Nonreactive atom/molecule-surface scattering within the finite basis wave packet method

A quantum wave packet code for studying nonreactive scattering of closed-shell atoms or diatomic molecules from a rigid surface is described. The time evolution relies on the Chebychev propagator. Up to 5 collider degrees of freedom, 3 in translation and 2 in rotation, are treated in a pseudospectral way with the momentum or finite basis representation as the primary space. Potential matrix elements are efficiently evaluated by means of sequential 1D transformations between momentum and coordinate spaces. Fast Fourier transforms are performed for the translational and azimuthal coordinates whereas a Gauss-Legendre transform is used for the polar coordinate. This pseudospectral strategy minimizes memory requirements because no off-diagonal Hamiltonian matrix elements need to be stored. In addition, a wide variety of physical systems can be studied since no particular functional form is imposed for the interaction potential.     

Direct Trajectory Optimization and Costate Estimation of Infinite-horizon Optimal Control Problems Using Collocation at the Flipped Legendre-Gauss-Radau Points

A pseudospectral method is presented for direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using global collocation at flipped Legendre-Gauss-Radau points which include the end point +1. A distinctive feature of the method is that it uses a new smooth, strictly monotonically decreasing transformation to map the scaled left half-open interval τ∈(-1, +1] to the descending time interval t ∈ (+∞, 0]. As a result, the singularity of collocation at point +1 associated with the commonly used transformation, which maps the scaled right half-open interval τ∈[-1, +1) to the increasing time interval [0,+∞), is avoided. The costate and constraint multiplier estimates for the proposed method are rigorously derived by comparing the discretized necessary optimality conditions of a finite-horizon optimal control problem with the Karush-Kuhn-Tucker conditions of the resulting nonlinear programming problem from collocation. Another key feature of the proposed method is that it provides highly accurate approximation to the state and costate on the entire horizon, including approximation at t = +∞, with good numerical stability. Numerical results show that the method presented in this paper leads to the ability to determine highly accurate solutions to infinite-horizon optimal control problems.      

欠驱动刚性航天器时间最优轨迹规划设计

应用伪谱法解决欠驱动刚性航天器的时间最优轨迹规划问题.首先建立欠驱动刚性航天器的动力学和运动学模型,对于给定的初末姿态,选取机动时间最短为待优化的性能指标,并考虑到实际控制输入受限,将其转化为优化过程中的不等式约束条件;然后应用Legendre伪谱法,将优化问题离散化为非线性规划问题进行求解.仿真结果表明,应用伪谱法规划得到的欠驱动航天器最优轨迹,能够较好地满足各种约束条件,而且计算精度高、速度快,具有良好的实时性.     

Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems

Pseudospectral (PS) computational methods for nonlinear constrained optimal control have been applied to many industrial-strength problems, notably, the recent zero-propellant-maneuvering of the international space station performed by NASA. In this paper, we prove a theorem on the rate of convergence for the optimal cost computed using a Legendre PS method. In addition to the high-order convergence rate, two theorems are proved for the existence and convergence of the approximate solutions. Relative to existing work on PS optimal control as well as some other direct computational methods, the proofs do not use necessary conditions of optimal control. Furthermore, we do not make coercivity type of assumptions. As a result, the theory does not require the local uniqueness of optimal solutions. In addition, a restrictive assumption on the cluster points of discrete solutions made in existing convergence theorems is removed.     

基于伪谱法的编队卫星队形重构防碰撞轨迹优化

提出了基于伪谱法的编队卫星队形重构最优轨迹规划方法.首先,应用Legendre伪谱法将队形重构问题离散化为非线性规划问题;然后,通过庞特里亚金极小值原理计算出不考虑碰撞规避问题时各星最优轨迹的解析形式,并由此计算出各星间的碰撞点;最后,在碰撞点附近设置符合高斯分布的测试点,进一步避免各星在配置点间发生碰撞.仿真结果表明,应用所提出方法得到的队形重构的最优轨迹能够较好地满足各种约束条件,计算精度高、速度快,使得编队卫星自主重构成为可能.     

