CENIC: Convex Error-controlled Numerical Integration for Contact

TL;DR

CENIC presents the first error-controlled, convex, continuous-time integrator tailored for contact-rich robotics, merging irrotational contact field theory with error-controlled integration to automatically adapt time steps to a user-defined accuracy $\varepsilon_{acc}$. By embedding external dynamical systems implicitly and modeling friction with static/dynamic coefficients in a convex framework, CENIC achieves real-time performance on challenging, stiff contact problems while guaranteeing convergence and consistency with the continuous model. Two integration strategies are offered (step-doubling for first-order accuracy and a second-order trapezoid variant), with extensive performance optimizations (warm starts, adaptive tolerances, Hessian reuse, cubic linesearch) and hardware validation demonstrating artifact-free, stable simulations that rival discrete-time robotics simulators in speed. This framework decouples numerical discretization error from modeling error, enabling more reliable sim-to-real transfer, policy learning, and model-based control in robotics.

Abstract

State-of-the-art robotics simulators operate in discrete time. This requires users to choose a time step, which is both critical and challenging: large steps can produce non-physical artifacts, while small steps force the simulation to run slowly. Continuous-time error-controlled integration avoids such issues by automatically adjusting the time step to achieve a desired accuracy. But existing error-controlled integrators struggle with the stiff dynamics of contact, and cannot meet the speed and scalability requirements of modern robotics workflows. We introduce CENIC, a new continuous-time integrator that brings together recent advances in convex time-stepping and error-controlled integration, inheriting benefits from both continuous integration and discrete time-stepping. CENIC runs at fast real-time rates comparable to discrete-time robotics simulators like MuJoCo, Drake and Isaac Sim, while also providing guarantees on accuracy and convergence.

Figures (19)

• Figure 1: A challenging bimanual manipulation scenario in simulation (top) and on hardware (bottom). A teleoperated robot drops several objects (spatula, spoon, mug) into a thin-walled bin. It then dumps the contents of the bin on a dish rack composed of thin wires. This scenario is particularly challenging: complex geometry, thin objects, heavy arms, and stiff joint controllers create a wide range of time scales that pose significant numerical difficulties. Discrete time-stepping methods struggle with passthrough, rattling, and unstable contact forces; CENIC eliminates these artifacts while completing the simulation at a real-time rate over 300%.
• Figure 2: CENIC's regularized friction model captures both static ($\mu_s$) and dynamic ($\mu_d$) friction regimes. Our default stiction tolerance is $v_s = 0.1 \,\text{mm/s}$, with a transition width of $\Delta = 10$.
• Figure 3: In practice, robotics simulations involve multibody dynamics connected to controllers, planners, and other components. We abstract these elements as a single external dynamical system. Unlike discrete formulations, CENIC integrates external systems implicitly, an effective strategy even for stiff feedback dynamics.
• Figure 4: Vertical position and velocity of a ball dropped on a table, showing differences between solutions computed with a 1 ms (solid line) and a 5 ms (dashed line) time step. Velocity error spikes, but position error does not.
• Figure 5: Solver convergence during hard clutter simulation, illustrating the impact of Hessian reuse. Red dashed lines indicate a recomputated Hessian, while light grey lines separate time steps. \ref{['eq:hessian_reuse_criterion']} allows us to reuse the same Hessian when possible (first ten steps). When necessary, a few fresh factorizations enable rapid Newton-type convergence (final step).

Theorems & Definitions (2)

• Remark 1
• Remark 2