Any-angle path planning

So far, four main any-angle path planning algorithms that are based on the heuristic search algorithm A*[2] have been developed, all of which propagate information along grid edges:

• Field D*[3][4] (FD*[5]) and 3D Field D*[6][7] - Dynamic pathfinding algorithms based on D* that use interpolation during each vertex expansion and find near-optimal paths through regular, nonuniform cost grids. Field D* therefore tries to solve the weighted region problem[8] and 3D Field D* the corresponding three-dimensional problem.
• Multi-resolution Field D*[9] – Extension of Field D* for multi-resolution grids.
• Theta*[5][10] - Uses the same main loop as A*, but for each expansion of a vertex ${\displaystyle s}$, there is a line-of-sight check between ${\displaystyle parent(s)}$ and the successor of ${\displaystyle s}$, ${\displaystyle s'}$. If there is line-of-sight, the path from ${\displaystyle parent(s)}$ to ${\displaystyle s'}$ is used since it will always be at least as short as the path from ${\displaystyle parent(s)}$ to ${\displaystyle s}$ and ${\displaystyle s}$ to ${\displaystyle s'}$. This algorithm works only on uniform-cost grids.[5] AP Theta*[5][10] is an optimization of Theta* that uses angle-propagation to decrease the cost of performing line-of-sight calculations to O(1), and Lazy Theta*[11] is another optimization of Theta* that uses lazy evaluation to reduce the number of line-of-sight calculations by delaying the line-of-sight calculations for each node from when it is explored to when it is expanded. Incremental Phi*[12] is an incremental, more efficient variant of Theta* designed for unknown 2D environments.[13]
• Block A* [14] - Generates a local distance database containing all possible paths on a small section of the grid. It references this database to quickly find piece-wise any-angle paths.
• ANYA[15] - Finds optimal any-angle paths by restricting the search space to the Taut paths; looking at an interval of points as a node rather than a single point.
• Strict Theta* and Recursive Strict Theta* [16] improves Theta* by restricting the search space to Taut Paths introduced by ANYA. This is an algorithm that returns near-optimal paths, while ANYA guarantees to return the optimal paths.
• CWave[17][18] - Uses geometric primitives (discrete circular arcs and lines) to represent the propagating wave front on the grid. For single-source path-planning on practical maps, it is demonstrated to be faster than graph search based methods. There are optimal and integer-arithmetic implementations.

Besides, for search in high-dimensional search spaces, such as when the configuration space of the system involves many degrees of freedom that need to be considered (see Motion planning), and/or momentum needs to be considered (which could effectively double the number of dimensions of the search space; this larger space including momentum is known as the phase space), variants of the rapidly-exploring random tree (RRT)[19] have been developed that (almost surely) converge to the optimal path by increasingly finding shorter and shorter paths:

• Rapidly-exploring random graph (RRG) and RRT*[20][21]
• Informed RRT*[22] improves the convergence speed of RRT* by introducing a heuristic, similar to the way in which A* improves upon Dijkstra's algorithm.

• Robot navigation
• Real-time strategy games