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This paper addresses the problem of tracking a prescribed geometric path by the end effector of a kinematically redundant manipulator at the control loop level. The constraints imposed on the robot actuator controls are taken into account. The Lyapunov stability theory and/or the calculus of variations is used to derive the control scheme. Through the use of an exterior penalty function approach, an additional objective to be fulfilled by the robot, that is, collision avoidance of the manipulator links with obstacles, is ensured. The extensive computer simulation results illustrate the trajectory performance of the proposed control scheme for a geometric end effector path given in both an obstacle-free work space and a work space including obstacles. KEY WORDS—redundant manipulator, end-effector path, stability, collision avoidance 1

Dynamical Multilayer Neural Networks That Learn Continuous Trajectories

er a bounded time interval or trains non-fixed-point attractors. Williams and Zipser [8], Meert and Ludik [9] have constructed a gradient descent learning rule which they call real-time recurrent learning (RTRL) and which can deal with time sequences of arbitrary length. A stochastic search method based on an adaptive simulated annealing algorithm has been used by Cohen et al. [10] to efficiently train recurrent neural networks with time delays (TDRNN). An effort was made in the above investigation to implement several benchmark tasks using minimum size networks. In all the above investigations, a dynamical neural network evolves in accordance with the following general equations 8 ( ) x x naf w x I H = - + + (1) where  n 'x=(x 1 ... x n ) T =x(t) is a state vector of the network; n denotes the number of all the neurons;