# Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systems

arXiv: https://arxiv.org/abs/2005.04374 · HTML: https://mlsanity.com/paper/2005.04374/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2020.

## Built on by (most cited first, 15 of 15)

- [Neural-Fly Enables Rapid Learning for Agile Flight in Strong Winds](https://mlsanity.com/paper/2205.06908/) (2022)
- [Agile But Safe: Learning Collision-Free High-Speed Legged Locomotion](https://mlsanity.com/paper/2401.17583/) (2024)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Optimal Exploration for Model-Based RL in Nonlinear Systems](https://mlsanity.com/paper/2306.09210/) (2023)
- [Learning-based Robust Motion Planning with Guaranteed Stability: A Contraction Theory Approach](https://mlsanity.com/paper/2102.12668/) (2021)
- [Infinite Time Horizon Safety of Bayesian Neural Networks](https://mlsanity.com/paper/2111.03165/) (2021)
- [Failure-Aware RL: Reliable Offline-to-Online Reinforcement Learning with Self-Recovery for Real-World Manipulation](https://mlsanity.com/paper/2601.07821/) (2026)
- [One Filter to Deploy Them All: Robust Safety for Quadrupedal Navigation in Unknown Environments](https://mlsanity.com/paper/2412.09989/) (2024)
- [Trajectory Optimization of Chance-Constrained Nonlinear Stochastic Systems for Motion Planning Under Uncertainty](https://mlsanity.com/paper/2106.02801/) (2021)
- [A Generalizable Physics-informed Learning Framework for Risk Probability Estimation](https://mlsanity.com/paper/2305.06432/) (2023)
- [Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems](https://mlsanity.com/paper/2407.08868/) (2024)
- [Safe Learning Control with Optimality and Stability Guarantees](https://mlsanity.com/paper/2501.15373/) (2025)
- [Learning to Control an Unstable System with One Minute of Data: Leveraging Gaussian Process Differentiation in Predictive Control](https://mlsanity.com/paper/2103.04548/) (2021)
- [Discrete fully probabilistic design: towards a control pipeline for the synthesis of policies from examples](https://mlsanity.com/paper/2112.11210/) (2021)
- [Digital Twin Calibration with Model-Based Reinforcement Learning](https://mlsanity.com/paper/2501.02205/) (2025)

## Abstract (arXiv)

Learning-based control algorithms require data collection with abundant supervision for training. Safe exploration algorithms ensure the safety of this data collection process even when only partial knowledge is available. We present a new approach for optimal motion planning with safe exploration that integrates chance-constrained stochastic optimal control with dynamics learning and feedback control. We derive an iterative convex optimization algorithm that solves an underlineInformation-cost underlineStochastic underlineNonlinear underlineOptimal underlineControl problem (Info-SNOC). The optimization objective encodes control cost for performance and exploration cost for learning, and the safety is incorporated as distributionally robust chance constraints. The dynamics are predicted from a robust regression model that is learned from data. The Info-SNOC algorithm is used to compute a sub-optimal pool of safe motion plans that aid in exploration for learning unknown residual dynamics under safety constraints. A stable feedback controller is used to execute the motion plan and collect data for model learning. We prove the safety of rollout from our exploration method and reduction in uncertainty over epochs, thereby guaranteeing the consistency of our learning method. We validate the effectiveness of Info-SNOC by designing and implementing a pool of safe trajectories for a planar robot. We demonstrate that our approach has higher success rate in ensuring safety when compared to a deterministic trajectory optimization approach.