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# Volume 21, Number 7-8, 2016 Nonlinear Dynamics & Mobile Robotics

 Ardentov A. A. Controlling of a Mobile Robot with a Trailer and Its Nilpotent Approximation Abstract This work studies a number of approaches to solving the motion planning problem for a mobile robot with a trailer. Different control models of car-like robots are considered from the differential-geometric point of view. The same models can also be used for controlling a mobile robot with a trailer. However, in cases where the position of the trailer is of importance, i.e., when it is moving backward, a more complex approach should be applied. At the end of the article, such an approach, based on recent works in sub-Riemannian geometry, is described. It is applied to the problem of reparking a trailer and implemented in the algorithm for parking a mobile robot with a trailer. Keywords: mobile robot, trailer, motion planning, sub-Riemannian geometry, nilpotent approximation Citation: Ardentov A. A., Controlling of a Mobile Robot with a Trailer and Its Nilpotent Approximation, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 775-791 DOI:10.1134/S1560354716070017
 Borisov A. V.,  Kuznetsov S. P. Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts Abstract For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations. Keywords: Chaplygin sleigh, nonholonomic mechanics, attractor, chaos, bifurcation Citation: Borisov A. V.,  Kuznetsov S. P., Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 792-803 DOI:10.1134/S1560354716070029
 Ivanov A. P. On Final Motions of a Chaplygin Ball on a Rough Plane Abstract A heavy balanced nonhomogeneous ball moving on a rough horizontal plane is considered. The classical model of a “marble” body means a single point of contact, where sliding is impossible. We suggest that the contact forces be described by Coulomb’s law and show that in the final motion there is no sliding. Another, relatively new, contact model is the “rubber” ball: there is no sliding and no spinning. We treat this situation by applying a local Coulomb law within a small contact area. It is proved that the final motion of a ball with such friction is the motion of the “rubber” ball. Keywords: Coulomb friction, Chaplygin ball, asymptotic dynamics Citation: Ivanov A. P., On Final Motions of a Chaplygin Ball on a Rough Plane, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 804-810 DOI:10.1134/S1560354716070030
 Kashchenko S. A. The Dynamics of Second-order Equations with Delayed Feedback and a Large Coefficient of Delayed Control Abstract The dynamics of second-order equations with nonlinear delayed feedback and a large coefficient of a delayed equation is investigated using asymptotic methods. Based on special methods of quasi-normal forms, a new construction is elaborated for obtaining the main terms of asymptotic expansions of asymptotic residual solutions. It is shown that the dynamical properties of the above equations are determined mostly by the behavior of the solutions of some special families of parabolic boundary value problems. A comparative analysis of the dynamics of equations with the delayed feedback of three types is carried out. Keywords: delay, feedback, nonlinear dynamics, boundary value problems Citation: Kashchenko S. A., The Dynamics of Second-order Equations with Delayed Feedback and a Large Coefficient of Delayed Control, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 811-820 DOI:10.1134/S1560354716070042
 Kozlov V. V. On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature Abstract This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure). Keywords: Lobachevsky plane, Fuchsian group, orbifold, Noether integrals Citation: Kozlov V. V., On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 821-831 DOI:10.1134/S1560354716070054
 Mashtakov A. P.,  Ardentov A. A.,  Sachkov Y. L. Relation Between Euler’s Elasticae and Sub-Riemannian Geodesics on $SE(2)$ Abstract In this note we describe a relation between Euler’s elasticae and sub-Riemannian geodesics on $SE(2)$. Analyzing the Hamiltonian system of the Pontryagin maximum principle, we show that these two curves coincide only in the case when they are segments of a straight line. Keywords: elastica, sub-Riemannian geodesic, group of rototranslations Citation: Mashtakov A. P.,  Ardentov A. A.,  Sachkov Y. L., Relation Between Euler’s Elasticae and Sub-Riemannian Geodesics on $SE(2)$, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 832-839 DOI:10.1134/S1560354716070066
 Pankratova E. V.,  Kalyakulina A. I. Environmentally Induced Amplitude Death and Firing Provocation in Large-scale Networks of Neuronal Systems Abstract We study the dynamics of multielement neuronal systems taking into account both the direct interaction between the cells via linear coupling and nondiffusive cell-to-cell communication via common environment. For the cells exhibiting individual bursting behavior, we have revealed the dependence of the network activity on its scale. Particularly, we show that small-scale networks demonstrate the inability to maintain complicated oscillations: for a small number of elements in an ensemble, the phenomenon of amplitude death is observed. The existence of threshold network scales and mechanisms causing firing in artificial and real multielement neural networks, as well as their significance for biological applications, are discussed. Keywords: network, dynamic environment coupling, viscosity, action potentials firing, amplitude death, stability Citation: Pankratova E. V.,  Kalyakulina A. I., Environmentally Induced Amplitude Death and Firing Provocation in Large-scale Networks of Neuronal Systems, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 840-848 DOI:10.1134/S1560354716070078
