In dynamic environments, robotic manipulators and especially cobots must be able to react to changing circumstances while in motion. This substantiates the need for quick trajectory planning algorithms that are able to cope with arbitrary velocity and acceleration boundary conditions. Apart from dynamic re-planning, being able to seamlessly join trajectories together opens the door for divide-and-conquer-type algorithms to focus on the individual parts of a motion separately. While geodesic motion planning has proven that it can produce very smooth and efficient actuator movement, the problem of incorporating non-zero boundary conditions has not been addressed yet. We show how a set of generalized coordinates can be used to transition between boundary conditions and free movement in an optimal way while still retaining the known advantages of geodesic planners. We also outline, how our approach can be combined with the family of time-scaling algorithms for further improvement of the generated trajectories.

Naturally, finding joint trajectories for robotic manipulators involves competing optimization goals. On the one hand, the end-effector should move along a predictable and short path while on the other hand joint movement and acceleration should be kept to a minimum. Obstacles in the workspace or joint limits complicate the situation even further. Constructing a metric that makes undesired configurations more expensive to travel through, we equip the joint space with a notion of cost. The motion planning task then reduces to the problem of finding cheapest paths connecting two configurations – so-called geodesics. We show how to construct suitable metrics for a variety of typical scenarios and present an efficient algorithm for the computation of the corresponding geodesics. Our approach makes it very easy to balance different optimization goals and produces natural and smooth manipulator movement.