  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
$SO\left( 3\right) \times {\Bbb R}^{3}$ 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 holonomicaly constrained MBS modeled with the 'absolute coordinate'
approach, i.e. using the Newton-Euler equations for the individual bodies
subjected to geometric constraints. 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\left( 3\right) $ 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
$SO\left( 3\right) \times {\Bbb R}^{3}$ yield the same order of accuracy.