自适应Radau伪谱法自由漂浮空间机器人轨迹规划

为解决Gauss伪谱法(GPM)计算速度和求解精度之间的矛盾,在多段Radau伪谱法的基础上,提出了求解自由漂浮空间机器人(FFSM)最优路径规划问题的hp自适应Radau伪谱法(hp-RPM).与传统的Gauss伪谱法不同,该方法并不是单纯通过增加节点数量来提高精度,而是在每次迭代的过程中对整个路径分段个数和各个路径子区间的宽度进行合理的分配,并能配置每个子区间内节点的数量.通过增加分段个数可以减小子区间内所需节点个数,以此降低多项式阶数、提高计算速度.基于上述理论,首先建立了多臂FFSM系统动力学模型,并给出了运动过程中系统模型更新方法;然后将连续最优轨迹规划问题离散化,完成了hp自适应Radau伪谱法的设计;最后利用hp-RPM解决两连杆FFSM系统轨迹规划问题并进行了仿真实验.结果表明:在初始条件相同的情况下,两种方法得到的位置、速度规划曲线相似,但hp-RPM在各个节点处的误差明显低于GPM计算误差;在精度要求较高,初始节点较多的情况下,hp-RPM可以在保证精度的同时有效的提高计算速度.     

考虑导引头多约束的滑翔制导炮弹弹道规划

为提高制导炮弹在大着角情况下导引头捕捉目标的速度，减小末制导起始点角度偏差，在传统制导炮弹方案弹道规划方法(trajectory programming method, TPM)的基础上考虑末制导段，提出一种考虑导引头多约束的弹道规划方法(trajectory programming method-constraints of seeker, TPM-CS)。根据导引头最大探测距离建立末制导起始点约束，根据弹目几何关系和导引头视场角建立攻击路径约束，并建立最小化前置角和控制变量幅值的目标函数。为实现制导炮弹初始弹道倾角、偏角、火箭点火时间、滑翔启控时间、导引头开启时间等参数的最佳匹配，建立了5阶段弹道规划模型，并采用多阶段Radau伪谱法将该弹道规划问题转化为非线性规划问题，最后调用非线性规划求解器SNOPT进行求解。选取不同性能参数的导引头进行仿真，分析了导引头最大探测距离和导引头视场角对方案弹道的影响。将文中提出的弹道规划方法与传统弹道规划方法进行对比仿真，结果表明，相比于传统方法，文中所提方法规划方案弹道的末制导初始角度偏差缩小71.590%,导引头对目标保持照射状态的时间延长6...     

An Adaptive Pseudospectral Method for Constrained Dynamic Optimization Problems in Chemical Engineering

An adaptive pseudospectral method with two novel strategies is proposed to solve constrained dynamic optimization problems in chemical engineering. The first strategy is to introduce slope information in designing appropriate subintervals, so that the approach becomes more efficient to track the optimal control profiles. The second strategy is to redistribute the collocation points based on the approximation error, thus ensuring the accuracy of the method. Two constrained dynamic optimization problems with multiple control variables are tested as illustrations, and the proposed approach is compared with other methods. The research results reveal the effectiveness of the proposed method in improving the solution accuracy of dynamic optimization problems.     

流体饱和多孔隙介质中弹性波运动方程的解—伪谱法

本文提出了一种解流体饱和多孔隙介质中波传播方程的数值方法，并详细推导了该方法的计算公式．用此公式对单界面模型中地震波的传播进行了模拟，结果与Ｂｉｏｔ理论预测的相符．     

Application of the chebyshev-fourier pseudospectral method to the eigenvalue analysis of circular mindlin plates with free boundary conditions

An eigenvalue analysis of the circular Mindlin plates with free boundary conditions is presented. The analysis is based on the Chebyshev-Fourier pseudospectral method. Even though the eigenvalues of lower vibration modes tend to convergence more slowly than those of higher vibration modes, the eigenvalues converge for sufficiently fine pseudospectral grid resolutions. The eigenvalues of the axisymmetric modes are computed separately. Numerical results are provided for different grid resolutions and for different thickness-to-radius ratios.