Abstract
This paper proposes some new polynomial solutions to the trajectory planning problem encountered in pick-and-place operations. When a robotic manipulator is used for such operations, it is possible to plan the required trajectory in joint space, provided that the inverse kinematic problem has been solved for the initial and final configurations — and possibly for a lift-off and a set-down configuration — and that the workspace is free of obstacles. Polynomial solutions to this problem can be found in the literature. However, they usually provide continuity up to the second derivative only, leading to a discontinuous jerk. The solutions derived in this paper preserve the continuity of the third derivative of the joint coordinates, thereby ensuring smooth trajectories with smooth variations of the actuator currents. Moreover, whenever possible, unique polynomial expressions valid between the initial and final configurations are used in order to simplify the logic. Polynomial formulations without lift-off and set-down configurations are first presented. Then, these intermediate configurations are introduced, leading to a new set of solutions. A global algorithm is then discussed in order to clearly indicate the relationship between the different solutions. Finally, an example illustrating the application to a pick-and-place operation is solved.
The Heisenberg ultrahyperbolic equation: $K$ -finite and polynomial solutions
