# Dynamics and Control of Multibody Systems

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*J. E. Marsden; P. S. Krishnaprasad; J. C. Simo*

The study of complex, interconnected mechanical systems with rigid and flexible articulated components is of growing interest to both engineers and
mathematicians. Recent work in this area reveals a rich geometry underlying
the mathematical models used in this context. In particular, Lie groups of
symmetries, reduction, and Poisson structures play a significant role in
explicating the qualitative properties of multibody systems. In engineering
applications, it is important to exploit the special structures of mechanical
systems. For example, certain mechanical problems involving control of
interconnected rigid bodies can be formulated as Lie-Poisson systems. The
dynamics and control of robotic, aeronautic, and space structures involve
difficulties in modeling, mathematical analysis, and numerical implementation.
For example, a new generation of spacecraft with large, flexible components are
presenting new challenges to the accurate modeling and prediction of the dynamic behavior of such structures. Recent developments in Hamiltonian dynamics and
coupling of systems with symmetries has shed new light on some of these issues,
while engineering questions have suggested new mathematical structures.

These kinds of considerations motivated the organization of the AMS-IMS-SIAM
Joint Summer Research Conference on Control Theory and Multibody Systems, held
at Bowdoin College in August, 1988. This volume contains the proceedings of
that conference. The papers presented here cover a range of topics, all of
which could be viewed as applications of geometrical methods to problems
arising in dynamics and control. The volume contains contributions from some
of the top researchers and provides an excellent overview of the frontiers of
research in this burgeoning area.

# Table of Contents

## Dynamics and Control of Multibody Systems

- Contents xi12 free
- Foreword xiii14 free
- Introduction xv16 free
- An enumerative theory of equilibrium rotations for planar kinematic chains 120 free
- Stability and stiffening of driven and free planar rotating beams 1130
- Characterization of Hamiltonian input-output systems 2746
- Robustness of distributed parameter systems 4766
- On the relationship between discrete-time optimal control and recursive dynamics for elastic multibody chains 6180
- Some solvable stochastic control problems in compact symmetric spaces of rank one 7998
- Slew-induced deformation shaping on slow integral manifolds 97116
- Feedback equivalence and symmetries of Brunowski normal forms 115134
- The application of total energy as a Lyapunov function for mechanical control systems 131150
- Classical adiabatic angles for slowly moving mechanical systems 159178
- Eulerian many-body problems 187206
- Morse theory for a model space structure 209228
- A unified approach for the control of multifingered robot hands 217236
- Tethered satellite system stability 241260
- Quantum control theory I 269288
- Cartan-Hannay-Berry phases and symmetry 279298
- Block diagonalization and the energy-momentum method 297316
- The dynamics of two coupled rigid bodies in three space 315334
- Nonsmooth optimization algorithms for the design of controlled flexible structures 337356
- Stability analysis of a rigid body with attached geometrically nonlinear rod by the energy-momentum method 371390
- Controllability of Poisson control systems with symmetries 399418
- Chaos in a rapidly forced pendulum equation 411430
- Accurate time critical control of many body systems 421440
- Hamiltonian control systems: decomposition and clamped dynamics 441460
- Graph-theoretical methods in multibody dynamics 459478