Free Rigid Body Research Articles

Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

If the three moments of inertia are distinct, the solution to the Euler equations for the free rigid body is given in terms of Jacobi elliptic functions. Using the arithmetic–geometric mean algorithm (Abramowitz and Stegun 1992 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover)), these functions can be calculated efficiently and accurately. Compared to standard numerical ODE and Lie–Poisson solvers, the overall approach yields a faster and more accurate numerical solution to the Euler equations. This approach is designed for mass asymmetric rigid bodies. In the case of symmetric bodies, the exact solution is available in terms of trigonometric functions, see Dullweber et al (1997 J. Chem. Phys. 107 5840–51), Reich (1996 Fields Inst. Commun. 10 181–91) and Benettin et al (2001 SIAM J. Sci. Comp. 23 1189–203) for details. In this paper, we consider the case of asymmetric rigid bodies subject to external forces. We consider a strategy similar to the symplectic splitting method proposed in Reich (1996 Fields Inst. Commun. 10 181–91) and Dullweber et al (1997 J. Chem. Phys. 107 5840–51). The method proposed here is time-symmetric. We decompose the vector field of our problem into a free rigid body (FRB) problem and another completely integrable vector field. The FRB problem consists of the Euler equations and a differential equation for the 3 × 3 orientation matrix. The Euler equations are integrated exactly while the matrix equation is approximated using a truncated Magnus series. In our experiments, we observe that the overall numerical solution benefits greatly from the very accurate solution of the Euler equations. We apply the method to the heavy top and the simulation of artificial satellite attitude dynamics.

Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket

Quadratic Hamiltonians with a linear Lie-Poisson bracket have a number of applications in mechanics. For example, the Lie-Poisson bracket e(3) includes the Euler-Poinsot model describing motion of a rigid body around a fixed point under gravity and the Kirchhoff model describes the motion of a rigid body in ideal fluid. Among the applications with a Lie-Poisson bracket so(4) and so(3, 1) is the description of free rigid body motion in a space of constant curvature. Advances in computer algebra algorithms, in implementations and hardware, together allow the computation of Hamiltonians with higher degree first integrals providing new results in the classic search for integrable models. A computer algebra module enabling related computations in a 3-dimensional vector formalism is described.

A symmetric splitting method for rigid body dynamics

It has been known since the time of Jacobi that the solution to the free rigid body Euler equations of motion is given in terms of a certain type of elliptic functions. Using the Arithmetic-Geometric mean algorithm, Abramowitz & Stegun (1992), these functions can be calculated efficiently and accurately. The overall approach yields a faster and more accurate numerical solution to the Euler equations compared to standard numerical ODE and symplectic solvers. In this paper we investigate the possibility of extending this approach to the case of rigid bodies subject to external forces. By using a splitting strategy similar to the one proposed in Reich (1996), we decompose the vector field of our problem in a free rigid body (FRB) problem and another completely integrable vector field. We apply the method to the simulation of the heavy top.

ACTIVE VIBRATION CONTROL FOR COUPLING SYSTEM OF FLEXIBLE STRUCTURES AND RIGID BODY

Spacecraft is a kind of complex coupling rigid and flexible system composed of free fly center rigid body connected with flexible appendages(such as sun plate,space robots etc.)by flexible joint,the problem of vibration will be caused when slewing or external disturbance excitation,which will affect the stability and the pointing accuracy of the system. Based on modal theory,the dynamics model of the coupling rigid and flexible structure including flexible link and flexible joint is built up and written in state space form,and controlla- bility is analyzed.The combining PPF (Positive position feed- back) and PD (Proportional dervative) control law is presented to suppress the vibration of the flexible structure,and the sta- bility and control performance are analyzed theoretically.The experimental setup of the coupling rigid and flexible structure is designed and built up,and the active vibration control re- searches are developed including set point vibration control and slewing vibration control.The experimental comparative research results verify that the presented methods can suppress the vibration quickly.

