Fast Approximation of Over-Determined Second-Order Linear Boundary Value Problems by Cubic and Quintic Spline Collocation
Fast Approximation of Over-Determined Second-Order Linear Boundary Value Problems by Cubic and Quintic Spline Collocation
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DOI：
10.3390/robotics9020048
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发表时间：
2020-06
期刊：
影响因子：
3.7
通讯作者：
Philipp Seiwald;D. Rixen
中科院分区：
文献类型：
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作者：
Philipp Seiwald;D. Rixen
本文提出了一种利用样条配点逼近二阶线性边值问题的高效通用算法。与大多数其他方法相比，我们的算法是专为超定问题。这些通常发生在控制理论中，其中系统，例如，机器人应该从某个初始状态转换到期望的目标状态，同时考虑特征系统动力学。我们的方法使用多项式的最大程度的3/5作为基函数，并产生一个三次/五次样条，这是C2/C4连续，并满足基本的常微分方程在用户定义的配置网站。此外，近似是被迫满足一组超定的两点边界条件，这是指定的给定的控制问题。该算法适用于对时间要求严格的应用，其中精度仅起次要作用。对于一致的边界条件，我们实验验证收敛到解析解，而对于不一致的边界条件，我们的算法仍然能够找到一个“合理的”近似。然而，为了避免分歧，搭配地点必须适当选择。通过与一个简单的测试系统的解析解进行比较，实验评估所提出的方案。此外，还提供了一个完整的C++实现文档，其中包含作为示例应用程序的单元测试。
We present an efficient and generic algorithm for approximating second-order linear boundary value problems through spline collocation. In contrast to the majority of other approaches, our algorithm is designed for over-determined problems. These typically occur in control theory, where a system, e.g., a robot, should be transferred from a certain initial state to a desired target state while respecting characteristic system dynamics. Our method uses polynomials of maximum degree three/five as base functions and generates a cubic/quintic spline, which is C 2 / C 4 continuous and satisfies the underlying ordinary differential equation at user-defined collocation sites. Moreover, the approximation is forced to fulfill an over-determined set of two-point boundary conditions, which are specified by the given control problem. The algorithm is suitable for time-critical applications, where accuracy only plays a secondary role. For consistent boundary conditions, we experimentally validate convergence towards the analytic solution, while for inconsistent boundary conditions our algorithm is still able to find a “reasonable” approximation. However, to avoid divergence, collocation sites have to be appropriately chosen. The proposed scheme is evaluated experimentally through comparison with the analytical solution of a simple test system. Furthermore, a fully documented C++ implementation with unit tests as example applications is provided.