Conservative integration of rigid body motion by quaternion parameters with implicit constraints

Abstract

SUMMARY An angular momentum and energy‐conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four‐component conjugate momentum vector via Hamilton's equations. The introduction of an extended mass matrix leads to a symmetric set of eight state‐space equations of motion. The extra inertial parameter serves as a multiplier on the kinematic constraint, and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments. External loads enter these equations via the set of momentum equations. Initially, the normalization of the quaternion array is introduced via a Lagrange multiplier. However, this Lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential, and elimination of the Lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient. An algorithm is developed by forming a finite increment of the Hamiltonian. This procedure identifies the proper selection of increments and mean values, and leads to an algorithm with conservation of momentum and energy. Implementation, conservation properties, and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top. Copyright © 2012 John Wiley & Sons, Ltd.

Rigid Body Time Integration by Convected Base Vectors With Implicit Constraints

“…This is similar to the result in [3] when eliminating the single scalar normalization constraint from the four-component quaternion representation of rigid-body rotation.…”

“…A fully conservative algorithm in terms of quaternion parameters can be obtained when the normalization condition is carried through the integration process via a Lagrange multiplier [2]. It was demonstrated in [3] that the rigid body dynamics problem can be formulated in such a way that the increment of the constraint is embedded in the kinematic evolution equation, and the Lagrange multiplier can be eliminated, leading to the introduction of a projection operator on the force potential gradient. An alternative formulation of the rigid body motion in terms of a set of convected base vectors has been introduced in [4].…”

“…In the present paper this problem is solved by extending the idea of 'implicit constraints' introduced in [3] to the formulation in terms of convected base vectors. The equations of motion are obtained from Hamilton's equations.…”

On the Numerical Influences of Inertia Representation for Rigid Body Dynamics

“…The top is represented as a cone with dimensions equivalent to those used in [16][17][18]. As illustrated in Figure 2, the parameters are height .…”

“…[1]. The initial conditions correspond to those used in [16][17][18] Recently, regular precession top is detailed discussed by Krenk and Nielsen, in their researches about energy-momentum conserving integrations of rigid body dynamics [17,18]. They observed that the numerical integrations are of significant nutation error in simulation of regular precession, whether the integrations are implemented in terms of unit quaternion or convected base vectors.…”