Double Spherical Pendulum Research Articles

Adaptive Anti-Swing Control for 7-DOF Overhead Crane With Double Spherical Pendulum and Varying Cable Length

Adaptive Anti-Swing Control for 7-DOF Overhead Crane With Double Spherical Pendulum and Varying Cable Length

Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

<abstract><p>This paper develops the theory of discrete Dirac reduction of discrete Lagrange–Dirac systems with an abelian symmetry group acting on the configuration space. We begin with the linear theory and, then, we extend it to the nonlinear setting using retraction compatible charts. We consider the reduction of both the discrete Dirac structure and the discrete Lagrange–Pontryagin principle, and show that they both lead to the same discrete Lagrange–Poincaré–Dirac equations. The coordinatization of the discrete reduced spaces relies on the notion of discrete connections on principal bundles. At last, we demonstrate the method obtained by applying it to a charged particle in a magnetic field, and to the double spherical pendulum.</p></abstract>

Load-swing suppression control using an omnidirectionally movable multi-rotor in 2D-horizontal plane

In this paper, a method for controlling the swing of loads is proposed by applying forces to a suspended load directly, instead of controlling a suspending machine itself, such as a helicopter and a crane. A special multi-rotor capable of moving in the level surface is developed to control the swing of loads. First, to create a model for controlling the swing of loads, a drone model is used to model a multi-rotor that can generate forces in (x, y)-directions and also produce a torque for yaw-rotation, and has four special coaxial rotors, while a double spherical pendulum is used as a rope model which hangs the multi-rotor. Moreover, in what is called a control allocation problem that determines the number of revolutions of each rotor, assume that the number of revolutions for each rotor consists of one for generating two forces in translational motion, and of one for generating one torque in rotational motion. Then, the number of revolutions for plane movement is first decided at the 1st step, and the number of revolutions for a torque required for yaw motion is next decided from the difference between the torque generated by the translational forces decided at the 1st step and the target torque. Finally, the validity of the proposed load-swing suppression control is verified with simulations.

Geometric modeling of the unfolding of a rod structure in the form of a double spherical pendulum in weightlessness

We investigated the geometric model of the new technique for unfolding a rod structure, similar to the double spherical pendulum, in weightlessness. Displacements of elements occur due to the pulses from pyrotechnic jet engines acting on the endpoints of links. The motion of the obtained inertial unfolding of a rod structure was described using a Lagrange equation of the second kind. Given the conditions of weightlessness, it was built applying only the kinetic energy of the system. The relevance of the chosen subject is emphasized by the need to choose and study the process of activation of the unfolding of a spatial rod structure. The proposed possible drivers are the pulse pyrotechnic jet engines installed at endpoints of the structure's links. They are lighter and cheaper compared, for example, to electric motors or spring devices. In addition, they are more efficient economically when the process of unfolding a structure in orbit is planned to be performed only once. We propose a technique for determining the parameters and initial conditions for initiating the oscillations of a double rod structure in order to obtain a cyclic trajectory of the endpoint of the second link. That makes it possible to avoid, when calculating the process of transformation, the chaotic movements of the structure's elements. We built the time-dependent charts of change in the functions of generalized coordinates, as well as the first and second derivatives from these functions. Therefore, there is a possibility to estimate the force characteristics of the system at the moment of braking (locking) the process of unfolding. The results are intended for the geometric modeling of one of the variants for unfolding the large-sized structures under conditions of weightlessness, for example, force frames for solar mirrors or space antennas, as well as other large-scale orbital infrastructures.

Cotangent bundle reduction and Poincaré–Birkhoff normal forms

In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré–Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincaré–Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.

Relative Critical Points

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic, Poisson, or variational - generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems - the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids - and generalizations of these systems.

On the topology of the double spherical pendulum

This paper studies the topology of the constant energy surfaces of the double spherical pendulum.

Discrete Routh reduction

This paper develops the theory of Abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with Abelian symmetry. The reduction of variational Runge–Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical J2 correction, as well as the double spherical pendulum. The J2 problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a non-trivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the non-canonical nature of the symplectic structure.

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.

Are Hamiltonian flows geodesic flows?

When a Hamiltonian system has a Kinetic + Potential structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure. We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the N-body problem. We show that the flow of the reduced planar N-body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.

Stabilization of relative equilibria

Discusses the problem of obtaining feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We show how to stabilize an internally unstable relative equilibrium using internal actuators. The methodology is that of potential shaping, but the system is allowed to be underactuated, i.e., have fewer actuators than the dimension of the shape space. The theory is illustrated with the problem of stabilization of the cowboy relative equilibrium of the double spherical pendulum.

Mechanical integrators derived from a discrete variational principle

Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct time-stepping algorithms that approximate the flow of continuous ODEs for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We first present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of Noether's theorem. We then use the discrete mechanics formulation to develop a procedure for constructing symplectic-momentum mechanical integrators for Lagrangian systems with holonomic constraints. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.

Equivariant constrained symplectic integration

We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for Lie-Poisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum.

Pattern evocation and geometric phases in mechanical systems with symmetry

This paper is concerned with the relation between the dynamics of a given Hamiltonian system with a given symmetry group and its reduced dynamics. We illustrate the process of visualization of reduced orbits using the double spherical pendulum. In this process of visualization, one sees certain patterns when the dynamics is viewed relative to rotating frames with certain critical angular velocities. By using the reduced dynamics, we also explain these patterns. We show that if the motion on the phase space reduced by a continuous symmetry group at a given momentum level is periodic, then there is a uniformly rotating frame, that is, a one parameter group motion, relative to which the unreduced trajectory is periodic with the same period. If the continuous symmetry group of the system is abelian, which corresponds to the system having cyclic variables, we derive an explicit expression for the required angular velocity in terms of the dynamic phase (an average of the mechanical connection) and the geometric phase (the holonomy of the mechanical connection). We show that one can also find such a frame if the reduced orbit is quasiperiodic and a KAM condition is satisfied. The almost periodic case is also discussed. An important aspect of this procedure is how to use it in the presence of discrete symmetries. We show that, under appropriate conditions, the visualized orbit has, relative to a suitable uniformly rotating frame, the same temporal behavior and discrete symmetries as the reduced orbit. Since these spatio-temporal patterns are not apparent with respect to most frames, we call the phenomenon pattern evocation.

Lagrangian reduction and the double spherical pendulum

This paper studies the stability and bifurcations of the relative equilibrium of the double spherical pendulum, which has the circle as its symmetry group. The example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group.