Lie group variational integrators for the full body problem

Abstract

We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative frame is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies.

Lagrangian mechanics and variational integrators on two‐spheres

“…Lie group methods are numerical integrators that preserve the Lie group structure of the configuration space [7]. Recently, these two approaches have been unified to obtain Lie group variational integrators that preserve the geometric properties of the dynamics as well as the Lie group structure of the configuration manifold [10].…”

“…However, it is not guaranteed that these methods preserve the geometric properties of the dynamics. In this paper, we focus on a Lagrangian mechanical system evolving on the homogeneous manifold, (S 2 ) n by extending the method of Lie group variational integrators [11,10]. The resulting integrator preserves the dynamic characteristics and the homogeneous manifold structure concurrently.…”

Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

“…The Lagrangian of the string pendulum is derived, and the corresponding action integral is defined. Due to the unique dynamic characteristics of the string pendulum, the variation of the action integral should be carefully developed: (i) since the unstretched length of the deployed portion of the string is not fixed, when deriving the variation of the corresponding part of the action integral, we need to apply Green's theorem; (ii) since the attitude of the rigid body is represented in the special orthogonal group, the variation of rotation matrices are carefully expressed by using the exponential map [18,19]; (iii) since the portion of the string on the guide way and the drum is inextensible, the velocity of the string is not continuous at the guide way entrance. To take account of the effect of this velocity discontinuity, an additional modification term, referred to as a Carnot energy loss term is incorporated [15].…”

“…When numerically simulating such systems, it is critical to preserve both the symplectic property of Hamiltonian flows and the Lie group structure for numerical accuracy and efficiency [17]. A geometric numerical integrator, referred to as a Lie group variational integrator, has been developed for a Hamiltonian system on an arbitrary Lie group and it has been applied to several multibody systems ranging from binary asteroids to articulated rigid bodies and magnetic systems in [18,19]. This paper develops a Lie group variational integrator for the proposed string pendulum model.…”

Geometric structure-preserving optimal control of a rigid body

“…This is achieved by considering the discrete Lagrange-d'Alembert variational principle [12] in combination with essential ideas from Lie group methods [10], which yields a Lie group variational integrator [17]. This integrator explicitly preserves the Lie group structure of the configuration space, and is similar to the integrators introduced in [14] for a rigid body in an external field, and in [15] for full body dynamics. These discrete equations are then imposed as constraints to be satisfied by the extremal solutions to the discrete optimal control problem, and we obtain the discrete extremal solutions in terms of the given terminal states.…”