Rapidly exploring random tree

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A visualization of a RRT graph after 45 and 390 iterations

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An animation of a RRT starting from iteration 0 to 10000

A rapidly exploring random tree (RRT) is an algorithm designed to efficiently search nonconvex, high-dimensional spaces by randomly building a space-filling tree. The tree is constructed incrementally from samples drawn randomly from the search space and is inherently biased to grow towards large unsearched areas of the problem. RRTs were developed by Steven M. LaValle and James J. Kuffner Jr. . They easily handle problems with obstacles and differential constraints (nonholonomic and kinodynamic) and have been widely used in autonomous robotic path planning.

RRTs can be viewed as a technique to generate open-loop trajectories for nonlinear systems with state constraints. An RRT can also be considered as a Monte-Carlo method to bias search into the largest Voronoi regions of a graph in a configuration space. Some variations can even be considered stochastic fractals.

Description

An RRT grows a tree rooted at the starting configuration by using random samples from the search space. As each sample is drawn, a connection is attempted between it and the nearest state in the tree. If the connection is feasible (passes entirely through free space and obeys any constraints), this results in the addition of the new state to the tree. With uniform sampling of the search space, the probability of expanding an existing state is proportional to the size of its Voronoi region. As the largest Voronoi regions belong to the states on the frontier of the search, this means that the tree preferentially expands towards large unsearched areas.

The length of the connection between the tree and a new state is frequently limited by a growth factor. If the random sample is further from its nearest state in the tree than this limit allows, a new state at the maximum distance from the tree along the line to the random sample is used instead of the random sample itself. The random samples can then be viewed as controlling the direction of the tree growth while the growth factor determines its rate. This maintains the space-filling bias of the RRT while limiting the size of the incremental growth.

RRT growth can be biased by increasing the probability of sampling states from a specific area. Most practical implementations of RRTs make use of this to guide the search towards the planning problem goals. This is accomplished by introducing a small probability of sampling the goal to the state sampling procedure. The higher this probability, the more greedily the tree grows towards the goal.

Algorithm

For a general configuration space C, the algorithm in pseudocode is as follows:

Algorithm BuildRRT
  Input: Initial configuration qinit, number of vertices in RRT K, incremental distance Δq)
  Output: RRT graph G

  G.init(qinit)
  for k = 1 to K
    qrand ← RAND_CONF()
    qnear ← NEAREST_VERTEX(qrand, G)
    qnew ← NEW_CONF(qnear, qrand, Δq)
    G.add_vertex(qnew)
    G.add_edge(qnear, qnew)
  return G

• "←" is a shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.
• "return" terminates the algorithm and outputs the value that follows.

In the algorithm above, "RAND_CONF" grabs a random configuration q_rand in C. This may be replaced with a function "RAND_FREE_CONF" that uses samples in C_free, while rejecting those in C_obs using some collision detection algorithm.

"NEAREST_VERTEX" is a straightforward function that runs through all vertices v in graph G, calculates the distance between q_rand and v using some measurement function thereby returning the nearest vertex.

"NEW_CONF" selects a new configuration q_new by moving an incremental distance Δq from q_near in the direction of q_rand. (According to in holonomic problems, this should be omitted and q_rand used instead of q_new.)

See also

• Probabilistic roadmap
• Space-filling tree
• Motion planning
• Randomized algorithm

External links

• Java visualizer of RRT and RRT* including map editor