Tailoring Solution Accuracy for Fast Whole-body Model Predictive Control of Legged Robots

TL;DR

The paper tackles real-time control for high-dimensional legged robots by reframing NMPC to tolerate low-accuracy inner-QP solutions. It combines an ADMM-based QP solver (via OSQP) with a first-timestep control barrier function to enforce self-collision constraints, enabling reliable, holistic planning at rates up to 90–200 Hz on the MIT Humanoid. Hardware experiments show the approach supports coordinated arm and crossed-leg motions, maintains joint limits, and recovers from substantial disturbances, while simulation highlights the influence of modeling errors, discretization, and delay on the benefits of higher solution accuracy. The results suggest that, in practical settings, rapidly computable, slightly inexact solutions can outperform precisely solved subproblems when dynamics and sensing imperfections dominate, with CBFs providing a robust safety layer.

Abstract

Thanks to recent advancements in accelerating non-linear model predictive control (NMPC), it is now feasible to deploy whole-body NMPC at real-time rates for humanoid robots. However, enforcing inequality constraints in real time for such high-dimensional systems remains challenging due to the need for additional iterations. This paper presents an implementation of whole-body NMPC for legged robots that provides low-accuracy solutions to NMPC with general equality and inequality constraints. Instead of aiming for highly accurate optimal solutions, we leverage the alternating direction method of multipliers to rapidly provide low-accuracy solutions to quadratic programming subproblems. Our extensive simulation results indicate that real robots often cannot benefit from highly accurate solutions due to dynamics discretization errors, inertial modeling errors and delays. We incorporate control barrier functions (CBFs) at the initial timestep of the NMPC for the self-collision constraints, resulting in up to a 26-fold reduction in the number of self-collisions without adding computational burden. The controller is reliably deployed on hardware at 90 Hz for a problem involving 32 timesteps, 2004 variables, and 3768 constraints. The NMPC delivers sufficiently accurate solutions, enabling the MIT Humanoid to plan complex crossed-leg and arm motions that enhance stability when walking and recovering from significant disturbances.

Figures (7)

• Figure 1: Hardware demonstration of the whole-body NMPC on the MIT Humanoid. Top row: walking with emergent arm swing. Bottom row: leg crossing with dynamic self-collision avoidance with right leg fully extended during abrupt halting during lateral walking.
• Figure 2: Two self-collision models. a) the simple collision model constrains the right foot to a minimum distance from a plane attached to the left foot. b) the complex collision model composed of 19 pairs of spheres. Each blue line corresponds to a distance constraint between a pair of spheres.
• Figure 3: Normalized number of successes for increasing ADMM iterations and inertial modeling errors without CBFs and computation delay for $dt$ = 25 ms. The potential success improvement decreases as the inertia randomization increases (i.e., 51, 46, 32, 27 and 21%, respectively).
• Figure 4: Normalized number of successes for increasing ADMM iterations and integration timestep without CBFs and computation delay. The potential success improvement decreases as the integration timestep increases (i.e., 67, 51, 41, 18, 7 and 1%, respectively).
• Figure 5: Normalized number of successes for increasing ADMM iterations with and without computation delay for $dt \in \{20, 25\}$ ms.