1742 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007 Energy-Based Nonlinear Control of Underactuated Euler–Lagrange Systems Subject to Impacts G. Hu, C. Makkar, and W. E. Dixon Abstract—In this note, Lyapunov-based methods are used to design a class of energy-based nonlinear controllers to globally asymptotically sta- bilize/regulate an underactuated mechanical system subject to an impact collision. The impact model is considered as an elastic contact with finite stiffness. One of the difficulties in controlling impact is that the equations of motion are quite different when the system status changes from a non- contact condition to a contact condition. Another difficulty arises when an impact occurs with an underactuated system because the impact may lead to instabilities or excessive transients. An energy coupling approach is de- veloped in this paper that is motivated by the desire to improve the transient response of the system. A Lyapunov stability analysis and numerical sim- ulations are provided to demonstrate the stability and performance of the developed controllers. Index Terms—Impact, Lyapunov methods, nonlinear systems. I. INTRODUCTION The control of mechanical systems subject to impact is a theoret- ically interesting problem with practical importance. Large stresses arise as a consequence of impact, demanding that the impact forces be properly recognized and controlled to prevent system failure. As described in [14], some useful short-duration effects such as high stresses, rapid dissipation of energy, and fast acceleration and decel- eration may be achieved from low-energy sources by controlling the impact of robots operating at low force levels. Some robotic examples in which controlled contacts are required include the impact between a walking robot and the ground, the interaction of a robot manipulator with an object, multifinger grasping, and the cooperation and contact of multirobots. One of the difficulties in controlling impact is that the equations of motion are quite different when the system status changes quickly from a noncontact condition to a contact condition. For the past decade, many researchers have addressed the modeling and control of impact [1]–[4], [8], [10]–[16]. In [2] and [11], switched controllers are given to control robotic manipulators during contact/ noncontact conditions separately. In [16], a switching control strategy is designed to guarantee the stability of the impact controller. In [12], a stable discontinuous transition controller is proposed to deal with the contact transition problem. In [10], the authors use a hybrid impedance/ time-delay controller that establishes a stable contact and achieves the desired dynamics for contact or noncontact conditions. In [13], a dis- continuous Lyapunov-based control scheme is introduced to regulate the impact of a hydraulic actuator coming in contact with a nonmoving environment. In [14], a continuous proportional derivative (PD) con- troller is proposed to control the impact of an underactuated system where the actuators are used to stabilize the contact coordinates, and the noncontact coordinates are indirectly stabilized. In [8], static and dynamic PD controllers are proposed to address the global asymptotic Manuscript received August 29, 2005; revised April 19, 2006 and March 15, 2007. Recommended by Associate Editor F. Bullo. This work was supported in part by the National Science Foundation under CAREER award CMS-0547448, by AFOSR under Contracts F49620-03-1-0381 and F49620-03-1-0170, and by AFRL under Contract FA4819-05-D-0011. The authors are with the Department of Mechanical and Aerospace En- gineering, University of Florida, Gainesville, FL 32611-6250 USA (e-mail: gqhu@ufl.edu; cmakkar@ufl.edu; wdixon@ufl.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2007.904319 stabilization problem of the underactuated mechanical system subject to an impact with an elastic surface with finite stiffness, but the contacts only happen at the equilibrium points. In this paper, we develop a new class of continuous energy-based controllers (e.g., [5] and [6]) that achieve global asymptotic stabiliza- tion/regulation of an underactuated Euler–Lagrange system subject to an elastic contact with finite stiffness. Motivation for this class of con- trollers is that when underactuated systems are subject to an impact, excessive transient performance may exist if natural damping is not present and the controller lacks a damping element. Hence, the devel- oped controllers exploit the system energy to couple all the states of the system so that transients in the unactuated states will be coupled to the actuated states. This energy coupling idea is in contrast to linear controllers that do not include any state coupling. The developed sta- bilization result is obtained regardless of which states (i.e., actuated or unactuated) undergo an impact collision. An extension is also provided that illustrates how the actuated states can be regulated to make con- tact, where the unactuated states converge to the resulting closed-loop equilibrium point. This paper is organized as follows. In Section II, the Euler–Lagrange dynamic model subject to impact is provided along with the related as- sumptions that are required for the control development. In Section III, the energy coupling controller is designed to globally asymptotically stabilize the generalized free motion and contact coordinates of an underactuated mechanical system subject to impact conditions, and a Lyapunov stability analysis is provided to demonstrate the stability of the developed controller. The regulation extension is provided in Section IV. In Sections V and VI, two examples and simulation re- sults are provided to demonstrate the performance of the developed controllers. Concluding remarks are provided in the last section. II. DYNAMIC MODEL The equations of motion of an -degrees-of-freedom (DOF) Euler–Lagrange system subject to an impact collision are assumed to have the following form [8], [14]: (1) In (1), , , denote the generalized position, velocity, and acceleration coordinates, denotes a positive definite inertia matrix, denotes the velocity-dependent force vector, denotes a conservative force vector (e.g., spring forces, gravity), denotes the control force/torque input, and is a transformation matrix defined as (2) that maps the actuator space into the generalized coordinates space, where denotes an matrix with all the elements equal to zero, and is an identity matrix. Also in (1), denotes an impact stiffness matrix defined as (3) where denotes the contact coordinates that are defined as where is a constant transformation matrix that maps the generalized coordinates space into the contact coordinates space and 0018-9286/$25.00 © 2007 IEEE