Two-stage spline-approximation in linear structure routing

In the article, computer design of routes of linear structures is considered as a spline approximation problem. A fundamental feature of the corresponding design tasks is that the plan and longitudinal profile of the route consist of elements of a given type. Depending on the type of linear structure, line segments, arcs of circles, parabolas of the second degree, clothoids, etc. are used. In any case, the design result is a curve consisting of the required sequence of elements of a given type. At the points of conjugation, the elements have a common tangent, and in the most difficult case, a common curvature. Such curves are usually called splines. In contrast to other applications of splines in the design of routes of linear structures, it is necessary to take into account numerous restrictions on the parameters of spline elements arising from the need to comply with technical standards in order to ensure the normal operation of the future structure. Technical constraints are formalized as a system of inequalities. The main distinguishing feature of the considered design problems is that the number of elements of the required spline is usually unknown and must be determined in the process of solving the problem. This circumstance fundamentally complicates the problem and does not allow using mathematical models and nonlinear programming algorithms to solve it, since the dimension of the problem is unknown. The article proposes a two-stage scheme for spline approximation of a plane curve. The curve is given by a sequence of points, and the number of spline elements is unknown. At the first stage, the number of spline elements and an approximate solution to the approximation problem are determined. The method of dynamic programming with minimization of the sum of squares of deviations at the initial points is used. At the second stage, the parameters of the spline element are optimized. The algorithms of nonlinear programming are used. They were developed taking into account the peculiarities of the system of constraints. Moreover, at each iteration of the optimization process for the corresponding set of active constraints, a basis is constructed in the null space of the constraint matrix and in the subspace – its complement. This makes it possible to find the direction of descent and solve the problem of excluding constraints from the active set without solving systems of linear equations. As an objective function, along with the traditionally used sum of squares of the deviations of the initial points from the spline, the article proposes other functions taking into account the specificity of a particular project task.