In this paper, we present a polynomial chaos-based framework for the trajectory optimization of an overhead crane system under uncertainty. The main research described in this paper is as follows. First, the deterministic trajectory optimization problem formulation of a two-dimensional overhead crane model is constructed. Based on this basic mathematical formulation, the uncertainty trajectory optimization problem is formed considering the uncertainty of initial state and system parameter. Then, to solve the uncertainty trajectory optimization problem efficiently, a robust trajectory optimization problem formulation is proposed. However, it is difficult to solve the robust trajectory optimization problem directly because it contains stochastic function terms, such as stochastic dynamic equations, constraint functions and objective functions. We consider both the system state and control input as functions of uncertainty and use polynomial chaos expansion to quantify these stochastic functions. An augmented deterministic trajectory optimization problem which can be solved directly is finally obtained. Based on the proposed robust trajectory optimization formation, the motion trajectory optimization of an overhead crane system under two different uncertainty types of is solved. All simulation results are compared with traditional sampling-based Monte Carlo simulations to demonstrate the feasibility and effectiveness of the proposed method.

在本文中，我们提出了一种基于不确定性的多项式混沌框架，用于桥式起重机系统的轨迹优化。本文介绍的主要研究如下。首先，构建了二维桥式起重机模型的确定性轨迹优化问题公式。在此基本数学公式的基础上，考虑了初始状态和系统参数的不确定性，形成了不确定性轨迹优化问题。然后，为有效地解决不确定性轨迹优化问题，提出了一种鲁棒的轨迹优化问题公式。但是，由于它包含随机函数项，例如随机动力学方程，约束函数和目标函数，因此直接解决鲁棒轨迹优化问题很困难。我们将系统状态和控制输入都视为不确定性的函数，并使用多项式混沌展开来量化这些随机函数。最终获得了可以直接解决的扩充确定性轨迹优化问题。基于提出的鲁棒轨迹优化结构，解决了两种不确定类型下桥式起重机系统的运动轨迹优化问题。将所有模拟结果与基于传统采样的蒙特卡洛模拟进行比较，以证明该方法的可行性和有效性。最终获得了可以直接解决的扩充确定性轨迹优化问题。基于提出的鲁棒轨迹优化结构，解决了两种不确定类型下桥式起重机系统的运动轨迹优化问题。将所有模拟结果与基于传统采样的蒙特卡洛模拟进行比较，以证明该方法的可行性和有效性。最终获得了可以直接解决的扩充确定性轨迹优化问题。基于提出的鲁棒轨迹优化结构，解决了两种不确定类型下桥式起重机系统的运动轨迹优化问题。将所有模拟结果与基于传统采样的蒙特卡洛模拟进行比较，以证明该方法的可行性和有效性。