Invariant sets in the Clebsch–Tisserand problem: Existence and stability

Questions of the existence and stability of invariant sets in the problem of the motion of a rigid body with a fixed point in an axi-symmetric force field with a quadratic potential (with respect to the direction cosines of the axis of symmetry of the field) are discussed. This problem is isomorphic with the problem of the motion of a free rigid body bounded by a simply connected surface in an ideal homogeneous incompressible fluid which performs irrotational motion and is at rest at infinity. [Kirchhoff F G.R. Über die Bewegung eines Rotationskörpers in eine Flüssigkeit, J Reine und angew Math, B. 71, 1870, S.237–S.262; Clebsch A. Über die Bewegung eines Köipers in eine Flüssigkeit, Math Anmnalen, Bd. 3, 1870, S.238–S.262] In particular, in the second case of the Clebsch integrability of a problem of the motion of a body in a fluid, this problem is isomorphic with the Tisserand problem on the motion of a body with a fixed point in the case of a quadratic potential of special form,[Tisserand M.F. Sur le mouvement des planètes autour du Soleil d’apreès la lol èlectrodynamique de Weber, C.R. Acad. Sci. Paris, 1872. V. 75, P. 760–763] which corresponds to the satellite approximation of the potential of the forces of Newtonian attraction.

Symplectic structure of poisson system

When the Poisson matrix of Poisson system is non-constant, classical symplectic methods, such as symplectic Runge-Kutta method, generating function method, cannot preserve the Poisson structure. The non-constant Poisson structure was transformed into the symplectic structure by the nonlinear transform. Arbitrary order symplectic method was applied to the transformed Poisson system. The Euler equation of the free rigid body problem was transformed into the symplectic structure and computed by the mid-point scheme. Numerical results show the effectiveness of the nonlinear transform.

Wave impact on elastically supported horizontal deck

Laboratory experiments supported by theoretical investigations were applied to study the wave impact on a horizontal deck, deck vibrations, and phenomena related to wave-induced vibrations. The analysis shows that one can distinguish four characteristic stages of deck vibrations between two consecutive wave impacts. In the first stage, the wave impact induces very high-frequency waves in the deck structure and in the wave flume. There are important components with acoustic frequencies. In the second stage, the deck vibrates in contact with water. The frequencies of vibrations in water are much lower than the corresponding frequencies of the free vibrations in air. The third stage of wave-induced vibrations is a transitional one. At the beginning of the third stage, the deck vibrates in contact with water and at the end of this stage it vibrates in air as a free rigid body. In this process water is shed from the deck. Sometimes the process of shedding is violent, which induces high-frequency vibrations with relatively large amplitudes. The fourth stage of wave-induced vibrations corresponds to the free vibrations of the deck in air. Similar four stages can be distinguished in the pressure records. Additional information regarding wave impact on the deck and the phenomena related to the impact can be obtained by the analysis of mechanical energy. The analysis of mechanical energy is especially helpful in understanding dissipation processes and phenomena in the transitional stage of wave-induced vibrations. In particular, the analysis of energy explains why the amplitudes of acceleration are increasing in the transitional stage of wave-induced vibrations.

The Discrete Moser?Veselov Algorithm for the Free Rigid Body, Revisited

In this paper we revisit the Moser–Veselov description of the free rigid body in body coordinates, which, in the 3 × 3 case, can be implemented as an explicit, second-order, integrable approximation of the continuous solution. By backward error analysis, we study the modified vector field which is integrated exactly by the discrete algorithm. We deduce that the discrete Moser–Veselov (DMV) is well approximated to higher order by time reparametrizations of the continuous equations (modified vector field). We use the modified vector field to scale the initial data of the DMV to improve the order of the approximation and show the equivalence of the DMV and the RATTLE algorithm. Numerical integration with these preprocessed initial data is several orders of magnitude more accurate than the original DMV and RATTLE approach.

