# Optimal Convex Cover as Collision-free Space Approximation for Trajectory Generation

**Authors:** Yuwei Wu, Igor Spasojevic, Pratik Chaudhari, Vijay Kumar  
**arXiv:** [2406.09631](https://arxiv.org/abs/2406.09631)  
**Date:** 2024-06-13  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2406.09631v1/

## Key points

- Authors propose an online iterative algorithm for finding a convex cover to under-approximate free space for autonomous navigation
- Convex cover comprises polytopes representing obstacle-free space for safe trajectories
- Objective is to determine Safe Flight Corridors (SFC) by identifying overlapping polytopes with maximum volumes encompassing specified waypoints
- Process involves addressing constraints at waypoints through two alternating stages in a joint optimization problem framework
- Authors draw inspiration from Alternating Direction Method of Multipliers (ADMM) and use partially distributed variables
- Extensive validation exercises demonstrate effectiveness of proposed algorithm in various environments
- Practical applications showcased in two-stage motion planning scenarios
- Approach enhances autonomous navigation capabilities and optimizes trajectory generation processes in dynamic environments, paving the way for more efficient and reliable autonomous systems

## Summary

In their paper titled "Optimal Convex Cover as Collision-free Space Approximation for Trajectory Generation," authors Yuwei Wu, Igor Spasojevic, Pratik Chaudhari, and Vijay Kumar propose an online iterative algorithm for finding a suitable convex cover to under-approximate the free space for autonomous navigation. This convex cover comprises a collection of polytopes that represent obstacle-free space and enable the identification of safe trajectories for robots. The key objective is to determine the Safe Flight Corridors (SFC) by identifying overlapping polytopes with maximum volumes that encompass specified waypoints set by geometric or kinematic planners. The process involves addressing constraints at these waypoints through two alternating stages within a joint optimization problem framework. To solve this challenge, the authors draw inspiration from the Alternating Direction Method of Multipliers (ADMM) and utilize partially distributed variables. Through extensive validation exercises in various environments, the effectiveness of their proposed algorithm is demonstrated. Additionally, its practical applications in two-stage motion planning scenarios are showcased. This innovative approach not only enhances autonomous navigation capabilities but also optimizes trajectory generation processes in dynamic environments. By leveraging advanced algorithms and iterative techniques, this research paves the way for more efficient and reliable autonomous systems in real-world applications.