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Generation of desired trajectory behavior using neural networks involves a particularly challenging spatio-temporal learning problem. This paper introduces a novel solution, i.e., designing a dynamic system whose terminal behavior emulates a prespecified spatio-temporal pattern independently of its initial conditions. The proposed solution uses a dynamic neural network (DNN), a hybrid architecture that employs a recurrent neural network (RNN) in cascade with a nonrecurrent neural network (NRNN). The RNN generates a simple limit cycle, which the NRNN reshapes into the desired trajectory. This architecture is simple to train. A systematic synthesis procedure based on the design of relay control systems is developed for configuring an RNN that can produce a limit cycle of elementary complexity. It is further shown that a cascade arrangement of this RNN and an appropriately trained NRNN can emulate any desired trajectory behavior irrespective of its complexity. An interesting solution to the trajectory modulation problem, i.e., online modulation of the generated trajectories using external inputs, is also presented. Results of several experiments are included to demonstrate the capabilities and performance of the DNN in handling trajectory generation and modulation problems.