In robotics, path planning and trajectory optimization are usually performed separately to optimize the path from the given starting point to the ending point in the presence of obstacles. In this paper, path planning and trajectory optimization for robotic manipulators are solved simultaneously by a newly developed methodology called Discrete Mechanics and Optimal Control (DMOC). In DMOC, the Lagrange–d'Alembert equation is discretized directly unlike the conventional variational optimization method in which first the Euler–Lagrange equations are derived and then discretization takes place. In this newly developed method, the constraints for optimization of a desired objective function are the forced discrete Euler–Lagrange equations. In this paper, DMOC is used for simultaneous path planning and trajectory optimization for robotic manipulators in the presence of known static obstacles. Two numerical examples, applied on a DELTA parallel robot, are discussed to show the applicability of this new methodology. The optimal results obtained from DMOC are compared with the other state-of-the-art techniques. The difficulties and problems associated in using the DMOC for Parallel Kinematic Machine (PKM) are also discussed in this paper.

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.