  This paper presents a unifying dynamics formulation for nonsmooth multibody
systems (MBSs) subject to changing topology and multiple impacts based on
linear projection operator. An oblique projection matrix ubiquitously derives
all characteristic variables of such systems as follow: i) The constrained
acceleration before jump discontinuity from projection of unconstrained
acceleration, ii) post-impact velocity from projection of pre-impact velocity,
iii) impulse during impact from projection of pre-impact momentum, iv)
generalized constraint force from projection of generalized input force, and v)
post-impact kinetic energy from pre-impact kinetic energy based on projected
inertia matrix. All solutions are presented in closed-form with elegant
geometrical interpretations. The formulation is general enough to be applicable
to MBSs subject to simultaneous multiple impacts with non-identical restitution
coefficients, changing topology, i.e., unilateral constraints becomes inactive
or vice versa, or even when the overall constraint Jacobian becomes singular.
Not only do the solutions always exist regardless the constraint condition, but
also the condition number for a generalized constraint inertia matrix is
minimized in order to reduce numerical sensitivity in computation of the
projection matrix to roundoff errors. The model is proven to be energetically
consistent if a global restitution coefficient is assumed. In the case of
non-identical restitution coefficients, the set of energetically consistent
restitution matrices is characterized by using Linear Matrix Inequality (LMI).
