LEGO Grammar. Version 1.7

Transcription

1 LEGO Grammar. Version 1.7 Parameterized context sensitive directional graph grammar designed to handle thee-dimensional structures assembled from LEGO blocks, axles, wheels, motors and batteries. Examples of the elements which can be handled by the grammar is shown below:

2 Grammar vocabulary is { MECHANISM, Module, Connect, Block, Element, Disk, Pole, PegPair, BorderPair, Snap, Insert, TInsert, GTrans, Beam, Brick, Plate, Battery, Motor, Wheel, Gear, Axle, PozX, PozY, SizeX, SizeY, Len, Diam, Teeth, Hole (, ), [, ], ;}. Starting word of the grammar is {MECHANISM}. Terminal words of the grammar are { Snap, Insert, TInsert, GTrans, Beam, Brick, Plate, Battery, Motor, Wheel, Gear, Axle, SizeX, SizeY, Len, PozX, PozY, Diam, Teeth, Hole (, ), [, ], ;}. Terminal words ( )[ ], and ; are used only to make sentences easier to read by people, so most often they will be ignored on the derivation and syntax trees. Terminals PozX, PozY, SizeX, SizeY, Len, Diam, Hole and Teeth are parameters. SizeX used to specify the number of pegs on the LEGO piece in the X direction. SizeY used to specify the number of pegs on the LEGO piece in the Y direction. Terminals PozX and PozY describe the location of the peg on the LEGO Block by giving its coordinates relative to the origin of the Block. PozX can alternate between 1 and SizeX of the Block. PozY can alternate between 1 and SizeY of the Block. SizeX and SizeY belong to the set {1, 2, 4, 6, 8, 10, 12, 16}. This set can be modified according to the LEGO brick standards. Len used to specify the number of pegs on the LEGO Beam or the length of an Axle measured in the peg sizes. It belongs to the set {2, 4, 6, 8, 10, 12, 16}. Hole defines the number of the hole in the Beam in to which Axle is inserted. It can alternate between 1 and (Len - 1). Diam reflects the diameter of the wheel and belongs to the set {17, 30, 43}. Teeth represent the number of teeth on the LEGO Gear. This number also uniquely defines the diameter of the Gear. It belongs to the set {8, 16, 24, 40} MECHANISM, Module, Element, Block, Disk, Pole, Beam, Brick, Plate, Battery, Motor, Wheel, Gear and Axle are the nodes of the graph grammar. Module is representing the number of Elements connected together. Element represents a single LEGO peace. Grammar includes three categories of LEGO elements, namely Block, Disk and Pole. Block represents a rectangular LEGO peace with pegs on one side. It is obvious that Brick and Plate are the LEGO Blocks with N*M size, where N and M can range from 1 to 16. However Brick is 3 times higher than Plate. Beam is the LEGO Block with 1*N size, where N can range from 2 to 16. Beam has the same height as a Brick but it also has holes in it. Battery and Motor also belong to the Block category, but they have some special properties. Battery size is (14,4) or (4,14) and it is 4 Bricks high (12 LEGO heights units). Motor size is (5,4) or (4,5) and it is 2 Bricks and one Plate high (12 LEGO heights units). Also Wheel and Gear can be attached to the Motor.

3 Relative sizes of the Battery and Motor shown on the second picture. Wheel and Gear are two types of Disks. I believe their names are self-explanatory. Only Axle belongs to the category of Pole, but this category may be extended when more LEGO pieces are added to the grammar. Examples of all Elements are shown on the first picture. Connect, Snap, Insert, TInsert and GTrans are the edges. Snap is directed edge. It represents the way Blocks are connected together. Arrow is pointing from the Block which provides pegs to the Block that provides connection surfaces. Snap connection defined by one or two PegPairs. If connection established over 1 peg than one BorderPair (special type of PegPair) is used to define the connection. If connection established over 2 or more pegs than two PegPairs are used to define the connection. PegPair is used to define the Pair of corresponding pegs in the connection [(PozX1, PozY1); (PozX2, PozY2)]. Illustration of the peg coordinates is shown in the third picture. (PozX1, PozY1) defines the peg on the Block witch provides pegs and (PozX2, PozY2) defines the peg on the Block which provides connection surface.. So peg on the Block Snap is pointing to always defined second in the PegPare.

4 Example of the Snap connection is shown on the forth picture. Insert is undirected edge. It represents the way Axle is connected to the Beam. First parameter of the Insert is number of Hole in which Axle is inserted. Second is the distance from the point of origin of an Axle to the connection in LEGO peg units. Example of the Insert connection is shown on the fifth picture. TInsert is undirected edge. It represents the way Axle or Motor is connected to the Gear or Wheel. TInsert stands for tight insert. Parameter of the TInsert is the distance from the point of origin of an Axle to the connection in LEGO peg units. Example of the TInsert connection is shown on the sixth picture.

