To compute robust 2D assembly plans, we present an approach that combines geometric planning with a deep neural network. We train the network using the Box2D physics simulator with added stochastic noise to yield robustness scores-the success probabilities of planned assembly motions. As running a simulation for every assembly motion is impractical, we train a convolutional neural network to map assembly operations, given as an image pair of the subassemblies before and after they are mated, to a robustness score. The neural network prediction is used within a planner to quickly prune out motions that are not robust. We demonstrate this approach on two-handed planar assemblies, where the motions are onestep translations. Results suggest that the neural network can learn robustness to plan robust sequences an order of magnitude faster than physics simulation.
Multi-Agent Path Finding (MAPF) is a fundamental motion coordination problem arising in multi-agent systems with a wide range of applications. The problem's intractability has led to extensive research on improving the scalability of solvers for it. Since optimal solvers can struggle to scale, a major challenge that arises is understanding what makes MAPF hard. We tackle this challenge through a fine-grained complexity analysis of time-optimal MAPF on 2D grids, thereby closing two gaps and identifying a new tractability frontier. First, we show that 2-colored MAPF, i.e., where the agents are divided into two teams, each with its own set of targets, remains NP-hard. Second, for the flowtime objective (also called sum-of-costs), we show that it remains NP-hard to find a solution in which agents have an individually optimal cost, which we call an individually optimal solution. The previously tightest results for these MAPF variants are for (non-grid) planar graphs. We use a single hardness construction that replaces, strengthens, and unifies previous proofs. We believe that it is also simpler than previous proofs for the planar case as it employs minimal gadgets that enable its full visualization in one figure. Finally, for the flowtime objective, we establish a tractability frontier based on the number of directions agents can move in. Namely, we complement our hardness result, which holds for three directions, with an efficient algorithm for finding an individually optimal solution if only two directions are allowed. This result sheds new light on the structure of optimal solutions, which may help guide algorithm design for the general problem.
Assembly planning, which is a fundamental problem in robotics and automation, aims to design a sequence of motions that will bring the separate constituent parts of a product into their final placement in the product. It is convenient to study assembly planning in reverse order, where the following key problem, assembly partitioning, arises: Given a set of parts in their final placement in a product, partition them into two sets, each regarded as a rigid body, which we call a subassembly, such that these two subassemblies can be moved sufficiently far away from each other, without colliding with one another (sliding of one subassembly over the other, namely motion in contact, is allowed). The basic assembly planning problem is further complicated by practical consideration such as how to hold the parts in a subassembly together. Therefore, a desired property of a valid assembly partition is that each of the two subassemblies will be connected.In this paper we show that even an utterly simple case of the connected-assembly-partitioning problem is hard: Given a connected set A of unit squares in the plane, each forming a distinct cell of the uniform integer grid, find a subset S ⊂ A such that S can be rigidly translated to infinity along a prescribed direction without colliding with A \ S, and both subassemblies S and A \ S are each connected. We show that this problem is NP-Complete, and by that settle an open problem posed by Wilson et al. a quarter of a century ago [16].We complement the hardness result with two positive results for the aforementioned problem variant of grid square assemblies. First, we show that it is fixed parameter tractable and give an O(2 k n 2 )-time algorithm, where n = |A| and k = |S|. Second, we describe a special case of this variant where a connected partition can always be found in linear time. Each of the positive results sheds further light on the special geometric structure of the problem at hand.
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