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Real-Time Dynamics-Based Torque-Sampling MPPI for Compliant and Force Aware Manipulation

Euncheol Im, Taehyun Kim, Yonghwan Oh, Myotaeg Lim, Yisoo Lee

arXiv:2609.02020v1cs.RO

TL;DR

Safe, force-aware manipulation needs real-time control that can handle nonlinear rigid-body dynamics and constraints. The paper develops torque-sampling MPPI with GPU-parallelized analytic dynamics and validates compliant motion-force behavior, reaching 166 Hz with a 0.18 s predictive window on a 7-DoF manipulator.

  • Problem

    Real-time MPC struggles to incorporate full rigid-body dynamics because manipulation systems are nonlinear, high-dimensional, and computationally expensive to optimize.

  • Method

    The framework directly samples joint torques and evaluates analytic rigid-body dynamics within GPU-parallelized MPPI rollouts, incorporating force objectives and constraints.

  • Results

    166 Hz update rate with a 0.18 s prediction horizon was achieved and validated through real-world experiments on a 7-DoF manipulator across motion, contact, force, and obstacle scenarios.

  • Takeaways & Limitations

    Direct torque sampling enables compliant and force-aware behavior while preserving dynamic fidelity and satisfying task-space constraints during physical interaction.

Abstract

from arXiv · show

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.

I. INTRODUCTION

Real-time manipulation requires safe force-aware interaction, but conventional dynamics-based and MPC methods face local-optimality, computational, and modeling limitations. The proposed MPPI framework directly samples torques with rigid-body dynamics on GPUs and demonstrates compliant motion-force control at high update rates.

  • Safe physical interaction in unstructured environments requires force control and compliance, especially for unexpected collisions and contact-rich tasks.
  • Explicitly incorporating full rigid-body dynamics into MPC is computationally costly because robotic systems are nonlinear and high-dimensional.
  • QP-based MPC commonly simplifies contact forces or dynamics, while DDP-based MPC remains sensitive to initialization, local minima, and differentiability requirements.
  • Sampling-based MPPI handles nonlinear dynamics and nonsmooth costs efficiently through GPU parallelization, but manipulation applications have largely remained at the kinematic level.
  • Full rigid-body dynamics and torque-level MPPI remain underexplored because simulator-based approaches can suffer discretization errors and high computational cost at short control timesteps.
  • The framework directly samples and optimizes joint torques, achieving a 0.18 s prediction horizon and 166 Hz update frequency while demonstrating hybrid motion-force control and compliance in real-world experiments.

II. PRELIMINARIES: MODEL PREDICTIVE PATH INTEGRAL

MPPI formulates finite-horizon stochastic optimal control by evaluating noisy control rollouts and updating a nominal sequence through importance-weighted sampling. The framework evaluates these rollouts in parallel while propagating states and dynamics quantities on GPU threads.

  • MPPI computes control inputs by minimizing a cost function over a finite time horizon using a stochastic system model.
  • The control input combines a nominal input u_t with zero-mean Gaussian disturbance δu_t having covariance Σ_u.
  • The optimal control problem minimizes terminal and running costs over a finite sequence, with running costs encoding effort, obstacle avoidance, state limits, or stability.
  • MPPI evaluates K noisy control sequences in parallel and updates the optimal sequence using an importance-weighted average rather than iteratively computing dynamics and cost derivatives.
  • Importance weights use shifted trajectory costs to prevent numerical overflow or underflow without changing optimality, while λ controls the temperature of the weighting.
  • After applying the first optimized input, the remaining sequence warm-starts the next optimization loop as its nominal control sequence.
  • Each GPU thread independently propagates rollout states and computes end-effector pose, Jacobian, and dynamics terms at every timestep.

III. PROPOSED METHOD

The proposed method integrates an analytic rigid-body model directly into torque-sampling MPPI, making joint torques primary control inputs for predictive motion and force control.

  • The framework embeds analytic rigid-body dynamics directly in the optimization process and treats joint torques as the primary control inputs.
  • Torque-level formulation supports force-related objectives and integrated motion-force control within one predictive architecture.

A. Analytic Dynamics for Torque-Sampling Rollouts

The framework samples joint torques and propagates each rollout using an analytic rigid-body model, recomputing dynamics throughout the prediction horizon. GPU-parallelized rollout integration supports high-throughput real-time control.

  • Torque-Sampling Rollouts: Torque-level sampling perturbs nominal torques directly, enabling compliance during interaction and integration of force objectives into predictive control.Each sampled sequence is evaluated by propagating robot configuration over the prediction horizon.
  • Torque-Sampling Rollouts: The projected external torque is obtained from the end-effector Jacobian and external wrench, which are assumed constant over the prediction horizon.τ ext = JT(q)Fext, with J(q) as the end-effector Jacobian and Fext as the external wrench.
  • State Propagation: Joint accelerations are explicitly computed for sampled torques, then joint velocities and positions are updated using explicit Euler integration.The integration uses time step dt.
  • Analytic Dynamics: M(q) and h(q, q˙) are recomputed at every timestep using the analytic rigid-body model to maintain prediction fidelity.The recursive integration runs inside custom CUDA kernels, with one GPU thread assigned to each rollout.
  • Analytic Dynamics: The inertia matrix is assembled from transformed link inertias and mass properties, while h is derived from link-wise Jacobians and their time derivatives.Fixed robot quantities are stored for reuse across rollouts that differ only in sampled torque sequences.

