We consider the problem of determining the trajectory for the tip of redundant mechanisms, by optimization of the given optimality criterion. The shortest distance between the initial and terminal position of the mechanism tip has been adopted as the optimality criterion. The multi-linked robotic mechanism with spherical joints is considered. The obstacles are given in the form of simple polyhedra. An algorithm for generation a set of admissible trajectories for the tip of robotic mechanism is given. On the basis of this set, using an optimization method based on the $\psi$-transformation, a near-optimal admissible trajectory for the mechanism tip is determined. Since the mechanism is redundant, we give a local deterministic criterion for motion of mechanism while tip is tracking the polygonal line.