Abstract
This article discusses models of the run method in the pursuit problem. The considered models are based on the correction of the direction vector. Let’s assume that the intended direction on ...a plane is the line of sight between the pursuer and the target. The direction correction consists in the rotation of the velocity vector until it coincides with the line of sight. When constructing trajectories on the surface, a line of sight is built on the horizontal projection plane. After calculating horizontal projections, all points are projected back onto the surface. On the basis of the research carried out proposed a mathematical model, proposed mathematical models of the method of pursuit on a plane and on a surface given in an explicit form. Mathematical models are the development of chase and parallel approach methods. A modification of these methods is that the speed of the pursuer and the target are directed at random. These models can be in demand by developers of autonomous unmanned vehicles equipped with artificial intelligence systems.
The article deals with the application of matrix modeling to the group pursuit of objects. The article applies matrix modeling to the group pursuit of objects based on models of actual objects' ...behavior and can be used in simulation modeling packages, virtual simulation of game processes, or transport logistics processes. The results of the article may be in demand in creating virtual reality models of delivery of postal goods by drones in creating an optimized hub - network. The author proposes to consider the model of group pursuit of multiple targets. Methods of pursuit by individual subjects are various modifications of methods of parallel approach, pursuit, and proportional approach. The author constructed a matrix reflecting the number of chasers and the number of targets. The conducted research contains a model of group pursuit of a set of targets. The model contains optimization based on the least time of simultaneous achievement of goals. These results are in the certificate of registration of programs for computers. Objectives assume that in the method of dynamic programming of matrices of distribution of pursuers, we will construct the matrix at each discrete moment, because the number of pursuers and objectives, and their strategies can change at any moment. The methods of forming matrices of distribution of pursuers and goals can be in demand in the design of virtual reality systems for game tasks. Such tasks simulate the process of group pursuit, running away, and evasion. The method of dynamic programming for the distribution matrix of pursuers on targets will allow us to move to an automated distribution process with optimization according to the specified parameters.
Abstract
In this article, the implementation of the method of parallel convergence in space in a computer mathematics system is considered and discussed. In this method, the pursuer’s velocity vector ...is directed arbitrarily. The pursuer’s trajectory gradually approaches movement in the plane formed by the line connecting the initial positions of the pursuer and the target, and the velocity vector. In this task, the target moves uniformly and rectilinearly. The pursuer moves evenly. The points of the pursuer’s trajectory are calculated sequentially. They are being the result of the intersection of the plane containing the line of sight, sphere and cone. As we approach the plane where the target is moving, the algorithm for calculating the trajectory points changes. Now the point of the pursuer’s trajectory is the result of the intersection of the sphere, the plane of movement of the target and the plane containing the line of sight.
Abstract
This article examines a kinematic model of a group pursuing several targets by the method of parallel approach. The model is based on the fact that pursuers try to adhere to pre-designed ...trajectories. The pursuers’ trajectories have curvature constraints. The initial directions of the pursuers’ velocities are arbitrary, which changes the well-known method of parallel approach. In our model, targets are chased by the pursuers simultaneously. This is due to the change in the lengths of the predicted trajectories in such a way as to synchronize the time to reach the target. The change in the lengths of the predicted trajectories occurs due to an increase in the radius of curvature in the initial segment of the trajectory.
