Safety Beyond the Training Data: Robust Out-of-Distribution MPC via Conformalized System Level Synthesis

1. Problem Statement

The paper addresses the critical challenge of safe planning and control for robotic systems when using learned dynamics models that operate Out-of-Distribution (OOD).

• The Core Issue: Data-driven planners often learn dynamics models from a specific dataset. When a robot encounters states or controls outside this training distribution, the model error can be unpredictable, leading to unsafe behavior (e.g., collisions).
• Limitations of Existing Methods:
  • Classical MPC: Assumes precise models and fails under uncertainty.
  • Standard Robust MPC: Often relies on fixed, conservative bounds or "trusted domains" that prevent the robot from ever leaving the training data region, limiting utility.
  • Existing Learning-based MPC: May lack formal safety guarantees or fail to actively steer the robot toward regions of lower uncertainty (low-error regions).
• Goal: Develop a framework that allows robots to operate temporarily OOD while maintaining probabilistic safety guarantees, actively reducing uncertainty, and optimizing closed-loop performance.

2. Methodology: CP-SLS-MPC

The authors propose CP-SLS-MPC, a framework combining Weighted Conformal Prediction (WCP) and System Level Synthesis (SLS).

A. Weighted Conformal Prediction (WCP) for Error Bounding

Instead of assuming a fixed error bound, the method learns a state-control-dependent covariance model ($\Sigma$) to capture spatially varying uncertainty.

• Model Training: A neural network learns the dynamics ($\hat{f}$) and a separate network learns the uncertainty covariance ($\Sigma$) using Multivariate Gaussian Negative Log-Likelihood (MGNLL).
• Calibration: The method uses WCP to convert raw non-conformity scores (errors between $\hat{f}$ and true $f$) into a calibrated threshold ($q_{1-\alpha}$).
• State-Dependent Ellipsoids: Unlike methods using fixed spherical bounds, CP-SLS-MPC constructs ellipsoidal prediction sets centered at the predicted state. The shape and size of these ellipsoids depend on the local state-control pair $(x, u)$.
• Handling Distribution Shift: WCP uses weighted calibration to account for spatio-temporal distribution shifts between training data and execution, providing a theoretical bound on the "coverage gap" (the probability that the true error falls outside the predicted set).

B. System Level Synthesis (SLS) for Robust Control

The method integrates these error bounds into an SLS-based Model Predictive Control (MPC) formulation.

• Decomposition: SLS decomposes the control problem into optimizing a nominal trajectory ($z, v$) and a disturbance-feedback controller ($\Phi_x, \Phi_u$).
• Reachable Tubes: The controller computes "reachable tubes" (overapproximations of the true closed-loop trajectory) by propagating the worst-case model errors (defined by the CP ellipsoids) over the prediction horizon.
• Constraint Tightening: The MPC optimization enforces constraints on the nominal trajectory such that, even with the worst-case error within the CP tube, the true system remains safe.
• Convexity: SLS provides a convex parameterization of the closed-loop response, allowing the non-linear MPC problem to be solved efficiently via Sequential Convex Programming (SCP).

C. Active Uncertainty Reduction

The framework includes a cost function term that penalizes trajectories far from the training data distribution. This encourages the planner to:

1. Stay in "in-distribution" regions where the model is accurate.
2. If OOD operation is necessary, choose paths that minimize the volume of the uncertainty tubes (reducing conservativeness).

3. Key Contributions

1. Novel Framework (CP-SLS-MPC): A fast, optimization-based planner that jointly learns dynamics, bounds model error via WCP, and computes robust control policies via SLS.
2. Theoretical Guarantees:
   • Proves finite-sample probabilistic safety for the true system under distributional drift.
   • Derives a bound on the coverage gap based on the density of calibration points and the Lipschitz continuity of the error distribution shift.
3. State-Dependent Uncertainty: Moves beyond fixed spherical bounds to use learned, state-control-dependent ellipsoidal bounds, allowing for less conservative planning in well-modeled regions.
4. Active Uncertainty Reduction: Introduces a mechanism to guide the robot toward low-error regions, improving both safety and performance.
5. Empirical Validation: Demonstrates success on high-dimensional nonlinear systems (4D car and 12D quadcopter) in challenging OOD scenarios.

4. Experimental Results

The method was evaluated on two systems: a 4D Dubins Car and a 12D Quadcopter.

• Baselines: Compared against "Vanilla MPC" (no robustness) and "CP-Ball" (fixed spherical error bounds).
• In-Distribution (ID): CP-SLS-MPC maintained performance comparable to baselines with minimal computational overhead.
• Out-of-Distribution (OOD) Performance:
  • Safety: In OOD scenarios (e.g., friction changes, attractive obstacles), Vanilla MPC and CP-Ball frequently failed (collisions). CP-SLS-MPC successfully reached goals by maintaining safe distances.
  • Adaptability: As the robot collected online data in OOD regions, the CP error bounds tightened, allowing the robot to become less conservative over time.
  • Tube Size: The state-dependent ellipsoids resulted in smaller, more accurate reachable tubes compared to fixed spherical bounds, leading to less conservative (more efficient) paths.
  • Active Uncertainty: When enabled, the planner successfully avoided high-uncertainty regions, resulting in smoother trajectories and lower prediction errors (Table 2 in Appendix).
• Scalability: The method successfully scaled to a 12D quadcopter, achieving a 80% success rate in OOD scenarios compared to 40% for the baseline, demonstrating applicability to complex, high-dimensional robotics.

5. Significance

This work bridges the gap between data-driven learning and formal safety guarantees in robotics.

• Beyond "Safe Domains": It challenges the paradigm that robots must stay strictly within training data. It enables temporary OOD operation with mathematically provable safety.
• Efficiency: By leveraging SLS and SCP, it achieves near-real-time replanning speeds suitable for dynamic environments.
• Practicality: The use of weighted conformal prediction allows the system to adapt to non-stationary environments and distribution shifts without requiring retraining, making it highly relevant for real-world deployment where data scarcity and environmental changes are inevitable.

In summary, CP-SLS-MPC provides a robust, theoretically grounded, and computationally efficient solution for safe robotic control in uncertain, changing environments.