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This paper formulates the dynamical equations of mechanics subject to holonomic constraints in terms of the states and controls using a constrained version of the Lagrange-d’Alembert principle. Based on a discrete version of this principle, a structure preserving time-stepping scheme is derived. It is shown that this respect for the mechanical structure (such as a reliable computation of the energy and momentum budget, without numerical dissipation) is retained when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced time-stepping equations serve as nonlinear equality constraints for the minimisation of a given cost functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. The resulting discrete optimal control algorithm is shown to have excellent energy and momentum properties, which are illustrated by two specific examples, namely reorientation and repositioning of a rigid body subject to external forces and the reorientation of a rigid body with internal momentum wheels.Copyright © 2007 by ASME

An articulated body can propel and steer itself in a perfect fluid by changing its shape only. Our strategy for motion planning for the submerged body is based on finding the optimal shape changes that produce a desired net locomotion; that is, motion planning is formulated as a nonlinear optimization problem.

The effect of the distribution of mass in triatomic reaction dynamics is analyzed using the geometry of the associated internal space. Atomic masses are appropriately incorporated into internal coordinates as well as the associated non-Euclidean internal space metric tensor after a separation of the rotational degrees of freedom. Because of the non-Euclidean nature of the metric in the internal space, terms such as connection coefficients arise in the internal equations of motion, which act as velocity-dependent forces in a coordinate chart. By statistically averaging these terms, an effective force field is deduced, which accounts for the statistical tendency of geodesics in the internal space. This force field is shown to play a crucial role in determining mass-related branching ratios of isomerization and dissociation dynamics of a triatomic molecule. The methodology presented can be useful for qualitatively predicting branching ratios in general triatomic reactions, and may be applied to the study of isotope effects.

This paper studies the dynamics of a spacecraft moving in the field of a binary asteroid. The asteroid pair is modeled as a rigid body and a sphere moving in a plane, while the spacecraft moves in space under the influence of the gravitational field of the asteroid pair, as well as that of the sun. This simple model captures the coupling between rotational and translational dynamics. By assuming that the binary dynamics is in a relative equilibrium, a restricted model for the spacecraft in orbit about them is constructed that also includes the direct effect of the sun on the spacecraft dynamics. The standard restricted three-body problem (RTBP) is used as a starting point for the analysis of the spacecraft motion. We investigate how the triangular points of the RTBP are modified through perturbations by taking into account two perturbations, namely, that one of the primaries is no longer a point mass but is an extended rigid body, and second, taking into account the effect of orbiting the sun. The stable zones near the modified triangular equilibrium points of the binary and a normal form of the Hamiltonian around them are used to compute stable periodic and quasi-periodic orbits for the spacecraft, which enable it to observe the asteroid pair while the binary orbits around the sun. Nomenclature A () = mechanical connection as = distance from the sun to the center of masses of the binary G = universal constant of gravitation Gr () = terms of degree r in the expansion of the generating function H () = Hamiltonian function Hr () = terms of degree r in the expansion of the Hamiltonian function Izz = inertia tensor of the rigid body

This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal referential variations are equivalent to those resulting from vertical spatial variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R 3 . Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call configurational inviscidity. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the DoyleEricksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density. © 2006 American Institute of Physics. DOI: 10.1063/1.2190827

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

Matching techniques are developed for discrete mechanical systems with symmetry. We describe new phenomena that arise in the controlled Lagrangian approach for mechanical systems in the discrete context. In particular, one needs to either make an appropriate selection of momentum levels or introduce a new parameter into the controlled Lagrangian to complete the matching procedure. We also discuss digital and model predictive control.

