# Provably Safe Trajectory Optimization in the Presence of Uncertain Convex Obstacles

arXiv: https://arxiv.org/abs/2003.07811 · HTML: https://mlsanity.com/paper/2003.07811/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2020.

## Built on by (most cited first, 8 of 8)

- [Position-Aware Relation Learning for RGB-Thermal Salient Object Detection](https://mlsanity.com/paper/2209.10158/) (2022)
- [Generalizing Multi-Objective Search via Objective-Aggregation Functions](https://mlsanity.com/paper/2509.22085/) (2025)
- [Improving the Resilience of Quadrotors in Underground Environments by Combining Learning-based and Safety Controllers](https://mlsanity.com/paper/2509.02808/) (2025)
- [Real-Time Tube-Based Non-Gaussian Risk Bounded Motion Planning for Stochastic Nonlinear Systems in Uncertain Environments via Motion Primitives](https://mlsanity.com/paper/2303.01631/) (2023)
- [Convex Risk Bounded Continuous-Time Trajectory Planning and Tube Design in Uncertain Nonconvex Environments](https://mlsanity.com/paper/2305.17291/) (2023)
- [MMD-OPT : Maximum Mean Discrepancy Based Sample Efficient Collision Risk Minimization for Autonomous Driving](https://mlsanity.com/paper/2412.09121/) (2024)
- [MoMo: Conditioned Contrastive Representation Learning for Preference-Modulated Planning](https://mlsanity.com/paper/2605.08512/) (2026)
- [Marginal Calibration Does Not Compose: Hidden Dependence in Modular Robot Navigation](https://mlsanity.com/paper/2609.23731/) (2026)

## Abstract (arXiv)

Real-world environments are inherently uncertain, and to operate safely in these environments robots must be able to plan around this uncertainty. In the context of motion planning, we desire systems that can maintain an acceptable level of safety as the robot moves, even when the exact locations of nearby obstacles are not known. In this paper, we solve this chance-constrained motion planning problem using a sequential convex optimization framework. To constrain the risk of collision incurred by planned movements, we employ geometric objects called ε-shadows to compute upper bounds on the risk of collision between the robot and uncertain obstacles. We use these ε-shadow-based estimates as constraints in a nonlinear trajectory optimization problem, which we then solve by iteratively linearizing the non-convex risk constraints. This sequential optimization approach quickly finds trajectories that accomplish the desired motion while maintaining a user-specified limit on collision risk. Our method can be applied to robots and environments with arbitrary convex geometry; even in complex environments, it runs in less than a second and provides provable guarantees on the safety of planned trajectories, enabling fast, reactive, and safe robot motion in realistic environments.