В статье рассматриваются математические модели и алгоритмы решения задач, возникающих при автоматизации проектирования трасс линейных сооружений. Принципиальная особенность этих задач состоит в том, что план и продольный профиль трассы состоят из элементов заданного вида. В зависимости от вида сооружения в проектной практике используются отрезки прямых, дуги окружностей, парабол и клотоид. В любом случае необходимо получить гладкую кривую, состоящую из нужной последовательности элементов заданного вида. На стыках элементы имеют общую касательную, а в наиболее сложном случае и общую кривизну. При этом параметры элементов должны удовлетворять техническим ограничениям, которые формализуются в виде системы неравенств. Такого рода кривые принято называть сплайнами. Важно подчеркнуть, что число элементов искомого сплайна, как правило, неизвестно и определяется в процессе решения задачи. Это обстоятельство существенно усложняет и не позволяет применить для решения методы нелинейного программирования, так как неизвестна ее размерность. Кроме того, искомый сплайн -это экстремаль некоторого функционала. Ранее была решена задача аппроксимации плоской кривой, заданной последовательностью точек, сплайном, состоящим из отрезков парабол. В данной статье рассматривается сплайн, включающий элементы различного вида, в том числе и наиболее сложная задача поиска сплайна, состоящего из последовательности отрезков: прямая+клотоида+окружность+клотоида+прямая и т.д. Этот сплайн в статье называется сплайн с клотоидами. Приводится оригинальная формализация задачи, которая для поиска сплайнов позволяет применять динамическое программирование, а также новый алгоритм.Ключевые слова: трасса, план и продольный профиль, сплайн, динамическое программирование, целевая функция, ограничения.Российский технологический журнал 2019 Том 7 № 3 Динамическое программирование как метод сплайн-аппроксимации в САПР линейных сооруженийUnder study is a problem of the line structure routing of roads, railways and other linear constructions. Designing a trace plan and longitudinal profile are considered as non-linear programming tasks. Since the number of elements of the plan and the longitudinal profile is not known, the problem is solved in three stages. First, a search is performed for a polyline consisting of short elements. On the second stage it is used to determine the initial approximation of the desired line, which is optimized at the last stage. The required line consists of a given type elements and it is a spline with a number of features:-In contrast to the polynomial elements considered in the theory of splines, when designing roads unknown spline is a sequence of elements: straight, clothoid, circle, clothoid, straight and so on.-In this task, the spline does not have to be a single-valued function.-The parameters of the elements of the desired spline must satisfy the constraints in the form of inequalities. These features of the task do not allow the use of non-linear programming methods to solve it. Converting a broken line to a spline is carrie...

Under study is the optimization problem of line structure (primarily railroad) routing. The improved mathematical models and algorithms of vertical alignment by set versions of the route plan are offered. The problem is solved in some stages in interrelation with other design problems. The original algorithm of descent is given for solving the arising problem of nonlinear programming. Structural features of constraints are used and so it is not required to solve any systems of linear equations.

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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.