As is well-known, the inviscid Burgers equation is prone to shock formation. A class of regularized Burgers equations is proposed that do not have an explicit artificial viscosity term. The steepening eect of the Burgers nonlinear convective term is mitigated by spatial averaging or low pass filtering of the convective velocity. Numerical simulations are performed to investigate the eect of this regularization. Specifically, the regularized Burgers equation is numerically shown to compare favorably with the viscous Burgers equation and to avoid spurious oscillations (unlike other conservative regularizations, such as the KdV equation). The numerics indicates that the width of the filter or the diameter of the spatial averaging domain controls the thickness of the regularized shock. While the governing equations of fluid flows (the Euler and Navier-Stokes equations) are well-known, their computation are still a challenge. There are two particular challenges associated with the computation and the understanding of solutions of these equations: one is turbulence modeling and the other is the formation of shocks and their regularization. Both challenges are attributed to the nonlinear convective term in the governing equations. In this article we will show that both diuculties can potentially be fixed by a particular class of averaging or low pass filtering. While the general case of the Euler or Navier-Stokes equations are not investigated here, we will show evidence of shock regularization in the Burgers equations. Lagrangian averaging is a technique for modeling the mean flow of incompressible turbulent flows. 8‐10 The Lagrangian Averaged Euler (LAE- ) equations for incompressible flow and their viscous counterpart are interesting from both the analytical and the numerical points of view. In these equations, is a spatial scale below which rapid fluctuations are smoothed by linear and nonlinear dispersion. The distinctive feature of the Lagrangian averaging approach is that averaging is carried out at the level of the variational principle and not at the level of the Euler or Navier-Stokes equations, which is the traditional averaging or filtering approach used for both the Reynolds averaged Navier-Stokes (RANS) and the large eddy simulation (LES) models. Therefore, the resulted LAE- equations possess conservation laws for energy and momentum, as well as Kelvin circulation theorem. The behavior of the LAE- solution approximates the behavior of Euler equations well to spatial scales of order , while truncating the energy spectrum for scales smaller than . This averaging or filtering is done without adding viscosity, but by a nonlinear dispersion from the large scales. The numerical simulations of the Lagrangian Averaged Navier-Stokes- (LANS- ) equations performed by Chen et al 4 and Mohseni et al 16 for isotropic homogenous turbulence demonstrated the good features of this model in reproducing large scales of turbulence. Recently, these techniques have been extended to wall bounded flows 18 and compressible barotropic flows. 2

The title of this paper is inspired by the work of Poincaré [1890, 1892], who introduced many key dynamical systems methods during his research on celestial mechanics and especially the three-body problem. Since then, many researchers have contributed to his legacy by developing and applying these methods to problems in celestial mechanics and, more recently, with the design of space missions. This paper will give a survey of some of these exciting ideas, and we would especially like to acknowledge the work of Michael Dellnitz, Frederic Gabern, Katalin Grubits, Oliver Junge, Wang-Sang Koon, François Lekien, Martin Lo, Sina Ober-Blöbaum, Kathrin Padberg, Robert Preis, and Bianca Thiere. One of the purposes of the AMS Current Events session is to discuss work of others. Even though we were involved in the research reported on here, this short paper is intended to survey many ideas due to our collaborators and others. This survey is by no means complete, and we apologize for not having time or space to do justice to many important and fundamental works. In fact, the results reported on here rely on and were inspired by important preceding work of many others in celestial mechanics, mission design and in dynamical systems. We mention just a few whose work had a positive influence on what is reported here: Brian Barden, Ed Belbruno, Robert Farquhar, Gerard Gómez, George Haller, Charles Jaffé, Kathleen Howell, Linda Petzold, Josep Masdemont, Vered Rom-Kedar, Radu Serban, Carles Simó, Turgay Uzer, Steve Wiggins, and Roby Wilson. In an upcoming monograph (see Koon, Lo, Marsden, and Ross [2005]), the dynamical systems and computational approach and its application to mission design are discussed in detail. One of the key ideas is that the competing gravitational pull between celestial bodies creates a vast array of passageways that wind around the Sun, planets and moons. The boundaries of these passageways are realized geometrically as invariant manifolds attached to equilibrium points and periodic orbits in interlinked three-body problems. In particular, tube-like structures form an interplanetary transport network which will facilitate the exploration of Mercury, the Moon, the asteroids, and the outer solar system, including a mission to assess the possibility of life on Jupiter’s icy moons. The use of these methods in problems in molecular dynamics of interest in chemistry is also briefly discussed.

