Theta*: Any-Angle Path Planning on Grids (1401.3843v1)

Abstract: Grids with blocked and unblocked cells are often used to represent terrain in robotics and video games. However, paths formed by grid edges can be longer than true shortest paths in the terrain since their headings are artificially constrained. We present two new correct and complete any-angle path-planning algorithms that avoid this shortcoming. Basic Theta* and Angle-Propagation Theta* are both variants of A* that propagate information along grid edges without constraining paths to grid edges. Basic Theta* is simple to understand and implement, fast and finds short paths. However, it is not guaranteed to find true shortest paths. Angle-Propagation Theta* achieves a better worst-case complexity per vertex expansion than Basic Theta* by propagating angle ranges when it expands vertices, but is more complex, not as fast and finds slightly longer paths. We refer to Basic Theta* and Angle-Propagation Theta* collectively as Theta*. Theta* has unique properties, which we analyze in detail. We show experimentally that it finds shorter paths than both A* with post-smoothed paths and Field D* (the only other version of A* we know of that propagates information along grid edges without constraining paths to grid edges) with a runtime comparable to that of A* on grids. Finally, we extend Theta* to grids that contain unblocked cells with non-uniform traversal costs and introduce variants of Theta* which provide different tradeoffs between path length and runtime.

• The paper presents two any-angle path planning algorithms, Basic Theta* and AP Theta*, that extend A* by incorporating flexible line-of-sight checks.
• It introduces a novel methodology that reduces rigid grid constraints, resulting in smoother and shorter paths closer to the true optimal routes.
• Experimental results demonstrate that while AP Theta* offers theoretical improvements, Basic Theta* achieves faster runtime and practical usability in robotics and gaming.

Overview of Theta*: Any-Angle Path Planning on Grids

The paper "Theta*: Any-Angle Path Planning on Grids" by Kenny Daniel, Alex Nash, Sven Koenig, and Ariel Felner introduces two new any-angle path-planning algorithms designed to address the inherent limitations of grid-based path planning. These algorithms, Basic Theta* and Angle-Propagation Theta* (AP Theta*), are variants of A* and aim to overcome the heading constraints imposed by traditional grid-based methods, which can lead to suboptimal longer paths.

Context and Motivation

Grid-based representations of terrain are prevalent in robotics and video games due to their simplicity and ease of implementation. However, standard A* path planning on grids restricts paths to grid edges, resulting in paths that are often significantly longer than necessary (45-degree increments). This paper addresses the need for algorithms that can produce high-quality paths without these artificial constraints, achieving shorter, more realistic paths.

Algorithms: Basic Theta* and AP Theta*

The authors present two algorithms:

• Basic Theta: This algorithm extends A by allowing the parent of a vertex to be any node, not just neighboring nodes, if it results in a shorter path and maintains the "line-of-sight" (LOS). The algorithm is simple to understand, easy to implement, and performs efficiently in finding short paths. However, it is not guaranteed to find the true shortest paths due to potential LOS constraints at intermediate steps.
• AP Theta: This more complex variant introduces angle ranges to handle LOS checks more efficiently. It reduces the worst-case complexity per vertex expansion from linear to constant. While this improvement can theoretically reduce runtime complexity, AP Theta is noted to be slower in practice and tends to generate slightly longer paths than Basic Theta*.

Performance Evaluation and Comparison

The algorithms were evaluated against A* on grids, A* with post-smoothed paths (A* PS), Field D* (FD*), and A* on visibility graphs. Key findings include:

• Path Quality: Theta* algorithms produce paths significantly closer to the true optimal paths with fewer unnecessary heading changes compared to A* on grids and FD*, displaying near-optimal performance with respect to path length.
• Runtime: Although Basic Theta* and AP Theta* are slower than A* on grids, they attain comparable runtime, particularly Basic Theta*. Moreover, Basic Theta* maintains simplicity in its implementation, contributing to its practical usability.
• Angle Propagation: While AP Theta* offers theoretical benefits with regard to vertex expansion complexity, Basic Theta* demonstrates superior runtime performance due to simpler operational mechanics and fewer computational overheads.

Extensions and Enhancements

The paper also discusses several extensions:

• Single Source Paths: Modifications to Theta* for finding paths from a given start vertex to all other vertices.
• Non-Uniform Traversal Costs: An enhancement to handle grids with non-uniform traversal costs, factoring in varying cell traversal costs into the path optimization.

Practical and Theoretical Implications

Practically, the Theta* algorithms are highly suitable for dynamic and real-time environments like robotics and gaming, where rapid recomputation of paths in changing terrains is essential. Theoretically, while Theta* finds paths close to the true shortest paths, it benefits from ongoing exploration into more efficient LOS computations to enhance practical runtime performance further.

Future Directions

The authors suggest future research areas, including:

• Establishing theoretical guarantees on the bound of path lengths for Theta* algorithms.
• Investigating constant-time LOS checks to further enhance AP Theta* and potentially surpass Basic Theta* in terms of both path quality and runtime.

In conclusion, Theta* represents a significant advancement in path-planning on grids, balancing path quality and computational efficiency. Its practical utility in domains requiring quick and accurate path computations marks it as a valuable contribution to the field.