A few mathematical tools can help guide the way toward defeating wobble, jitter, cogging, and similar effects that can prevent positioning systems from working smoothly.
Edited by Leland Teschler
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:
Vn = 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, Fp.
Fp = 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, Ed, using the following equations. The Mathcad tool uses these equations to compute estimates of s and Ed.
Knowing the maximum allowable Ed of the machine for CV applications can lead the machine designer to a general system configuration to reduce internal noise as follows:
Ed< 0.1 μm — Use air bearings, brushless linear-motor stages, laser interferometer.
0.1 μm <Ed< 0.3 μm — Use cross rollers, brushless linear-motor stages, linear encoder.
1.0 μm <Ed< 50 μm — Use recirculating bearings, ball-screw-driven stages, rotary encoder.
50 μm < Ed — Consider roller bearings, belt-driven stages with rotary encoder or stepper motor.
CV requirements also impact the feedback-device resolution, r. The feedback device must be able to resolve changes of less than one-fifth of the maximum allowable dynamic error, Ed. In this example, r is a parameter that the Mathcad software calculates.
Maximum value of encoder resolution for CV, r = 0.3 μm.
Other machine requirements, such as positioning accuracy, maximum travel, maximum velocity, and environmental conditions, also affect the choice of parameters.
The typical way of testing CV is to observe and count the encoder pulses between fixed time intervals. But this method does not detect the dynamic error at the point of interest, which can arise because of vibrations caused by resonance or various disturbance sources.
A more-precise way of testing CV is through use of an external source such as a laser interferometer with precise positioning measurement at the point of interest. The actual velocity can then be determined by dividing the difference between two consecutive position samples and the sampling period. The difference between the actual and nominal velocity divided by the nominal velocity determines the CV value.
It is, thus, also important to specify an acceptable sampling frequency, Fs, at which CV will be tested. A typical sampling rate for CV testing by laser interferometer is 1 kHz. This value is considerably higher than mechanical-resonance frequencies typical of positioning systems, which are normally on the order of several hundred hertz.
The solution to the CV problem thus reduces to finding the output velocity at the point of interest as a function of input-noise disturbance.
There are three important goals when designing any sort of high-performance, high-accuracy machine: Make the machine as stiff as possible, minimize moving weight, and provide enough structural damping to limit the amplitude of vibration. Stiffness depends primarily on the rails, the actuator, and the structure of the machine. High-stiffness values are on the order of 100 N/μm . Low values are on the order of 1 N/μm. For our example, assume a machine stiffness k = 15 N/ μm and moving mass M = 70 kg.
The resulting lowest natural frequency, Fn, of the machine part or positioning stage is best calculated during the design phase using finite-element analysis. The analysis will typically indicate not only the natural frequency of the slide but also its modes of vibration. The designer typically chooses the location of the encoder read head to be on a node. The nodes are areas on the positioning slide that have zero amplification in resonance conditions.
For the purpose of our simplified CV analysis, we can estimate the lowest natural frequency Fn as follows:
Designers can use an advanced tool such as Matlab to design and simulate the machine, once they know the natural frequency of the positioning stage as well as other structural parameters such as mass and damping coefficient, transmissibility, and servo characteristics of the machine. The analysis can also determine the effect of various internal and external disturbances on the required CV.
For the purpose of our simplified CV analysis, we will estimate the closed-loop position bandwidth of the positioning system Fb. Typically, for a robust design (meaning one having high enough gain margin and phase margin), the position bandwidth is designed to be one-third of the natural frequency. This will assist in assuring that the CV will be insensitive to variations in system parameters and disturbances. Thus, estimated servo bandwidth Fb = Fn/3. For our example system, Fb = 24,571 Hz.
The servo bandwidth is an important figure of merit for positioning systems. It determines how closely the servosystem can follow a commanded position signal. A positioning bandwidth above 50 Hz is considered high, while a position bandwidth below 10 Hz is considered low. The accompanying Bode diagrams show a low-performance system with 3.2-Hz bandwidth. In this case the low bandwidth arises because stainless-steel belts drive the stage and the belts typically have a low stiffness.
Position bandwidth also sets the limit at which the servosystem can filter out disturbance frequencies. We can expect the system to filter out disturbance noise that the encoder picks up, below the closed-loop servo bandwidth. Beyond the servo bandwidth, the noise will either be amplified by structural resonance or attenuated by the system inertia.
Structural damping and frequency response
Structural damping is also important to assure smooth motion that satisfies desired CV requirements. Typical positioning structures of steel and aluminum have damping coefficients on the order of 0.04. For composite material, it is around 0.1. We’ll assume for our example that the servo-damping coefficient, ξ, is 0.04.