The method of controlled Lagrangians for discrete mechanical systems is extended to include potential shaping in order to achieve complete state-space asymptotic stabilization. New terms in the controlled shape equation that are necessary for matching in the discrete context are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline. We also discuss digital and model predictive control., Comment: IEEE Conference on Decision and Control, 2006 6 pages, 4 figures

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 $J_2$ correction, as well as the double spherical pendulum. The $J_2$ 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 nontrivial 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 noncanonical nature of the symplectic structure., 24 pages, 7 figures, numerous minor improvements, references added, fixed typos

We employ set oriented methods in combination with graph partitioning algorithms to identify key dynamical regions in the Sun-Jupiter-particle three-body system. Transport rates from a region near the 32 Hilda resonance into the realm of orbits crossing Mars' orbit are computed. In contrast to common numerical approaches, our technique does not depend on single long term simulations of the underlying model. Thus, our statistical results are particularly reliable since they are not affected by a dynamical behavior which is almost nonergodic (i.e., dominated by strongly almost invariant sets).

This paper proves a discrete analogue of the Poincar´e lemma in the context of a discrete exterior calculus based on simplicial cochains. The proof requires the construction of a generalized cone operator, p : Ck(K) -> Ck+1(K), as the geometric cone of a simplex cannot, in general, be interpreted as a chain in the simplicial complex. The corresponding cocone operator H : Ck(K) -> Ck−1(K) can be shown to be a homotopy operator, and this yields the discrete Poincar´e lemma. The generalized cone operator is a combinatorial operator that can be constructed for any simplicial complex that can be grown by a process of local augmentation. In particular, regular triangulations and tetrahedralizations of R2 and R3 are presented, for which the discrete Poincar´e lemma is globally valid.

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.

The geometric mechanics of a pair of asteroids in orbit about each other under mutual gravitational attraction was analyzed using Lagrangian reduction. The technique rely on a specific choice of connection, whose curvatures leads to Coriolis effects, as well as critical for carrying out sharp stability analysis and a geometric phase analysis. The configuration space of a single asteroid was the Euclidean group. It was observed that the technique is essential in carrying out the dynamical system analysis of tube and lobe dynamics. Fil: Cendra, Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina Fil: Marsden, Jerrold, E.. California Institute of Technology; Estados Unidos

This paper studies the effect of uncertainty, using random perturbations, on area preserving maps of R/sub 2/ to itself. We focus on the standard map and a discrete Duffing oscillator as specific examples. We relate the level of uncertainty to the large-scale features in the dynamics in a precise way. We also study the effect of such perturbations on bifurcations in such maps. The main tools used for these investigations are a study of the eigenfunction and eigenvalue structure of the associated Perron-Frobenius operator along with set oriented methods for the numerical computations.

This paper is concerned with the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of ℝn or of an n-dimensional manifold M). It focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives, that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume-preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.

A dynamic procedure for the Lagrangian Averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$) equations is developed where the variation in the parameter $\alpha$ in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where $\alpha$ is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-$\alpha$ procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic energy decay are captured more accurately as compared with the LANS-$\alpha$ simulations using a fixed $\alpha$. In order to evaluate the applicability of the dynamic LANS-$\alpha$ model in anisotropic turbulence, a priori test of a turbulent channel flow is performed. It is found that the parameter $\alpha$ changes in the wall normal direction. Near a solid wall, the length scale $\alpha$ is seen to depend on the distance from the wall with a vanishing value at the wall. On the other hand, away from the wall, where the turbulence is more isotropic, $\alpha$ approaches an almost constant value. Furthermore, the behavior of the subgrid scale stresses in the near wall region is captured accurately by the dynamic LANS-$\alpha$ model. The dynamic LANS-$\alpha$ model has the potential to extend the applicability of the LANS-$\alpha$ equations to more complicated anisotropic flows., Comment: 17 pages, 17 figures

Techniques using gyroscopic forces and scalar potentials are used to create swarming behaviors for multiple agent systems. The methods result in collision avoidance between the agents as well as with obstacles.

