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A method to construct the normal modes for a class of piecewise linear vibratory systems is developed in this study. The approach utilizes the concepts of Poincar6 maps and invariant manifolds from the theory of dynamical systems. In contrast to conventional methods for smooth systems, which expand normal modes in a series form around an equilibrium point of interest, the present method expands the normal modes in a series form of polar coordinates in a neighborhood of an invariant disk of the system. It is found that the normal modes, modal dynamics and frequency-amplitude dependence relationship are all of piecewise type. A two degree of freedom example is used to demonstrate the method.

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This work is concerned with the contouring control problem of robotic manipulators using the method of equivalent errors. The method was previously proposed for multi-axis motion systems with emphasis on machine tools applications. It is shown in this paper that the method of equivalent errors is equally applicable to robot manipulators. For robotic systems, a desired path usually is described in terms of the position of the end-effector (task space), but the system dynamics are described in generalized coordinates (joint space). The proposed method utilizes the forward kinematics to transform the desired path in the task space into that in the joint space. As such, the equivalent errors can be defined easily. Two examples contouring free-form paths are studied numerically to demonstrate the effectiveness of the proposed method. One is a two-link robot manipulator, and the other is a 6-DOF Stewart platform. Three controllers for the system with and without uncertainty are considered. The simulation results demonstrate the excellent performance of the proposed method and verify the effectiveness of its application to the contouring control problem of robotic systems, even with uncertainty and higher degrees of freedom.

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