Optimal Control of the Sphere Sn Rolling on En

This paper investigates the optimal control problem concerning the sphere S n rolling without slipping on n-dimensional Euclidean space E n , n≥2. The differential equations governing the behaviour of the sphere constitute a sub-Riemannian distribution on the Lie group G=ℝ n ×SO n +1. Minimizing over the lengths of paths traced by the point of contact of the sphere yields an optimal control problem which is exploited using Noether’s Theorem to derive a family of integrals of motion for the system. This family is then employed to prove that all optimizing trajectories are projections of normal extremals. The Lax form of the extremal equations gives rise to an additional family of integrals which is reminiscent of the Manakov integrals of motion for the free rigid body problem. Both families are required to show complete integrability if n=4.

Relative equilibria for the generalized rigid body

This paper gives necessary and sufficient conditions for the ( n-dimensional) generalized free rigid body to be in a state of relative equilibrium. The conditions generalize those for the case of the three-dimensional free rigid body, namely that the body is in relative equilibrium if and only if its angular velocity and angular momentum align, that is, if the body rotates about one of its principal axes. For the n-dimensional rigid body in the Manakov formulation, these conditions have a similar interpretation. We use this result to state and prove a generalized Saari’s Conjecture (usually stated for the N-body problem) for the special case of the generalized rigid body.

Analysis of Fully Reversed Sequences of Rotations of a Free Rigid Body

A fully reversed (FR) sequence of rotations is defined as a series of rotations of a free rigid body about its body-fixed axes such that the rotation about each axis is fully reversed at the end of the sequence. Due to the non-commutative property of finite rigid body rotations, an FR sequence can effect non-zero changes in orientation of the rigid body even though the net rotation about each axis is zero. The FR sequences are useful for attitude maneuvers of miniature spacecraft that use elastic deformation-based microactuators and for other airborne or neutrally buoyant underwater vehicles where actuations effecting orientation change are restricted. This paper considers the kinematics of six-rotation FR sequences and discusses if each of them is capable of producing any desired change in orientation. It is concluded that out of a total of 24 six-rotation sequences, 12 are capable of this but not the remaining 12. The results are proved using mathematical formalism and are also interpreted using numerical computations and graphical visualization.

Preserving poisson structure and orthogonality in numerical integration of differential equations

We consider the numerical integration of two types of systems of differential equations. We first consider Hamiltonian systems of differential equations with a Poisson structure. We show that symplectic Runge-Kutta methods preserve this structure when the Poisson tensor is constant. Using nonlinear changes of coordinates this structure can also be preserved for nonconstant Poisson tensors, as exemplified on the Euler equations for the free rigid body. We also consider orthogonal flows and the closely related class of isospectral flows. To numerically preserve the orthogonality property we take the approach of formulating an equivalent system of differential-algebraic equations (DAEs) and of integrating the system with a special combination of a particular class of Runge-Kutta methods. This approach requires only matrix-matrix products and can preserve geometric properties of the flow such as reversibility.

Control of Rotation and Swing of a Rigid Body Suspended Load Using Inertia Rotors.

This paper describes rotational control and swing suppression of a crane suspended load model. A rotational free rigid body suspended by a single rope is controlled using three inertia rotors. Fiber optical gyros installed on the suspended load are used to measure the load orientation angles. Controlling angles and angular velocities are obtained from measured data using the conversion between zxy and zxz Euler angles and the driving system of the inertia rotors are developed based on the coupling dynamics. Experiments and simulations show that the rotational orientation is controlled without problems, but swing control is possible under some limit conditions of the load mass size and the rope length, since the gravity force act on the swing motion.

Geometric integration on spheres and some interesting applications

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ODE) and a discretization of micromagnetics (a dissipative PDE). We emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms; this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions.

