  The simulation of multi-body systems with frictional contacts is a
fundamental tool for many fields, such as robotics, computer graphics, and
mechanics. Hard frictional contacts are particularly troublesome to simulate
because they make the differential equations stiff, calling for computationally
demanding implicit integration schemes. We suggest to tackle this issue by
using exponential integrators, a long-standing class of integration schemes
(first introduced in the 60's) that in recent years has enjoyed a resurgence of
interest. We show that this scheme can be easily applied to multi-body systems
subject to stiff viscoelastic contacts, producing accurate results at lower
computational cost than classic explicit schemes. In our tests with quadruped
and biped robots, our method demonstrated stable behaviors with large time
steps (10 ms) and stiff contacts ($10^5$ N/m). Its excellent properties,
especially for fast and coarse simulations, make it a valuable candidate for
many applications in robotics, such as simulation, Model Predictive Control,
Reinforcement Learning, and controller design.