The purpose of this paper is to describe the general setting for the application of techniques from geometric mechanics and dynamical systems to the problem of asteroid pairs. The paper also gives some preliminary results on transport calculations and the associated problem of calculating binary asteroid escape rates. The dynamics of an asteroid pair, consisting of two irregularly shaped asteroids interacting through their gravitational potential is an example of a full-body problem or FBP in which two or more extended bodies interact. One of the interesting features of the binary asteroid problem is that there is coupling between their translational and rotational degrees of freedom. General FBPs have a wide range of other interesting aspects as well, including the 6-DOF guidance, control, and dynamics of vehicles, the dynamics of interacting or ionizing molecules, the evolution of small body, planetary, or stellar systems, and almost any other problem in which distributed bodies interact with each other or with an external field. This paper focuses on the specific case of asteroid pairs using techniques that are generally applicable to many other FBPs. This particular full two-body problem (F2BP) concerns the dynamical evolution of two rigid bodies mutually interacting via a gravitational field. Motivation comes from planetary science, where these interactions play a key role in the evolution of asteroid rotation states and binary asteroid systems. The techniques that are applied to this problem fall into two main categories. The first is the use of geometric mechanics to obtain a description of the reduced phase space, which opens the door to a number of powerful techniques, such as the energy-momentum method for determining the stability of equilibria and the use of variational integrators for greater accuracy in simulation. Second, techniques from computational dynamic systems are used to determine phase space structures that are important for transport phenomena and dynamic evolution.

The paper demonstrates how discrete exterior calculus (DEC) tools may be useful in computer vision and graphics. A variational approach provides a link with mechanics. Our development of DEC includes discrete differential forms, discrete vector fields and the operators acting on these. This development of a discrete calculus, when combined with the methods of discrete mechanics and other recent work is likely to have promising applications in a field like computer vision which offers such a rich variety of challenging variational problems to be solved computationally. As a specific example we consider the problem of template matching and show how numerical methods derived from a discrete exterior calculus are starting to play an important role in solving the equations of averaged template matching. We also show some example applications using variational problems from computer graphics and mechanics to demonstrate that formulating the problem discretely and using discrete methods for solution can lead to efficient algorithms.

Increasingly accurate remote sensing techniques are available today, and methods such as modal analysis are used to transform, interpolate, and regularize the measured velocity fields. Until recently, the modes used did not incorporate flow across an open boundary of the domain. Open boundaries are an important concept when the domain is not completely closed by a shoreline. Previous modal analysis methods, such as those of Lipphardt et al. (2000), project the data onto closed-boundary modes, and then add a zero-order mode to simulate flow across the boundary. Chu et al. (2003) propose an alternative where the modes are constrained by a prescribed boundary condition. These methods require an a priori knowledge of the normal velocity at the open boundary. This flux must be extrapolated from the data or extracted from a numerical model of a larger-scale domain, increasing the complexity of the operation. In addition, such methods make it difficult to add a threshold on the length scale of open-boundary processes. Moreover, the boundary condition changes in time, and the computation of all or some modes must be done at each time step. Hence real-time applications, where robustness and efficiency are key factors, were hardly practical. We present an improved procedure in which we add scalable boundary modes to the set of eigenfunctions. The end result of open-boundary modal analysis (OMA) is a set of time and data independent eigenfunctions that can be used to interpolate, extrapolate and filter flows on an arbitrary domain with or without flow through segments of the boundary. The modes depend only on the geometry and do not change in time.