Control of Micro-Satellite Orientation Using Bounded-Input, Fully-Reversed MEMS Actuators

We present a novel technique for controlling the attitude of micro-satellites using MEMS-based microactuators. In addition to being restricted in the magnitudes of inputs, most microactuators need to be returned to their original inactive state at the end of the attitude control maneuver in order to reduce energy consumption. This type of bounded and fully-reversed actuation on a free rigid body leads to a special type of system in which nonholonomic effects have been incorporated by design . We discuss issues of controllability and develop motion planning algorithms for such a system considering the nonintegrable nature of rigid body rotations. We use approximate methods to develop two algorithms for controlling satellite orientation. The first method uses a series of pair-wise actuation steps and is guaranteed to converge to the desired orientation; the second uses fewer steps, and in certain cases uses less input power. We also present preliminary results of a practical implementation using electro-thermal microactuation.

Reconstruction phases for Hamiltonian systems on cotangent bundles

Reconstruction phases describe the motions experienced by dynamical systems whose symmetry-reduced variables are undergoing periodic motion. A well known example is the non-trivial rotation experienced by a free rigid body after one period of oscillation of the body angular momentum vector. Here reconstruction phases are derived for a general class of Hamiltonians on a cotangent bundle {\mathrm T}^*Q possessing a group of symmetries G , and in particular for mechanical systems. These results are presented as a synthesis of the known special cases Q=G and G Abelian, which are reviewed in detail.

On the dynamics and Lyapunov stability of constrained and embedded rigid bodies

Lyapunov stability of constrained and embedded rigid bodies is considered. The constraints are of the equality type where the desired motion is to take place on an a priori defined submanifold of movement. Special and augmented state spaces for the representation of systems of rigid bodies are presented. A systematic method of stabilizing these augmented systems and a procedure for constructing Lyapunov functions are presented. The representation is applicable to augmented as well as reduced state spaces of the system defined by the constraints. The augmented state space results from the embedding of the free rigid body system in the larger state space of free rigid body and position control states, and in which the Lyapunov function is constructed. The reduced state space results when the system is restricted and is reduced to lie on the submanifold of movement. It is shown that, for the class of rigid bodies and the physical constraints considered, the projected feedback structures, and the reduced Lyapunov function constitute appropriate stabilizing structures for the constrained system. It is shown that the method applies equally to holonomically constrained and visco-elastically coupled rigid bodies. Digital computer simulations of one single rigid body system are presented to demonstrate the feasibility and effectiveness of the method. Applications to natural systems and the role of cartilage, ligaments and muscles in maintaining the integrity and stability of the joints are noted.

Dynamics of the n-dimensional Suslov problem

In this paper we study the dynamics of a constrained generalized rigid body (the Suslov problem) on its full phase space. We use reconstruction theory to analyze the qualitative dynamics of the system and discuss differences with the free rigid body motion. Use is made of the so-called quasi-periodic Floquet theory.

LIE–TROTTER FORMULA AND POISSON DYNAMICS

We construct via the Lie–Trotter formula some explicit Poisson integrators for the Maxwell–Bloch equations from laser-matter dynamics, the Euler equations of the free rigid body and the equations of the rigid body with a spinning rotor.

Generalized Serret-Andoyer transformation and applications for the controlled rigid body

The Serret-Andoyer transformation is a classical method for reducing the free rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimensional Hamiltonian flow. First, we show that this transformation is the computation, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein) reduction associated with the lifted left-action of SO(3) on T*SO(3)—a generalization and extension of Noether's theorem for Hamiltonian systems with symmetry. In fact, we go on to generalize the Serret-Andoyer transformation to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperregular Hamiltonian functions. Interpretations of the Serret-Andoyer variables, both as Eulerian coordinates and as canonical coordinates of the co-adjoint orbit, are given. Next, we apply the result obtained to the controlled rigid body with momentum wheels. For the class of Hamiltonian controls that preserve the symmetry on T*SO(3), the closed-loop motion of the main body can again be reduced to canonical form. This simplifies the stability proof for relative equilibria , which then amounts to verifying the classical Lagrange-Dirichlet criterion. Additionally, issues regarding numerical integration of closed-loop dynamics are also discussed. Part of this work has been presented in LumBloch:97a.