Аннотация. В настоящей статье рассматривается компьютерная модель квазидискретной игры группового преследования. В которой присутствуют преследователи, цели и защитники. В моделях статьи задачей ...преследователей является достижение статических целей. Достижение одной цели возможно несколькими преследователями в разное время. Задачей защитников является поражение преследователей. Выигрышем для преследователей можно считать достижение хотя бы одним из них своей цели. Выигрышем для защитников считается поражение всех целей. Для защитников количество преследователей не является определенным. В модели статьи формируется единая среда обнаружения преследователей. Единая среда обнаружения есть результат теоретикомножественных операций над точками пространства. В моделях статьи единая среда обнаружения является односвязным множеством. В случае многосвязных множеств были бы предложены алгоритмы моделей с иной логикой. Преследователь считается обнаруженным, если входит в данную область. Назначение обнаруженному преследователю защитника цели производится по нескольким оптимизационным критериям. Защитник может назначаться из предполагаемого времени достижения. В одной из реализаций модели это минимальное время из выборки для данного защитника. Как вариант фактора оптимизации, защитник для преследователя может выбираться по минимальному расстоянию до него. В статье также рассматриваются варианты локализаций защитников в одной точке. В статье к рассмотрению предлагается четыре модели. Модель группового преследования с оптимизацией по предполагаемому времени достижения преследователей защитниками. Модель группового преследования с оптимизацией по начальному расстоянию от момента обнаружения преследователей защитниками. Разработаны дополнительные алгоритмы вышеописанных моделей, учитывающих ограничения нахождения защитников в одной локации. Рассматривается различие алгоритмов в случае наличия или отсутствия информации для защитников о количестве преследователей, их целях и т.д. Результаты, полученные в статье, могут быть востребованы разработчиками робототехнических комплексов с автономным управлением. В идеале виделось бы следующее. Группа преследователей и группа защитников (каждый участник имеет свой алгоритм действий) имели бы свою программную среду, где каждый участник выполняет свою задачу, руководствуясь правилами. То есть данная программная среда напоминает сетевую игру стратегии реального времени.
Introduction.
It is obvious that in the near future, the issues of equipping moving robotic systems with autonomous control elements will remain relevant. This requires the development of models of ...group pursuit. Note that optimization in pursuit tasks is reduced to the construction of optimal trajectories (shortest trajectories, trajectories with differential constraints, fuel consumption indicators). At the same time, the aspects of automated distribution by goals in group pursuit were not considered. To fill this gap, the presented piece of research has been carried out. Its result should be the construction of a model of automated distribution of pursuers by goals in group pursuit.
Materials and Methods.
A matrix was formed to study the multiple goal group pursuit. The control parameters for the movement of the pursuers were modified according to the minimum curvature of the trajectory. The methods of pursuit and approach were considered in detail. The possibilities of modifying the method of parallel approach were shown. Matrix simulation was used to build a scheme of multiple goal group pursuit. The listed processes were illustrated by functions in the given coordinate systems and animation. Block diagrams of the phase coordinates of the pursuer at the next step, the time and distance of the pursuer reaching the goal were constructed as a base of functions. In some cases, the location of targets and pursuers was defined as points on the circle of Apollonius. The matrix was formed by samples corresponding to the distribution of pursuers by goals.
Results.
Nine variants of the pursuit, parallel, proportional and three-point approach on the plane and in space were considered. The maximum value of the goal achievement time was calculated. There were cases when the speed vector of the pursuer was directed arbitrarily and to a point on the Apollonius circle. It was noted that the three-point approach method was convenient if the target was moving along a ballistic trajectory. To modify the method of parallel approach, a network of parallel lines was built on the plane. Here, the length of the arc of the line (which can be of any shape) and the array of reference points of the target trajectory were taken into account. An equation was compiled and solved with these elements. On an array of samples with corresponding time values, the minimum time was found, i.e., the optimal time for simultaneous group achievement of multiple goals was determined. For unified access to the library, the control vector was expressed through a one-parameter family of parallel planes. A library of calculations of control vectors was formed. An example of applying matrix simulation to group pursuit was shown. A scheme of group pursuit of multiple goals was presented. For two goals and three pursuers, six samples corresponding to the distribution of pursuers by goals were considered. The data was presented in the form of a matrix. Based on the research results, the computer program was created and registered – “Parallel Approach on Plane of Group of Pursuers with Simultaneous Achievement of the Goal”.
Discussions and Conclusion.