A dynamic procedure for the Lagrangian Averaged Navier-Stokes-α (LANS-α) equations is developed where the variation in the parameter α in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. In order to derive this model, the incompressible Navier-Stokes equations are Helmholtz-filtered at the grid and a test filter levels. A Germano type identity is derived by comparing the filtered subgrid scale stress terms with those given in the LANS-α equations. Assuming constant α in homogenous directions of the flow and averaging in these directions, results in a nonlinear equation for the parameter α, which determines the variation of α in the non-homogeneous directions or in time. Consequently, the parameter α is calculated during the simulation instead of a pre-defined value. As an initial test, the dynamic LANS-α model is used to compute isotropic homogenous forced and decaying turbulence, where α is constant over the computational domain, but is allowed to vary in time. The resulting simulations are compared with direct numerical simulations and with the LANS-α simulations using fixed value of α. As expected, α is found to change rapidly during the first eddy turn-over time during the simulations. It is also observed that by using the dynamic LANS-α procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic energy decay are captured more accurately as compared with the LANS-α simulations using a fixed α. The current results suggest some promising applications of this dynamic LANS-α model, such as to a spatially varying turbulent flow, which we hope to undertake in future research.

This paper describes a procedure for incorporating artificial gyroscopic forces and physical dissipation in the method of controlled Lagrangians. Energy-conserving gyroscopic forces provide additional freedom to expand the basin of stability and tune closed-loop system performance. We also study the effect of physical dissipation on the closed-loop dynamics and discuss conditions for stability in the presence of natural damping. We apply the technique to the inverted pendulum on a cart,a case study from previous papers. We develop a controller that asymptotically stabilizes the inverted equilibrium at a specific cart position for the conservative dynamic model. The region of attraction contains all states for which the pendulum is elevated above the horizontal plane. We also develop conditions for asymptotic stability in the presence of linear damping.

The long-term goal of this research is to enable a cooperating group of underwater gliders to perform efficiently and robustly as an autonomous, adaptive sampling network in a three-dimensional, dynamic ocean environment. This will involve developing the mathematical infrastructure and design tools for effecting feedback-controlled, schooling-like network behavior in an unsteady flow field while exploiting the natural dynamics of the ocean.

The method of controlled Lagrangians is a technique for stabilizing underactuated mechanical systems which involves modifying a system's energy and dynamic structure through feedback. These modifications can obscure the effect of physical dissipation in the closed-loop. For example, generic damping can destabilize an equilibrium which is closed-loop stable for a conservative system model. We consider the effect of damping on Euler-Poincare (special reduced Lagrangian) systems which have been stabilized about an equilibrium using the method of controlled Lagrangians. We describe a choice of feedback dissipation which asymptotically stabilizes a sub-class of controlled Euler-Poincare systems subject to physical damping. As an example, we consider intermediate axis rotation of a damped rigid body with a single internal rotor.

The invariant manifold structures of the collinear libration points particular, the stable and unstable invariant manifold "tubes" associated to for the spatial restricted three-body problem provide the framework for understanding complex dynamical phenomena from a geometric point of view. In libration point periodic orbits are phase space structures that provide a conduit for orbits between the primary bodies in separate three-body systems. These invariant manifold tubes can be used to construct new spacecraft trajectories, such as a "Petit Grand Tour" of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. The current work extends the results to the spatial case.

We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system.A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm

We extend the method of controlled Lagrangians to include potential shaping for complete state-space stabilization of mechanical systems. The method of controlled Lagrangians deals with mechanical systems with symmetry and provides symmetry-preserving kinetic shaping and feedback-controlled dissipation for state-space stabilization in all but the symmetry variables. Potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables. The approach also extends the method of controlled Lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline.

In this paper we discuss the stabilization of a nonholonomic system consisting of a unicycle with rider. We show in particular that one can achieve stability of slow steady vertical motions by imposing a feedback control force on the rider's limb.

