Input description: A polygonal-shaped robot in a given starting position in a room containing polygonal obstacles, with a desired ending position t.
Problem description: Find the shortest path in the room taking to t without intersecting any of the obstacles.
Discussion: The difficulty of motion planning will be obvious to anyone who has ever had to move a large piece of furniture into a small apartment. The problem of motion planning also arises in systems for molecular docking. Many drugs are small molecules that act by binding to a given target model. The problem of identifying which binding sites are accessible to a candidate drug is clearly an instance of motion planning. Plotting paths for mobile robots is another canonical motion-planning application.
Motion planning also provides a tool for computer animation. Given a set of object models that appear in two different scenes and , a motion planning algorithm can construct a short sequence of intermediate motions to transform to . These motions can serve to fill in the intermediate scenes between and , with such scene interpolation greatly reducing the amount of work the animator has to do.
There is a wide range in the complexity of motion planning problems, with many factors to consider:
A brute-force algorithm to construct the visibility graph tests each candidate edge against the obstacle edges for a total time of time. By weighting each edge of the visibility graph with its length and using Dijkstra's shortest-path algorithm (see Section ) on this graph, we can find the shortest path from to t in time bounded by the time to construct the graph.
The most practical approach to motion planning involves randomly sampling the configuration space of the robot. The configuration space defines the set of legal positions for the robot and has one dimension for each degree of freedom. For a planar robot capable of translation and rotation, the degrees of freedom are the x- and y-coordinates of a reference point on the robot and the angle relative to this point. Certain points in this space represent legal positions, while others intersect obstacles.
Construct a set of legal configuration-space points by random sampling. For each pair of points and , decide whether there exists a direct, nonintersecting path between them. Construct a graph with vertices for each legal point and edges for each such traversable pair. The problem of finding a motion between two arbitrary positions reduces to seeing if there is a direct path from the initial/final position to some vertex in the graph, and then solving a shortest-path problem in the graph.
There are lots of ways to enhance this basic technique for specific applications, such as adding additional vertices to regions of particular interest. This is a nice, clean approach for solving problems that would get very messy otherwise.
Implementations: An implementation of collision detection (not really motion planning) is the I_COLLIDE collision detection library. For more information, check out the I_COLLIDE WWW page: http://www.cs.unc.edu/ geom/I_COLLIDE.html.
O'Rourke [O'R94] gives a toy implementation of an algorithm to plot motion for a two-jointed robot arm in the plane. See Section .
Notes: Motion planning was originally studied by Schwartz and Sharir as the ``piano mover's problem.'' Their solution constructs the complete free space of robot positions which do not intersect obstacles, and then finds the shortest path within the proper connected component. These free space descriptions are very complicated, involving arrangements of higher-degree algebraic surfaces. The fundamental papers on the piano mover's problem appear in [HSS87], with [SS90] being a survey of current results. The best general result for this free space approach to motion planning is due to Canny [Can87], who showed that any problem with d degrees of freedom can be solved in , although faster algorithms exist for special cases of the general motion planning problem.
Latombe's book [Lat91] describes practical approaches to motion planning, including the random sampling method described above. The expanded obstacle approach to motion planning is due to Lozano-Perez and Wesley [LPW79]. The heuristic, sightless man's approach to motion planning discussed above has been studied by Lumelski [LS87].
The time complexity of algorithms based on the free-space approach to motion planning depends intimately on the combinatorial complexity of the arrangement of surfaces defining the free space. Algorithms for maintaining arrangements are presented in Section . Davenport-Schintzl sequences often arise in the analysis of such arrangements. Sharir and Agarwal [SA95] provide a comprehensive treatment of Davenport-Schintzl sequences and their relevance to motion planning.
Kedem and Sharir [KS90] give an time algorithm to find a path (not necessarily shortest) to translate a convex k-gon from to t. Vegter [Veg90] gives an optimal algorithm for moving a line segment (often called a ladder) in the plane with both translation and rotation.
Related Problems: Shortest path (see page ), Minkowski sum (see page ).