Near-Optimal Edge Evaluation in Explicit Generalized Binomial Graphs
Robotic motion-planning problems, such as a UAV flying fast in a partially-known environment or a robot arm moving around cluttered objects, require finding collision-free paths quickly. Typically, this is solved by constructing a graph, where vertices represent robot configurations and edges represent potentially valid movements of the robot between these configurations. The main computational bottlenecks are expensive edge evaluations to check for collisions. State of the art planning methods do not reason about the optimal sequence of edges to evaluate in order to find a collision free path quickly. In this paper, we do so by drawing a novel equivalence between motion planning and the Bayesian active learning paradigm of decision region determination (DRD). Unfortunately, a straight application of existing methods requires computation exponential in the number of edges in a graph. We present BISECT, an efficient and near-optimal algorithm to solve the DRD problem when edges are independent Bernoulli random variables. By leveraging this property, we are able to significantly reduce computational complexity from exponential to linear in the number of edges. We show that BISECT outperforms several state of the art algorithms on a spectrum of planning problems for mobile robots, manipulators, and real flight data collected from a full scale helicopter. Open-source code and details can be found here:
https://github.com/sanjibac/matlab_learning_collision_checking
Overview and Summary
This paper presents a method for motion planning where the cost of evaluating transitions between robot configurations is high.  The problem is formulated as a graph-search algorithm, where the order of graph expansion has a large impact on the performance of the algorithm due to the edge expansion cost.  The paper uses ideas from optimal test selection in order to derive the resulting algorithm.  The algorithm is tested on a number of synthetic datasets, a simulator, and a real-world helicopter planning problem.  
Detailed Comments
This paper extended work within the well-studied domain of robotic motion planning, extending and combining prior work in the area to construct a new algorithm.  The idea of using optimal test selection to address the graph expansion ordering is a natural, and reasonable approach.  The originality of this paper is moderate, and it does address a problem area of interest to many researchers.
There are some issues with clarity in the paper.  Specifically, starting in section 2, the term P(o) is first defined as an outcome vector, but then in equation (1) is used as a set to be drawn from (probabilistically?).  Next, and more significantly, lines 129-130 define the set of subregions, consisting of a single consistent subregion and a number of disjoint subregions that are each invalid.  The graph as a whole consists of nodes as subregions, and edge costs that are the product of the probabilities of each subregion.  I can see two issues with this formulation that are not made clear within the context of the paper:
- Given that the invalid subregions will expand with each test performed, the graph as a whole will have to change as testing progresses.  
- Because every edge contains at least one invalid subregion, every edge cost must contain at least one test where P(S_i) = 0, and therefore w must also equal 0.  
Due to this, either the problem formulation, or the clarity of the description must be changed.  
Also in terms of clarity, the index i is used on equations 3 and 5 on both the left and right hands of equations with different meanings.
Finally, within equation 5, the last term of the top equation does not equate to the last term of the bottom equation.  
Within the results, the text states that there are 13 datasets, but the results table lists 14.  Line 215 states that there are 8 tests with exclusively best performance, but the table lists only 7.  The claim on line 218 of a factor of 20 improvement is not possible to verify due to the normalization of all results.  Finally, and most importantly, the results are not compared against any of the other motion planning algorithms discussed in the related work, only against self-constructed heuristics used within the same general framework.  We have no idea how these algorithms compare to the state of the art.
Post-feedback:
Thanks to the authors for their explanation, additional material, and commitments to update the paper.  In light of this interaction, I have modified my score.