Settling down

Air bearings can include motor, encoder, and switch connections as well as crossed roller ways.

Our command trajectory is a step function, but because a stage cannot instantly respond to a step command, following error jumps to equal the move size. So how does following-error decay (or settle) over time?

Unlike conventional stages, which suffer from friction and other issues, direct-drive air bearing stages are a model physics package, without friction or physical contact. Performance is mathematically easy to replicate, and the models match real-world results well. In short, following-error decay after a small step move is desribed well by simple exponential decay, for which the time constant is a function of servo bandwidth. Because these stages lack friction, the integrator term can be zero or very small; the time constant is determined by the servo loop proportional term — assuming that the derivative term is properly set for adequate damping.

The proportional time constant τ for this decay is 1/ω0 or ½ πf0 … for a typical servo bandwidth of 50 Hz, this is about 3.2 msec. Following-error time behavior starting with a step function equal to move size is:

X · e-t/τ

where X = Move size

and τ = Time constant ½ πf0

The following error is zero just before the command move is entered. The error then spikes and quickly decays — more quickly than in other systems.
Select figure to enlarge.

Following error drops by a factor of e (2.718) every τ. If we permit close-count approximation, following error falls by a factor of three every 3 msec. Consider a 10-µm move: At time t = 0, our command position is +10 microns, the stage has not yet moved, and following error is by definition 10 microns.

At t = 3 msec, error has fallen by a factor of three to 3 µm — remember, close counts. After 6 msec, error is 1 µm; in 9 msec it has dropped to 300 nm, and continuing in this manner, we drop to within 10 nm in a mere 18 msec.

In other words, this machine can make 50 10-micron moves per second, settling to 10 nm. The real world being as it is, a more realistic settling window with moderate-cost encoders would be below 25 nm. That is a dramatic improvement over any other positioning technology, and highlights the advantages of frictionless air-bearing stages.

Servo bandwidth is one factor critical to stage dynamics. Consider a servo system command with a small amplitude sine wave, and assume we vary the frequency. The resulting amplitude versus frequency graph has a constant value from dc out through a certain frequency, where the amplitude will peak slightly and then decline as 1/f ² if properly tuned.
Select figure to enlarge.

As detailed in the sidebar Absolute value of force were the proportional term the only term in the servo-loop filter, following error would remain trapped at this level, never reaching the target position. The proportional term of the servo loop can be thought of as continuously asking the position counter, Where am I? Upon obtaining the position error, it calculates what it recons appropriate restoring force, which it then writes to the output DACs.

The problem is that this value is less than system friction, and no motion ensues. A high-performance digital motion controller may have an impressive sample rate of 5 kHz, but the proportional term is not particularly sophisticated, and just doesn't register some changes. Each second it accurately calculates five thousand output values, all equal and inadequate to move the stage. If motion in one direction is considered, this failure of the proportional term will cause the stage to stop well short of its destination; if we include moves in both directions, the error is doubled, as shown by the deadband distance in plots of absolute force values.