Fast and accurate clothoid fitting
For computer-aided design and robotics, this provides an efficient and accurate method for clothoid fitting, which is a known bottleneck in path planning and geometric modeling.
The paper presents a fast and robust algorithm for Hermite G1 interpolation with clothoid curves, reducing the problem to a single nonlinear equation solved with Newton-Raphson iterations and using asymptotic expansions for Fresnel integrals to avoid precision loss.
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.