= Unilateral contact
{title2=$d\ge0,\quad N\ge0,\quad dN=0$}
<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> conditions
$$
\boxed{d\ge0,\qquad N\ge0,\qquad dN=0.}
$$
An 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.<Underactuated Robotics by Russ Tedrake>[Multi-Body Dynamics, Time-stepping LCP: Normal force, nonnegative gap and normal reaction and their complementarity product]{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> is
$$
m\ddot d=u+N-mg.
$$
If 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> is
$$
\ddot d\ge0,\qquad N\ge0,\qquad N\ddot d=0,
$$
which yields
$$
\boxed{N=\max(mg-u,0),\qquad
\ddot d=\max((u-mg)/m,0).}
$$
These 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 $g=9.81\,\mathrm{m/s^2}$. 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 $2.69\,\mathrm{m/s^2}$. At $u=19.62\,\mathrm N$, both the reaction and the initial <acceleration> vanish, while touching remains possible.
\Image[/unilateral-contact-load-release.png]
{title=<Unilateral contact> allows a closed unloaded contact and releases a support when a tensile reaction would be required}
{description=Original ideal-model illustration. The left panel shows allowed gap/reaction pairs; the right shows the initial reaction and acceleration for a stationary 2 kg block as an upward pull is increased. The two panels illustrate geometry and force balance respectively; they are not measured loading data.}
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 example
$$
N=k\max(-d,0),\qquad k>0.
$$
Here a small negative gap represents an allowed modeled overlap relative to undeformed reference surfaces. It is not an exact solution of $d\ge0$. For the $14.62\,\mathrm N$ supported load above and chosen penalty <stiffness> $k=10^5\,\mathrm{N/m}$, 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.<Underactuated Robotics by Russ Tedrake>[Multi-Body Dynamics, Time-stepping LCP dynamics for block collision, perfectly inelastic collision hint]{ref}
Back to article page
