incorporated in the flight phase controller that steers the robot to land in a pre-determined configuration, while respecting conservation of angular momentum about the robot's center of mass. This hybrid controller creates a one DOF HZD that allows the stability of a running motion to be analyzed in closed form on the basis of a one-dimensional Poincaré map.…”

“…1(a)), has provided a means to test conceptually new approaches to underactuated, active, dynamic walking and running. Since March 2003, RABBIT has been used to experimentally verify a mathematical framework for the systematic design, analysis, and optimization of controllers that induce stable walking gaits in N -link planar bipedal robots with one degree of underactuation [9,21]. Walking controllers designed within this framework act by enforcing virtual constraints, which are holonomic constraints on the robot's configuration that are asymptotically imposed through feedback control.…”