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Point-clothoid distance and projection computation. (English) Zbl 07124609
A clothoid (or Euler spiral, or Cornu spiral) is a planar curve whose curvature is a linear function of the arc length. It appears as the solution of an optimal control problem for a car-like robot that has to find the shortest path connecting two points in the plane with given initial and final angles and curvatures, driving at constant speed. The authors propose an algorithm to compute the minimum distance between a given point in the plane and an assigned clothoid spiral curve. The projection of the point on the curve is also found. The solution is relevant in many applications ranging from robotics to autonomous vehicles. The minimization is formulated as a root-finding problem which typically has multiple solutions associated to local minima. A proper interval for the curvilinear abscissa, that contains the global solution is recognized. Due to its spiraling path, the clothoid has a low curvature region near the inflection point and a high curvature region around points at infinity, where the revolving curve shows many potential solutions. The transition from a clothoid to an arc of circle and from an arc of circle to a straight line (which are particular cases of clothoids) is smoothly computed. The algorithm is validated with extensive numerical tests and is proved to be much better than brute force algorithms in terms of accuracy, convergence and computational time.

MSC:
65D17 Computer-aided design (modeling of curves and surfaces)
65H05 Numerical computation of solutions to single equations
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65S05 Graphical methods in numerical analysis
Software:
DLMF; Clothoids; GitHub
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References:
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