Nonsmooth Variational Approach to Trajectory Optimization for Electrical Energy Recuperations

Abstract Energy saving plays an important role and stacker cranes make up a remarkable share of the energy consumption of a warehouse. The power flow within the two power trains of a stacker crane is described by intricate nonlinear functions to model their technical features. For a general functional relation between the kinematic status of a stacker crane and the resultant power (power flow model), we formulate the task of optimizing the recuperation between both gears (lifting and running gear) as a constrained nonsmooth variational problem and derive a necessary optimality condition in terms of a differential equation applicable to a variety of vehicles. The constraints are given in terms of boundaries of velocity, acceleration and jerk. To not reduce the performance of the stacker crane and therefore the throughput of the warehouse, one of the gears executes a minimum-time maneuver; the other one minimizes the overall net power flow. For a specified type of power flow model, the shapes of optimal trajectories are categorized such that each storage task is predictively assigned to a certain kind of trajectory. A particular type of trajectory was found where maximizing the recuperation forces the trajectory to form oscillations which are both mechanically and electrically highly undesirable.