Spline approximation of multivalued functions in linear structures routing

Abstract

Objectives. The theory and methods of spline approximation of plane curves given by a sequence of points are currently undergoing rapid development. Despite fundamental differences between used splines and those considered in the theory and its applications, results published earlier demonstrate the possibility of using spline approximation when designing routes of linear structures. The main difference here consists in the impossibility of assuming in advance the number of spline elements when designing the routes. Here, in contrast to widely use polynomial splines, the repeating element is the link “segment of a straight line + arc of a circle” or “segment of a straight line + arc of a clothoid + arc of a circle + arc of a clothoid.” Previously, a two-stage scheme consisting of a determination of the number of elements of the desired spline and subsequent optimization of its parameters was proposed. Although an algorithm for solving the problem in relation to the design of a longitudinal profile has been implemented and published, this is not suitable for designing a route plan, since, unlike a profile, a route plan is generally a multivalued function. The present paper aims to generalize the algorithm for the case of spline approximation of multivalued functions making allowance for the design features of the routes of linear structures.Methods. At the first stage, a novel mathematical model is developed to apply the dynamic programming method taking into account the constraints on the desired spline parameters. At the second stage, nonlinear programming is used. In this case, it is possible to analytically calculate the derivatives of the objective function with respect to the spline parameters in the absence of its analytical expression through these parameters.Results. An algorithm developed for approximating multivalued functions given by a discrete series of points using a spline consisting of arcs of circles conjugated by line segments for solving the first stage of the problem is presented. An additional nonlinear programming algorithm was also used to optimize the parameters of the resulting spline as an initial approximation. However, in the present paper, the first stage is considered only, since the complex algorithm of the second stage and its justification require separate consideration.Conclusions. The presented two-stage spline approximation scheme with an unknown number of spline elements is also suitable for approximating multivalued functions given by a sequence of points on a plane, in particular, for designing a route plan for linear structures.