Nonlinear Analysis: Hybrid Systems 2 (2008) 231–241 www.elsevier.com/locate/nahs Control of a class of underactuated mechanical systems Tatsuo Narikiyo a,∗,1 , Junichiro Sahashi b , Kazutomo Misao a a Department of Advanced Science and Technology, Toyota Technological Institute, 2-12-1 Hisakata, Nagoya 468-8511, Japan b Technical Administration Division, Toyota Motor Corporation, 1.Toyota-cho, Toyota 471-8572, Japan Received 8 April 2006; accepted 15 April 2006 Abstract In this paper a new control methodology for underactuated mechanical systems is proposed. The basic idea of this method is to combine passive velocity field control with decoupling vector field. In order that the underactuated mechanical systems can be stabilized at the desired position after settling on the desired velocity vector field, novel control strategies are proposed. Proposed control strategies are applied to the underactuated planar three-link manipulator and underactuated planar body. Simulations demonstrate the usefulness and effectiveness of the proposed control methodology. c  2006 Elsevier Ltd. All rights reserved. Keywords: Underactuated mechanical systems; Decoupling vector fields; Passive velocity field control 1. Introduction In recent years, the problem of finding the control laws for underactuated mechanical systems (UAMS) has been attracting great interest since it was shown in [1,2] that the dynamics of UAMS is characterized by nonholonomic constraints. Examples of such systems include planar manipulator with passive joints and free flying space robot. Since these systems are considered to be useful in engineering fields and attractive in scientific fields, there are many researchers treating trajectory planning problems for planar underactuated manipulators and free flying bodies. For example, in order to find the path which can be time scaled without violating the underactuation constraints, the decoupling vector field has been developed in [3,4]. Feedforward control method through end-effector generalized forces has been developed in [5]. Also several feedback control methods have been proposed. For example, in [6] the equations of motion of the underactuated planar manipulator have been transformed into second order chained form and feedback control laws have been proposed through a backstepping approach. Since each control method depends on its own dynamic equations of motion, any proposed control method cannot be applied to another class of the UAMS. On the other hand, in [7] the feedback control method for the general classes of the UAMS through the port-controlled Hamiltonian system has been developed. However solutions of the two partial differential equations (PDE) should be required to synthesize the proposed control laws. It is well known that solving the PDE is not easy. ∗ Corresponding author. E-mail addresses: n-tatsuo@toyota-ti.ac.jp (T. Narikiyo), sahashi@jun.tec.toyota.co.jp (J. Sahashi). 1 Visiting researcher at Bio-mimetic Control Research Center (RIKEN), Shimoshidami, Nagoya 463-0003, Japan. 1751-570X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2006.04.009