快速且精确的clothoid拟合
Fast and accurate clothoid fitting
AI总结：
提出一种将Hermite G1插值问题简化为单变量非线性方程求解的方法，结合牛顿-拉夫森迭代与渐近展开，实现快速鲁棒的clothoid曲线拟合。
摘要：
针对clothoid曲线的Hermite $G^1$插值问题，提供了一种有效解法。首先，该问题自然表述为具有多个解的非线性方程组，通常难以数值求解。该非线性系统的所有解被简化为单个非线性方程的零点计算。一个简单的策略，结合一个良好且简单的猜测函数，允许通过牛顿-拉夫森方法的几次迭代求解该单个非线性方程。clothoid曲线的计算需要计算菲涅尔及相关积分。这些积分在临界值附近需要渐近展开以避免精度损失。例如，当插值问题的解接近直线或圆弧时，这是必要的。此外，推导出一些特殊的递推关系，用于高效计算渐近展开。将问题简化为单个单变量非线性函数以及使用渐近展开，使得求解算法快速且鲁棒。
英文摘要：
An effective solution to the problem of Hermite $G^1$ interpolation with a clothoid curve is provided. At the beginning the problem is naturally formulated as a system of nonlinear equations with multiple solutions that is generally difficult to solve numerically. All the solutions of this nonlinear system are reduced to the computation of the zeros of a single nonlinear equation. A simple strategy, together with the use of a good and simple guess function, permits to solve the single nonlinear equation with a few iterations of the Newton--Raphson method. The computation of the clothoid curve requires the computation of Fresnel and Fresnel related integrals. Such integrals need asymptotic expansions near critical values to avoid loss of precision. This is necessary when, for example, the solution of interpolation problem is close to a straight line or an arc of circle. Moreover, some special recurrences are deduced for the efficient computation of asymptotic expansion. The reduction of the problem to a single nonlinear function in one variable and the use of asymptotic expansions make the solution algorithm fast and robust.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
