Analytical models are the most accurate method of geometric information representation. Parameterized smooth curves cannot be used in the field of analytical geometry, which explains the necessity for finding of analytical representation of such curves. The article considered the construction of a smooth curve presented in an analytical form and some approaches to finding an analytical model for a parametric Bezier curve. А presentation of a function in the form of its functional areas was chosen as prototype of the analytical model. The selected representation formed on the basis of the De Casteljau's method of constructing the Bezier curve and set-theoretic modeling. The Rvachev functions (R-functions) are used as the mathematical apparatus of set-theoretic operations on function areas. The functional-voxel method makes it possible to simplify the computation of R-functional procedures. An algorithm for constructing the functional area of the Bezier curve is developed on the basis of the presented combined R-voxel approach. The obtained results allow for the conclusions about the adequacy of this approach and its development protentional to construct more complicated structures.

One of the approaches to the construction of graphic images of the stress state for the force vector applied to a point is considered in this work. Has been proposed a geometric model for a continuous medium, formed by a bunch of projection planes for each point of the examined object’s space. This permits to obtain a model for a volume vector in the form of a distributed decomposition into stress components at each point specified by a bunch of projection planes. The building a model for a volume vector, defined as a set of specified laws of direction and length, in the context of modeling stress from an applied force vector to a selected point, is based on strength of materials’ classical laws for calculation the stress state values at an inclined section. Such approach allows use a voxel graphic structure for computer representation of the simulated stress, rather than a finite element mesh. In such a case, there is no obtained result’s error dependence on the spatial position of the mesh nodal points, which is often a problem in FEM calculations. The resulting functional-voxel computer model of the volume stress vector is a structural unit for modeling the distributed load on areas of complex configuration. In this case, the elementary summation of such vectors allows any uneven distribution of the load relative to each point on the specified area. The considered approach works well with geometric models initially represented analytically in the form of a function space (for example, models obtained by the R-functional modelling – RFM-method), and reduced to functional-voxel computer models. A method for deformation modeling based on obtained stresses by means of local transformations of the function space, describing the investigated geometric object, is demonstrated.

Рассматривается пример описания одного из алгоритмов децентрализованной навигации больших групп агентов ORCA средствами функционально-воксельного метода, а также случай построения области допустимых скоростей для предотвращения столкновений для двух мобильных роботов. На основе графических образ-моделей разработан подход к построению области допустимых скоростей роботов на плоскости.

The analytical representation most accurately describes the geometry of the simulated objects. However, its application is associated with a number of difficulties. In particular, R-functional modeling imposes high requirements to the qualification of the researcher and may require considerable time for modeling due to recursive nesting of calculations. The application of features of functional-voxel models to simplify R-functional modeling of complex contours is considered. The Function of Local Zeroing is proposed as the main tool for iterative modeling of complex contours, including parametric curves. The method of determining the negative area of FLOZ-constructed contour models for further construction of predicate complex functions by means of R-functional operations is described.

The problem of avoiding a collision between moving agents constantly arises in multi-agent systems with decentralized control. The various algorithms for solving this problem are accompanied by computational complexity and increasing computational power requirements as the number of agents in question increases. There are difficulties in adapting these algorithms to practical applications on mobile platforms. It is necessary to develop simpler computational schemes and to apply appropriate models. The most computationally expensive step in the classical collision avoidance algorithm ORCA is to calculate the mutual half-planes of possible collision for each pair of robots and use linear programming to calculate the new velocity from them. The application of the functional-voxel method will simplify the necessary calculations by storing in graphical images the local geometric characteristics of the searched domain. Moreover, the application of such models will make it possible to perform most of the necessary calculations in advance, which will accelerate the work of the algorithm. This paper proposes the construction of a functional-voxel model of a required geometric domain by interpolating the contour of the domain using Bezier curves. The local geometric modelling by means of local zeroing function is used as a tool for functional-voxel curve modelling. The obtained functional-voxel model represents a static case of possible mutual positioning of two agents. A four-dimensional graphical model is proposed to solve the dynamic case. This model performs the distribution of the static case modelling results in the space-time characteristics.