 Smirnov L. A.,  Kryukov A. K.,  Osipov G. V.,  Kurths J. Bistability of Rotational Modes in a System of Coupled Pendulums Abstract The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an in-phase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the in-phase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable in-phase cycle, but also two out-of phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the out-ofphase limit regimes are discussed as well. Keywords: coupled elements, bifurcation, multistability Citation: Smirnov L. A.,  Kryukov A. K.,  Osipov G. V.,  Kurths J., Bistability of Rotational Modes in a System of Coupled Pendulums, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 849-861 DOI:10.1134/S156035471607008X
 Anastassiou  S. Dynamical Systems on the Liouville Plane and the Related Strictly Contact Systems Abstract We study vector fields of the plane preserving the Liouville form. We present their local models up to the natural equivalence relation and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given according to a relation stricter than contact equivalence. In addition, we discuss their relation with strictly contact vector fields in dimension three. Analogous results for diffeomorphisms are also given. Keywords: systems preserving the Liouville form, strictly contact systems, classification, bifurcations Citation: Anastassiou  S., Dynamical Systems on the Liouville Plane and the Related Strictly Contact Systems, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 862-873 DOI:10.1134/S1560354716070091
 Vetchanin E. V.,  Kilin A. A.,  Mamaev I. S. Control of the Motion of a Helical Body in a Fluid Using Rotors Abstract This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on $SE(3)$ are obtained. Keywords: ideal fluid, motion of a helical body, Kirchhoff equations, control of rotors, gaits, optimal control Citation: Vetchanin E. V.,  Kilin A. A.,  Mamaev I. S., Control of the Motion of a Helical Body in a Fluid Using Rotors, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 874-884 DOI:10.1134/S1560354716070108
 Borisov A. V.,  Kazakov A. O.,  Pivovarova E. N. Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top Abstract This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of period-doubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic. Keywords: Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact Citation: Borisov A. V.,  Kazakov A. O.,  Pivovarova E. N., Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901 DOI:10.1134/S156035471607011X
 Quillen A. C.,  Askari H.,  Kelley D. H.,  Friedmann T.,  Oakes P. W. A Coin Vibrational Motor Swimming at Low Reynolds Number Abstract Low-cost coin vibrational motors, used in haptic feedback, exhibit rotational internal motion inside a rigid case. Because the motor case motion exhibits rotational symmetry, when placed into a fluid such as glycerin, the motor does not swim even though its vibrations induce steady streaming in the fluid. However, a piece of rubber foam stuck to the curved case and giving the motor neutral buoyancy also breaks the rotational symmetry allowing it to swim. We measured a 1 cm diameter coin vibrational motor swimming in glycerin at a speed of a body length in 3 seconds or at 3 mm/s. The swim speed puts the vibrational motor in a low Reynolds number regime similar to bacterial motility, but because of the vibration it is not analogous to biological organisms. Rather the swimming vibrational motor may inspire small inexpensive robotic swimmers that are robust as they contain no external moving parts. A time dependent Stokes equation planar sheet model suggests that the swim speed depends on a steady streaming velocity $V_{stream} \sim Re_s^{1/2} U_0$ where $U_0$ is the velocity of surface vibrations, and streaming Reynolds number $Re_s = U_0^2/(\omega \nu)$ for angular vibrational frequency $\omega$ and fluid kinematic viscosity $\nu$. Keywords: swimming models, hydrodynamics, nonstationary 3-D Stokes equation, bio-inspired micro-swimming devices Citation: Quillen A. C.,  Askari H.,  Kelley D. H.,  Friedmann T.,  Oakes P. W., A Coin Vibrational Motor Swimming at Low Reynolds Number, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 902-917 DOI:10.1134/S1560354716070121
 Karavaev Y. L.,  Kilin A. A.,  Klekovkin A. V. Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot Abstract In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid. Keywords: screwless underwater robot, experimental investigations, helical body Citation: Karavaev Y. L.,  Kilin A. A.,  Klekovkin A. V., Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 918-926 DOI:10.1134/S1560354716070133
 Klenov A. I.,  Kilin A. A. Influence of Vortex Structures on the Controlled Motion of an Above-water Screwless Robot Abstract This paper is devoted to an experimental investigation of the motion of a rigid body set in motion by rotating two unbalanced internal masses. The results of experiments confirming the possibility of motion by this method are presented. The dependence of the parameters of motion on the rotational velocity of internal masses is analyzed. The velocity field of the fluid around the moving body is examined. Keywords: self-propulsion, PIV, vortex formation, above-water screwless robot Citation: Klenov A. I.,  Kilin A. A., Influence of Vortex Structures on the Controlled Motion of an Above-water Screwless Robot, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 927-938 DOI:10.1134/S1560354716070145
 Borisov A. V.,  Kazakov A. O.,  Sataev I. R. Spiral Chaos in the Nonholonomic Model of a Chaplygin Top Abstract This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn about the influence of “strangeness” of the attractor on the motion pattern of the top. Keywords: nonholonomic constraint, spiral chaos, discrete spiral attractor Citation: Borisov A. V.,  Kazakov A. O.,  Sataev I. R., Spiral Chaos in the Nonholonomic Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954 DOI:10.1134/S1560354716070157

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