The methods of using matrices in modeling group pursuit were investigated. The possibility of modifying the method of parallel approach was shown. Matrix simulation of group pursuit enabled to build its scheme for a set of purposes. The matrix of the distribution of pursuers by goals would be generated at each moment of time. Methods of forming matrices of the distribution of pursuers and targets are of interest in the design of virtual reality systems, for tasks with simulating the process of group pursuit, escape, evasion. The dynamic programming method opens up the possibility of automating the distribution with optimization according to the specified parameters under the formation of the matrix of the distribution of pursuers by goals.
Introduction.
A kinematic model of group pursuit of a set of targets on a plane is considered. Pursuers use a technique similar to parallel approach method to achieve goals. Unlike the parallel ...approach method, the speed vectors of pursuers and targets are directed arbitrarily. In the parallel approach method, the instantaneous directions of movement of the pursuer and the target intersect at a point belonging to the circle of Apollonius. In the group model of pursuing multiple goals, the pursuers try to adhere to a network of predictable trajectories.
Materials and Methods.
The model sets the task of achieving goals by pursuers at designated points in time. This problem is solved by the methods of multidimensional descriptive geometry using the Radishchev diagram. The predicted trajectory is a composite line that moves parallel to itself when the target moves. On the projection plane “Radius of curvature — speed value”, the permissible speed range of the pursuer is displayed in the form of level lines (these are straight lines parallel to one of the projection planes). Images of speed level lines are displayed on the projection plane “Radius of curvature — time to reach the goal”. The search for points of intersection of the speed line images and the appointed time level line is being conducted. Along the communication lines, the values of the intersection points are lowered to the plane “Radius of curvature — speed value”. Using the obtained points, we construct an approximating curve and look for the intersection point with the line of the assigned speed. As a result, we get values of the radius of the circle at the predicted line of the trajectory of the pursuer.
Results.
Based on the results of the conducted research, test programs have been created, and animated images have been made in the computer mathematics system.
Discussion and Conclusions.
This method of constructing trajectories of pursuers to achieve a variety of goals at a given time values can be in demand by developers of autonomous unmanned aerial vehicles.
Protein‐protein interactions play a central role in numerous processes in the cell and are one of the main fields of functional proteomics. This review highlights the methods of bioinformatics and ...functional proteomics of protein‐protein interaction investigation. The structures and properties of contact surfaces, forces involved in protein‐protein interactions, kinetic and thermodynamic parameters of these reactions were considered. The properties of protein contact surfaces depend on their functions. The contact surfaces of permanent complexes resemble domain contacts or the protein core and it is reasonable to consider such complex formation as a continuation of protein folding. Characteristics of contact surfaces of temporary protein complexes share some similarities with active sites of enzymes. The contact surfaces of the temporary protein complexes have unique structure and properties and they are more conservative in comparison with active site of enzymes. So they represent prospective targets for a new generation of drugs. During the last decade, numerous investigations were undertaken to find or design small molecules that block protein dimerization or protein(peptide)‐receptor interaction, or, on the contrary, to induce protein dimerization.
Abstract
The article considers the kinematic model of pursuit, transferred from plane to surface, by the method of parallel approach. Simulation of iterative pursuit processes is a topical ...continuation of autonomous unmanned vehicles. The aim of the article is to develop a model in which the trajectory of the pursuer is the result of following the predicted routes at each discrete moment in time. The research was carried out in the computer mathematics system Math CAD on a surface defined by a point basis. Situations with different initial states were simulated. Detailed results in the form of animated images based on materials, program codes can be found on the authors’ website and channel.
This article considers a kinematic, geometric model of the pursuit problem on a plane by the chase method, where the pursuer cannot instantly change the direction of movement, while moving at a ...constant modulo speed. The initial speed of the pursuer is not directed at the target when the pursuit begins. In order for the speed vector of the pursuer to be directed at the target after some time, we have developed a method that is based on following the trajectory that connects the pursuer and the target. This trajectory takes into account the inertia of the pursuer in the sense that the radius of curvature of the trajectory is not less than a certain threshold value. Based on the materials of this article, test programs were written and an animated image was made.
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