Freedom from friction

By KEVIN McCARTHY
Chief Technology Officer
Danaher Precision Systems
Salem, N.H.

Edited by Leland E. Teschler

The advantages of air bearings are well known among those working in motion control. These devices are most noted for providing a means of movement without the ill effects of friction. The benefits they provide come in handy when realizing the nanometer-scale resolution that typifies the assembly of photonics devices such as lasers and fiber-optic pigtails.

However, there are subtleties that are not widely appreciated when it comes to the numerous small moves in photonic alignment. These subtleties have an impact on how air bearings are deployed. They also illustrate why conventional bearing systems are likely to fall short in providing such small moves.

It is interesting to compare the approach used to realize nanometer-scale moves with that for large moves. Positioning systems generally "profile" such moves, with velocity carefully shaped as a function of time so as to minimize higher derivatives and avoid exciting system resonances. The resulting graph of velocity versus time is the familiar trapezoidal move curve where velocity ramps up to some level and holds steady for a specified time, then ramps down to zero at the end of the move.

As it happens, such careful move profiling is of no utility whatsoever for the numerous small (under 100 micron) moves typical of photonic alignment. In this case, the overall energy is small and the servoloop itself is a perfectly adequate trajectory shaper. The position servoloop functions as a low pass filter. The position command can be a simple step function; At time t = 0, we simply command the position loop to be at the destination. Despite the discontinuity of the step command, the actual stage motion follows a smooth curve.

The goal, of course, is to minimize the move and settle time so as to make as many small moves per second as possible. We may also impose fairly tight values on the settling tolerance. A "move and settle time" is meaningless without a definition of the settling window, which is the acceptable difference between the target position and the actual position. In cutting-edge photonic alignment, this settling window may be as small as 10 nm.

It is interesting to graph the following error, the difference between commanded and actual position, for such a system. It is zero prior to the move, then jumps to equal the move size at t = 0 because the stage cannot respond instantly to the step command. It eventually decays back to zero. A key performance issue is this decay or settling time. It relates to servo bandwidth.

Predicting dynamic response
Consider the case where the position loop receives commands in the form of small-amplitude sine waves. A plot of position-loop response versus the frequency of the sine wave will be flat from dc out to some frequency where it will peak slightly (if the loop is properly tuned). It will then decline at a rate of f ^-2.

The point at which the amplitude has fallen by 3 dB is termed the system servo bandwidth. It corresponds to the point where there is a 90 phase relationship between the command and the response. It is also the most important parameter in predicting the dynamic response.

The goal, normally, is to maximize servo bandwidth. Doing so brings the highest rejection of outside perturbations and provides highest dynamic performance. There are a number of factors that can limit servo bandwidth, but the phase lag from the first structural resonance usually sets the practical limit. Attempts to boost servo bandwidth beyond this limit turns the positioner into an oscillator.

The natural form for the servo bandwidth is ₀, expressed in radians/sec, but the more familiar term for servo bandwidth is f₀ expressed in Hz. A conservative value for the realistic servo bandwidth of either air or mechanical bearing stages is about 50 Hz.

The lack of friction acting on airbearing stages make them easy to model mathematically, and the fidelity of the models to real-world results is quite good.

A detailed model for the decay of following error after a step move is beyond the scope of a magazine article. Nevertheless, it resembles a simple exponential decay whose time constant is a direct function of the servo bandwidth. Lack of friction in the stage means the integrator term can be zero or small. The time constant is then determined by the servoloop proportional term (assuming, of course, that the derivative term has been set to provide adequate damping).

The proportional time constant for this decay is ( ₀)^-1 or (2 f₀)^-1. For a typical servo bandwidth of 50 Hz, this is about 3.2 msec. The time behavior of the following error, starting with a step function equal to the move size is

where E = following error, X = move size, and = the time constant (2 f₀)^-1. Accordingly, the following error drops by a factor of e (2.718) every . If we permit a "close counts" approximation, the following error will fall by a factor of three every 3 msec.

