# How to make a "Virtual" Pallet

K

#### Kris Grindstaff

I am trying to figure out how to create a frame to use in palletizing. If I have a tray with 12 rows and 18 columns, and I know the X,Y,Z, and Theta coordinates (in millimeters) for the position in each of the four corners, how can I calculate the X,Y,Z and Theta positions for each position on the tray? There should be 216 total positions.

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#### Peter McWha

Kris, this should be very straightforward, but I'm unclear as what you mean by the "theta" address. Please e-mail me the definition
thereof.

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#### Rob Taylor

The theta is the angle of rotation of the robot which you would palletize with; it is not needed for this problem. Take a very simple case: a 2x1 pallet on the XY plane.
- Number the the parts as on a "BattleShip" board
Let's assume the coordinates (X,Y) are:

A0 A1 (0,0) (200,0)
No problem -both positions are known.

- Next simplest case, expand it to 3x1
A0 A1 A2 (0,0) (?) (200,0)

- you know the 2 ends, so you only need the X for A1:

X(A1) = [X(A2) - X(A0)]/2

Divide by 2 as it is halfway between the first part and the last part in that row. Easy so far!

- Now go to a generic pallet with 'N' columns, (START THE NUMBERING AT ZERO,
it makes the math easier) and figure out the position of part 'n':
X(An) = [X(AN) - X(A0)]/N

Extrapolate this and you've the formula for any size pallet, in any number of dimensions.

X(An) = [X(AN) - X(A0)]/N
X(Bn) = [X(BN) - X(B0)]/N
Y(An) = [Y(AN) - Y(A0)]/N
Y(Bn) = [Y(BN) - Y(B0)]/N
Z(An) = [Z(AN) - Z(A0)]/N
Z(Bn) = [Z(BN) - Z(B0)]/N
etc.

Rob

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#### dooley

Do you mean that z,theta are the polar coordinates? If so then the coordinates for the position xn,ym are given by: xn = x1+(n-1)*(x18-x1)/17 ym = y1+(m-1)*(y12-y1)/11 znm = SQRT(xn^2+ym^2) Thetanm = ARCTAN(ym/xn) If you mean that you want to determine coordinates for numbered positions then there is a preliminary step to determine the x,y coordinate positions. Suppose the positions are numbered from 1 to 216 starting at the bottom left hand corner and progressing left to right, bottom to top then the coordinates for any position P
are:
x coordinate position n = 1+REMAINDER((P-1)/18))
y coordinate position m = 1+INT((P-1)/18))

I hope that's what you wanted.... I hope it's right!! .... you'd better check it.

Vince