We consider the theoretical model of Crystalline robots, which have been introduced and prototyped by the robotics community. These robots consist of independently manipulable unit-square atoms that can extend contract arms on each side and attach detach from neighbors. These operations suffice to reconfigure between any two given (connected) shapes. The worstcase number of sequential moves required to transform one connected configuration to another is known to be ( n). However, in principle, atoms can all move simultaneously. We develop a parallel algorithm for reconfiguration that runs in only O(logn) parallel steps, although the total number of operations increases slightly to ( n logn). The result is the first (theoretically) almost-instantaneous universally reconfigurable robot built from simple units.
When @math strength is required but velocities are allowed to grow arbitrarily over time, reconfiguration takes @math steps @cite_11 . Note that this is for 2D robots and the third dimension is used as an intermediate.
@cite_11
1981392728 1487445608 1534588514 2593751597
We study the localization problem in the ring: a collection of @math n anonymous mobile robots are deployed in a continuous ring of perimeter one. All robots start moving at the same time with arbitrary velocities, starting in clockwise or counterclockwise direction around the ring. The robots bounce against each other according to the principles of conservation of energy and momentum. The task of each robot is to find out, in finite time, the initial position and the initial velocity of every deployed robot. The only way that robots perceive the information about the environment is by colliding with their neighbors; robots have no control of their walks or velocities moreover any type of communication among them is not possible. The configuration of initial positions of robots and their speeds is considered feasible, if there is a finite time, after which every robot starting at this configuration knows initial positions and velocities of all other robots. It was conjectured in (2012) that if the principles of conservation of energy and momentum were assumed and the robots had arbitrary velocities, the localization problem might be solvable. We prove that this conjecture is false. We show that if @math v0,v1,?,vn-1 are the velocities of a given robot configuration @math S, then @math S is feasible if and only if @math vi?v¯ for all @math 0≤i≤n-1, where @math v¯=v0+?+vn-1n. To figure out the initial positions of all robots no more than @math 2min0≤i≤n-1|vi-v¯| time is required. In this paper we propose novel algorithms for reconfiguring modular robots that are composed of n atoms. Each atom has the shape of a unit cube and can expand contract each face by half a unit, as well as attach to or detach from faces of neighboring atoms. For universal reconfiguration, atoms must be arranged in 2×2×2 modules. We respect certain physical constraints: each atom reaches at most unit velocity and (via expansion) can displace at most one other atom. We require that one of the atoms can store a map of the target configuration. Our algorithms involve a total of O(n 2) such atom operations, which are performed in O(n) parallel steps. This improves on previous reconfiguration algorithms, which either use O(n 2) parallel steps [8,10,4] or do not respect the constraints mentioned above [1]. In fact, in the setting considered, our algorithms are optimal, in the sense that certain reconfigurations require Ω(n) parallel steps. A further advantage of our algorithms is that reconfiguration can take place within the union of the source and target configurations. A self-reconfigurable (SR) robot is one composed of many small modules that autonomously act to change the shape and structure of the robot. In this paper we consider a general class of SR robot modules that have rectilinear shape that can be adjusted between fixed dimensions, can transmit forces to their neighbors, and can apply additional forces of unit maximum magnitude to their neighbors. We present a kinodynamically optimal algorithm for general reconfiguration between any two distinct, 2D connected configurations of n SR robot modules. The algorithm uses a third dimension as workspace during reconfiguration. This entire movement is achieved within O( √ n) movement time in the worst case, which is the asymptotically optimal time bound. The only prior reconfiguration algorithm achieving this time bound was restricted to linearly arrayed start and finish configurations (known as the “x-axis to y-axis problem”). All other prior work on SR robots assumed a constant velocity bound on module movement and so required at least time linear in n to do the reconfiguration. We consider the case in which a robot has to navigate in an unknown environment but does not have enough on-board power or payload to carry a traditional depth sensor (e.g., a 3D lidar) and thus can only acquire a few (point-wise) depth measurements. We address the following question: is it possible to reconstruct the geometry of an unknown environment using sparse and incomplete depth measurements? Reconstruction from incomplete data is not possible in general, but when the robot operates in man-made environments, the depth exhibits some regularity (e.g., many planar surfaces with only a few edges); we leverage this regularity to infer depth from a small number of measurements. Our first contribution is a formulation of the depth reconstruction problem that bridges robot perception with the compressive sensing literature in signal processing. The second contribution includes a set of formal results that ascertain the exactness and stability of the depth reconstruction in 2D and 3D problems, and completely characterize the geometry of the profiles that we can reconstruct. Our third contribution is a set of practical algorithms for depth reconstruction: our formulation directly translates into algorithms for depth estimation based on convex programming. In real-world problems, these convex programs are very large and general-purpose solvers are relatively slow. For this reason, we discuss ad-hoc solvers that enable fast depth reconstruction in real problems. The last contribution is an extensive experimental evaluation in 2D and 3D problems, including Monte Carlo runs on simulated instances and testing on multiple real datasets. Empirical results confirm that the proposed approach ensures accurate depth reconstruction, outperforms interpolation-based strategies, and performs well even when the assumption of structured environment is violated.