Exponential integration for efficient and accurate multibody simulation with stiff viscoelastic contacts

Abstract

The simulation of multibody 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 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 1960s) that in recent years has enjoyed a resurgence of interest. This scheme can be applied to multibody systems subject to stiff viscoelastic contacts, leading to integration errors similar to implicit Euler, but at much lower computational costs (between 2 to 100 times faster). In our tests with quadruped and biped robots, our method demonstrated a stable behavior 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.

Appendix:  Implicit Euler

This Appendix reports some details about our implementation of implicit Euler. The state of our robots lies in a Lie group, so we have taken into account the derivatives of the integrate and difference functions, which need to be used in place of the simple + and − operators:

$$\begin{aligned} \text{difference}(x^{+}, \text{integrate}(x, \delta t f(x^{+}, u))) = 0. \end{aligned}$$

(35)

We have computed the dynamics Jacobian using the appropriate functions of the Pinocchio library. In particular, we have used the analytical derivatives of the ABA function (28 μs for the Solo quadruped robot), and we have added to it the derivatives of our contact model (which are fast to compute because they depend on already computed quantities). For the line search, we have used the ABA function (8 μs for the Solo quadruped robot). This resulted in a rather efficient implementation of implicit Euler.

For solving the nonlinear equations, we have implemented Newton’s method with line search and regularization. For solving the linear system of equations, we have used the LU decomposition with partial pivoting provided by the Eigen library. We initialized the search with the solution of an explicit Euler integration (which proved to be better than initializing with the current state).

We have measured where computation time is spent in implicit Euler to make sure that our implementation was efficient and to reason about potential improvements. For the “solo-squat” test, with an integration step $\delta t =10$ ms, the average computation time was 0.92 ms. Expo was roughly 10 times faster for the same test and gave roughly the same integration error. The Newton method took on average 8 iterations to converge to an error norm below $10^{-6}$. Computation time was distributed in the following way:⁴

• Prepare linear system: 51%

  • Dynamics Jacobian: 29%

    • ⁎ ABA derivatives: 24%

    • ⁎ Contact model derivatives: 4%

  • Lie-group derivatives: 21%

• Line search: 29%

• Solve linear system: 12%

• Other: 8%

Most of the time was spent preparing the linear system (51%) and performing the line search (29%). Given the high efficiency of the Pinocchio library, we believe that there is not much room for improvement in the steps “ABA derivatives” and “line search” (which mainly consists of calls to ABA, integrate, and difference). The only significant room for improvement is in the “Lie-group derivatives” step, which includes two matrix–matrix multiplications of the dynamics Jacobian and the sparse Jacobians of the integrate and difference functions. Since these Jacobians are sparse, these multiplications could be computed more efficiently than what we currently do. However, this could lead to a computational gain of at most 15%. Therefore there seems to be little hope to make implicit Euler 10 times faster (i.e., as efficient as Expo).