Reconfiguration-Complete Motion Primitives with Constructive Planning for Deformable Planar Modular Robots
Authors: Jie Gu, Tingting Wang, Hongrun Gao, Yirun Sun, Zhihao Xia, Chunxu Tian, Dan Zhang
Organizations: Institute of AI and Robotics, Academy for Engineering & Technology, Fudan University, Shanghai 200433, China · Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China
The continuously deformable geometry of modular robots makes it difficult to define a fixed representation for reconfiguration planning and analysis. This letter introduces a square-cell abstraction that maps deformable rhombus modules to fixed-size grid cells while retaining physically interpretable local motions through two primitives, pivoting and shearing. Under this abstraction, we prove that every non-straight edge-connected configuration with N≥7 can be transformed to a fixed canonical staircase using only admissible primitive motions. Since these motions are reversible, any two configurations in this class are mutually reconfigurable. The proof is constructive and directly yields a staircase-canonicalization planner that transports removable boundary modules while preserving connectivity. As a practical enhancement, we further introduce a boundary-to-delivery lookahead selector that ranks admissible high level choices without affecting the completeness guarantee. Experiments demonstrate the constructive reconfiguration process and show that the selector substantially reduces planning time, while reference comparisons indicate lower planning times than the prior framework over the shared module counts.
Figures & tables
Fig. 1: Continuous morphing and the corresponding primitive-motion abstractions. (a) Example morphing of the deformable rhombus platform. (b) Representative shearing sequence. (c) Representative pivoting sequence. (d) Geometric notation for a fully occupied 2×2 patch and its neighboring regions. (e) Shearing primitive and neighbor condition. (f) Pivoting primitive about a lattice vertex.
Fig. 2: Illustration of the configuration decomposition. The overall configuration is disjointly partitioned into a staircase prefix ( P ), the remainder ( R ), and a mobile gadget ( G ) for module transportation.
Fig. 3: Primitive templates underlying the high level gadget motions used by the constructive planner.
Fig. 4: Illustration of the constructive proof for Lemma 5. A standard carrying gadget G∙ is reconfigured into the final segment of the target staircase SN via a sequence of primitive motions.
Fig. 5: Reconfiguration demonstration for a 15-module example. The sequence shows the planned boundary transport process from the initial configuration to the target staircase configuration, where green, blue, and pink cells denote the gadget G , remainder R , and staircase prefix P , respectively. The dashed red cell marks the first staircase site ωk1 in the lookahead window, and the arrows indicate the executed primitive motions.
Fig. 6: Full-reconfiguration planning-time comparison with respect to N . Baseline denotes the prior initial-to-target planner. Ablation Study and Ours denote the proposed two-stage protocol through SN , without and with the lookahead selector, respectively.