# A Small Step Is a Big Step

T

#### Tom Bullock

A SMALL STEP IS A BIG STEP

In 1969, when Neil A. Armstrong stepped onto the surface of the moon, he declared, “One small step for a man, one giant leap for mankind.” This concept of a small step being a big step applies to servos as well. For a servo to run smoothly at low speeds or to position accurately, it must be able to respond to very small increments (or steps) in position.

My friend and partner, George Younkin, is permitting me to summarize one of the chapters in his book, Industrial Servo Control Systems—Fundamentals and Applications. George is retired after 42 years in industry. During that time he established himself as one of the foremost authorities on servos in the United States. He now helps us with our servo seminars when he’s not in Florida or on a cruise. He can balance theory with practice better than anyone I know.

George presents a formula in his book for computing the least step (in command to which a servo will respond). When one is specifying an accuracy for a machine, the machine must be able to respond to steps smaller than the specified accuracy. This formula was developed after many years of theoretical contemplation between George and Dr. John Bollinger at the University of Wisconsin and after much testing and verification by George. It works! It is not my intention to present the formula here because there is no space to do it justice, so I’ll relate the concept instead.

Resolution in a positioning servo is the ability to respond to a small step in position. To begin with, we would hope that the electronics would work in increments well below the machine resolution so it would not be the limiting factor. This is a matter of scaling. Should you require an accuracy of one-thousandth of an inch (.001 inch) in a system, you would be advised to scale the command and feedback to work in increments of .0001inch. This is the ten-to-one rule of thumb that is considered ideal. There are many real-life applications where this ideal is hard to achieve. I know of cases where four-to-one and even two-to-one got the job done.

Once the electronic resolution is sufficient, we are faced with the machine itself. Let’s begin by measuring the machine resolution. This can be done by continuing to increase the command by its least increment (.0001 inch in the above example) until something happens. Let’s say we incremented the command five times (.0005 inch); we will find that when the machine finally moves, it will jump by .0005 inch also. If we increment the command five more times, the machine will move again. The amount by which the machine jumps when we very slowly increment the command is the resolution.

This value can be computed with George’s formula, which has shown excellent correlation with the above-described measurement. The motor constants, armature resistance, and position loop gain are important factors, but the ratio of static friction to running friction is by far the most critical. Static friction is the force that it takes to initiate motion, and running friction is the force that it takes to maintain motion. Theoretically, if the static friction could be made equal to the running friction, there would be infinitesimal resolution. This says that friction isn’t bad if it can be properly controlled. As a matter of fact, friction will help to dampen a servo, thereby making it less touchy.

Usually, with hydrostatic bearings, air bearings, or roller bearings, the static-to-running ratio is fairly close to 1.0. With a metal-to-metal horizontal slide, it is not uncommon for the static friction to exceed the running friction by 50% to 100%. To achieve good resolution (and accuracy), something must be done.

George was involved with a number of studies where Teflon-like way-lining materials were experimented with. A few were found that had static-to-running friction ratios very close to 1. These are commonly being used on large machine tools today where resolutions of .0002 inch and less are being achieved.

Should you want to chat with George or me about resolution, our number is 920/929-6544, or you can e-mail us at [email protected] or [email protected].

Achieving small steps with metal-to-metal slides on large machine tools has certainly been a big step forward. It may not be a walk on the moon, but it makes us servo guys smile!