[Paper Review] Fast and accurate clothoid fitting

Enrico Bertolazzi, Marco Frego|arXiv (Cornell University)|Sep 5, 2012

Advanced Numerical Analysis TechniquesEngineering11 references17 citations

TL;DR

This paper presents a fast, robust, and accurate algorithm for G1 Hermite interpolation using a single clothoid curve by reducing the problem to solving one nonlinear equation via Newton-Raphson with an optimal initial guess. The method ensures high precision through asymptotic expansions of Fresnel integrals near degenerate cases (e.g., straight lines or circles), enabling stable computation even when curvature approaches zero or infinity, with numerical validation showing rapid convergence in few iterations.

ABSTRACT

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.

Motivation & Objective

• To develop a numerically stable and efficient method for G1 Hermite interpolation using a single clothoid segment.
• To eliminate case-based branching in existing algorithms by treating degenerate cases (straight lines, circular arcs) naturally.
• To ensure high accuracy in computing Fresnel integrals, especially near critical parameter values where standard evaluation fails.
• To provide a reliable initial guess for the Newton–Raphson solver to achieve fast convergence in few iterations.
• To enable robust and precise clothoid fitting for applications in road design, robotics, and CAD by handling all parameter regimes uniformly.

Proposed method

• Reduce the three-variable nonlinear system of G1 Hermite interpolation to a single nonlinear equation in one variable using geometric and parametric transformations.
• Use a high-quality initial guess function to ensure quadratic convergence of the Newton–Raphson method in just a few iterations.
• Implement asymptotic expansions for Fresnel integrals when parameters are small (e.g., near straight-line or circular limits), replacing unstable recurrence relations.
• Derive and apply recurrence relations for generalized Fresnel integrals Xk(a,b) and Yk(a,b) when |a| ≥ ε, with special handling for |a| < ε using series expansions.
• Integrate Lommel function-based explicit formulas for the degenerate case a = 0 to maintain numerical stability.
• Design a hybrid algorithm that switches between recurrence and asymptotic series based on a threshold ε, ensuring arbitrary precision across all parameter regimes.

Experimental results

Research questions

• RQ1 How can G1 Hermite interpolation with a single clothoid be solved efficiently without case splitting for degenerate configurations?
• RQ2 What initial guess strategy ensures fast and reliable convergence of the Newton–Raphson method in the context of clothoid fitting?
• RQ3 How can Fresnel integrals be computed accurately when parameters approach values that cause loss of precision in standard evaluation?
• RQ4 What asymptotic expansions are required to maintain numerical stability near straight-line or circular arc limits in clothoid fitting?
• RQ5 How can the generalized Fresnel integrals Xk(a,b) and Yk(a,b) be computed with arbitrary precision across all parameter ranges?

Key findings

• The algorithm reduces the G1 Hermite interpolation problem to a single nonlinear equation, enabling efficient and unified solution across all cases.
• A well-chosen initial guess ensures convergence of the Newton–Raphson method in only 2–4 iterations, significantly improving performance.
• Asymptotic expansions for small |a| (e.g., |a| < 10−4) reduce numerical error to below 6.26×10−18 with just p=2 terms, ensuring high precision.
• The use of Lommel functions and series expansions for a=0 avoids unstable recurrence relations, enabling accurate computation in degenerate cases.
• The hybrid algorithm, combining recurrence and asymptotic series based on a threshold ε, achieves robust and accurate results across all parameter regimes.
• The method maintains high accuracy even when curvature approaches zero or infinity, making it suitable for practical applications in CAD and trajectory planning.