Email the author THE ACCURATE COMPUTING OF CLOTHOID COORDINATE VALUES AND OF THE DISTANCE BETWEEN A POINT AND A CLOTHOID
Abstract
The purpose of this paper is to present two important problems related to the numerical studies of clothoid curves: the exact calculation of their coordinates and the deduction of the distance between a point and the clothoid.
The paper begins with a brief presentation of the history of this curve, names of great mathematicians involved over time in the research and a lot of contemporary problems such as: the use of clothoids as transition curves between a line and a circle arc, between two lines or between two circle arcs, with the direct use in the construction of railways and highways, in the trajectory path-planning of mobile robots and autonomous vehicles, CAGD and so on , with many references to the main existing works.
The clothoid curve has a very interesting kinematic property: in the case of a mobile moving along the curve with a constant velocity, the normal acceleration of the mobile varies linearly, depending on the curvilinear abscissa corresponding to the curve point, because the product between the curvilinear abscissa and the curvature radius is constant: s ρ(s) = A2.
The paper solves the problem of calculating the exact coordinates of the clothoid points, using numerical quadrature formulas, in the context of Romberg procedure, which is based on the principle of extrapolation of Richardson. These calculations are necessary because the accuracy of the coordinates calculus induces the accuracy of all subsequent calculations for: the point-clothoid distance, the connection points coordinates, the parameters that determine the position of a clothoid in the Oxy axis system, etc.
In the second part of the paper is presented a new method to obtain the nonlinear equation whose solution is the curvilinear abscissa of the point where the perpendicular line on the clothoid from an outer point intersects it.
Many numerical results, presented mainly as graphs, obtained with our own C programs, are given in the final part of the paper, certifying the correctness of the proposed calculation methods.
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