Automatica 49 (2013) 17–29 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Viability control for a class of underactuated systems ✩ Dimitra Panagou a,1 , Kostas J. Kyriakopoulos b a Coordinated Science Laboratory, College of Engineering, University of Illinois at Urbana-Champaign, 1308 W. Main St., Urbana, IL, 61801, USA b Control Systems Lab, School of Mechanical Engineering, National Technical University of Athens, 9 Heroon Polytechneiou Str., Zografou, Athens, Greece article info Article history: Received 27 October 2011 Received in revised form 21 May 2012 Accepted 20 July 2012 Available online 9 October 2012 Keywords: Constraint satisfaction problems Constrained control Underactuated robots Nonholonomic control Robot control Convergent control Invariance Disturbance rejection Robust control abstract This paper addresses the feedback control design for a class of nonholonomic systems which are subject to inequality state constraints defining a constrained (viability) set K . Based on concepts from viability theory, the necessary conditions for selecting viable controls for a nonholonomic system are given, so that system trajectories starting in K always remain in K . Furthermore, a class of state feedback control solutions for nonholonomic systems are redesigned by means of switching control, so that system trajectories starting in K converge to a goal set G in K , without ever leaving K . The proposed approach can be applied in various problems, whose objective can be recast as controlling a nonholonomic system so that the resulting trajectories remain forever in a subset K of the state space, until they converge into a goal (target) set G in K . The motion control for an underactuated marine vehicle in a constrained configuration set K is treated as a case study; the set K essentially describes the limited sensing area of a vision-based sensor system, and viable control laws which establish convergence to a goal set G in K are constructed. The robustness of the proposed control approach under a class of bounded external perturbations is also considered. The efficacy of the methodology is demonstrated through simulation results. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction Nonholonomic control has been and still remains a highly challenging and attractive problem from a theoretical viewpoint. Related research during the past two decades has attributed various control design methodologies addressing stabilization, path following and trajectory tracking problems for nonholonomic systems of different types, which nowadays feature a solid framework within control theory. From a practical viewpoint, nonholonomic control is of particular interest within the fields of robotics and multi-agent systems, since a significant class of robotic systems is subject to nonholonomic constraints. The control design in this case typically pertains to realistic, complex systems, which should perform efficiently and reliably; in this sense, the robustness of control solutions with respect to (w.r.t.) uncertainty and additive disturbances is a desirable property, ✩ The material in this paper was partially presented at the 50th IEEE Conference on Decision and Control and European Control Conference (CDC–ECC 2011), December 12–15, 2011, Orlando, Florida, USA. This paper was recommended for publication in revised form by Associate Editor Warren E. Dixon under the direction of Editor Andrew R. Teel. E-mail addresses: dpanagou@mail.ntua.gr, dpanagou@illinois.edu (D. Panagou), kkyria@mail.ntua.gr (K.J. Kyriakopoulos). 1 Tel.: +30 2107723656; fax: +30 2107722417. which highly affects the performance, or even safety, of the considered systems. In part for this reason, the development of robust nonholonomic controls w.r.t. vanishing, as well as non- vanishing perturbations has received special attention, see Dixon, Dawson, Zergeroglu, and Behal (2001), Floquet, Barbot, and Perruquetti (2003), Ge, Wang, Lee, and Zhou (2001), Guo (2005), Lizarraga, Aneke, and Nijmeijer (2003), Lucibello and Oriolo (2001), Morin and Samson (2003), Prieur and Astolfi (2003), Valtolina and Astolfi (2003), Zhu, Dong, Cai, and Hu (2007) and the references therein. Non-vanishing perturbations are typically more challenging, since a single desired configuration might no longer be an equilibrium for the system (Khalil, 2002). In this case one should rather pursue the ultimate boundedness of state trajectories; this problem is often addressed as practical stabilization. In a similar context, the development of input-to- state stability (iss) as a fundamental concept of modern nonlinear feedback analysis and design, has allowed the formulation of robustness considerations for nonholonomic systems in the iss framework Aguiar, Hespanha, and Pascoal (2007), Laila and Astolfi (2005), Liberzon, Sontag, and Wang (2002), Tanner (2004). Furthermore, one often cannot neglect that control systems are subject to hard state constraints, encoding safety or perfor- mance criteria. An illustrative paradigm of hard state constraints is encountered in the case of agents that have limited sensing ca- pabilities while accomplishing a task. For instance, consider an underactuated robotic vehicle equipped with sensors (e.g. cam- eras) with limited range and angle-of-view, which has to surveil 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.09.002