# Planning with Learned Dynamics: Probabilistic Guarantees on Safety and Reachability via Lipschitz Constants

arXiv: https://arxiv.org/abs/2010.08993 · HTML: https://mlsanity.com/paper/2010.08993/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2020.

## Built on by (most cited first, 25 of 25)

- [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)
- [Robust Uncertainty Bounds in Reproducing Kernel Hilbert Spaces: A Convex Optimization Approach](https://mlsanity.com/paper/2104.09582/) (2021)
- [Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems](https://mlsanity.com/paper/2104.08695/) (2021)
- [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)
- [Local Linearity of LLMs Enables Activation Steering via Model-Based Linear Optimal Control](https://mlsanity.com/paper/2604.19018/) (2026)
- [How Safe Am I Given What I See? Calibrated Prediction of Safety Chances for Image-Controlled Autonomy](https://mlsanity.com/paper/2308.12252/) (2023)
- [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)
- [Reimagination with Test-time Observation Interventions: Distractor-Robust World Model Predictions for Visual Model Predictive Control](https://mlsanity.com/paper/2506.16565/) (2025)
- [Formal Verification of Unknown Dynamical Systems via Gaussian Process Regression](https://mlsanity.com/paper/2201.00655/) (2021)
- [Promises of Deep Kernel Learning for Control Synthesis](https://mlsanity.com/paper/2309.06569/) (2023)
- [Q-Distribution guided Q-learning for offline reinforcement learning: Uncertainty penalized Q-value via consistency model](https://mlsanity.com/paper/2410.20312/) (2024)
- [Pixels to Proofs: Probabilistically-Safe Latent World Model Control via Parallel Conformal Robust MPC](https://mlsanity.com/paper/2606.15594/) (2026)
- [Learning Contraction Policies from Offline Data](https://mlsanity.com/paper/2112.05911/) (2021)
- [Certified Training with Branch-and-Bound for Lyapunov-stable Neural Control](https://mlsanity.com/paper/2411.18235/) (2024)
- [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)
- [Data-driven Acceleration of MPC with Guarantees](https://mlsanity.com/paper/2511.13588/) (2025)
- [Parallel Differentiable Reachability for Learning and Planning with Certified Neural Dynamics and Controllers](https://mlsanity.com/paper/2605.25346/) (2026)
- [Robust Model Based Reinforcement Learning Using L1 Adaptive Control](https://mlsanity.com/paper/2403.14860/) (2024)
- [How Safe Will I Be Given What I Saw? Calibrated Prediction of Safety Chances for Image-Controlled Autonomy](https://mlsanity.com/paper/2508.09346/) (2025)
- [Scalable Data-Driven Reachability Analysis and Control via Koopman Operators with Conformal Coverage Guarantees](https://mlsanity.com/paper/2601.01076/) (2026)

## Abstract (arXiv)

We present a method for feedback motion planning of systems with unknown dynamics which provides probabilistic guarantees on safety, reachability, and goal stability. To find a domain in which a learned control-affine approximation of the true dynamics can be trusted, we estimate the Lipschitz constant of the difference between the true and learned dynamics, and ensure the estimate is valid with a given probability. Provided the system has at least as many controls as states, we also derive existence conditions for a one-step feedback law which can keep the real system within a small bound of a nominal trajectory planned with the learned dynamics. Our method imposes the feedback law existence as a constraint in a sampling-based planner, which returns a feedback policy around a nominal plan ensuring that, if the Lipschitz constant estimate is valid, the true system is safe during plan execution, reaches the goal, and is ultimately invariant in a small set about the goal. We demonstrate our approach by planning using learned models of a 6D quadrotor and a 7DOF Kuka arm. We show that a baseline which plans using the same learned dynamics without considering the error bound or the existence of the feedback law can fail to stabilize around the plan and become unsafe.