Published in IET Control Theory and Applications Received on 3rd September 2010 Revised on 9th February 2011 doi: 10.1049/iet-cta.2010.0512 ISSN 1751-8644 Two-level control scheme for stabilisation of periodic orbits for planar monopedal running N. Sadati 1,2 G.A. Dumont 2 K.Akbari Hamed 1 W.A. Gruver 3 1 Intelligent Systems Laboratory, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran 2 Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4 3 School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 E-mail: sadati@sharif.edu; sadati@ece.ubc.ca Abstract: This study presents an online motion planning algorithm for generating reference trajectories during ﬂight phases of a planar monopedal robot to transfer the conﬁguration of the mechanical system from a speciﬁed initial pose to a speciﬁed ﬁnal one. The algorithm developed in this research is based on the reachability and optimal control formulations of a time-varying linear system with input and state constraints. A two-level control scheme is developed for asymptotic stabilisation of a desired period- one orbit during running of the robot. Within-stride controllers, including stance and ﬂight phase controllers, are employed at the ﬁrst level. The ﬂight phase controller is a feedback law to track the reference trajectories generated by the proposed algorithm. To reduce the dimension of the full-order model of running, the stance phase controller is chosen to be a parameterised time-invariant feedback law that produces a family of two-dimensional ﬁnite-time attractive and invariant submanifolds. At the second level, the parameters of the stance phase controller are updated by an event-based update law to achieve hybrid invariance and stabilisation. To illustrate the analytical results developed for the behaviour of the closed-loop system, a detailed numerical example is presented. 1 Introduction This paper presents an analytical approach for designing a two-level control law to asymptotically stabilise a desired period-one orbit during running by a planar monopedal robot. The monopedal robot is a three-link, two-actuator planar mechanism in the sagittal plane with point foot. It is assumed that the model of monopedal running can be expressed by a hybrid system with two continuous phases, including stance phase (one leg on the ground) and ﬂight phase (no leg on the ground), and discrete transitions between the continuous phases, including takeoff and landing (impact) [1, 2, Chap. 9]. The conﬁguration of the mechanical system is speciﬁed by the absolute orientation with respect to an inertial world frame and by the joint angles determining the shape of the robot. During the ﬂight phase, the angular momentum of the mechanical system about its centre of mass (COM) is conserved. To reduce the dimension of the full-order hybrid model of running, which in turn simpliﬁes the stabilisation problem of the desired orbit, as proposed by Chevallereau et al. [1], we desire that the conﬁguration of the mechanical system can be transferred from a speciﬁed initial pose (immediately after the takeoff) to a speciﬁed ﬁnal pose (immediately before the landing) during ﬂight phases. This problem is referred to as ‘landing in a ﬁxed conﬁguration or conﬁguration determinism at landing’ [1, 2, p. 252]. However, the ﬂight time and angular momentum about the COM may differ during consecutive steps. Consequently, the reconﬁguration problem must be solved online. A number of control problems for reconﬁguration of a planar multilink robot with zero angular momentum have been considered in the literature, for example [3–6]. For the case that the angular momentum is not necessarily zero, Kolmanovsky et al. [7] presented a method based on the averaging theorem [8, Theorem 2.1] such that for any value of the angular momentum, joint motions can reorient the multilink arbitrarily over an arbitrary time interval. However, when the angular momentum is not zero, this method cannot be employed online for solving the reconﬁguration problem for monopedal running. For this reason, we present an online reconﬁguration algorithm that solves this problem for given ﬂight times and angular momentums. The algorithm proposed in this paper is expressed using the methodology of reachability and optimal control for time-varying linear systems with input and state constraints. This algorithm can also be utilised for online generation of C 2 trajectories for free open kinematic chains, conserving angular momentum about their COM. Probably the most basic tool for analysing the stability of periodic orbits of time-invariant dynamical systems described by ordinary differential equations is the Poincare ´ return map. Grizzle et al. [9] showed that the Poincare ´ return map can be applied to systems with impulse effects for analysing the stability of periodic orbits. To reduce the dimension of the Poincare ´ return map during bipedal 1528 IET Control Theory Appl., 2011, Vol. 5, Iss. 13, pp. 1528–1543 & The Institution of Engineering and Technology 2011 doi: 10.1049/iet-cta.2010.0512 www.ietdl.org