We consider a left-invariant sub-Riemannian problem on the Lie group of rotations of a threedimensional space. We find the cut locus numerically, in fact we construct the optimal synthesis numerically, i. e., the shortest arcs. The software package nutopy designed for the numerical solution of optimal control problems is used. With the help of this package we investigate sub-Riemannian geodesics, conjugate points, Maxwell points and diffeomorphic domains of the exponential map. We describe some operating features of this software package.

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the orbit space of the unstable saddle separatrix in the manifold $\mathbb{S}^2\times \mathbb{S}^1$, is a complete invariant of the topological conjugacy of the system. In this paper we distinguish a class of three-dimensional Morse – Smale diffeomorphisms for which the complete invariant of topological conjugacy is the equivalence class of a link in $\mathbb{S}^2\times \mathbb{S}^1$.
We prove that, if $M$ is a link complement in $\mathbb{S}^3$, or a handlebody $H_g^{}$ of genus $g\geqslant 0$, or a closed, connected, orientable 3-manifold, then the set of equivalence classes of tame links in $M$ is countable. As a corollary, we prove that there exists a countable number of equivalence classes of tame links in $\mathbb{S}^2\times \mathbb{S}^1$. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration.

This paper presents a practical approach to fully automated kinematic calibration of an industrial manipulator. The approach is based on the principle of plane constraint. The electrical signal is used to fix the moment of contact between the conductive tool and the flat surface. The measurement data are manipulator configurations (joint angles) at the moment of contact. A modification of the algorithm to deal with the scaling problem is also proposed. This approach provides both high calibration accuracy and lower cost of the experimental setup compared to coordinate measuring machines (CMMs), laser trackers, and vision systems. The article examines the impact of various methods of kinematic parameterization of manipulators: the Denavit – Hartenberg agreement (DH), product of exponentials (POE), as well as the complete and parametrically continuous model (CPC) on the calibration accuracy. A comparison is made of the open-loop and the proposed closed-loop calibration methods on the Puma 560 model known in the literature. POE parameters were converted to DH and CPC to compare accuracy after calibration based on these parameterizations. The method of computing POE-CPC transformation as a solution to a certain optimization problem is proposed. The problem of identifying geometric parameters in the presence of restrictions is solved by gradient optimization methods. Experiments have been carried out on an ABB IRB 1600 industrial manipulator with an installed conductive probe and an ABB IRBP A-500 robotic positioner with a conductive metal flat surface. A technique for indirectly checking the accuracy of calibration of kinematic parameters is proposed based on a study of the accuracy of manipulation when using these parameters. A comparison is made of the manipulation accuracy when using four sets of parameters: nominal parameters obtained during factory calibration with the Leica AT901B laser tracker and two sets of parameters obtained by applying the proposed calibration method. The kinematic parameters obtained from the experiment determine more accurately the position of the manipulator TCP for part of the configuration working space, even for areas that were not used for calibration.

In this paper, the conditions of nonregular precession with a constant ratio of the velocities of precession and proper rotation for a gyrostat in the superposition of two homogeneous and one axisymmetric field are obtained. The case where the gyrostat has axial dynamical symmetry and the proper rotation axis coincides with the body’s symmetry axis is singled out. It is shown that in the case where the gyrostatic momentum is collinear to the symmetry axis, nonregular precession is possible with a precession velocity equal to, twice as large as, or twice as small as the proper rotation velocity. In each of these cases, the condition expressing the ratio of the axial and equatorial inertia moments of the body in terms of the nutation angle coincides with the corresponding condition obtained earlier for the nonregular precession of a solid in three homogeneous fields. In the particular case of the gyrostat’s spherical symmetry, when the precession speed is half or twice as large as its proper rotation speed, the cosine of the nutation angle is equal to one fourth; at equal speeds, the nutation angle should be equal to sixty degrees. The sets of admissible positions of the forces’ centers for the general case of nonorthogonal fields are found. The precession of a gyrostat whose gyrostatic momentum is deflected from the symmetry axis is considered. The possibility of nonregular precession is shown for the case where the precession velocity is twice as large as the proper rotation velocity. The solution is expressed in terms of elementary functions. The rotation of the gyrostat is either periodic or the rotation velocity tends to zero and the carrier body of the gyrostat approaches the equilibrium position.

