Conventional stages with rolling steel bearings exhibit friction and other mechanical nonlinearities; they are also dependent on servo-loop filters (including sluggish integrator terms) which all limit dynamics, throughput, and precision. In contrast, air bearings — those that carry loads on a layer of air — are nonfriction devices, so they can boost throughput 10 times versus rolling bearings.

Systems that make lots of small moves benefit most from the higher productivity of air-bearing ways. More specifically, small air-bearing stages can follow the rapid exponential decay of the proportional servo — down into noise — boosting throughput and accuracy. These stages are only limited by encoder resolution, amplifier linearity, and external vibration.

First some background: When a normal positioner makes large moves, profiling (the careful shaping of velocity as a function of time) minimizes higher derivatives and avoids exciting system resonances. However, move profiling is useless for small (under 100 µm) moves. Here, overall energy is very small, and the servo loop itself is a perfectly adequate trajectory shaper. The position servo loop functions as a low-pass filter, allowing a simple step-function position command. At time t = 0, we simply command the position loop to be at the destination. Despite step-command discontinuity, actual stage motion follows a smooth curve.

Where the goal is to minimize move and settle time, and make as many small moves per second as possible, tight settling tolerance is paramount. A move-and-settle time is meaningless without a definition of the settling window — the acceptable difference between target and actual position.

For example, in cutting-edge photonics-based alignment, this settling window may be as small as 10 to 20 nanometers. Following error — the difference between commanded and actual position — is zero just prior to the move.

### Bandwidth

Servo bandwidth is critical to stage dynamics. Consider a servo system command with a small amplitude sine wave, and assume that we vary the frequency. Resulting amplitude versus frequency has a constant value from DC to a certain frequency; the amplitude peaks slightly (if properly tuned) and then declines as *1/f ^{2}*. The point at which amplitude has fallen by 3 dB (which is also the point of a 90° phase relationship between the command and response) is the servo bandwidth of the servo system, and the most important predictor of dynamic response.

Normally, it's best to set servo bandwidth as high as possible, for highest rejection of outside perturbations, and greatest dynamic performance. The servo bandwidth is limited by a number of factors, but usually it is the *phase lag* from the first structural resonance that sets a practical limit. Why? Attempts to increase servo bandwidth beyond the limit set by structural resonances make positioners oscillate, with a poor prognosis for its service life.

The natural form for servo bandwidth is ω_{0}, expressed in radians per second, but the more familiar term for the servo bandwidth is *f*_{0}, expressed in Hertz. A conservative (and realistic) servo bandwidth value for either air or mechanical bearing stages is about 50 Hz.

### Settling down

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 π*f*0 … 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 π*f*0

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.

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.

The magnitude of the problem can be readily calculated. To do so we simply divide the friction (in Newtons) by the stiffness, in Newtons per meter, to obtain the error in meters. This result can then be doubled if we want to consider bi-directional motion. Taking advantage of the stiffness formula provided above, and using the more familiar Hertz value for the servo bandwidth:

**Error** (in meters) = *F*/m·π2*f*02

where *F* = Friction, N-m

*m* = Mass, kg

*f*0 = Servo bandwidth, Hz

If friction is 2 N, moving mass 1 kg, and servo bandwidth 50 Hz, the friction-boundary value is a whopping 81 microns. For moves of this size or smaller, the proportional term might as well be turned off.

These values are typical of mechanical bearing stages; friction would be a bit greater for a recirculating bearing stage, and a bit smaller for a crossed roller stage — but not by much. For the micron and sub-micron sized moves required during alignment operations, our servo loop simply doesn't work. This is clearly not an acceptable situation.

In the above example, we assume a direct drive for our mechanical bearing stage. It's worth asking if the use of a leadscrew can improve the situation. In fact, there are numerous reasons to avoid leadscrews in high-precision mechanisms, but let us see how they address the issue of friction. Reformulating the above equation to reflect the case of a leadscrew-based system, the angular error, in meters, is:

**Error** = *LT*/2J · π3*f*02

Where *L* = Screw lead - — *advancement per revolution*, m

*T* = Torque, Nm

ℱ = Total rotary moment of inertia, kg-m2

Rotary inertia is dominated by the motor rotor, followed by the leadscrew, and in a distant third, reflected payload inertia. If we plug in typical values (a screw lead of 0.002 m, leadscrew torque of 0.05 N-m, total rotary inertia of 5 × 10-5 kg-m2, and a servo bandwidth of 50 Hz) the friction boundary is 13 microns.

Conclusion: The mechanical advantage of leadscrews helps a bit here, but the inability to make moves smaller than 13 microns is of little value.

### Integrator term help?

In a typical servocontroller PID loop, however, there are two additional terms present. The D term supplies a force or torque that opposes motion, proportional to velocity. While this term provides damping for stability, it is of no use once motion has ceased at the friction boundary, and the force it produces is of the wrong sign, anyway. If conventional stages with friction are to close to final position at all, they must turn to the last of the three terms in the PID loop — the *I* or integrator term.

The good news is that the integrator term, unlike the proportional term, gets the picture, and slowly sums the errors of past samples to produce a growing output command that eventually get the system to zero steady-state position error.

The bad news is that the introduction of the integrator term degrades stability, and for stable systems (a reasonable expectation, after all) the integrator τ or system time constant is five to ten times that of the proportional τ. Unlike the proportional τ*p* of 3.2 milliseconds, integrator τ*i* is on the order of 25 msec.

The ten-micron move in our frictionless air bearing example, which took a mere 18 milliseconds, is about 150 msec with a conventional stage. While there are a series of tricks (gain scheduling, friction bias, backlash compensation, and so on) that can be used to try to correct the problem, the issue remains.

*For more information on air bearings, call (800) 227-1066 or visit* dovermotion.com.

### Traditional system: Absolute force value

Friction is the main drawback of conventional stages, largely because it makes proportional servocontrol terms useless for small moves. To better understand the problem, consult the graph below. Here, the vertical axis is absolute value of force (which would be torque in a rotary-driven system) and the horizontal axis is position error, both positive and negative.

If we look at the proportional term of a servo loop, it produces a force that is linearly proportional to the error — hence its name. For example, if an error of 1,000 counts results in 100 Newtons of force, then an error of 500 counts would produce 50 Newtons, and so on.

The response of the proportional term is shown by the red line; its slope is the **servo stiffness**, in Newtons per meter; as it happens, this can be readily calculated - it is equal to *m*·ω02/4, where m is the mass in kilograms, and ω0 is the servo bandwidth in radians per second. The more familiar servo bandwidth in Hz. is simply ω0/2π.

Returning to the graph, note the horizontal line just above the X axis. This corresponds to the friction in the system, in units of Newtons. The force developed by the proportional term acts to drive the moving element of the stage towards zero position error. A problem arises, however, when the force due to the proportional term is less than or equal to the frictional force. At this point, we're stuck: the stage is, say, 50 µm from the target position. The proportional term responds with 5 N of force, but with 6 N of frictional force, nothing happens. The stage motion has encountered the friction boundary, at which the proportional term fails.