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Efficient high-quality motion planning by fast all-pairs r-nearest-neighbors


Sampling-based motion-planning algorithms typically rely on nearest-neighbor (NN) queries when constructing a roadmap. Recent results suggest that in various settings NN queries may be the computational bottleneck of such algorithms. Moreover, in several asymptotically-optimal algorithms these NN queries are of a specific form: Given a set of points and a radius r report all pairs of points whose distance is at most r. This calls for an application-specific NN data structure tailored to efficiently answering this type of queries.

Randomly transformed grids (RTG) were recently proposed by Aiger, Kaplan and Sharir as a tool to answer such queries and have been shown to outperform common implementations of NN data structures in this context.

In this work we employ RTG for sampling-based motion-planning algorithms and describe an efficient implementation of the approach. We show that for motion-planning, RTG allow for faster convergence to high-quality solutions when compared with existing NN data structures. Additionally, RTG enable significantly shorter construction times for batched-PRM variants; specifically, we demonstrate a speedup by a factor of two to three for some scenarios.



Comparison to state-of-the-art nearest-neighbor libraries

To evaluate our implementation we compared our RTG implementation with the following nearest-neighbor implementations: FLANN kd-tree, ANN kd-tree, and LSH in Euclidean metric spaces (E2LSH).

For each method we measured the time for answering all-pairs r-nearest-neighbors queries for n random uniform samples from the unit d-dimensional hypercube.

The radius r = r(n) was defined as follows:


We used point sets, of increasing sizes, of dimensions d = 3,6,9, and 12.

We performed the same experiment in environment cluttered with obstacles.

The following plots present our results (averaged over ten runs) for different dimensions:


                      No obstacles - 3D                                                      No obstacles - 9D                                                                       With obstacles - 6D

nn_comp_3d         NN_comp_9D         nn_comp_cubicles_6D




Motion-planning experiments


For the following experiments we used OMPL

, and compared several sampling-based motion-planning algorithms with two possible structures for  nearest-neighbors queries: (i) RTG and (ii) GNAT  (OMPL's default).


The tested scenarios:

(a)  Z-tunnel                                                                                      (b) 3D grid                                                                   (c) Cubicles

   (based on a scenario from Parasol motion-planning group)                                                                                            (taken from OMPL)  

z_tunnel     3d_grid   cubicles


The roadmap construction time as a function of the number of samples for the PRM* algorithm (circle marks) on the Z-tunnel scenario in a 3D C-space (one translating robot in space) using both RTG and GNAT.

A similar experiment was performed for Lazy Batch-PRM* (marked in squares).

For these two experiments we used the Euclidean metric for distance computations.




The cost as a function of time for the MPLB algorithm on the 3D Grid scenario in a 6D C-space (two translating robots in space).

The distance metric computes the sum of the distances that each robot travels (Non-Euclidean metric).

The same experiment was performed while using the Euclidean metric instead.

                                        Non-Euclidean metric                                                                     Euclidean Metric

3d_grid_MR   sd_grid_eucl


The success rate of finding a solution as a function of time for the MPLB algorithm on the Cubicles scenario in a 6D C-space (using both the Euclidean and the non-Euclidean metrics).


                                  Euclidean metric                                                        Non-Euclidean metric





Our C++ implementation is available here.

The reference manual can be found here.



  • Michal Kleinbort, Oren Salzman and Dan Halperin
    Efficient high-quality motion planning by fast all-pairs r-nearest-neighbors
    In International Conference on Robotics and Automation (ICRA), 2015 [link]


Michal Kleinbort
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Dan Halperin
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