Real-time motion planning of robots in a dynamic environment requires a continuous evaluation of the determined trajectory so as to avoid moving obstacles. This is even more challenging when the robot also needs to perform a task optimally while avoiding the obstacles due to the limited time available for generating a new collision-free path. In this paper, we propose the sequential expanded Lagrangian homotopy (SELH) approach, which is capable of determining the globally optimal robot's motion sequentially while satisfying the task constraints. Through numerical simulations, we demonstrate the capabilities of the approach by planning an optimal motion of a redundant mobile manipulator performing a complex trajectory. Comparison against existing optimal motion planning approaches, such as genetic algorithm (GA) and neural network (NN), shows that SELH is able to perform the planning at a faster rate. The considerably short computational time opens up an opportunity to apply this method in real time; and since the robot's motion is planned sequentially, it can also be adjusted to accommodate for dynamically changing constraints such as moving obstacles.

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