In the case of a typical 10-micron move, this means following error drops from 10 microns to within 10 nm after a mere 18 msec. Viewed from another perspective, it is possible to make 50 such 10-micron moves every second, settling to within 10 nm. The real world being what it is a more realistic settling window with moderate cost encoders would be below 20 nm, still better than any other positioning technology.

Proportional-term failure
For small moves, friction is the dominant problem plaguing conventional stages. This leads to the important point that the proportional term in a PID (proportional integral differential) control loop is of no use whatsoever for small moves with this class of stage. This becomes obvious by considering that proportional term of a servoloop produces a force (or torque in a rotary system) linearly proportional to the error. For example, if 1,000 counts of position error produces 100 N of force, 500 counts of error produces 50 N, and so on.

In the graph of position error versus applied force, the slope of the proportional term represents servo stiffness, usually in Newtons-per-meter. It is readily calculated from

where S = stiffness, m = mass, kg, and ₀ = servo bandwidth, radians/sec. Friction is a constant force. A problem arises, however, when the force due to the proportional term is less than or equal to the frictional force. Say, for example, the stage is 50 microns from the target. The proportional term responds with 5 N of force. If the frictional force is 6 N, nothing happens. The point at which the proportional term fails is called the friction boundary. If the proportional term were the only one in the servoloop filter, the following error would remain trapped at this level, never reaching the target position.

Put another way, there is no motion if the restoring force written to the output d/a converters is less than system friction. This is true regardless of how high the sample rate might be.

The amount of position error is calculated by dividing friction by stiffness. Doubling this figure yields the error for a bidirectional motion system:

where E = error, meters; F = friction, N; m = mass, kg; and f₀ = servo bandwidth, Hz. In a typical case with 2 N of friction, a 1-kg moving mass, and a servo bandwidth of 50 Hz, the value for the friction boundary is 81 microns. This value is typical of mechanical bearing stages. Friction would be a little larger for a recirculating bearing stage and a little less for a crossed roller stage. But errors on the order of 80 microns are large compared to the dimensions involved in photonic alignment.

It is interesting to run the same analysis for a leadscrew stage. The angular error for a leadscrew is

where E = error, m; L = screw lead (advance/rev), m; T = torque, N-m; J = total rotary moment of inertia, kg-m²;

and f₀ = servo bandwidth, Hz. The motor rotor dominates the rotary inertia. Other significant contributors are the leadscrew and (coming in a distant third) the reflected payload inertia.

Typical values for screw lead (0.002 m), leadscrew torque (0.05 N-m), total rotary inertia (5 10^-5 kg-m²), and servo bandwidth (50 Hz) yield a friction boundary of 13 microns. This is still too large to be useful in photonic alignment tasks.

The I in PID
It is also interesting to examine whether the integral or differential terms in the typical servocontroller PID loop can cure the friction boundary problem. The differential term supplies a force or torque which opposes motion and which is proportional to velocity. It provides damping necessary to ensure stability. But it is of no use once motion has ceased at the friction boundary. Moreover, it generates a force with the wrong sign to be of any help.

If conventional stages with friction get at all close to final position, they must turn to the integrator term. It slowly sums the errors of past samples to produce a growing output command that will eventually get the system to zero steady state position error.

Unfortunately the introduction of the integrator term degrades stability. For stable systems, the integrator time constant will be five to ten times that of the time constant of the proportional term. So with a system having the 3.2 msec decay time of the earlier example, the integrator time constant would be on the order of 25 msec.

Similarly, the time needed for the 10-micron move in the example of the frictionless air bearing would expand from 18 msec to about 150 msec if carried out using a conventional stage with friction. Tricks such as gain scheduling, acceleration feed forward, friction bias, and backlash compensation can mitigate the problem somewhat. Nevertheless, the resulting systems have a limited performance as well as substantially less throughput and precision.

In contrast, frictionless air-bearing stages can follow the rapid exponential decay of the proportional servo down into the noise. The ultimate limits on performance for these stages are set only by encoder resolution, amplifier linearity, and external vibration.