\({\mathrm {RRT^{X}}}\): Real-Time Motion Planning/Replanning for Environments with Unpredictable Obstacles

We present \({\mathrm {RRT^{X}}}\), the first asymptotically optimal sampling-based motion planning algorithm for real-time navigation in dynamic environments (containing obstacles that unpredictably appear, disappear, and move). Whenever obstacle changes are observed, e.g., by onboard sensors, a graph rewiring cascade quickly updates the search-graph and repairs its shortest-path -to-goal subtree. Both graph and tree are built directly in the robot’s state space, respect the kinematics of the robot, and continue to improve during navigation. \({\mathrm {RRT^{X}}}\) is also competitive in static environments—where it has the same amortized per iteration runtime as RRT and RRT* \(\varTheta \left( \log {n}\right) \) and is faster than RRT^# \(\omega \left( \log ^2{n}\right) \). In order to achieve \(O\left( \log {n}\right) \) iteration time, each node maintains a set of \(O\left( \log {n}\right) \) expected neighbors, and the search graph maintains \(\epsilon \)-consistency for a predefined \(\epsilon \).

1. 1.

   “\(\epsilon \)-consistency” means that the cost-to-goal stored at each node is within \(\epsilon \) of its look-ahead cost-to-goal, where the latter is the minimum sum of distance-to-neighbor plus neighbor’s cost-to-goal.

2. 2.

   (1) Allows graph inconsistency. (2) Prevents the practical realization of some paths.

3. 3.

   Replanning algorithms find a sequence of solutions to the same goal state “on-the-fly” versus an evolving obstacle configuration and start state, and are distinct from multi-query algorithms (e.g., PRM [6]) and single-query algorithms (e.g., RRT [10]).

4. 4.

   The use of the term “dynamic” to indicate that an environment is “unpredictably changing” comes from the artificial intelligence literature. It should not be confused with the “dynamics” of classical mechanics.

5. 5.

   For example, if \(\mathcal {X}\subset \mathbb {R}^d\) space, \(\mathbb {T}\) is time, and obstacle movement is known a priori, obstacles are stationary with respect to \({\hat{\mathcal {X}} \subset \left( \mathbb {R}^d \times \mathbb {T}\right) }\) space-time.

6. 6.

   In particular, if a node \(u\) receives an \(\epsilon \)-cost decrease \(> \epsilon \) via another node \(v\), then \(u\) agrees to take responsibility for the runtime associated with that exchange (i.e., including it as part \(u\)’s next propagation time).

7. 7.

   i.e., because the number of neighbors of a node converges to the function \(\log {n}\) with probability 1 (as explained in Lemma 5).

8. 8.

   Note that neighbors that are not removed during a cull are touched again during the RRT*-like rewiring operation that necessarily follows a cull operation.

This work was supported by the Air Force Office of Scientific Research, grant #FA-8650-07-2-3744.

Authors and Affiliations

1. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

   Michael Otte & Emilio Frazzoli

Corresponding author

Correspondence to Michael Otte .

Editors and Affiliations

1. Department of Computer Engineering, Bogazici University, Istanbul, Turkey

   H. Levent Akin

2. Department of Computer Science and Engineering, Texas A&M University, College Station, Texas, USA

   Nancy M. Amato

3. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, Minnesota, USA

   Volkan Isler

4. Department of Information and Computing Sciences, Utrecht University, Utrecht, Utrecht, The Netherlands

   A. Frank van der Stappen

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