The mathematics of probe formula in multi-point metrology fixtures can be quite confusing. In many measuring systems these formula are hidden away quite intransparent to the customer. They are treated as secrets by the fixture manufacturer and protected by passwords, even cryptology. This does not help when checking the fixture as these formula can be one source of deviation when measurements are compared to the results of the metrology laboratory.

The aim of this article is to explain the mathematics behind these formula and provide an aid for customers to systematically check multi-point fixtures and understand the underlying measuring principles applied.

To help us break down the problem into manageable steps we will start with some definitions:

• we will assume that the probes touch the part directly. They will register a positive deviation when the surface of the part moves towards the probe and displaces it. We will also assume that all probes have the same leverage (1:1). This will simplify our formula and make it easier for us to visualise the measurement 1.
• Our first formula will evaluate a 2 dimensional plane as we have to walk before we run!
• Let us assume that the setting master and part have the same profile.

Use the same axis system as the production

To help us visualise the problem we can cut out a part from a part drawing displaying the plane which we wish to analyse or sketch a roughly to scale profile. On a second sheet of paper and to the same scale we draw the fixture and mark the points of contact of the probes and their direction of deflection. Do not forget to note the part axis, preferably using the same convention used in the manufacturing machine.

Armed with two pieces of paper and a list of inspection characters (preferably in a spread sheet) we can start generating our formula.

First of all we can knock off all the simple cases:

• Opposing probes (same probe axis opposite direction) measure the distance between them as the sum of their deviations [(Px + Py)].
• Probes on the same axis pointing in the same direction have to be subtracted [(Px - Py)].
• A point on the part axis may lie halfway between opposing probes and is half the difference between the two [(Px - Py)/2].

These formula may represent some of the inspection characters required - diameters or lengths. They also represent some of the raw values required for form or position measurement. Think of them as the basic building blocks. It is therefore important that we understand why these basic formula work and where their limitations lie.

Test the base formula before proceeding

To test the basic building blocks and understand the underlying mathematics just place the profile of the part onto our paper fixture. Align its axis with the fixture axis and position it as centrally as possible. Assume the first position of the part to be the position of our setting master during "calibration" and set your "probes" in this position to zero. You can do this in the spread sheet by adding cells for each of your probes and including the cells in the formula. Now move the part one unit (1 mm or 10 mm depending upon the size of your part drawing) along any one of its two axis. Do not rotate the part. Estimate the deflection of each probe and transfer the value to the spread sheet. You should see that the lengths and diameters remain unchanged. The points upon the parts axis should have moved, the magnitude AND direction depending upon the direction of movement relative to the direction of the probes involved.

We utilise the relative strength of small angles

Why did we not rotate the part? What about limitations? Rotation is not calculated for in this simplified model and it is normally not required. Most multi-point fixtures just assume that part rotation is kept to a minimum. An acceptable minimum will differ for every fixture and almost every inspection character with it. It is a fact that small distances in combination with small angles will have very little influence on the final measurement value.

For example, if a spindle is rotated by 0,1° off its axis, the simple Px + Py measurement of a diameter of 30 mm will be measured too large if we ignore rotation. The mistake is however small as the formula below demonstrates:

• correct length = 30.000 mm
• rotated distance between probes = sqrt((30.000 * 30.000) + (30.000 * sin(0,1) * 30.000 * sin(0,1))) = 30.000045 mm

Once you are convinced that the basic building blocks are working (i.e. the values generated by the formula, both magnitude and direction, are plausible) we can continue to more ambitious inspection characters.

To be continued

1. Of course this is normally not the case - the probes should touch the part indirectly, often using a lever, however all system manufacturers allow the use of factors to change the direction of the probe and also if required change the leverage applied.