The optimal velocity planning problem has been widely investigated in the past. The literature mainly addresses minimum-time velocity planning problems: a time optimal trajectory is planned subject to constraints on the maximum velocity, acceleration, and jerk. Sometimes, constraints deriving from dynamic solicitations, such as forces or torques, are also considered. On the contrary, the optimal velocity planning problem with assigned travelling time has been scarcely investigated. In this case, the target is the generation of a velocity profile which fulfills assigned kinematic and/or dynamic constraints and guarantees, at the same time, an exact travelling time and the minimization of an appropriate performance index. The problem is motivated by several applications where trajectory travelling time needs to be imposed. This is the case, e.g., of a robot which must intercept, or avoid, a moving object: any error in the time scheduling will lead to miss the appointment with the moving object or, in case of obstacle avoidance, to an undesired collision. The optimal, assigned-time, planning problem poses feasibility issues which do not appear in the case of minimum-time problems: if the imposed constraints are too restrictive, then the solution could even not exist or the feasible region could be hardly found.
A first solution was proposed in [1], where an assigned-time velocity profile was online generated by only minimizing the maximum jerk. In a subsequent paper, a constraint on the maximum admissible velocity was added [2], while in [3] an improved solution, fulfilling assigned constraints on the maximum velocity and acceleration, was proposed. In all the cases, the assigned bounds were supposed to be constant along the path. This limitation was dropped in [4], where maximum allowable velocities and accelerations were not assumed constant but, on the contrary, they were correlated to the path curvature. Feasibility issues have been considered in all the papers.
