 Penalty approximation of unilateral contact

A penalty approximation of unilateral contact allows a small modeled overlap and uses a stiff restoring force instead of enforcing an exact rigid nonpenetration constraint. With signed gap $d$, positive in separation, a simple nonadhesive law isAn open or just-touching contact has zero force in this spring law; a sustained compressive load requires negative $d$. Thus it approximates unilateral contact rather than satisfying its rigid gap condition exactly. Tedrake's primary compliant-contact treatment uses stiff springs and damping to resist penetration and explains the resulting numerical stiffness.^[ref]

For an original gravity-supported example, a $2\mathrm{kg}$ mass with no other vertical load has $N=mg=19.62\mathrm{N}$. With $k=2\times 10^5\mathrm{N}{\mathrm{m}}^{-1}$, its static overlap is $mg/k=98.1\mu \mathrm{m}$. Increasing $k$ by a factor of 100 reduces that overlap to $0.981\mu \mathrm{m}$, but also makes the contact dynamics faster.

While the contact stays compressed, linearization about equilibrium givesThe normal contact penalty time scale is therefore proportional to $\sqrt{m/k}$. For the first stiffness above, ${\omega }_c\approx 316.2\mathrm{rad}{\mathrm{s}}^{-1}$; after increasing stiffness it is ten times larger. At fixed numerical step size, a smaller overlap error need not mean a more accurate trajectory.

For example, the Forward Euler method applied to this undamped oscillator has amplification factors $1\pm i{\omega }_c\Delta t$, whose magnitudes exceed one for every positive step. Merely shrinking the step does not make that method asymptotically stable for indefinite undamped oscillation, although it can reduce error on a finite interval. A suitable integration method, damping model and convergence check must be chosen. The source notes specifically identify stiff differential equations as a challenge for compliant contact simulation.^[ref]

A penalty stiffness is a chosen model parameter, not automatically the actual material stiffness. Impact dissipation and tangential friction still need their own constitutive choices; a purely elastic normal spring does not imply a perfectly inelastic collision.