Knowing the natural frequency and the damping coefficient of the structure, let’s write the complex number expression of gain, Gf, for the amplification of noise amplitude, as a function of frequency f
The accompanying diagrams show typical frequency-response curves (magnitude and phase angle, respectively) of a single-degree-of-freedom frame, having mass, M; stiffness, k; and damping coefficient, ξ. These diagrams will be used for our CV estimates. Of course, real systems are more complex, typically including multiple resonances at higher frequencies. However, with the ratio of three between the lowest natural frequency, Fn, and the bandwidth Fb, as described earlier, we assume that the servosystem is stable and robust with sufficiently high gain margin (>45°) and phase margin (>10 dB).
The gain diagram also shows the amplification of a disturbance noise in the event the encoder does not detect the position error. If the encoder does indeed detect the position error, the servosystem will filter out the noise up to the frequency of the servo bandwidth. For this reason it is important to mount the encoder components using stiff brackets, as close as possible to the point of interest. This will minimize relative motion between the encoder and the point of interest on the machine. As a result, the servosystem will detect and filter all noise vibrations at frequencies up to that of the system position bandwidth.
Also important in these calculations is the velocity error due to noise. Recall the meaning of the gain as a function of frequency, Gf. It relates the amplitude of the input disturbance, or noise, X, to the actual output of position noise, Y. We can write it as Gf= Y/X. This means that for any reference sinusoidal input noise of amplitude, X, and a frequency, f, there will be output position noise with an amplitude, Y, at the same frequency, f.
The output noise velocity, Vy, at any frequency, f, can be determined by multiplying the output noise amplitude, Y, by its frequency, f. At any frequency, f, an input sinusoidal noise with an amplitude, X, will generate an output noise velocity, Vy. The value of Vy can be determined as follows:
(1) Vy= X × Gf × f × 6.28
Knowing Vy, we can now determine the constant velocity, CV:
(2) CV(%) = 100× (Vy × 10-6/Vn)
where Vy= output velocity, μ /sec; Vn= nominal velocity, m/sec; CV= constant velocity error, %.
Substituting equation (1) in (2), we get, for every given noise position, X, measured in microns, at any given frequency, f, measured in hertz, the final value of the actual constant-velocity error, CVa, measured in %, as given by
For our example, consider a ball-screw stage moving at 0.33 m/sec with pitch of 10 mm and lead-angle accuracy of ±15 μm. As the stage moves, the ball screw injects noise into the positioning stage of ±15-μm amplitude at a frequency of 0.3/0.010 = 30 Hz. To estimate the CV in this case, we plug a disturbance amplitude, X = 15 × 10-6m, and a disturbance frequency, f = 30 Hz into (3). The result for our example is an actual constant-velocity error CVa = 1.13%, and an allowable constant-velocity error CV = 5%.
Notice that by using equation (3), we conservatively assume that the encoder does not detect errors at the point of interest and, therefore, the servosystem does not filter them out.
Other disturbance sources such as rotating unbalance, ground vibration, and rail jitter can be checked in a similar quick way. The design parameters can then be changed to meet the CV requirements. With a 1-kHz disturbance at 10 μm, as might be caused by jerks, the analysis tool shows a CV = 0.17%.
Note that the examples assume the position of the disturbance is known. If the disturbance is just an applied force on the stage structure, the position disturbance can be estimated by dividing the force amplitude by the structural stiffness.
This Parker Daedal simplified analysis tool employs the equations we’ve described to interactively estimate the CV of a given positioning stage. The tool can also be used to determine a general way of configuring the position-system components — including rail, actuator, and a feedback device — for CV requirements. The tool further lets the user estimate by trial-and-error structural qualities of the positioning system to get a required CV for a given noise disturbance.
|Mathcad to the rescue|
To show how the Parker Daedal simplified analysis Mathcad tool would handle this task, we’ll show entered numerical values in red and observed results in blue.
Nominal velocity, Vn = 0.3 m/sec
Constant velocity, CV = 5 %
Process frequency, Fp = 10,000 Hz
Sampling interval, s = 30 μm
Maximum allowable dynamic error, Ed = 1.5 μm
Maximum value of encoder resolution for CV, r = 0.3 μm
Sampling test frequency CV, Fs = 1,000 Hz
Machine stiffness, k = 15 N/μm
Moving mass, M = 70 kg
Natural frequency, Fn = 73.712 Hz
Estimated servo bandwidth, Fb = 24.571 Hz
Structural-damping coefficient, ξ = 0.04
Disturbance amplitude, X = 15 μm
Disturbance frequency, f = 30 Hz
Actual constant-velocity error, CVa = 1.295 %
Allowable constant-velocity error, CV = 1.13 %