Technical Summary: CORD-SLS for Certified Deformable Manipulation

Problem Statement

The manipulation of deformable objects, such as ropes and cloth, presents significant challenges for robotics due to their lack of compact state representations, high-dimensional nonlinear dynamics, and the complexity of contact interactions. Existing approaches face a trade-off: classical methods using geometric abstractions lack robustness and are often open-loop, while learned policies lack safety guarantees and require extensive training. Furthermore, differentiable simulators, which enable gradient-based planning, often suffer from discontinuous gradients caused by naive contact formulations, hindering efficient contact discovery. Additionally, ensuring safety under model and sensing uncertainty in high-dimensional, contact-rich environments remains an open problem, particularly for output-feedback control where full state knowledge is unavailable.

Methodology

The authors propose CORD-SLS (Contact-smoothed Output-feedback Robust control of Deformables via System Level Synthesis), a framework integrating four core components:

1. GPU-Parallel Differentiable Simulator with Contact Smoothing

To address the gradient discontinuity issue in contact-rich simulation, the authors develop a simulator based on an implicit time-stepping scheme.

• Dynamics Formulation: The system models elastic energy and constraints (holonomic and contact) using a two-stage solver. The first stage solves for an intermediate configuration under holonomic constraints, and the second stage resolves contact interactions via a convex relaxation of Anitescu's formulation.
• Differentiation: Instead of unrolling iterative solvers, the method employs implicit differentiation through the KKT conditions of the optimization problems, allowing for efficient gradient computation without additional forward pass costs.
• Contact Smoothing: To prevent vanishing gradients during contact transitions, the authors introduce surrogate gradients. They smooth the activation of holonomic constraints and the complementarity conditions of contact constraints using exponential functions and relaxed complementarity conditions. This allows the planner to receive informative gradient signals to "seek" contact, while the forward rollouts utilize the original nonsmooth dynamics to preserve physical fidelity.

2. GPU-Parallel Output-Feedback Robust MPC

The control strategy utilizes System Level Synthesis (SLS) to synthesize robust output-feedback policies in real-time.

• Robust Formulation: The method formulates a reachability-constrained optimization problem that accounts for bounded process disturbances and observation errors. It computes "reachable tubes" to ensure that closed-loop trajectories remain within safe sets despite uncertainty.
• Parallelization: Extending previous GPU-SLS work (limited to state-feedback), the authors adapt the solver for output-feedback by expressing observer recursions as associative operators. This enables the use of GPU-parallel prefix scans to solve the Kalman recursions and assemble closed-loop response operators in logarithmic time, making real-time planning feasible for high-dimensional systems.

3. Accelerated Model-Based RL (APG)

The differentiable simulator is leveraged to accelerate Model-Based Reinforcement Learning (MBRL). By treating the simulator as a differentiable layer, the authors compute Analytical Policy Gradients (APG). This allows neural policies to be trained with high sample efficiency, guiding them to discover and maintain contact without relying solely on stochastic exploration.

4. Conformal Prediction (CP) for Safety Calibration

To bridge the gap between simulation and reality, the authors use Conformal Prediction to calibrate uncertainty bounds.

• Calibration: Using a held-out calibration set of perceived-state transitions, the method computes nonconformity scores to determine a worst-case prediction error bound.
• Guarantees: This calibrated bound is used to inflate the reachable tubes computed by the robust SLS controller. The authors prove that this approach provides high-probability safety guarantees for the true state, even when only noisy, delayed observations are available.

Key Contributions

1. Simulator: A GPU-parallel, differentiable, real-time deformable object simulator featuring contact smoothing to enable gradient-based planning through intermittent contact.
2. Control Algorithm: A GPU-parallel method for real-time synthesis of robust output-feedback control policies via reachability-constrained SLS, capable of handling high-dimensional deformable dynamics.
3. Learning Acceleration: A model-based RL strategy (APG) that utilizes analytical policy gradients from the differentiable simulator for sample-efficient training of neural manipulation policies.
4. Safety Calibration: A real-world vision-based implementation that uses conformal prediction to calibrate perception and model-error bounds, yielding probabilistic safety guarantees for robust MPC.

Results

The framework was evaluated on high-dimensional rope and cloth manipulation tasks (lifting, dragging with obstacle avoidance, folding, and flattening) in both simulation and hardware.

• Performance: CORD-SLS achieved millisecond-speed planning (e.g., <1 second per MPC update for 300-DoF cloth tasks).
• Efficiency: The simulator accelerated forward simulation and Jacobian computation by up to 700× and 2000×, respectively, compared to prior differentiable simulators.
• Safety and Success: In simulation, CORD-SLS achieved 100% safety rates and lower goal errors compared to baselines like MPPI and PPO. For instance, in the "Drag Rope" task, MPPI achieved only a 1.0% safety rate, while CORD-SLS achieved 99.9%.
• Hardware Validation: On physical KUKA LBR iiwa manipulators, the system successfully performed constrained rope manipulation over obstacles and cloth folding. The robust tubes computed by the controller effectively accounted for dynamical and measurement uncertainties, ensuring safe execution.
• Comparison: The method outperformed baselines in safety, speed, and task success. Notably, PPO required warm-starting with expert demonstrations to succeed in contact-rich tasks, whereas CORD-SLS solved them directly via trajectory optimization.

Significance and Claims

The paper claims that CORD-SLS closes the gap between scalable planning, deformable contact dynamics, and robustness. By unifying differentiable simulation, contact smoothing, and output-feedback robust MPC, the framework enables real-time, safe control of deformable objects in contact-rich environments without bespoke skills or extensive training. The integration of conformal prediction allows the system to provide certified, high-probability safety in real-world settings despite model and perception uncertainties. The authors emphasize that their approach generalizes across diverse tasks (obstacle avoidance, routing, folding) without task-specific retraining, demonstrating efficacy in both simulation and hardware.

Limitations

The authors acknowledge that contact smoothing introduces errors near contact transitions, though these are manageable. They note that deformable-object perception remains challenging under occlusion and complex topologies, suggesting future work could involve multi-view sensing or learned output equations. Finally, they identify memory usage for higher-dimensional systems and long horizons as a bottleneck, motivating the use of low-rank SLS response operators.