Modeling lattice modular systems with space groups Nicolas Brener, - - PowerPoint PPT Presentation

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

• Use of crystallographic knowledge

– Discrete Motion Chiral space groups – Connectors position Wyckoff sets

• Lattice Robot

–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

Outline

• Discrete motion spaces

– Point groups – Lattice groups – Space groups

• Wyckoff positions
• Group hierarchy
• Classification of lattice robots
• Design of lattice robots
• Next steps

Transformation Groups taxonomy

Continuous

Isometry (length preserved) Motion (length and orientation preserved)

Discrete

230 Crystallographic Space groups types SE3 SO3 32 Crystallographic point groups 3D Isometry group 65 Chiral Space Groups types (Sohncke Groups) 14 Lattice groups types 11 Crystallographic rotation groups

Subgroup Subset

Discrete rotation groups point groups Translations group

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 }

4-fold rotation 3-fold rotation 2-fold rotation

Geometric elements

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, a3/2, 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

Discret motion spaces

Geometric elements of P432, except screw axes

4-fold 3-fold 2-fold

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 identity 2-fold

Space Group Hierarchy

Space Group Hierarchy

P1 P622 P432 I432 F432 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

Connector Lattice System

• rientation

Symmetry Wyckof position

Space Group None 4-fold None None 4-fold 4-fold j e j,j a,a,a e e F432 F432 F432 P1 P432 F432 (1,0,0) Molecule NA Atron NA Telecube (1,0,0) M-tran NA 3D universal (1,0,0) I-Cube

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)

Node Rotation axis Connexion plate 4-fold 2-fold 3-fold

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.