# Deep Learning Tubes for Tube MPC

arXiv: https://arxiv.org/abs/2002.01587 · HTML: https://mlsanity.com/paper/2002.01587/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2020.

## Built on by (most cited first, 14 of 14)

- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [NeBula: Quest for Robotic Autonomy in Challenging Environments; TEAM CoSTAR at the DARPA Subterranean Challenge](https://mlsanity.com/paper/2103.11470/) (2021)
- [STEP: Stochastic Traversability Evaluation and Planning for Risk-Aware Off-road Navigation](https://mlsanity.com/paper/2103.02828/) (2021)
- [Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems](https://mlsanity.com/paper/2104.08695/) (2021)
- [Learning Risk-aware Costmaps for Traversability in Challenging Environments](https://mlsanity.com/paper/2107.11722/) (2021)
- [Demonstration-Efficient Guided Policy Search via Imitation of Robust Tube MPC](https://mlsanity.com/paper/2109.09910/) (2021)
- [Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems](https://mlsanity.com/paper/2212.06874/) (2022)
- [Dual Online Stein Variational Inference for Control and Dynamics](https://mlsanity.com/paper/2103.12890/) (2021)
- [Real-Time Decision-Making for Digital Twin in Additive Manufacturing with Model Predictive Control using Time-Series Deep Neural Networks](https://mlsanity.com/paper/2501.07601/) (2025)
- [Bridging Model-based Safety and Model-free Reinforcement Learning through System Identification of Low Dimensional Linear Models](https://mlsanity.com/paper/2205.05787/) (2022)
- [Risk-aware Meta-level Decision Making for Exploration Under Uncertainty](https://mlsanity.com/paper/2209.05580/) (2022)
- [Dynamic Tube MPC: Learning Tube Dynamics with Massively Parallel Simulation for Robust Safety in Practice](https://mlsanity.com/paper/2411.15350/) (2024)
- [Dropout MPC: An Ensemble Neural MPC Approach for Systems with Learned Dynamics](https://mlsanity.com/paper/2406.02497/) (2024)
- [A Continual Validation, Updating, and Decision-Making Framework for Self-Adaptive Digital Twins via Robust Model Predictive Control: A Case Study in Additive Manufacturing](https://mlsanity.com/paper/2607.18164/) (2026)

## Abstract (arXiv)

Learning-based control aims to construct models of a system to use for planning or trajectory optimization, e.g. in model-based reinforcement learning. In order to obtain guarantees of safety in this context, uncertainty must be accurately quantified. This uncertainty may come from errors in learning (due to a lack of data, for example), or may be inherent to the system. Propagating uncertainty forward in learned dynamics models is a difficult problem. In this work we use deep learning to obtain expressive and flexible models of how distributions of trajectories behave, which we then use for nonlinear Model Predictive Control (MPC). We introduce a deep quantile regression framework for control that enforces probabilistic quantile bounds and quantifies epistemic uncertainty. Using our method we explore three different approaches for learning tubes that contain the possible trajectories of the system, and demonstrate how to use each of them in a Tube MPC scheme. We prove these schemes are recursively feasible and satisfy constraints with a desired margin of probability. We present experiments in simulation on a nonlinear quadrotor system, demonstrating the practical efficacy of these ideas.