B. Cost Function Design for Hybrid Control

The cost design combines motion and force objectives with joint, collision, and other physical constraints. Dynamically consistent force estimation and GPU-parallelized sampling support integrated hybrid control at real-time rates.

  • Force and Motion Costs: Force-based cost terms explicitly incorporate end-effector force objectives alongside motion control objectives.This extends MPPI formulations described here as focusing solely on motion control.
  • Force and Motion Costs: The task cost decomposes into motion and force components, while the overall cost combines task-related objectives with physical constraints.Motion cost penalizes deviations from the desired end-effector trajectory in SE(3).
  • Force and Motion Costs: The force estimate Fee = ¯JTτ uses the transpose of a dynamically consistent generalized inverse, accounting for system inertia and decoupling null-space torques.Diagonal weighting matrices specify axis selection and priority in the motion and force costs.
  • Safety Constraints: Joint costs penalize position and velocity limit violations and regularize redundant motions to promote balanced postures and avoid biased local solutions.Collision costs address self-collisions and collisions with external objects when distances fall below safety thresholds.
  • GPU-Parallelized Control: At each timestep, GPU-generated torque-noise samples propagate analytic dynamics in parallel before costs and the control input are updated by importance sampling.The loop follows the sampling strategy described in.

A. Experimental Setup

Experiments use a 7-DoF Franka Research 3 with GPU-based MPPI computation and a separate real-time communication bridge. The robot interface operates at 1 kHz.

  • Hardware and Communication: The experimental platform is a 7-DoF Franka Research 3 (FR3) connected through a real-time communication interface.The GPU controller and communication bridge run on separate computing platforms.
  • Hardware and Communication: An i9-10900KF desktop with an RTX 2070 Super performs MPPI computation, while an Intel NUC13 with 32 GB RAM bridges the robot and controller.The NUC connects to the robot through the Franka Control Interface.
  • Hardware and Communication: The Franka Control Interface connection runs at 1 kHz between the NUC and the robot.The NUC communicates with the MPPI computing PC.

B. Control Performance Verification

The experiments test hybrid motion-force tracking under motion blocking and external perturbations. The controller tracks pose and force, responds compliantly to disturbances, and returns to the desired state after disturbance removal.

  • Hybrid Motion-Force Experiment: The hybrid experiment commands Fx,d = 10 N while tracking yd = 0.0 m and zd = 0.38 m with fixed initial orientation.External perturbations assess compliance and force-control capabilities.
  • Experimental Parameters: Table I is identified as reporting MPPI parameters for different experimental conditions.The supplied table passage provides its title but no parameter values or condition-specific comparisons.
  • Hybrid Motion-Force Experiment: Motion blocking is applied along the force-controlled x-axis at approximately t = 3–4.5 s, followed by manual forces in +z, +y, and −y directions.The applied forces occur at approximately t = 7 s, 11.5 s, and 17 s, respectively.
  • Tracking Performance: 0.0137 m mean position error and 0.0208 rad mean orientation error are measured over the converged interval t = 4.5–6 s without external disturbances.Errors are computed in the position-controlled y and z axes.
  • Tracking Performance: Under external disturbances, the manipulator responds compliantly in the applied-force direction and promptly reconverges to the desired force and pose after removal.End-effector force is estimated through torque projection.

C. Hybrid Motion–Force Control under Maintained Contact

The framework maintains contact while controlling force and pose on a curved cylindrical surface, despite lacking prior geometric information and dedicated force sensors.

  • Fz,d = −4 N was applied along the z-axis while the y-axis followed a linear motion trajectory during cylindrical-surface contact.
  • 7.67 s marked the approximate establishment of sustained contact, which the controller reliably maintained throughout the contact phase.
  • 1.80 N mean absolute force error was measured during maintained contact, primarily attributed to unmodeled surface curvature and absent explicit force feedback.
  • 0.012 m position error and 0.0921 rad orientation error were obtained during hybrid control while maintaining contact force.
  • The framework achieved stable contact and pose tracking on the curved surface without prior geometric information or dedicated force sensors.

D. Dynamic Obstacle Avoidance

The framework performs hybrid motion–force control while avoiding a moving obstacle, prioritizing collision safety over the primary force command during real-time interaction.

  • Fy,d = −4 N was commanded while the robot maintained its posture during dynamic obstacle avoidance.
  • 20 Hz was the approximate obstacle-position tracking rate provided by an Intel RealSense L515.
  • The robot redirected motion toward +y to prevent collision when a mustard container moved into its path, temporarily opposing the −y force command.
  • A high collision-cost weight wcollision assigned safety priority over task execution during obstacle avoidance.
  • The obstacle and robot links were modeled as simplified cylinders with a safety threshold of di = 0.2 m for efficient collision detection.

V. CONCLUSIONS

The paper concludes that torque-sampling MPPI can integrate rigid-body dynamics into real-time optimization while supporting compliant, force-aware manipulation and safety-constrained task-space behavior.

  • Analytic rigid-body dynamics were evaluated within GPU-parallelized torque-sampling rollouts, combining small time steps with a prolonged prediction horizon.
  • 166 Hz solver updates and a 0.18 s predictive window were achieved on a 7-DoF manipulator.
  • The framework was validated across disturbance rejection in free-space motion and contact-rich manipulation scenarios.
  • Direct rigid-body dynamics integration into torque sampling enabled compliant and force-aware behavior while satisfying task-space constraints.
  • Future work will incorporate object dynamics and address dynamic uncertainty and physical-contact effects for in-hand and multi-arm tasks.
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