Collision-free trajectory planning for 2D and 3D robotic arm manipulators in the presence of mobile wandering obstacles

This paper presents a generic method for optimal motion planning for 2D and 3D multi-link robotic manipulators. The operation of the manipulator, which may involve its transfer and payload transportation/ capture/release, can be a subject to the minimisation of the user-defined objective function, enabling for example minimisation of the actuation efforts or the time transfer. More importantly, the following additional constraints can be included in the task formulation: (a) box constraints on the actuation torques/forces, (b) box constraints on the generalised displacement variables (angular and translational displacements), (c) arbitrary time-variable path constraints. The solutions of the associated nonlinear differential equations of motion are obtained numerically using the direct transcription method. The direct method seeks to transform the continuous optimal control problem into a discrete mathematical programming problem, which in turn is solved using a non-linear programming algorithm. By discretizing the state and control variables at a series of nodes, the integration of the dynamical equations of motion is not required. The Chebyshev-pseudospectral method, due to its high accuracy and fast computation times, was chosen as the direct optimization method to be employed to solve the problem. To illustrate the capabilities of the methodology, maneuvering of the two- and three-link 2D manipulators and RRR 3D robot manipulators were considered in detail. Their collision-free optimal operations were simulated in the presence of the multiple mobile obstacles, continuously changing their positions and shapes with time. For the visualization of the results of simulations, Virtual Reality animations are used. They enable demonstration of the features of the resulting complex motion of the robotic arm systems and their positioning relative to the moveable obstacles in a 3D environment and provide enormous insight into the nature of the complex phenomena involved.