# Convex Risk Bounded Continuous-Time Trajectory Planning and Tube Design in Uncertain Nonconvex Environments

arXiv: https://arxiv.org/abs/2305.17291 · HTML: https://mlsanity.com/paper/2305.17291/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 15 of 15)

- [MultiPath: Multiple Probabilistic Anchor Trajectory Hypotheses for Behavior Prediction](https://mlsanity.com/paper/1910.05449/) (2019)
- [DenseTNT: End-to-end Trajectory Prediction from Dense Goal Sets](https://mlsanity.com/paper/2108.09640/) (2021)
- [MultiPath++: Efficient Information Fusion and Trajectory Aggregation for Behavior Prediction](https://mlsanity.com/paper/2111.14973/) (2021)
- [Monte Carlo Motion Planning for Robot Trajectory Optimization Under Uncertainty](https://mlsanity.com/paper/1504.08053/) (2015)
- [Provably Safe Robot Navigation with Obstacle Uncertainty](https://mlsanity.com/paper/1705.10907/) (2017)
- [KEMP: Keyframe-Based Hierarchical End-to-End Deep Model for Long-Term Trajectory Prediction](https://mlsanity.com/paper/2205.04624/) (2022)
- [Provably Safe Trajectory Optimization in the Presence of Uncertain Convex Obstacles](https://mlsanity.com/paper/2003.07811/) (2020)
- [Non-Gaussian Risk Bounded Trajectory Optimization for Stochastic Nonlinear Systems in Uncertain Environments](https://mlsanity.com/paper/2203.03038/) (2022)
- [Chance Constrained Motion Planning for High-Dimensional Robots](https://mlsanity.com/paper/1811.03073/) (2018)
- [Convex Risk Bounded Continuous-Time Trajectory Planning in Uncertain Nonconvex Environments](https://mlsanity.com/paper/2106.05489/) (2021)
- [Moment-Sum-Of-Squares Approach For Fast Risk Estimation In Uncertain Environments](https://mlsanity.com/paper/1810.01577/) (2018)
- [Chance-constrained optimization with tight confidence bounds](https://mlsanity.com/paper/1711.03747/) (2017)
- [Real-Time Risk-Bounded Tube-Based Trajectory Safety Verification](https://mlsanity.com/paper/2110.00233/) (2021)
- [Time-Varying Semidefinite Programs](https://mlsanity.com/paper/1808.03994/) (2018)
- [Piecewise-Linear Motion Planning amidst Static, Moving, or Morphing Obstacles](https://mlsanity.com/paper/2010.08167/) (2020)

## Abstract (arXiv)

In this paper, we address the trajectory planning problem in uncertain nonconvex static and dynamic environments that contain obstacles with probabilistic location, size, and geometry. To address this problem, we provide a risk bounded trajectory planning method that looks for continuous-time trajectories with guaranteed bounded risk over the planning time horizon. Risk is defined as the probability of collision with uncertain obstacles. Existing approaches to address risk bounded trajectory planning problems either are limited to Gaussian uncertainties and convex obstacles or rely on sampling-based methods that need uncertainty samples and time discretization. To address the risk bounded trajectory planning problem, we leverage the notion of risk contours to transform the risk bounded planning problem into a deterministic optimization problem. Risk contours are the set of all points in the uncertain environment with guaranteed bounded risk. The obtained deterministic optimization is, in general, nonlinear and nonconvex time-varying optimization. We provide convex methods based on sum-of-squares optimization to efficiently solve the obtained nonconvex time-varying optimization problem and obtain the continuous-time risk bounded trajectories without time discretization. The provided approach deals with arbitrary (and known) probabilistic uncertainties, nonconvex and nonlinear, static and dynamic obstacles, and is suitable for online trajectory planning problems. In addition, we provide convex methods based on sum-of-squares optimization to build the max-sized tube with respect to its parameterization along the trajectory so that any state inside the tube is guaranteed to have bounded risk.