Projection-Retraction MPPI: Exact Constraint-Manifold Control for Manipulators

2026-08-04

AI Summary

Traditional Model Predictive Path Integral (MPPI) control approximates constraints via soft penalties, which often fails to enforce hard constraints—such as closed-chain kinematics, joint limits, and collision avoidance—accurately under high task costs. This work proposes PR-MPPI, a novel approach that explicitly embeds constraint manifold geometry within the sampling dynamics. At each timestep, sampled velocities are projected onto the intersection of the equality-constraint subspace and inequality-defined half-spaces, followed by a retraction step that precisely maps control inputs back onto the constraint manifold. PR-MPPI is the first method to jointly satisfy both equality and inequality constraints exactly within the MPPI framework, achieving numerical-tolerance-level precision without requiring task-weight tuning. Experiments demonstrate stable maintenance of closed-chain constraints on a 14-DOF dual-arm system under extreme joint limits and random obstacles, and enable efficient real-time dynamic obstacle avoidance on the Unitree H1-2 humanoid robot.

Abstract

Model Predictive Path Integral (MPPI) control is widely used in manipulation for its gradient-free, parallel handling of non-convex costs. Manipulation tasks, however, often impose constraints that hold throughout the motion: a closed kinematic chain that two grasping arms keep exactly, or joint limits and obstacle clearances that are never crossed. MPPI handles such constraints only through the cost, as soft penalties that hold approximately and fail under a strong task cost. To address this, we propose Projection-Retraction MPPI (PR-MPPI), which enforces the constraints inside the sampled dynamics. At every rollout step, the sampled velocity is projected to satisfy both constraint types: the equality restricts it to a subspace, and each inequality to a half-space within that subspace, so inequality handling never breaks the equality. This projection, however, satisfies the constraints only to first order, and a finite step leaves a small drift off the equality. Therefore, we retract the returned command back onto the constraint to numerical tolerance and independent of task weighting. We validate PR-MPPI on 14-DoF dual-arm systems. In simulation, the returned commands satisfy the closed-chain equality to numerical tolerance through a joint-limit stress test and randomized obstacle avoidance. On real hardware, the arms of a Unitree H1-2 humanoid reactively avoid a moving obstacle. Code and experiment videos are available at https://rcilab.github.io/prmppi.

Problem

• constraint enforcement
• manipulation
• closed kinematic chain
• joint limits
• obstacle avoidance

Innovation

• Projection-Retraction MPPI
• constraint-manifold control
• closed kinematic chain
• hard constraints
• model predictive path integral