Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction

In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems. Lagrangian reduction can do in one step what one can alternatively do by applying the Pontryagin Maximum Principle followed by an application of Poisson reduction. The idea of using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and Crouch [1995a,b] in a somewhat different context and the general idea is closely related to those in Montgomery [1990] and Vershik and Gershkovich [1994]. Here we develop this idea further and apply it to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). However, one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of Bloch, Krishnaprasad, Marsden and Murray [1995]. The snakeboard is used to illustrate the method

The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems

This paper compares the Hamiltonian approach to systems with nonholonomic constraints (see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993], van der Schaft and Maschke [1994] and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski [1996] and Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in the approaches and each has its own advantages; some structures have been discovered on one side and their analogues on the other side are interesting to clarify. For example, the momentum equation and the reconstruction equation were first found on the Lagrangian side and are useful for the control theory of these systems, while the failure of the reduced two form to be closed (i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on the Hamiltonian side. Clarifying the relation between these approaches is important for the future development of the control theory and stability and bifurcation theory for such systems. In addition to this work, we treat, in this unified framework, a simplified model of the bicycle (see Getz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimum phase) control system

The geometric structure of nonholonomic mechanics

Many important problems in multibody dynamics, the dynamics of wheeled vehicles and motion generation, involve nonholonomic mechanics. Many of these systems have symmetry, such as the group of Euclidean motions in the plane or in space and this symmetry plays an important role in the theory. Despite considerable advances on both Hamiltonian and Lagrangian sides of the theory, there remains much to do. We report on progress on two of these fronts. The first is a Poisson description of the equations that is equivalent to those given by Lagrangian reduction, and second, a deeper understanding of holonomy for such systems. These results promise to lead to further progress on the stability issues and on locomotion generatio

Poisson reduction for nonholonomic mechanical systems with symmetry

This paper continues the work of Koon and Marsden [1997b] that began the comparison of the Hamiltonian and Lagrangian formulations of nonholonomic systems. Because of the necessary replacement of conservation laws with the momentum equation, it is natural to let the value of momentum be a variable and for this reason it is natural to take a Poisson viewpoint. Some of this theory has been started in van der Schaft and Maschke [1994]. We build on their work, further develop the theory of nonholonomic Poisson reduction, and tie this theory to other work in the area. We use this reduction procedure to organize nonholonomic dynamics into a reconstruction equation, a nonholonomic momentum equation and the reduced Lagrange d’Alembert equations in Hamiltonian form. We also show that these equations are equivalent to those given by the Lagrangian reduction methods of Bloch, Krishnaprasad, Marsden and Murray [1996]. Because of the results of Koon and Marsden [1997b], this is also equivalent to the results of Bates and Sniatycki [1993], obtained by nonholonomic symplectic reduction. Two interesting complications make this effort especially interesting. First of all, as we have mentioned, symmetry need not lead to conservation laws but rather to a momentum equation. Second, the natural Poisson bracket fails to satisfy the Jacobi identity. In fact, the so-called Jacobiizer (the cyclic sum that vanishes when the Jacobi identity holds), or equivalently, the Schouten bracket, is an interesting expression involving the curvature of the underlying distribution describing the nonholonomic constraints. The Poisson reduction results in this paper are important for the future development of the stability theory for nonholonomic mechanical systems with symmetry, as begun by Zenkov, Bloch and Marsden [1997]. In particular, they should be useful for the development of the powerful block diagonalization properties of the energy-momentum method developed by Simo, Lewis and Marsden [1991]

Intramolecular energy transfer and the driving mechanisms for large-amplitude collective motions of clusters

This paper uncovers novel and specific dynamical mechanisms that initiate large-amplitude collective motions in polyatomic molecules. These mechanisms are understood in terms of intramolecular energy transfer between modes and driving forces. Structural transition dynamics of a six-atom cluster between a symmetric and an elongated isomer is highlighted as an illustrative example of what is a general message. First, we introduce a general method of hyperspherical mode analysis to analyze the energy transfer among internal modes of polyatomic molecules. In this method, the (3n−6) internal modes of an n-atom molecule are classified generally into three coarse level gyration-radius modes, three fine level twisting modes, and (3n−12) fine level shearing modes. We show that a large amount of kinetic energy flows into the gyration-radius modes when the cluster undergoes structural transitions by changing its mass distribution. Based on this fact, we construct a reactive mode as a linear combination of the three gyration-radius modes. It is shown that before the reactive mode acquires a large amount of kinetic energy, activation or inactivation of the twisting modes, depending on the geometry of the isomer, plays crucial roles for the onset of a structural transition. Specifically, in a symmetric isomer with a spherical mass distribution, activation of specific twisting modes drives the structural transition into an elongated isomer by inducing a strong internal centrifugal force, which has the effect of elongating the mass distribution of the system. On the other hand, in an elongated isomer, inactivation of specific twisting modes initiates the structural transition into a symmetric isomer with lower potential energy by suppressing the elongation effect of the internal centrifugal force and making the effects of the potential force dominant. This driving mechanism for reactions as well as the present method of hyperspherical mode analysis should be widely applicable to molecular reactions in which a system changes its overall mass distribution in a significant way

The Genesis Trajectory and Heteroclinic Cycles

Genesis will be NASA's first robotic sample return mission. The purpose of this mission is to collect solar wind samples for two years in an L_1 halo orbit and return them to the Utah Test and Training Range (UTTR) for mid-air retrieval by helicopters. To do this, the Genesis spacecraft makes an excursion into the region around L_2 . This transfer between L_1 and L_2 requires no deterministic maneuvers and is provided by the existence of heteroclinic cycles defined below. The Genesis trajectory was designed with the knowledge of the conjectured existence of these heteroclinic cycles. We now have provided the first systematic, semi-analytic construction of such cycles. The heteroclinic cycle provides several interesting applications for future missions. First, it provides a rapid low-energy dynamical channel between L_1 and L_2 such as used by the Genesis Discovery Mission. Second, it provides a dynamical mechanism for the temporary capture of objects around a planet without propulsion. Third, interactions with the Moon. Here we speak of the interactions of the Sun-Earth Lagrange point dynamics with the Earth-Moon Lagrange point dynamics. We motivate the discussion using Jupiter comet orbits as examples. By studying the natural dynamics of the Solar System, we enhance current and future space mission design

J_2 Dynamics and Formation Flight

We study the dynamics of the relative motion of satellites in the gravitational field of the Earth, including the effects of the bulge of the Earth (the J_2 effect). Using Routh reduction and dynamical systems ideas, a method is found that locates orbits such that a cluster of satellites remains close with very little dispersing, even with no controls

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena

A Nonequilibrium Rate Formula for Collective Motions of Complex Molecular Systems

We propose a compact reaction rate formula that accounts for a non‐equilibrium distribution of residence times of complex molecules, based on a detailed study of the coarse‐grained phase space of a reaction coordinate. We take the structural transition dynamics of a six‐atom Morse cluster between two isomers as a prototype of multi‐dimensional molecular reactions. Residence time distribution of one of the isomers shows an exponential decay, while that of the other isomer deviates largely from the exponential form and has multiple peaks. Our rate formula explains such equilibrium and non‐equilibrium distributions of residence times in terms of the rates of diffusions of energy and the phase of the oscillations of the reaction coordinate. Rapid diffusions of energy and the phase generally give rise to the exponential decay of residence time distribution, while slow diffusions give rise to a non‐exponential decay with multiple peaks. We finally make a conjecture about a general relationship between the rates of the diffusions and the symmetry of molecular mass distributions