Stability of Equilibria for the $\mathfrak{so}(4)$ Free Rigid Body

The stability for all generic equilibria of the Lie–Poisson dynamics of the \mathfrakso(4)\mathfrak{so}(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of \mathfrakso(n)\mathfrak{so}(n) are equilibrium points for the rigid body dynamics. In the case of \mathfrakso(4)\mathfrak{so}(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in \mathfrakso(3)\mathfrak{so}(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.     

Regularization of the Amended Potential and the Bifurcation of Relative Equilibria

In the context of simple mechanical systems with symmetry, we give a method based on blowing up the amended potential for obtaining symmetry-breaking branches of relative equilibria bifurcating from a given set of symmetric relative equilibria. The general method is illustrated with two concrete mechanical examples, the double spherical pendulum and the symmetric coupled rigid bodies.     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Dynamical systems with variable dissipation: approaches, methods, and applications

This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can arouse the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under streamline flow-around conditions. The author obtains a full spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases of integrability, each of the first integrals is expressed through a finite combination of elementary functions and is a transcendental function of its variables, simultaneously. In this case, the transcendence is meant in the complex analysis sense; i.e., after the continuation of the functions considered to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling limit sets in the system considered (for example, attracting and repelling foci). The author obtains new families of phase portraits of systems with variable dissipation on lowerand higher-dimensional manifolds. He discusses the problems of their absolute or relative roughness, He discovers new integrable cases of rigid body motion, including those in the classical problem of motion of a spherical pendulum placed in an over-running medium flow.     

A stochastic differential equation from friction mechanics

The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented. To cite this article: F. Bernardin et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Stabilization of the non-trivial relative equilibria of a gyrostat with an elastic element in a circular orbit

The motion of a gyrostat, regarded as a rigid body, in a circular Kepler orbit in a central Newtonian force field is investigated in a limited formulation. A uniformly rotating statically and dynamically balanced flywheel is situated in the rigid body. A uniform elastic element, which, during the motion of the system, is subjected to small deformations, is rigidly connected to the rigid body-gyrostat body. The problem is discretized without truncating the corresponding infinite series, based on a modal analysis or using a certain specified system of functions, for example, of the assumed forms of the oscillations, which depend on the spatial coordinates and which satisfy appropriate boundary-value problems of the linear theory of elasticity. The elastic element is specified in more detail (a rod, plate, etc.), as well as its mass and stiffness characteristics and the form of the fastening, and the choice of the system of functions is determined. Non-trivial relative equilibria of the system (the state of rest with respect to an orbital system of coordinates when the elastic element is deformed) is sought approximately on the basis of a converging iteration method, described previously. It is shown, using Routh's theorem, that by an appropriate choice of the gyrostatic moment and when certain conditions, imposed on the system parameters are satisfied, one can stabilize these equilibria (ensure that they are stable).     

Coupling 3D and 1D Skeletal Muscle Models

This work introduces a modelling framework towards a forward dynamics simulation of skeletal muscle mechanics that couples three-dimensional (3D) continuum-mechanical-based Finite Element (FE) simulations to rigid body simulations. In this regard, this is a methodological approach, which incorporates different methods to realise simulations of the musculoskeletal system. Such simulations are at present computationally not feasible. To set up such a modelling framework the upper limp is selected. Here, the upper limb consists of an antagonistic muscle pair, the elbow (a simple hinge joint) and an external load. The skeletal muscles are represented by a 3D continuum-mechanical model. The tendons are, for now, assumed to be rigid. The results demonstrate the ability of the system to converge to a physiological realistic position. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Poisson variations in the problem of the stability of equilibria in rigid body mechanics

The existence and stability of equilibria in rigid body mechanics is considered. A class of variations is indicated which satisfy the analogue of Poisson's equations, suitable for use when investigating both the sufficient and necessary conditions for the stability of such equilibria and which, in particular, enable the invariance of the equations of motion of the system and their first integrals to be effectively used when the phase variables and parameters of the problem are interchanged. The result is illustrated using the example of the problem of the motion of a gyrostat far from attracting objects.     

