Lazy Theta*: Any-Angle Path Planning and Path Length Analysis in 3D

Abstract

Grids with blocked and unblocked cells are often used to represent continuous 2D and 3D environments in robotics and video games. The shortest paths formed by the edges of 8-neighbor 2D grids can be up to 8% longer than the shortest paths in the continuous environment. Theta* typically finds much shorter paths than that by propagating information along graph edges (to achieve short runtimes) without constraining paths to be formed by graph edges (to find short "any-angle" paths). We show in this paper that the shortest paths formed by the edges of 26-neighbor 3D grids can be 13% longer than the shortest paths in the continuous environment, which highlights the need for smart path planning algorithms in 3D. Theta* can be applied to 3D grids in a straight-forward manner, but it performs a line-of-sight check for each unexpanded visible neighbor of each expanded vertex and thus it performs many more line-of-sight checks per expanded vertex on a 26-neighbor 3D grid than on an 8-neighbor 2D grid. We therefore introduce Lazy Theta*, a variant of Theta* which uses lazy evaluation to perform only one line-of-sight check per expanded vertex (but with slightly more expanded vertices). We show experimentally that Lazy Theta* finds paths faster than Theta* on 26-neighbor 3D grids, with one order of magnitude fewer line-of-sight checks and without an increase in path length.

Towards Time-Optimal Any-Angle Path Planning With Dynamic Obstacles

“…checking for the collisions with the dynamic obstacles, computing the earliest arrival time to a destination that avoids collision with the latter etc. This is conceptually similar to performing lazy expansions like in Lazy Theta* (Nash, Koenig, and Tovey 2010). If the transition is valid and…”

“…The most widely known algorithm for (sub-optimal) anyangle path finding in a static environment (represented as a grid) is apparently Theta* (Nash et al 2007). This algorithm was modified and enhanced in numerous works (Nash, Koenig, and Tovey 2010;Oh and Leong 2016), etc. Other prominent algorithms of that kind are Block A* (Yap et al 2011) and Field D* (Ferguson and Stentz 2006).…”

Speeding Up A* Search on Visibility Graphs Defined Over Quadtrees to Enable Long Distance Path Planning for Unmanned Surface Vehicles

“…However, paths produced by these methods may not be optimal. Notable methods belonging to this class are Incremental Phi* (Nash, Koenig, and Likhachev 2009), Theta* (Daniel et al 2010) and its variants (Nash, Koenig, and Tovey 2010;Uras and Koenig 2015b;Tovey, Koenig, and Nash 2015), and Field D* (Ferguson and Stentz 2006).…”

Modeling of a wheeled humanoid robot and hybrid algorithm-based path planning of wheel base for the dynamic obstacles avoidance

“…The introduction of path planning using A star algorithm is explained by Hart et al (1968). The different variants of A star algorithms such as Theta star, Lazy theta star, Any angle propagation A star are demonstrated by De Filippis et al (2012), Nash et al (2010) and Daniel et al (2010). Path planning algorithm using updated version of A star algorithm [Duchoň et al (2014)] is carried out.…”