# Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems

arXiv: https://arxiv.org/abs/2212.06874 · HTML: https://mlsanity.com/paper/2212.06874/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 36 references that are in the map.
- First posted on arXiv 13 Dec 2022.

## Builds on (most cited first, 26 of 26)

- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Learning Certified Control using Contraction Metric](https://mlsanity.com/paper/2011.12569/) (2020)
- [Learning Stability Certificates from Data](https://mlsanity.com/paper/2008.05952/) (2020)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes](https://mlsanity.com/paper/1603.04915/) (2016)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularization](https://mlsanity.com/paper/1907.13122/) (2019)
- [Funnel Libraries for Real-Time Robust Feedback Motion Planning](https://mlsanity.com/paper/1601.04037/) (2016)
- [Planning with Learned Dynamics: Probabilistic Guarantees on Safety and Reachability via Lipschitz Constants](https://mlsanity.com/paper/2010.08993/) (2020)
- [Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach](https://mlsanity.com/paper/2006.04361/) (2020)
- [Safe Output Feedback Motion Planning from Images via Learned Perception Modules and Contraction Theory](https://mlsanity.com/paper/2206.06553/) (2022)
- [Deep Learning Tubes for Tube MPC](https://mlsanity.com/paper/2002.01587/) (2020)
- [Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems](https://mlsanity.com/paper/2104.08695/) (2021)
- [Safe Feedback Motion Planning: A Contraction Theory and L1-Adaptive Control Based Approach](https://mlsanity.com/paper/2004.01142/) (2020)
- [Learning Density Distribution of Reachable States for Autonomous Systems](https://mlsanity.com/paper/2109.06728/) (2021)
- [Uncertainty-Aware Constraint Learning for Adaptive Safe Motion Planning from Demonstrations](https://mlsanity.com/paper/2011.04141/) (2020)
- [Technical Report: Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures](https://mlsanity.com/paper/1305.7484/) (2013)
- [Non-Gaussian Risk Bounded Trajectory Optimization for Stochastic Nonlinear Systems in Uncertain Environments](https://mlsanity.com/paper/2203.03038/) (2022)
- [Gaussian Process Constraint Learning for Scalable Chance-Constrained Motion Planning from Demonstrations](https://mlsanity.com/paper/2112.04612/) (2021)
- [Nonlinear Stabilization via Control Contraction Metrics: a Pseudospectral Approach for Computing Geodesics](https://mlsanity.com/paper/1607.04340/) (2016)
- [Contraction Metrics in Adaptive Nonlinear Control](https://mlsanity.com/paper/1912.13138/) (2019)

## Built on by (most cited first, 11 of 11)

- [Fighting Uncertainty with Gradients: Offline Reinforcement Learning via Diffusion Score Matching](https://mlsanity.com/paper/2306.14079/) (2023)
- [Local Linearity of LLMs Enables Activation Steering via Model-Based Linear Optimal Control](https://mlsanity.com/paper/2604.19018/) (2026)
- [Synthesizing Stable Reduced-Order Visuomotor Policies for Nonlinear Systems via Sums-of-Squares Optimization](https://mlsanity.com/paper/2304.12405/) (2023)
- [Safety Beyond the Training Data: Robust Out-of-Distribution MPC via Conformalized System Level Synthesis](https://mlsanity.com/paper/2602.12047/) (2026)
- [Safe Large-Scale Robust Nonlinear MPC in Milliseconds via Reachability-Constrained System Level Synthesis on the GPU](https://mlsanity.com/paper/2604.07644/) (2026)
- [Pixels to Proofs: Probabilistically-Safe Latent World Model Control via Parallel Conformal Robust MPC](https://mlsanity.com/paper/2606.15594/) (2026)
- [VISION-SLS: Safe Perception-Based Control from Learned Visual Representations via System Level Synthesis](https://mlsanity.com/paper/2604.24894/) (2026)
- [Formal Safety Verification and Refinement for Generative Motion Planners via Certified Local Stabilization](https://mlsanity.com/paper/2509.19688/) (2025)
- [Parallel Differentiable Reachability for Learning and Planning with Certified Neural Dynamics and Controllers](https://mlsanity.com/paper/2605.25346/) (2026)
- [Scalable Data-Driven Reachability Analysis and Control via Koopman Operators with Conformal Coverage Guarantees](https://mlsanity.com/paper/2601.01076/) (2026)
- [Statistical Contraction for Chance-Constrained Trajectory Optimization of Non-Gaussian Stochastic Systems](https://mlsanity.com/paper/2603.07092/) (2026)

## Abstract (arXiv)

We present a method for providing statistical guarantees on runtime safety and goal reachability for integrated planning and control of a class of systems with unknown nonlinear stochastic underactuated dynamics. Specifically, given a dynamics dataset, our method jointly learns a mean dynamics model, a spatially-varying disturbance bound that captures the effect of noise and model mismatch, and a feedback controller based on contraction theory that stabilizes the learned dynamics. We propose a sampling-based planner that uses the mean dynamics model and simultaneously bounds the closed-loop tracking error via a learned disturbance bound. We employ techniques from Extreme Value Theory (EVT) to estimate, to a specified level of confidence, several constants which characterize the learned components and govern the size of the tracking error bound. This ensures plans are guaranteed to be safely tracked at runtime. We validate that our guarantees translate to empirical safety in simulation on a 10D quadrotor, and in the real world on a physical CrazyFlie quadrotor and Clearpath Jackal robot, whereas baselines that ignore the model error and stochasticity are unsafe.