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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.

Unilateral contact permits bodies to separate while preventing them from passing through each other. In an ideal rigid, nonadhesive model, the supporting normal contact force can push but cannot pull. It is therefore different from a permanently attached support: an upward pull may lift a body away instead of producing a downward support reaction.

Let $d$ be the signed normal gap, positive when the bodies are separated, and $N$ the outward compressive normal contact force. The nonpenetration constraint and absence of adhesion give the unilateral contact complementarity conditionsAn open gap requires $N=0$, while $N>0$ requires a closed gap. The pair $d=N=0$ is also allowed: touching does not by itself imply a loaded contact.^[ref] Tangential friction requires an additional relation such as Coulomb friction; it is not specified by the normal-gap conditions.

The equation of motion is still needed to determine the reaction. Consider a block of mass $m$ on a fixed horizontal floor, initially at rest at $d=0$, with an upward applied force $u$ through its center of mass. With upward positive, Newton's second law isIf the contact remains closed, $\ddot{d}=0$ would require $N=mg-u$. That is feasible for $u\le mg$. For $u>mg$, a negative normal reaction would violate unilateral contact, so the contact opens with $N=0$ and $\ddot{d}=(u-mg)/m>0$. At a closed contact with zero normal velocity, the appropriate acceleration-level contact complementarity iswhich yieldsThese acceleration conditions are being applied at a closed, initially stationary contact; they are not a rule forbidding downward acceleration during separated free flight.

For an original numerical example, let $m=2\mathrm{kg}$ and $g=9.81m/{\mathrm{s}}^2$. Pulling upward with $u=5\mathrm{N}$ leaves a normal reaction of $14.62\mathrm{N}$. At $u=25\mathrm{N}$, the reaction is zero and the initial upward acceleration is $2.69m/{\mathrm{s}}^2$. At $u=19.62\mathrm{N}$, both the reaction and the initial acceleration vanish, while touching remains possible.

A simple apparatus test places the block on a calibrated load cell and applies the upward pull through a spring balance, keeping the pull vertical through the center of mass to avoid an extra overturning torque. The support reading should decrease with the pull while contact persists, then vanish at lift-off. Recording the gap as well distinguishes an unloaded touch from a genuinely open contact. Real support elastic deformation and adhesion must be accounted for before treating that apparatus as the ideal rigid, nonadhesive model.

A penalty approximation of unilateral contact replaces perfect rigidity by a chosen stiff response, for exampleHere a small negative gap represents an allowed modeled overlap relative to undeformed reference surfaces. It is not an exact solution of $d\ge 0$. For the $14.62\mathrm{N}$ supported load above and chosen penalty stiffness $k=10^{5}N/m$, equilibrium requires $d=-0.1462\mathrm{mm}$. Increasing $k$ reduces that overlap at the same load. The penalty method and the ideal constraint are different approximations of the contact response.

Finally, impact requires a rule for the velocity change. The position-level inequalities alone do not decide whether an approaching body sticks or rebounds. A coefficient of restitution or another specified impact law determines that behavior, and the impulsive contact force is summarized by its impulse. Tedrake's particular time-stepping formulation identifies its block collisions as perfectly inelastic; that is a feature of the stated model, not a consequence of the three gap/reaction inequalities alone.^[ref]