Obtains feedback stabilization of an inverted pendulum on a rotor arm by the “method of controlled Lagrangians”. This approach involves modifying the Lagrangian for the uncontrolled system so that the Euler-Lagrange equations derived from the modified or “controlled” Lagrangian describe the closed-loop system. For the closed-loop equations to be consistent with available control inputs, the modifications to the Lagrangian must satisfy “matching” conditions. The pendulum on a rotor arm requires an interesting generalization of our earlier approach which was used for systems such as a pendulum on a cart.

Matching techniques are applied to the problem of stabilization of uniformly accelerated motions of mechanical systems with symmetry. The theory is illustrated with a simple model-a wheel and pendulum system.

The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. The derivation involves an averaging procedure that combines ideas from both the material (Lagrangian) and spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I. Taylor's "frozen turbulence" hypothesis as the foundation for the model equations; more precisely, the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow. In this article, we use this hypothesis to derive the averaged Lagrangian for the theory, and all the terms up to and including order α^2 are accounted for. The equations come in both an isotropic and anisotropic version. The anisotropic equations are a coupled system of PDEs (partial differential equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works by Foias, Holm & Titi [10], and ourselves [16], an analysis of the isotropic equations has been given. In the second part of this paper, we establish local in time well-posedness of the anisotropic LANS-α equations using quasilinear PDE type methods.

This paper develops the foundations of the multisymplectic formulation of nonsmooth continuum mechanics. It may be regarded as a PDE generalization of previous techniques that developed a variational approach to collision problems. These methods have already proved of value in computational mechanics, particularly in the development of asynchronous integrators and efficient collision methods. The present formulation also includes solid—pfluid interactions and material interfaces and, in addition, lays the groundwork for a treatment of shocks.

We propose an algorithmic approach to stabilization of Lagrangian systems. The first step involves making admissible modifications to the Lagrangian for the uncontrolled system, thereby constructing what we call the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system where new terms are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. The procedure is demonstrated for the problem of stabilization of an inverted pendulum on a cart and for the problem of stabilization of rotation of a rigid spacecraft about its unstable intermediate axis using a single internal rotor. Similar results hold for the dynamics of an underwater vehicle.

We consider the problem of estimating the time-varying root of a time-dependent nonlinear map. We introduce a "dynamic inverse" of a map, another generally time-dependent map which one composes with the original map to form a nonlinear vector-field. The flow of this vector field decays exponentially to the root. We then show how a dynamic inverse may be determined dynamically while being used simultaneously to find a root. We construct a continuous-time analog computational paradigm around the dynamic inverse.

We present a controller for a class of robotics manipulators which provides exponential convergence to a desired end-effector trajectory using gains specified in joint-space. This is accomplished without appeal to the use of discrete inverse-kinematics algorithms, allowing the controller to be posed entirely in continuous time.

In this paper we consider the Hamiltonian formulation of the equations of incompressible ideal fluid flow from the point of view of optimal control theory. The equations are compared to the finite symmetric rigid body equations analyzed earlier by the authors. We discuss various aspects of the Hamiltonian structure of the Euler equations and show in particular that the optimal control approach leads to a standard formulation of the Euler equations – the so-called impulse equations in their Lagrangian form. We discuss various other aspects of the Euler equations from a pedagogical point of view. We show that the Hamiltonian in the maximum principle is given by the pairing of the Eulerian impulse density with the velocity. We provide a comparative discussion of the flow equations in their Eulerian and Lagrangian form and describe how these forms occur naturally in the context of optimal control. We demonstrate that the extremal equations corresponding to the optimal control problem for the flow have a natural canonical symplectic structure.

In this paper we extend the previous work on the asymptotic stabilization of pure Euler-Poincaré mechanical systems using controlled Lagrangians to the study of asymptotic stabilization of Euler-Poincaré mechanical systems such as the heavy top.

