Semi-analytical Approach to Trajectory Optimization for Stacker Cranes Regarding Energy Saving

TL;DR

This paper addresses energy-aware trajectory optimization for stacker cranes in high-bay warehouses, aiming to reduce energy consumption and improve recuperation under power-flow constraints. It adopts an indirect variational approach, deriving optimality conditions and implementing a targeted numerical strategy using a nonuniform time grid to compute and classify energy-efficient trajectories. Key findings show that up-travel optimizations yield similar trajectories for energy recuperation and energy consumption, while down-travel optimizations differ, and that maximal recuperation can be technically undesirable. The approach enables real-time generation of hundreds of trajectories per second and offers a data-rich basis for disposition planning in warehouse logistics, with implications for general power-flow driven machines.

Abstract

The aim of this study is to give insights into the trajectory optimization w.r.t. energy consumption and recuperation for stacker cranes in a high-bay warehouse. Based on an analytical necessary optimality condition, a targeted numerical implementation is set up to perform systematic computations of optimal trajectories which are further categorized. Particularly, the differences between energy consumption and recuperation as well as for up and down movements are pointed out. Although examined for a concrete, experimentally validated model of stacker cranes, the methodical approach could be adapted to other electrical machines possessing a power flow model, i.e. a functional relation between the kinematics (velocity, acceleration for instance) and the resultant power. In addition, boundaries of the velocity, the acceleration and the jerk are incorporated. Such a systematic analysis of energy optimal trajectories can be further used for improving the job scheduling in a warehouse.

Figures (7)

• Figure 1: Geometric aspects of the trajectory optimization, where the load attachment device starts from point $A(x_A,y_A)$ and moves to point $B(x_B,y_B)$, in such a way that one of the drives -- exactly that one which needs more time to overcome the prescribed distance -- behaves time minimally and the other one makes optimal use of the energy recuperation (\ref{['3.1']}) or minimizes the energy consumption (\ref{['3.11']}). The respective distance of this drive is assigned to the parameter $s_0$ in (\ref{['3.2']}).
• Figure 2: Scheme of the implemented optimization strategy.
• Figure 3: Contour plots of the power $P=P(v,a)$ in kW as a function of the velocity and the acceleration for a load mass 1000 kg (panel (a): running gear, panel (b): lifting gear). Hereby, negative power means power supply.
• Figure 4: Panel (a): Standardized reference points for experimental measurements of the grid connection power and power output of the motor. The angular momentum is denoted by $M$ (nominal momentum $M_{\mathrm{N}} = 19~\mathrm{Nm}$) and $n$ stands for the rotational speed (nominal value $n_{\mathrm{N}} = 3480~\mathrm{min}^{-1}$). Panel (b): Comparison of the experimental values (boxplots) of the grid connection power with the values according to the employed power flow model Schutzhold (red crosses). The green crosses indicate the power output of the motor equal to $2\pi nM$.
• Figure 5: Contour plot of the energy saving in % comparing the energy optimal movement with the fully time minimal movement for up travels. The black curve separates the cases, where the movement is dominated by the running gear (below) or the lifting gear (above). The light green area indicates those combinations of distances $s_x$ and $s_y$, where the energy optimal trajectory can be approximated fairly well by a velocity profile achieving the minimal constant velocity. The island of optimality surrounded by the red curve is formed because of two effects: on the one hand, a longer horizontal movement allows more flexibility for a better recuperation, but a longer travel needs more energy on the other hand.