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Rigid body dynamics with a scalable body, quaternions and perfect constraints

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Abstract

In this paper, we present a formulation of the quaternion constraint for rigid body rotations in the form of a standard perfect bilateral mechanical constraint, for which the associated Lagrangian multiplier has the meaning of a constraint force. First, the equations of motion of a scalable body are derived. A scalable body has three translational, three rotational, and one uniform scaling degree of freedom. As generalized coordinates, an unconstrained quaternion and a displacement vector are used. To the scalable body, a perfect bilateral constraint is added, restricting the quaternion to unit length and making the body rigid. This way a quaternion based differential algebraic equation (DAE) formulation for the dynamics of a rigid body is obtained, where the 7×7 mass matrix is regular and the unit length restriction of the quaternion is enforced by a mechanical constraint. Finally, the equations of motion in the form of a DAE are linked to the Newton–Euler equations of motion of a rigid body. The rigid body DAE formulation is useful for the construction of (energy) consistent integrators.

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Authors and Affiliations

  1. Institute of Mechanical Systems, Center of Mechanics, Department of Mechanical and Process Engineering, ETH Zurich, 8092, Zurich, Switzerland

    Michael Möller & Christoph Glocker

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  1. Michael Möller
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  2. Christoph Glocker
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Correspondence to Michael Möller.

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Möller, M., Glocker, C. Rigid body dynamics with a scalable body, quaternions and perfect constraints. Multibody Syst Dyn 27, 437–454 (2012). https://doi.org/10.1007/s11044-011-9276-5

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  • DOI: https://doi.org/10.1007/s11044-011-9276-5

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