## Cartesian coordinates a heresy

Has anyone every seriously wondered why CNC machines always work using the Cartesian coordinate system? Why are X-, Y- and Z-axes always used? Why, when such machines are so difficult to build? The linear guides must be absolutely parallel, because otherwise the carriage will jam. The axes must be at exactly 90 degrees to one another, or else everything goes askew. The table must be absolutely true and the whole machine must be solidly fixed to a base.

These are all disadvantages. But the greatest disadvantage is the linkage between the axes. Consider the X-Y table, the original form of hand-operated milling machine. This has two handwheels, one to move the table in the X-direction, the other to move the table in the Y-direction. There are thus two linear guides, one fixed at an angle of 90 degrees to the other, supporting one another. If the lower mechanism has play in it, this is transferred to the upper one, even if the upper one is absolutely precise. And the lower guide also has to bear the weight of the upper one.

This traditional mechanism stems from a time before computers, when positioning along the axes was controlled manually by a technician using handwheels. Technical drawings are normally marked up with XY coordinates so that the successively required positions can easily be reached by use of the handwheels. In the age of automation the technician is no longer employed and the handwheels are replaced by motors under computer control. But the coordinate system has not changed: in the human imagination everything has a length, a breadth and a height. Curious, when most human actions are polar: 'take three steps in this direction and then turn right'!

Imagine now how you would drill a circuit board by hand without a pillar drill. With the fingers of one hand you would hold the board steady and with the other hand you would hold the drill. Your drilling arm would be rather higher at the elbow than the other arm, since the mini-drill has a certain height. But you do not move your arms in the X- and Y-directions — no, you turn your drilling arm about the pivot at the elbow and turn and slide the circuit board to suit. You optimise your movements using your visual system — not perfectly, however, as sometimes you might miss a hole hidden by swarf. You do not need a firmly fixed base on which to work; your drilling arm is fixed at the elbow pivot, and what is between this pivot and the circuit board does not matter. Even a small tool between the two makes no difference. Your arms and your sitting position need not be absolutely parallel, or even anywhere near, and there are no 90 degree angles to be seen: two pivots are enough!

fit smoothly and without play in the two holes we have drilled. The peg for the reference point is fixed, while the peg for the rotation point can be slid along a line allowing for various distances between the two points. The coordinates of the fixed peg are known, as is the angle of the sliding rotation-point peg. The drilling data comprise the X- and Y-coordinates of the two points, and so the distance between the two and the angle they make with the Cartesian axes can be calculated using a simple program on the PC. After a translation and a rotation the position of all the other points can be determined, no matter how askew the film was placed on the circuit board during exposure or how the board lies on the CNC table.

### This method of registration is suit-

able for the manufacture of one-off circuit boards, or for a number of different boards. For small-quantity production runs a slightly different procedure can be followed.

The system uses two pivots, one for the workpiece (the circuit board) and one for the drilling arm. This allows any desired point on the circuit board to be brought into range on the turntable. This system has the big advantage over a linear construction that only two bearing points are needed whose exact separation is the only quantity that needs to be known. This requires no expensive specialist components: the bearings simply have to remain vertical and free of play. To align an axle to professional standards of accuracy, two so-called taper bearings are used. These can withstand enormous forces and are

The Concept

Figure 2. The well-known principle of the 'axle drive'.

expertly made to remain solid and permanently free of play. This is the main reason why our machine is so economical to build.

A rather significant disadvantage ought to be pointed out. Normally, in conventional linear machines the axles are long threaded rods supported by bearings and turned by a motor. A nut is fixed to the moving part and moves backwards or forwards according to the direction of rotation of the screw. This mechanism naturally gives a high effective gear ratio. Suppose that the screw gives a linear movement of 4 mm per rotation and is driven by a stepper motor with an angular resolution of 200 steps per revolution. The linear motion corresponding to one step is 4/200=0.02 mm. That is ideal for this kind of machine. Gearing is therefore completely unnecessary.

Our machine is not driven by a threaded rod, but rather works directly with angular motion. The 240 mm long tool arm has a travel of

240 mm x 2 x 3.14 = 1510 mm (circumference of circle with arm length as radius)

A stepper motor driving this directly would take 200 steps to make one revolution; the distance corresponding to one step is therefore 1510 mm/200=7.55 mm, rather too great for a CNC machine. For a desired resolution of less than 0.04 mm we need to gear the motor down by a factor of at least 7.55 mm/0.04 mm=190:1. That is not exactly straightforward.

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