Optimal exploration and control for a robotic pick-up and delivery problem

In this paper we address a problem where a robot moving on a line has to find and collect a finite number of objects and move them to a specified point. The robot is modeled as a second-order system and the task has to be completed in minimum time. Both the robot and the objects are represented by point masses. The objects are located at unknown places within a given interval and their pick-up and drop-off leads to a switching of the dynamics. The corresponding hybrid Optimal Control Problem (OCP) is investigated for the worst-case and a probabilistic case assuming a uniform distribution of the objects over the interval. We first derive optimal solutions for a single object. Then, we show that an optimal solution for the multi-object case consists of complete exploration followed by a deterministic optimal pick-up and drop-off (with possible intermediate drop-offs) of all objects. Thus, the computation of the exploration and the exploitation part of the control can be decoupled, similar to the single object case. The worst- and the probabilistic case optimal solutions are compared in a numerical example. The proposed methods are particularly relevant for different robotic applications like automated cleaning, search and rescue, harvesting, manufacturing etc.