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This paper considers real-time energy-optimal trajectory generation for a servomotor system that performs a single-axis point-to-point positioning task for a fixed time interval. The servomotor system is subject to acceleration and speed constraints. The trajectory generation is formulated as a linear-constrained optimal control problem (LCOCP), and the Pontryagin's maximum principle is applied to derive necessary optimality conditions. Instead of solving multipoint boundary value problems directly, this paper proposes a novel real-time algorithm based on two realizations: solving the LCOCP is equivalent to determining an optimal time interval of the speed-constrained arc and solving a specific acceleration-constrained optimal control problem (ACOCP), and solving an ACOCP is equivalent to determining optimal switch times of acceleration-constrained arcs and solving a specific two-point boundary value problem (TBVP). The proposed algorithm constructs sequences of time intervals, ACOCPs, switch times, and TBVPs, such that all sequences converge to their counterparts of an optimal solution of the LCOCP. Numerical simulation verifies that the proposed algorithm is capable of generating energy-optimal trajectories in real time. Experiments validate that the use of energy-optimal trajectories as references in a servomotor system does not compromise tracking performance but leads to considerable less energy consumption.
Wang et al. (Wed,) studied this question.