# Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems

arXiv: https://arxiv.org/abs/2104.08695 · HTML: https://mlsanity.com/paper/2104.08695/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 29 references that are in the map.
- First posted on arXiv 18 Apr 2021.

## Builds on (most cited first, 17 of 17)

- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Evaluating the Robustness of Neural Networks: An Extreme Value Theory Approach](https://mlsanity.com/paper/1801.10578/) (2018)
- [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)
- [Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularization](https://mlsanity.com/paper/1907.13122/) (2019)
- [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)
- [Deep Learning Tubes for Tube MPC](https://mlsanity.com/paper/2002.01587/) (2020)
- [Safe Feedback Motion Planning: A Contraction Theory and L1-Adaptive Control Based Approach](https://mlsanity.com/paper/2004.01142/) (2020)
- [GuSTO: Guaranteed Sequential Trajectory Optimization via Sequential Convex Programming](https://mlsanity.com/paper/1903.00155/) (2019)
- [Asymptotically Optimal Planning by Feasible Kinodynamic Planning in State-Cost Space](https://mlsanity.com/paper/1505.04098/) (2015)
- [Nonlinear Stabilization via Control Contraction Metrics: a Pseudospectral Approach for Computing Geodesics](https://mlsanity.com/paper/1607.04340/) (2016)

## Built on by (most cited first, 13 of 13)

- [Safe Output Feedback Motion Planning from Images via Learned Perception Modules and Contraction Theory](https://mlsanity.com/paper/2206.06553/) (2022)
- [Fighting Uncertainty with Gradients: Offline Reinforcement Learning via Diffusion Score Matching](https://mlsanity.com/paper/2306.14079/) (2023)
- [Probabilities Are Not Enough: Formal Controller Synthesis for Stochastic Dynamical Models with Epistemic Uncertainty](https://mlsanity.com/paper/2210.05989/) (2022)
- [Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems](https://mlsanity.com/paper/2212.06874/) (2022)
- [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)
- [Bridging Model-based Safety and Model-free Reinforcement Learning through System Identification of Low Dimensional Linear Models](https://mlsanity.com/paper/2205.05787/) (2022)
- [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)
- [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)
- [Guaranteed Trajectory Tracking under Learned Dynamics with Contraction Metrics and Disturbance Estimation](https://mlsanity.com/paper/2112.08222/) (2021)

## Abstract (arXiv)

We present a method for contraction-based feedback motion planning of locally incrementally exponentially stabilizable systems with unknown dynamics that provides probabilistic safety and reachability guarantees. Given a dynamics dataset, our method learns a deep control-affine approximation of the dynamics. To find a trusted domain where this model can be used for planning, we obtain an estimate of the Lipschitz constant of the model error, which is valid with a given probability, in a region around the training data, providing a local, spatially-varying model error bound. We derive a trajectory tracking error bound for a contraction-based controller that is subjected to this model error, and then learn a controller that optimizes this tracking bound. With a given probability, we verify the correctness of the controller and tracking error bound in the trusted domain. We then use the trajectory error bound together with the trusted domain to guide a sampling-based planner to return trajectories that can be robustly tracked in execution. We show results on a 4D car, a 6D quadrotor, and a 22D deformable object manipulation task, showing our method plans safely with learned models of high-dimensional underactuated systems, while baselines that plan without considering the tracking error bound or the trusted domain can fail to stabilize the system and become unsafe.