Evgeniya Mikishanina

This study investigates the rolling along the horizontal plane of two coupled rigid bodies: a spherical shell and a dynamically asymmetric rigid body which rotates around the geometric center of the shell. The inner body is in contact with the shell by means of omniwheels. A complete system of equations of motion for an arbitrary number of omniwheels is constructed. The possibility of controlling the motion of this mechanical system along a given trajectory by controlling the angular velocities of omniwheels is investigated. The cases of two omniwheels and three omniwheels are studied in detail. It is shown that two omniwheels are not enough to control along any given curve. It is necessary to have three or more omniwheels. The quaternion approach is used to study the dynamics of the system.

The problem of controlling the rolling of a spherical robot with a pendulum actuator pursuing a moving target by the pursuit method, but with a minimal control, is considered. The mathematical model assumes the presence of a number of holonomic and nonholonomic constraints, as well as the presence of two servo-constraints containing a control function. The control function is defined in accordance with the features of the simulated scenario. Servo-constraints set the motion program. To implement the motion program, the pendulum actuator generates a control torque which is obtained from the joint solution of the equations of motion and derivatives of servo-constraints. The first and second components of the control torque vector are determined in a unique way, and the third component is determined from the condition of minimizing the square of the control torque. The system of equations of motion after reduction for a given control function is reduced to a nonautonomous system of six equations. A rigorous proof of the boundedness of the distance function between a spherical robot and a target moving at a bounded velocity is given. The cases where objects move in a straight line and along a curved trajectory are considered. Based on numerical integration, solutions are obtained, graphs of the desired mechanical parameters are plotted, and the trajectory of the target and the trajectory of the spherical robot are constructed.

This article examines the dynamics of the movement of a wheeled vehicle consisting of two links (trolleys). The trolleys are articulated by a frame. One wheel pair is fixed on each link. Periodic excitation is created in the system due to the movement of a pair of masses along the axis of the first trolley. The center of mass of the second link coincides with the geometric center of the wheelset. The center of mass of the first link can be shifted along the axis relative to the geometric center of the wheelset. The movement of point masses does not change the center of mass of the trolley itself. Based on the joint solution of the Lagrange equations of motion with undetermined multipliers and time derivatives of nonholonomic coupling equations, a reduced system of differential equations is obtained, which is generally nonautonomous. A qualitative analysis of the dynamics of the system is carried out in the absence of periodic excitation and in the presence of periodic excitation. The article proves the boundedness of the solutions of the system under study, which gives the boundedness of the linear and angular velocities of the driving link of the articulated wheeled vehicle. Based on the numerical solution of the equations of motion, graphs of the desired mechanical parameters and the trajectory of motion are constructed. In the case of an unbiased center of mass, the solutions of the system can be periodic, quasi-periodic and asymptotic. In the case of a displaced center of mass, the system has asymptotic dynamics and the mobile transport system goes into rectilinear uniform motion.

This article is devoted to the study of the dynamics of movement of an articulated n-trailer wheeled vehicle with a controlled leading car. Each link of the vehicle can rotate relative to its point of fixation. It is shown that, in the case of a controlled leading car, only nonholonomic constraint equations are sufficient to describe the dynamics of the system, which in turn form a closed system of differential equations. For a detailed analysis of the dynamics of the system, the cases of movement of a wheeled vehicle consisting of three symmetric links are considered, and the leading link (leading car) moves both uniformly along a circle and with a modulo variable velocity along a certain curved trajectory. The angular velocity remains constant in both cases. In the first case, the system is integrable and analytical solutions are obtained. In the second case, when the linear velocity is a periodic function, the solutions of the problem are also periodic. In numerical experiments with a large number of trailers, similar dynamics are observed.

This work is devoted to the study of the dynamics of the Chaplygin ball with variable moments of inertia, which occur due to the motion of pairs of internal material points, and internal rotors. The components of the inertia tensor and the gyrostatic momentum are periodic functions. In general, the problem is nonintegrable. In a special case, the relationship of the problem under consideration with the Liouville problem with changing parameters is shown. The case of the Chaplygin ball moving from rest is considered separately. Poincaré maps are constructed, strange attractors are found, and the stages of the origin of strange attractors are shown. Also, the trajectories of contact points are constructed to confirm the chaotic dynamics of the ball. A chart of dynamical regimes is constructed in a separate case for analyzing the nature of strange attractors.

The dynamics of a body with a fixed point, variable moments of inertia and internal rotors are considered. A stability analysis of permanent rotations and periodic solutions of the system is carried out. In some simplest cases the stability analysis is reduced to investigating the stability of the zero solution of Hill’s equation. It is shown that by periodically changing the moments of inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the oscillation frequency of the moments of inertia increases, the dynamics of the system becomes close to an integrable one.