Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration

Abstract

SUMMARYIn the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unitlength constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential-algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non-singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion-based energy-momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion-based variational integrator, a director-based energy-momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations.

On the Numerical Influences of Inertia Representation for Rigid Body Dynamics

“…In the (real) matrix representation [16], unit quaternion can be thought of as a 4-dimensional vector…”

“…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.…”

Rigid Body Time Integration by Convected Base Vectors With Implicit Constraints

“…with V (q) = 0, are considered by application to free rotation of a rigid body. The moment of inertia tensor with respect to the center of mass is chosen as J = diag[13, 5, 10], which is equivalent to a box with side lengths [1,3,2] and mass 12. The motion is initiated by the initial angular velocity ω 0 = [0, 0.05, 10] T leading to non-trivial rotation in which the body is reversed at regular intervals, see e.g.…”

“…In contrast to earlier methods based on asymptotic properties, the conservative algorithms depend in an essential way on the parameter representation of the problem. 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.…”