We consider the situation where three heavy gravitational bodies form the Lagrange configuration rotating in a fixed plane and the fourth body of negligible mass moves in this plane. We present three cases of so-called libration points and we study their stability using linear approximation and KAM theory. In some situations we prove the Lyapunov stability for generic values of some parameter of the problem.

In this paper, we study the symmetry of a bicycle moving on a flat, level ground. Applying the Gibbs – Appell equations to the bicycle dynamics, we previously observed that the coefficients of these equations appeared to depend on the lean and steer angles only, and in one such equation, a term quadratic in the rear wheel’s angular velocity and a pseudoforce term would always vanish. These properties indeed arise from the symmetry of the bicycle system. From the point of view of the geometric mechanics, the bicycle’s configuration space is a trivial principal fiber bundle whose structure group plays the role of a symmetry group to keep the Lagrangian and constraint distribution invariant. We analyze the dimension relationship between the space of admissible velocities and the tangent space to the group orbit, and then employ the reduced nonholonomic Lagrange – d’Alembert equations to directly prove the previously observed properties of the bicycle dynamics. We then point out that the Gibbs – Appell equations give the local representative of the reduced dynamic system on the reduced constraint space, whose relative equilibria are related to the bicycle’s uniform upright straight or circular motion. Under the full rank condition of a Jacobian matrix, these relative equilibria are not isolated, but form several families of one-parameter solutions. Finally, we prove that these relative equilibria are Lyapunov (but not asymptotically) stable under certain conditions. However, an isolated asymptotically stable equilibrium may be achieved by restricting the system to an invariant manifold, which is the level set of the reduced constrained energy.

The problem of the existence and stability of relative equilibria (libration points) of a uniformly rotating gravitating body, which is a homogeneous ball with a spherical cavity, is considered. It is assumed that the rotation is carried out around an axis perpendicular to the axis of symmetry of the body and passing through its center of mass. The libration points located inside the cavity are investigated. Families of both isolated and nonisolated relative equilibria are found. Their stability and bifurcations are investigated. Realms of possible motion are constructed.

A simple model of a capsule robot is studied. The device moves upon a rough horizontal plane and consists of a capsule with an embedded motor and an internal moving mass. The motor generates a harmonic force acting on the bodies. Capsule propulsion is achieved by collisions of the inner body with the right wall of the shell. There is Coulomb friction between the capsule and the support, it prevents a possibility of reversal motion. A periodic motion is constructed such that the robot gains the maximal average velocity.

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.

The orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev – Steklov case is investigated. In particular, a nonlinear study of the orbital stability is performed for the so-called case of degeneracy, where it is necessary to take into account terms of order six in the Hamiltonian expansion in a neighborhood of the unperturbed periodic orbit.

According to the Nielsen – Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_1^{}$) periodic homeomorphism; $T_2^{}$) reducible non-periodic homeomorphism of algebraically finite order; $T_3^{}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_4^{}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_1^{}$, $T_2^{}$, $T_4^{}$ only. Moreover, all representatives of the class $T_4^{}$ have chaotic dynamics, while in each homotopy class of types $T_1^{}$ and $T_2^{}$ there are regular diffeomorphisms, in particular, Morse – Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse – Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse – Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_1^{}$. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse – Smale diffeomorphisms belong to types $T_1^{}$ or $T_2^{}$ is uniquely determined by the total intersection index of such knots.

We study the effect of an emerging virus mutation on the evolution of an epidemic, inspired by the appearance of the delta variant of SARS-CoV-2. We show that if the new variant is markedly more infective than the existing ones the epidemic can resurge immediately. The vaccination of the population plays a crucial role in the evolution of the epidemic. When the older (and more vulnerable) layers of the population are protected, the new infections concern mainly younger people, resulting in fewer hospitalisations and a reduced stress on the health system. We study also the effects of vacations, partially effective vaccines and vaccination strategies based on epidemic-awareness. An important finding concerns vaccination deniers: their attitude may lead to a prolonged wave of epidemic and an increased number of hospital admissions.

Any obstacle-free path planning algorithm, in general, gives a sequence of waypoints that connect start and goal positions by a sequence of straight lines, which does not ensure the smoothness and the dynamic feasibility to maneuver the MAV. Kinodynamic-based motion planning is one of the ways to impose dynamic feasibility in planning. However, kinodynamic motion planning is not an optimal solution due to high computational demands for real-time applications. Thus, we explore path planning followed by kinodynamic smoothing while ensuring the dynamic feasibility of MAV. The main difference in the proposed technique is not to use kinodynamic planning when finding a feasible path, but rather to apply kinodynamic smoothing along the obtained feasible path. We have chosen a geometric-based path planning algorithm “RRT*” as the path finding algorithm. In the proposed technique, we modified the original RRT* introducing an adaptive search space and a steering function that helps to increase the consistency of the planner. Moreover, we propose a multiple RRT* that generates a set of desired paths. The optimal path from the generated paths is selected based on a cost function. Afterwards, we apply kinodynamic smoothing that will result in a dynamically feasible as well as obstacle-free path. Thereafter, a b-spline-based trajectory is generated to maneuver the vehicle autonomously in unknown environments. Finally, we have tested the proposed technique in various simulated environments. According to the experiment results, we were able to speed up the path planning task by 1.3 times when using the proposed multiple RRT* over the original RRT*.

This paper addresses the problem of balancing an inverted pendulum on an omnidirectional platform in a three-dimensional setting. Equations of motion of the platform – pendulum system in quasi-velocities are constructed. To solve the problem of balancing the pendulum by controlling the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is investigated under different initial conditions taking into account a necessary stop of the platform or the need for continuation of the motion at the end point of the trajectory. It is shown that the solution of the problem in a two-dimensional setting is a particular case of three-dimensional balancing.

In this paper we present a study of the dynamics of a mobile robot with omnidirectional wheels taking into account the reaction forces acting from the plane. The dynamical equations are obtained in the form of Newton – Euler equations. In the course of the study, we formulate structural restrictions on the position and orientation of the omnidirectional wheels and their rollers taking into account the possibility of implementing the omnidirectional motion. We obtain the dependence of reaction forces acting on the wheel from the supporting surface on the parameters defining the trajectory of motion: linear and angular velocities and accelerations, and the curvature of the trajectory of motion. A striking feature of the system considered is that the results obtained can be formulated in terms of elementary geometry.