This paper is concerned with a mechanical system consisting of a rigid body (outer body) placed on a horizontal rough plane and of an internal moving mass moving in a circle lying in a vertical plane, so that the radius vector of the point has a constant angular velocity. The interaction of the outer body and the horizontal plane is modeled by the Coulomb –Amonton law of dry friction with anisotropy (the friction coefficient depends on the direction of the body’s motion). The equation of the body’s motion is a differential equation with a discontinuous right-hand side. Based on the theory of A. F. Filippov, it is proved that, for this equation, the existence and right-hand uniqueness of the solution takes place, and that there exists a continuous dependence on initial conditions. Some general properties of the solutions are established and possible periodic regimes and their features are considered depending on the parameters of the problem. In particular, the existence of a periodic regime is proved in the case where the motion occurs without sticking of the outer body, and conditions for the existence of such a regime are shown. An analysis is made of the final dynamics of how the system reaches a periodic regime in the case where the outer body sticks twice within a period of revolution of the internal mass. This periodic regime exhibits sticking in the so-called upper and lower deceleration zones. The outer body comes twice to a stop and is at rest in these zones for some time and then continues its motion. This paper gives a complete description of the solution pattern for such motions. It is shown that, in the parameter space of the system where such a regime exists, the solution reaches this regime in finite time. All qualitatively different solutions in this case are described. In particular, special attention is devoted to the terminal motion, namely, to the solution of the system during the last period of motion of the internal mass before reaching the periodic regime. Existence regions of such solutions are found and the boundaries of the regions of initial conditions determining the qualitatively different dynamics of the system are established.

We investigate discrete-time dynamical systems generated by an infinite-dimensional nonlinear operator that maps the Banach space $l_1^{}$ to itself. It is demonstrated that this operator possesses up to seven fixed points. By leveraging the specific form of our operator, we illustrate that analyzing the operator can be simplified to a two-dimensional approach. Subsequently, we provide a detailed description of all fixed points, invariant sets for the two-dimensional operator and determine the set of limit points for its trajectories. These results are then applied to find the set of limit points for trajectories generated by the infinite-dimensional operator.

When an articulated drinking straw is slid over a rod vibrated by a motor, the straw moves up and down continuously. To clarify the mechanism for this motion, a straw was created that was narrower at the center than at the ends and the mechanism for the up-and-down motion as well as its reversal at the rod's tip was examined. The angle $\psi$ between the rod and the straw is constant because they are in contact at three points, i.e., at both ends and the center. Therefore, when the straw rotates around the rod, it will move unidirectionally according to the sign of $\psi$. This velocity is determined as a function of the straw half-length and the number of rotations of the nut to which the rod is attached. As the straw rises, the rod's tip enters the straw, and the ascent stops when $\psi$ reaches zero because the rate of ascent is proportional to $\psi$. The sign of $\psi$ then reverses to start the downward motion, and the straw returns. When it reaches a reflector disk, the sign of $\psi$ reverses again and the straw rises. Experiments were conducted to measure the velocities of three straws with different lengths, and the results showed that the theoretical velocities were greater than the experimental ones. The reason for this is assumed to be that although in theory the straw does not slip at the point of contact between its center and the rod, in reality it does slip. However, the theoretical and experimental velocities decreased in similar ways with increasing straw half-length, and the agreement between them was determined by the relative error. For half-lengths below $1.0$ cm, the average agreement was approx. $80\,\%$, and for all lengths it was approx. $73\,\%$. The agreement would have been even better if it had not been for the effects of slippage between the straw and the rod, the presence of nodes and antinodes of the rod vibration, and deformation in the central portion of straw. Considering these effects, the experimental values support the validity of the mechanistic considerations of the straw motion and the theoretically determined velocity.

In this work we focus on the chaotic diffusion in the phase space of a triaxial potential resembling an elliptical galaxy. The transport process is studied in two different action-like starting spaces in order to cope with circulating and noncirculating orbits. Estimates of the diffusion rate obtained by means of the variance approach are discussed in detail and their limitations are exposed. After revisiting the Shannon-entropy-based method from a conceptual point of view in the framework of simple arguments taken from the information theory, we apply it to measure changes in the unperturbed actions or integrals of motion of the system for different sets of small ensembles of random initial conditions. For such sets of ensembles, estimates of the Lyapunov times are also provided. The results show that, within the chaotic component of the phase space, the Lyapunov times are shorter than any physical time scale as the Hubble time, but the diffusion times are much larger than the latter. Thus, we conclude that stable chaos dominates the dynamics of realistic galactic models.