Designing Positioning Systems for Constant Velocity

Take the covers off such precision equipment as inkjet printers, scribers, and scanning microscopes, and you’ll likely find a finely tuned positioning system. It is paramount that the positioning mechanics in such applications deliver smooth, constant motion despite vibration noise in the surrounding environment.

But designing such gear for smooth motion can be tricky. One key element in devising robust positioning systems is to ensure the stage has a high natural frequency. This goal, in turn, involves minimizing sources of internal and external vibration, which may excite structural resonance, and equipping the servosystem with properly designed noise-rejection filters.

Internal vibration sources include, among other things, motor cogging, bearing jitter, ball-screw wobble, rail straightness and flatness, cable track rattling, amplifier commutation, and encoder noise. External vibration sources can include ground vibration, acoustic noise, tool forces, and other rotating parts such as fans and pumps. Each of the vibration sources can be characterized in a simplified way by two parameters: disturbance amplitude, X, measured in microns, and disturbance frequency, f, measured in hertz.

If the disturbance frequency coincides with the resonance frequency, the disturbance amplitude will be amplified by a transmissibility factor, sometimes called magnification or simply gain. The resonance frequency depends on the stiffness, mass and structural damping of the resonating structural member.

Motion designs must undergo a constant-velocity (CV) analysis to see how their systems will behave when running at steady states. But a full constant-velocity analysis may become a bit complex, requiring finite-element analysis for natural frequency calculations and complex analysis for noise filtering and servo compensation.

There are a few simplified equations that can give an intuitive feel and quick estimates for the level of smoothness that various positioning-stage configurations can realize. These equations have also been built into an interactive and educational tool built on the widely used Mathcad program.

We will consider an example to illustrate how a simplified constant-velocity analysis might proceed. It should be noted that the following calculations are approximate. They should be confirmed by testing and by other means of system analysis before using them to design real machinery.

Most engineering specifications of constant velocity include the nominal process velocity, Vn, and the allowable variations of this velocity, CV, as follows:

V_n = 0.3 m/sec
CV = 5 %

Another important specification is the process frequency at which CV is needed. For example, consider a drug-discovery machine, where an inkjet printer must place sample droplets of 50-μm diameter next to each other 100 μm apart with an accuracy of ± 5% at a constant velocity of 0.5 m/sec. This means the printing frequency is equal to (500,000 μm/sec)/(50 μm) = 10 kHz. The generic term for this parameter is process frequency, F_p.

F_p = 10 kHz

Encoder feedback can conceivably be used to trigger the inkjet pulses precisely. But the inkjet assembly could be vibrating as the inkjet prepares to fire, with amplitudes the encoder can’t detect. The result would be a dynamic error. So it is better to design and test the system for CV at the point of interest (in this case, at the inkjet head) and not at the encoder.

Knowing the process frequency, we can determine the sampling interval, s, and the allowable dynamic error, E_d, using the following equations. The Mathcad tool uses these equations to compute estimates of s and E_d.