In this paper we extend the matching technique to a class of nonholonomic systems with symmetries. Assuming that the momentum equation defines an integrable distribution, we introduce a family of reduced systems. The method of controlled Lagrangians is then applied to these systems resulting in a smooth stabilizing controller.

In this paper, we introduce a new method of model reduction for nonlinear control systems. Our approach is to construct an approximately balanced realization. The method requires only standard matrix computations, and we show that when it is applied to linear systems it results in the usual balanced truncation. For nonlinear systems, the method makes use of data from either simulation or experiment to identify the dynamics relevant to the input}output map of the system. An important feature of this approach is that the resulting reduced-order model is nonlinear, and has inputs and outputs suitable for control. We perform an example reduction for a nonlinear mechanical system.

This paper analyzes stabilization of a nonholonomic system consisting of a unicycle with rider. It is shown that one can achieve stability of slow steady vertical motions by imposing a feedback control force on the rider’s limb.

Transition states in phase space are identified and shown to regulate the rate of escape of asteroids temporarily captured in circumplanetary orbits. The transition states, similar to those occurring in chemical reaction dynamics, are then used to develop a statistical semianalytical theory for the rate of escape of asteroids temporarily captured by Mars. Theory and numerical simulations are found to agree to better than 1%. These calculations suggest that further development of transition state theory in celestial mechanics, as an alternative to large-scale numerical simulations, will be a fruitful approach to mass transport calculations.

We present a study of the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory specific to this problem. The construction of Lyapunov functions is based on the fact that a non-degenerate Keplerian orbit is uniquely described by its angular momentum and Laplace (- Runge-Lenz) vectors. We suggest a Lyapunov function, which gives a feedback controller such that the target elliptic orbit becomes a locally asymptotically stable periodic orbit in the closed-loop dynamics. We show how to perform a global transfer between two arbitrary elliptic orbits based on the local transfer result. Finally, a second Lyapunov function is presented that works only for circular target orbits.

We develop discrete mechanics and variational integrators for a class of degenerate Lagrangian systems, and apply these integrators to a system of point vortices. Excellent numerical behavior is observed. A longer term goal is to use these integration methods in the context of control of mechanical systems, such as coordinated groups of underwater vehicles. In fact, numerical evidence given in related problems, such as those in Kane et al. (2000) shows that in the presence of external forces, these methods give superior predictions of energy behavior.

The 1:1 resonance for perturbed Hamiltonian systems with small dissipative and energy injection terms has been studied. These perturbations of the 1:1 resonance exhibit dissipation induced instabilities. This mechanism allows one to show that an optical cavity with small pumping is unstable when one takes into account the dissipative effects. The Maxwell-Bloch equations are the asymptotic normal form that describe this instability when energy is injected through forcing at zero frequency. A simple mechanical system close to the 1:1 resonance has been displayed, which is a mechanical analog of the laser.

One of the significant challenges to successful formation flight of spacecraft is maintenance of the formation, i.e., control of the motion of the individual spacecraft to maintain the overall formation. This includes both stabilization of a given formation and reconfiguring of the formation-While the dynamics and control a single spacecraft; is well understood, a formation of spacecraft effectively acts a deformable body due to control forces which restore it to its desired formation, As a deformable body, the formation is capable of exhibit complex dynamic behavior. Effective control strategies must exploit this behavior as well as the natural dynamics of the system to achieve goals such as formation error minimization and minimal fuel consumption during formation reconfiguration. An additional concern is the impact a decentralized control structure would have control algorithm design and formation controllability. Spacecraft dynamics are mechanical, meaning they admit a Lagrangian or Hamiltonian formulation. We have investigated the dynamics and control of formation flight by exploiting the mechanical structure of the dynamical systems in conjunction with proven methods of linearization and structured uncertainty. Our work built on previous analytical tools developed at Caltech, such as the energy-momentum method for assessing the stability of a mechanical system, as well as methodologies for control of mechanical systems. One important property of mechanical systems is the ability of small changes in the internal shape of the system to effect global motion of the system.