Projection-Retraction MPPI: Exact Constraint-Manifold Control for Manipulators
Authors: Seulchan Lee, Leesai Park, Minhyeong Kang, Sanghyun Kim
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.
Sampling-based model predictive control methods, such as Model Predictive Path Integral (MPPI), offer derivative-free optimization and robustness in complex robotic systems. However, standard MPPI relies on cost-based soft penalties that cannot guarantee hard-constraint satisfaction, severely limiting its applicability to highly constrained tasks such as closed-chain manipulation. To address this, we propose Manifold-Constrained MPPI (MC-MPPI), a real-time sampling-based control framework that enforces manifold-based equality constraints while preserving the computational advantages of MPPI. The key idea is to decouple the constrained optimal control problem into latent-space planning and execution-level correction. At the planning stage, a Variational Autoencoder (VAE) learns a low-dimensional latent representation of the constraint manifold, enabling MPPI to efficiently generate near-feasible candidate trajectories without per-sample modification. Since this reference enables accurate linearization of the equality constraints, an execution-level Quadratic Programming (QP) controller resolves the residual manifold mismatch in a single solve rather than through iterative projection. Experiments on a 14-DoF closed-chain dual-arm system in both simulation and real-world settings demonstrate that MC-MPPI operates stably at 100 Hz, reliably navigates dynamic environments while effectively maintaining hard equality constraints, and significantly outperforms baseline methods in tracking accuracy. Supplementary videos and implementation details are available at https://rcilab.github.io/mcmppi.
This study proposes a novel Model Predictive Path Integral (MPPI)-based task-space control framework. The proposed framework explicitly solves rigid-body dynamics within a real-time MPC formulation and enforces safety constraints, enabling accurate motion and force control that yields compliant behaviors for safe and effective physical interaction of robotic manipulators in unstructured environments. By leveraging MPPI, the proposed framework efficiently handles nonlinear dynamics that are difficult to solve with conventional MPC approaches in real-time. Furthermore, we develop a torque-sampling-based control architecture that enables efficient exploitation of GPU-based parallelization, resulting in effective compliant and force-aware behaviors. As a result, the proposed framework achieves a solver update rate of over 166 Hz with a 0.18 s prediction horizon, and its performance is validated through real-world experiments on a 7-DoF manipulator.
Humanoid robots require whole-body controllers that are both robust and precise in contact-rich environments. While deep reinforcement learning (RL) achieves robust stability, its behavior is tightly coupled to the training objective and command interface, making it difficult to add new feedback objectives without retraining. In this study, we propose an RL guided whole-body model predictive path integral (MPPI) framework that acts as an add-on feedback controller on top of a pretrained RL policy. Instead of using RL policy as the final controller, we use it as a sampling prior that biases MPPI rollouts toward dynamically feasible behaviors. Task objectives are specified through modular MPPI cost terms, and MPPI closes the loop by continuously correcting the RL prior online to satisfy these objectives without retraining the policy. Simulations on a 29-DoF Unitree G1 humanoid in MuJoCo demonstrate stable high-rate control (average 280~Hz). The proposed method improves task-level precision over a pure RL baseline under the same command interface. This is achieved by correcting systematic drift during straight walking and tracking additional whole-body reference signals imposed through the cost.