5 GTrans is undirected edge. It represents the way two Gears are connected together. GTrans stands for gear transmission. GTrans does not have any parameters, but it constrains relative positioning of the Axles to which gears are attached. In order to validate GTrans one should calculate the distance between candidate Axles and than locate the closest value in the Distance table. Connection is valid if and only if the difference between these values less than 1 mm. Table of Distances between axles for different gear connections: Gear 1 (#Teeth) Gear 2 (#Teeth) Distance (mm) Example of the GTrans connection is shown on the seventh picture. Distances between axles can be calculated from the chart presented below. All valid Axle locations shown as a red dot and X*Y distances specified on the axis. Couple valid positionings is shown for 40 to 40 and 8 to 16 teeth Gear connections.

6 Axles positioning chart: Distance Between Holes in Y Direction *40 teeth 8*16 teeth Distance Between Holes in X Direction

7 This Grammar has following roles: MECHANISM => Module Module => Module or Element Connect Module Module => Module or Module Connect Module Module Module Element => Block or Disk or Pole Block => Block or Block Connect Snap [BorderPair] Snap [PegPair, PegPair] Block Block Block Disk => Gear (Teeth, 1) or Gear (1, Teeth) or Connect GTrans GTrans Disk Gear (Teeth, 1) Gear (1, Teeth) Element => Beam (Len, 1) or Beam (1, Len) or Connect Insert [Hole,Len] Insert [Hole, Len] Pole Axle (1, Len) Axle (Len, 1) Wheel (Diam, 1) or Wheel (1, Diam) or TInsert [Len] TInsert [Len] Axle (1, Len) Axle (Len, 1) Gear (Teeth, 1) or Gear (1, Teeth) TInsert [Len] TInsert [Len] Axle (1, Len) Axle (Len, 1) Disk => Motor (4, 5) or Motor (5, 4) or Connect TInsert [1] TInsert [1] Motor Wheel (Diam, 1) Wheel (1, Diam) Motor (4, 5) or Motor (5, 4) or TInsert [1] TInsert [1] Gear (Teeth, 1) Gear (Teeth, 1) Pole => Axle (Len, 1) where Len {2, 4, 6, 8, 10, 12, 16} or Axle (1, Len) where Len {2, 4, 6, 8, 10, 12, 16} Disk => Wheel (Diam, 1) where Diam {17, 30, 43} or Wheel (1, Diam) where Diam {17, 30, 43} or Gear (1, Teeth) where Teeth {8, 16, 24, 40} or Gear (Teeth,1) where Teeth {8, 16, 24, 40}

8 Block => Beam (Len, 1) where Len {2, 4, 6, 8, 10, 12, 16} or Beam (1, Len) where Len {2, 4, 6, 8, 10, 12, 16} or Motor (4,5) or Motor (5,4) or Battery (14,4) or Battery (4,14) or Beam (1, Len) where Len {2, 4, 6, 8, 10, 12, 16} or Plate (SizeX, SizeY) where SizeX, SizeY {1, 2, 4, 6, 8, 10, 12, 16} or Brick (SizeX, SizeY) where SizeX, SizeY {1, 2, 4, 6, 8, 10, 12, 16} PegPair => (PozX, PozY); (PozX, PozY) where PozX {1.. SizeX}, PozY {1.. SizeY} for each Block BorderPair => (1, PozY); (SizeX, PozY) or (SizeX, PozY); (1, PozY) or (PozX, 1); (PozX, SizeY) or (PozX, SizeY); (PozX, 1) where SizeX, SizeY {1, 2, 4, 6, 8, 10, 12, 16} PozX {1.. SizeX}, PozY {1.. SizeY} for each Block

9 Example of the Mechanism with a Assembly Graph. MECHANISM LEGEND Beam Plate Brick Wheel Gear Axle Motor Battery Assembly Graph GTrans TInsert[1] Gear(40) Gear(8) Motor(5,4) Insert[7] Wheel(44) Snap[(6,1)(5,1);(2,1)(1,1)] Snap[(1,1)(1,13);(1,2)(1,14)] TInsert[1] TInsert[8] Snap[(1,1)(1,1);(1,2)(1,2)] Snap[(6,1)(5,1);(2,1)(1,1)] Wheel(30) Plate(6,8) Battery(4,14) Axle(8,1) Insert[2] Snap[(1,1)(6,1);(2,1)(2,6)] Snap[(1,1)(4,13);(1,2)(4,14)] TInsert[8] TInsert[1] Insert[2] Insert[7] Wheel(30) TInsert[10] Axle(10,1) Wheel(44)

10 ASSEMBLY GRAPH DERIVATION LEGEND Derives Derives (several steps) Connect Snap Mechanism Module Element Block Disk Pole Beam Plate Brick Wheel Gear Axle Motor Battery

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12 GTrans TInsert[1] Gear(40) Gear(8) Motor(5,4) Insert[7] Wheel(44) Snap[(6,1)(5,1);(2,1)(1,1)] Snap[(1,1)(1,13);(1,2)(1,14)] TInsert[1] TInsert[8] Snap[(1,1)(1,1);(1,2)(1,2)] Snap[(6,1)(5,1);(2,1)(1,1)] Wheel(30) Plate(6,8) Battery(4,14) Axle(8,1) Insert[2] Snap[(1,1)(6,1);(2,1)(2,6)] Snap[(1,1)(4,13);(1,2)(4,14)] TInsert[8] TInsert[1] Insert[2] Insert[7] Wheel(30) TInsert[10] Axle(10,1) Wheel(44)