Modeling lattice modular systems with space groups Nicolas Brener, Faiz Ben Amar, Philippe Bidaud Laboratoire de Robotique de Paris Université de Paris 6

Lattice Robot vs Crystal Connectors have discrete positions in a lattice Mobilities act on the position of the connectors Mobility are discrete motions Crystal Atoms have discrete positions in a lattice Symmetries act on the atoms positions Symmetries are discrete isometries Use of crystallographic knowledge Discrete Motion  Chiral space groups Connectors position  Wyckoff sets

Outline Discrete motion spaces Wyckoff positions Group hierarchy Point groups Lattice groups Space groups Wyckoff positions Group hierarchy Classification of lattice robots Design of lattice robots Next steps

Transformation Groups taxonomy Isometry (length preserved) Motion (length and orientation preserved) Continuous 3D Isometry group SE3 SO3 Translations group Subgroup point groups 230 Crystallographic Space groups types Discrete rotation groups Subset 65 Chiral Space Groups types (Sohncke Groups) Discrete 32 Crystallographic point groups 11 Crystallographic rotation groups 14 Lattice groups types

Discrete motion spaces Discrete rotation  chiral point group Discrete translation  lattice group Discrete motion  chiral space group

Discrete motion spaces 11 chiral point groups (rotation groups) Finite sets Point group 622 12 rotations 1 6-fold axes 6 2-fold axes Generator: { 180° rotation along x, 60° rotation along z } Geometric elements 6-fold rotation 2-fold rotation

Discrete motion spaces Point group 432 24 rotations 3 4-fold axes 4 3-fold axes 6 2-fold axes Generator: { 90° rotation along x, 90° rotation along y } Geometric elements 4-fold rotation 3-fold rotation 2-fold rotation

Discrete motion spaces Lattice groups Infinite sets Generators: 3 translations 14 types Cubic lattice {(a,0,0),(0,a,0),(0,0,a)} Cubic centered lattice {(-a,a,a), (a,-a,a), (a,a,-a)} Cubic face centered lattice {(a,a,0),(a,0,a), (0,a,a)} Hexagonal lattice {(a,0,0),(a/2, a√3/2, h)} Triclinic lattice {A,B,C}

Discrete motion spaces Chiral Space Group Infinite sets Have rotations and translations (and screw) 65 types Described in the « International Tables for Crystallography » edition th. Hahn

Discrete motion spaces Geometric elements of P622, except screw axes 6-fold 3-fold 2-fold

Geometric elements of P432, except screw axes Discret motion spaces Geometric elements of P432, except screw axes 4-fold 3-fold 2-fold

Geometric elements of F432, except screw axes Discret motion spaces Geometric elements of F432, except screw axes 4-fold 3-fold 2-fold

Wyckoff positions and Connectors identity 2-fold 3-fold 4-fold 6-fold Orthogonal rotation symmetry Tangential rotation symmetry

Space Group Hierarchy

Space Group Hierarchy Classification Minimum space group 60 other chiral space groups Maximum chiral space groups Minimum chiral space group Classification Minimum space group Design in Maximum groups

Classification M-tran Mobilities match 4-fold axes of F432 Connectors match Wyckoff position e The connectors have 4-fold symmetry

Classification Lattice System Space Group Connector M-tran F432 e Wyckof position Symmetry orientation M-tran F432 e 4-fold (1,0,0) I-Cube P432 Telecube P1 a,a,a None NA Atron j,j Molecule 3D universal j

Designing Choose a space, and corresponding wyckoff position(s) and orientation(s). A module is constructed by selecting some mobilities and connectors and gluing them together Example: in F432 with 1 position in (a)

Designing Choose a space, and corresponding wyckoff position(s) and orientation(s). A module is constructed by selecting some mobilities and connectors and gluing them together Example: in F432 with 1 position in (a)

Designing Choose a space, and corresponding wyckoff position(s) and orientation(s). A module is constructed by selecting some mobilities and connectors and gluing them together Example: in F432 with 1 position in (a)

Designing Choose a space, and corresponding wyckoff position(s) and orientation(s). A module is constructed by selecting some mobilities and connectors and gluing them together Example: in F432 with 1 position in (a)

Designing Choose a space, and corresponding wyckoff position(s) and orientation(s). A module is constructed by selecting some mobilities and connectors and gluing them together Example: in F432 with 1 position in (a) 4-fold 2-fold 3-fold Node Connexion plate Rotation axis

Conclusion Chiral Space groups give: All possible mobility All possible orientation, position and symmetries for connectors Lattice design but chain type product (ex: M-tran) Provides a framework to design several compatible systems, or extend existing systems with new modules. But: it does not provide automatically an operational modular system… Collision Joint stops …

Further steps Formalize and automate classification. Find rules on connector positions and mobilities that provide systems that efficiently reconfigure. Develop a design interface.