The article deals with the decomposition of nonlinear differential equations based on the group-theoretic approach. At the beginning, the decomposition of differential equations of linear systems using a transition matrix of state is presented, and then, based on the theory of continuous groups (Lie groups), the process of decomposition of differential equations of nonlinear systems is shown. The decomposition approach is based on the isomorphism theorem of the space of vector fields and Lie derivatives, which allows us to consider vector fields as differential operators of smooth functions. A formula is derived about the adjoin representation of a Lie group in its Lie algebra, which actually determines the finding of a vector field that characterizes the interaction of two or more vector fields. The Lie algebra of derivatives makes it possible to determine the infinitesimal action of the Lie group, i.e. the linearization of this action is carried out (transformation of the points of the trajectory space of the original system in a small neighborhood). Decomposition allows, as in the linear case, to separate the finding of an action (only locally) of a group of transformations from the transformed points themselves. For linear systems, this separation is global. It is also shown that the decomposition of linear equations is a particular case of the decomposition of nonlinear equations. An algorithm of the method of model predictive control with Gramian weighting using this decomposition is presented. A practical example of decomposition and application of the model predictive control for stabilization of a nonstationary nonlinear system is considered.

This paper considers one of the problems that arise in the developing of the ergatic brain-computer interfaces. This technology allows a person to control various mechatronic systems through the "power of thought", i.e. based on the registration of electrical activity of the brain. The problem is the complexity and poor knowledge of the brain. To describe the electrical activity of the brain, various models of neural ensembles are used, one of which is the neural mass model proposed by Jansen and Rit in 1995. To tune the parameters of this model according to real data, it is proposed to use an adaptive parameter identifier. An important condition for the synthesis of an adaptive identifier is that only the system output, which is the potential difference between two points of the head, can be measured. At the beginning, it is assumed that the entire state vector of the neural mass model is available for measurement. An identifier is synthesized to tune the parameters of such a system and its convergence is proved using the Lyapunov function method. Further, the obtained identifier is refined in such a way that it uses only the output of the system. To do this, using the finite difference method, the output derivative of the neural mass model is approximately calculated, which is used to make several replacements of the unknown components of the state vector. It is very difficult to analytically prove the convergence of the obtained adaptive parameter identifier, therefore, the possibility of using it to estimate the parameters of a neural mass model is checked using simulation. The synthesized identifier uses only the system output to tune the parameters, which in the future will allow us to consider real data instead of the system output. Thus, this identifier can be used to tune the parameters of the neural mass model based on real data.

The article is devoted to the analysis of features and evaluation of the prospects for using the RRT method in the problem of motion planning for autonomous robots. It is noted that the expansion of the areas of application of modern robotics is inextricably linked with an increase in the level of functionality and improvement of the designs of the created samples, for which the placement of a manipulator on a mobile platform becomes a typical layout. Based on a review of the literature and generalization of experimental data, it is shown that the use of the RRT method and its modifications opens up the fundamental possibility of developing a universal motion planner for mobile and manipulative robots, including systems with a manipulator on a mobile platform, as well as systems with a redundant or reconfigurable structure. Based on the results of the analysis, it was found that the effectiveness of the RRT method is largely determined by the declared parameter of the growth factor. A decentralized variant of the counter-growth RRT method is proposed, which makes it possible to plan the movements of autonomous mobile robots in the process of their mutual approach and subsequent docking. The fundamental possibility of automatic docking of autonomous robots in an environment with obstacles is confirmed by the results of a full-scale experiment.

This paper gives the analysis of the structure and characteristics of the mathematical model of the impact projectile’s flight dynamics. The model is designed for being used as an element of the projectile’s digital twin. The model is based on differential motion equations of a gyro-stabilized solid with an axisymmetric mass distribution. Different types of angle variables were chosen for describing aerodynamics and formulating equations of motion. Non-linear (considering the nutation angle) dependences for aerodynamic coefficients are proposed. They are created by applying proven scientific concepts and research methods in aerodynamics of axisymmetric body and by comparing with known numerical and experimental results obtained in exterior ballistics of gyro-stabilized aviation and artillery projectiles. Special aspects of initial conditions for angles and angular velocity were also studied. Since the impact projectile is considered as an axially symmetric body, its self-rotation angle is not of practical inte rest. Using algebraic manipulations, the differential equation for this angle was eliminated from the set of equations. This has made it possible to significantly reduce stiff of the remaining system of differential equations. The Dormand-Prince method is recommended as a method of numerical integration. The method of the eight-order (with seventh-order uncertainty estimate) allows getting the high accurate solution of the differential equations set under relatively small computing costs. The model allows computing the projectile trajectory under various initial conditions, including the flight with high nutation angles up to 87°—89°. As a result, there is a possibility to determine the nature of the interaction between impact projectiles and typical targets (ricochet, surface effect, after-penetration effect) within a wide range of approach angles to the target’s surface (skin) unattainable during the full-scale tests. The possibility of solving similar problems allows to recommend the designed model as an element of the impact projectile’s digital twin intended for testing its exterior ballistics on the digital (virtual) test range. All testing calculations and final modeling were made by using the "GNU Octave" computational software package.

The paper is dedicated to the development of a method for the synthesis of the control system of a swarm multicopter. The motion of the agents in the swarm is organised by the thermal motion equivalent method. The idea of the method is the behavioral similarity of the thermal motion of the atoms by the agents. In the practical implementation of the thermal motion equivalent method, it is important to ensure constancy of velocity and isotropy of agent dynamics. Violating these properties will cause the swarm to fail a mission, such as an area exploration, by reducing the RMS speed of the agents to zero. The proposed solution to these problems is to synthesize a modal controller for the agent-boundaries test system for the slowest control channel, thereby ensuring RMS velocity constancy of the agent. The synthesized controller is used as a filter in the fast-acting channels, the second horizontal channel and the vertical channel. In the fast-acting channels, an additional filter is proposed to bring their dynamics to the slowest channel, thereby ensuring isotropy. The inclusion of a limit on the maximum length of the equivalent field vector ensures isotropy. The synthesis was carried out using a simplified multicopter dynamics mathematical model, obtained with small UAV deviations from the vertical and without considering the Coriolis force. The methodology for the synthesis of a multicopter control system for functioning as part of a swarm is developed using the obtained results. Numerical simulation results of both a single vehicle in closed space and a swarm using a more complete nonlinear dynamic quadcopter model are presented. The proposed method has the advantage of simple synthesis using a linear model. Numerical simulation results confirm the operability of the developed methodology.