The significance of the configuration space Lie group for the constraint satisfaction in numerical time integration of multibody systems

Abstract

The dynamics simulation of multibody systems (MBS) using spatial velocities (non-holonomic velocities) requires time integration of the dynamics equations together with the kinematic reconstruction equations (relating time derivatives of configuration variables to rigid body velocities). The latter are specific to the geometry of the rigid body motion underlying a particular formulation, and thus to the used configuration space (c-space). The proper c-space of a rigid body is the Lie group SE (3), and the geometry is that of the screw motions. The rigid bodies within a MBS are further subjected to geometric constraints, often due to lower kinematic pairs that define SE (3) subgroups. Traditionally, however, in MBS dynamics the translations and rotations are parameterized independently, which implies the use of the direct product group SO (3)×ℝ3 as rigid body c-space, although this does not account for rigid body motions. Hence, its appropriateness was recently put into perspective.In this paper the significance of the c-space for the constraint satisfaction in numerical time stepping schemes is analyzed for holonomically constrained MBS modeled with the ‘absolute coordinate’ approach, i.e. using the Newton–Euler equations for the individual bodies subjected to geometric constraints. The numerical problem is considered from the kinematic perspective. It is shown that the geometric constraints a body is subjected to are exactly satisfied if they constrain the motion to a subgroup of its c-space. Since only the SE (3) subgroups have a practical significance it is regarded as the appropriate c-space for the constrained rigid body. Consequently the constraints imposed by lower pair joints are exactly satisfied if the joint connects a body to the ground. For a general MBS, where the motions are not constrained to a subgroup, the SE (3) and SO (3)×ℝ3 yield the same order of accuracy. Hence an appropriate configuration update can be selected for each individual body of a particular MBS, which gives rise to tailored update schemes. Several numerical examples are reported illustrating this statement.The practical consequence of using SE (3) is the use of screw coordinates as generalized coordinates. To account for the inevitable singularities of 3-parametric descriptions of rotations, the kinematic reconstruction is additionally formulated in terms of (dependent) dual quaternions as well as a coordinate-free ODE on the c-space Lie group. The latter can be solved numerically with Lie group integrators like the Munthe-Kaas integration method, which is recalled in this paper.