Constant-sign Lyapunov functionals in the problem of the stability of a functional differential equation

The stability of the zero solution of a non-autonomous functional differential equation of the delayed type is investigated by means of limiting equations and a constant-sign Lyapunov functional, which has a constant-sign derivative. Special cases when the Lyapunov functional and its derivative are explicitly independent of time and the case of an almost periodic equation are also considered. The problem of stabilizing a pendulum in the upper unstable position and the problem of stabilizing the rotational motion of a rigid body are solved as examples.     

Hybrid control for tracking of invariant manifolds

This paper presents a hybrid control method that controls to unstable equilibria of nonlinear systems by taking advantage of systems’ free dynamics. The approach uses a stable manifold tracking objective in a computationally efficient, optimization-based switching control design. Resulting nonlinear controllers are closed-loop and can be computed in real-time. Our method is validated for the cart–pendulum and the pendubot inversion problems. Results show the proposed approach conserves control effort compared to tracking the desired equilibrium directly. Moreover, the method avoids parameter tuning and reduces sensitivity to initial conditions. The resulting feedback map for the cart–pendulum has a switching structure similar to existing energy based swing-up strategies. We use the Lyapunov function from these prior works to numerically verify local stability for our feedback map. However, unlike the energy based swing-up strategies, our approach does not rely on pre-derived, system-specific switching controllers. We use hybrid optimization to automate switching control synthesis on-line for nonlinear systems.     

Planar motion and stability for a rigid body with a beam in a field of central gravitational force

Planar motion for a rigid body with an elastic beam in a field of central gravitational force was investigated, and both of the orbital motion and attitude motion were under consideration. The equations of motion of the system were derived by the variational principle, and on view point of generalized Hamiltonian dynamics, the sufficient conditions for the stability of one class of relative equilibria were given by the energymomentum method.     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.     

KAM dynamics and stabilization of a particle sliding over a periodically driven curve

The dynamics of a bead sliding without friction along a periodically pulsating wire is under consideration. If the arc length of the wire is taken as the relevant coordinate, the motion of the bead is described by a periodic newtonian equation. Sufficient conditions are derived assuring that a given equilibrium is of twist type, a property that implies its nonlinear stability as well as a KAM scenario around it. Special attention is paid to the stabilization of unstable equilibria, in parallel with the stabilization of the inverted pendulum.     

Integrable variational equations of non-integrable systems

Paper is devoted to the solvability analysis of variational equations obtained by linearization of the Euler-Poisson equations for the symmetric rigid body with a fixed point on the equatorial plain. In this case Euler-Poisson equations have two pendulum like particular solutions. Symmetric heavy top is integrable only in four famous cases. In this paper is shown that a family of cases can be distinguished such that Euler-Poisson equations are not integrable but variational equations along particular solutions are solvable. The connection of this result with analysis made in XIX century by R. Liouville is also discussed.     

Bifurcation of slow motions in a stiff spring pendulum

We consider free oscillations of a double pendulum, where one of the pendula is modelled as a very stiff spring. Contrary to a single spring pendulum numerical simulations show an unexpected large influence of the fast longitudinal oscillations on the slow pendulum oscillations even for extremely large values of the stiffness. The transition from the regular motion, which is governed by the dynamics of a rigid double pendulum close to a periodic orbit, to the irregular motion with large contributions from the longitudinal oscillations occurs due to a subcritical symmetry breaking bifurcation of the periodic solution. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relative equilibria in the Bjerknes problem

We consider a motion of a rigid body of an arbitrary shape in a vibrating irrotational flow. A sufficient condition is established for the existence of relative equilibria of the body, i.e., of equilibria of the averaged system.     

Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics

We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikov’s method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.     

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.