Alex Nash

AIJ Journal 2021 Journal Article

Path-length analysis for grid-based path planning

• James P. Bailey
• Alex Nash
• Craig A. Tovey
• Sven Koenig

In video games and robotics, one often discretizes a continuous 2D environment into a regular grid with blocked and unblocked cells and then finds shortest paths for the agents on the resulting grid graph. Shortest grid paths, of course, are not necessarily true shortest paths in the continuous 2D environment. In this article, we therefore study how much longer a shortest grid path can be than a corresponding true shortest path on all regular grids with blocked and unblocked cells that tessellate continuous 2D environments. We study 5 different vertex connectivities that result from both different tessellations and different definitions of the neighbors of a vertex. Our path-length analysis yields either tight or asymptotically tight worst-case bounds in a unified framework. Our results show that the percentage by which a shortest grid path can be longer than a corresponding true shortest path decreases as the vertex connectivity increases. Our path-length analysis is topical because it determines the largest path-length reduction possible for any-angle path-planning algorithms (and thus their benefit), a class of path-planning algorithms in artificial intelligence and robotics that has become popular.

SoCS Conference 2010 Conference Paper

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

• Alex Nash
• Sven Koenig
• Craig A. Tovey

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.

AAAI Conference 2010 Conference Paper

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

• Alex Nash
• Sven Koenig
• Craig Tovey

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 8neighbor 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.

AAAI Conference 2010 Conference Paper

Model AI Assignments

• Todd Neller
• John DeNero
• Dan Klein
• Sven Koenig
• William Yeoh
• Xiaoming Zheng
• Kenny Daniel
• Alex Nash

IJCAI Conference 2009 Conference Paper

• Alex Nash
• Sven Koenig
• Maxim Likhachev

We study path planning on grids with blocked and unblocked cells. Any-angle path-planning algorithms find short paths fast because they propagate information along grid edges without constraining the resulting paths to grid edges. Incremental pathplanning algorithms solve a series of similar pathplanning problems faster than repeated single-shot searches because they reuse information from the previous search to speed up the next one. In this paper, we combine these ideas by making the anyangle path-planning algorithm Basic Theta* incremental. This is non-trivial because Basic Theta* does not fit the standard assumption that the parent of a vertex in the search tree must also be its neighbor. We present Incremental Phi* and show experimentally that it can speed up Basic Theta* by about one order of magnitude for path planning with the freespace assumption.

AAAI Conference 2007 Conference Paper

Theta*: Any-Angle Path Planning on Grids

• Alex Nash
• Sven Koenig

Grids with blocked and unblocked cells are often used to represent terrain in computer games and robotics. However, paths formed by grid edges can be sub-optimal and unrealistic looking, since the possible headings are artificially constrained. We present Theta*, a variant of A*, that propagates information along grid edges without constraining the paths to grid edges. Theta* is simple, fast and finds short and realistic looking paths. We compare Theta* against both Field D*, the only other variant of A* that propagates information along grid edges without constraining the paths to grid edges, and A* with post-smoothed paths. Although neither path planning method is guaranteed to find shortest paths, we show experimentally that Theta* finds shorter and more realistic looking paths than either of these existing techniques.