Abstract-In this paper, we present algorithms for selfreconfiguration of modular robots that move by pivoting. The modules are cubes that can pivot about their edges along thê x,ŷ, orẑ axes to move on a 3-dimensional substrate. This is a different model from prior work, which usually considers modules that slide along their faces. We analyze the pivoting cube model and give sufficient conditions for reconfiguration to be feasible. In particular, we show that if an initial configuration does not contain any of three subconfigurations, which we call rules, then it can reconfigure into a line. We provide provably correct algorithms for reconfiguration for both 2-D and 3-D systems, and we verify our algorithms via simulation on randomly generated 2-D and 3-D configurations.I. INTRODUCTION Modular robots consist of multiple connected modules, each with limited capabilities, that can be reconfigured to produce complex functionality as required by a task. Among the self-reconfigurable modular systems that have been developed [1]- [7], pivoting has emerged as a simple but powerful motion predicate [7]. In this paper, we describe a model for pivoting cubes in 3-D. We consider reconfiguration in both 2-D and 3-D settings and demonstrate that barring certain subconfigurations allows us to guarantee reconfiguration in O(n 2 ) moves. We provide provably correct algorithms for performing such reconfiguration. These are not optimal but are the first correct algorithms for the 3-D pivoting cube model. We perform simulations on random 2-D and 3-D configurations and show that in many cases, reconfiguration does not require the upper bound of n 2 moves. Pivoting modules, although prevalent in hardware [5], [7], are not well-studied. Pivoting modules sweep out a volume that extends past their initial and final positions, and any reconfiguration algorithm must take this motion constraint into account. An O(n 2 ) algorithm for 2-D pivoting modules was given in [8] but the model relaxed connectivity constraints compared to what many modular robots require and allowed movements that are often not physically realizable. Nguyen et al. [9] considered 2-D pivoting hexagons while requiring strong connectivity (connectivity through faces) and provided sufficient conditions for reconfiguration in O(n 5/2 ) moves. In [10], pivoting squares were reconfigured using a stochastic approach, and work in [5] analyzed the same system using meta-modules. A model for pivoting cubes in 3-D was introduced in [7], but the planning problem was not addressed. To our knowledge, there have been no studies of reconfiguration for pivoting cubes in 3-D.

Compliance is a defining characteristic of biological systems. Understanding how to exploit soft materials as effectively as living creatures do is consequently a fundamental challenge that is key to recreating the complex array of motor skills displayed in nature. As an important step towards this grand challenge, we propose a model-based trajectory optimization method for dynamic, cable-driven soft robot locomotion. To derive this trajectory optimization formulation, we begin by modeling soft robots using the Finite Element Method. Through a numerically robust implicit time integration scheme, forward dynamics simulations are used to predict the motion of the robot over arbitrarily long time horizons. Leveraging sensitivity analysis, we show how to efficiently compute analytic derivatives that encode the way in which entire motion trajectories change with respect to parameters that control cable contractions. This information is then used in a forward shooting method to automatically generate optimal locomotion trajectories starting from high-level goals such as the target walking speed or direction. We demonstrate the efficacy of our method by generating and analyzing locomotion gaits for multiple soft robots. Our results include both simulation and fabricated prototypes.

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