 Acceleration-level contact complementarity

Acceleration-level contact complementarity decides whether a smooth, closed contact with zero normal velocity must carry a compressive reaction or begin to separate. At $d=0$ and $\dot{d}=0$, the normal conditions areThey do not constrain the sign of acceleration at a positive gap, and they do not resolve an impact with negative incoming normal velocity. The distinction between smooth constrained dynamics and time-stepped contact/impact models is explicit in the primary dynamics treatment.^[ref]

For a frictionless multibody model with generalized velocity $v$, invertible mass matrix $M$ and contact Jacobian $J$ defined by $\dot{d}=Jv$, writewhere $b=J{M}^{-1}f+\dot{J}v$ is the free normal acceleration and $W=J{M}^{-1}{J}^{\mathsf{T}}$. The normal reactions then solve the linear complementarity problemThe matrix couples contacts: choosing each reaction from a separate scalar formula generally fails when it changes acceleration at the other contacts. This formulation assumes the displayed velocity-coordinate kinematics and active closed contacts; additional friction forces modify the dynamics.

For an original coupled example, take free normal acceleration $b=(-3,-3{)}^{\mathsf{T}}\mathrm{m}{\mathrm{s}}^{-2}$ andThe reaction $N=(1,1{)}^{\mathsf{T}}\mathrm{N}$ gives zero normal acceleration at both contacts. Ignoring the off-diagonal coupling would instead choose $1.5\mathrm{N}$ at each; inserted into the full system, that produces $1.5\mathrm{m}{\mathrm{s}}^{-2}$ of outward acceleration at both loaded contacts and violates complementarity. Positive reaction and positive separating acceleration cannot coexist in this smooth active-contact model.

Because these are acceleration-level conditions, a numerical trajectory can still accumulate position or velocity drift. They must be paired with an integration/contact-update procedure that maintains the assumed state. During separated free flight, a negative acceleration can be perfectly admissible until the next contact event. At impact, an impulse and an appropriate velocity-level law are needed instead of applying the smooth equations across a discontinuity.