It’s all too common to forget about abbe errors when designing linear positioning systems. An ounce of prevention, though, can be better than a pound of cure.
Each component in a system imposes errors on the total system accuracy. One particularly pesky, and usually overlooked, inaccuracy is abbe error. For the best multiaxis linear performance, engineers should analyze the effect bearings, position feedback systems, and drive mechanisms may have on this type of error.
Every axis of motion has six degrees of freedom. The objective in designing a positioning system is to precisely control one degree — the direction of motion — and minimize the effects of the other five. The remaining degrees often produce unwanted motions that reduce system performance.
These motions come from several sources, including imperfections in the bearings, deflections due to loads, and thermal distortions. Motions caused by angular deflections are the most troublesome because they contribute to a phenomenon known as abbe error. This inaccuracy is a combination of an axis' angular deflection and offset distance from the position sensor (abbe error is the product of offset distance and the sine of the offset angle). Its effect is to make a feedback unit's indicated position read as either longer or shorter than the actual position.
Budgeting for errors
In X-Y or multiaxis systems, all errors statistically combine into a total error budget. Designers use this number to allocate allowable values for each error source. They then select the linear components that do not exceed those values.
In some designs, abbe error is the largest component of the error budget. Even a relatively small angular deflection with a moderate offset can result in a significant error. For example, an angle of only 5 arc-sec and an offset distance of 100 mm results in an abbe error of 2.5 μm. This means that a linear encoder scale provides correct positioning information only at the point where the measuring head attaches to the machine.
The usual suspects
The main components that contribute to abbe errors are position measurement devices, bearings, and the drive mechanism.
Position measurement. Measurement devices are available for either indirect or direct determination of offset or angular displacement. Indirect measurement, however, can cause positioning errors because it requires a conversion factor between the indicated measurement and the actual position. A typical example is a rotary encoder mounted to a motor driving a lead screw. The conversion will not be exact because of the mechanical tolerances of components such as lead-screw pitch and deflections in the couplings.
Direct measurement gauges the position without a conversion factor. Devices capable of such measurement include linear encoders, laser interferometers, and 2-D grid encoders, with the last two devices capable of doing the best job at minimizing abbe errors.
Bearings. Linear systems typically use one of four types of bearings: linear crossed rollers, linear ball bearings, recirculating ball bearings, and air bearings.
Because of their line contact, crossed roller bearings can handle high loads. Deformations under load, however, may add to abbe error.
Linear ball bearings do not recirculate. Instead, they move half the distance that the slide moves so they need a long length to accommodate their motion. Because they move relative to the load, the slide may experience large cantilevers at the travel extremes, leading to angular displacements.
In recirculating bearings, the bearing reaction forces remain in one place with respect to the load, so the system uses a smaller footprint. These bearings can support heavy loads because weight distributes over several rows of balls in the nut assembly. One of the drawbacks of this system, though, is that the motion is not as smooth as that from linear bearings. As the balls enter and exit the raceway, they can create vibrations that may affect position measurement.
Air bearings offer the highest mechanical precision because the noncontact drive eliminates wear and friction. Also, since an air bearing can float in two directions, it enables single-plane stage configurations, minimizing abbe errors. However, these bearings can be the most costly alternative.
Drive choices. In general, the typical drive mechanisms for linear systems include lead screw or ball screw with rotary servo motor and encoder; lead screw or ball screw with open-loop microstepper; linear servomotor; and piezo ceramic linear motors, a type of friction drive.
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Precision applications commonly use linear-screw drives. The screw pitch offers a mechanical advantage that is useful for positioning heavy loads. However, mechanical backlash, pitch errors, and wind up are common limitations. In applications requiring long travels and high speeds, accuracy and repeatability can be reduced if the screw reaches its critical speed, which causes vibration along the shaft. The solution is to increase the diameter of the screw, but that requires additional power and heat dissipation.
Linear servomotors provide linear thrust to position the load directly – without any intervening mechanical elements. They possess high dynamic stiffness, which keeps following error low. Additionally, because there is no linkage to mechanical components, there are no hysteresis, wind up, or pitch errors. With these motors, accuracy depends entirely on the bearings and feedback control system. Thus they can offer an advantage in positioning accuracy over ball screws and rack and pinion systems.
There is a drawback to the lack of mechanical links: Linear motors are not well suited to widely varying loads, nor vertical loads without a counterbalance.
There are many types of linear motors available today. The appropriate choice depends on the application. The smooth non-ferrous (slotless) motor designs have no magnetic cogging, so these motors can deliver the levels of accuracy and smooth motion required in applications such as scanning.
Piezo ceramic linear motors are a relatively new choice for precision motion. A class of friction drive, these motors operate by driving matchstick sized piezo electric elements in a cross polarized configuration. Each movement of an element imparts a few nanometers of tangential drive motion to the driven slide. Resonant actuation at 40 kHz offers continuously adjustable velocities to 300 mm/sec with accelerations to more than 1 g, depending on the load. Mimicking servo motor dynamics, these motors are compact, generate almost no heat, and provide good position-hold stability (less than 10 nm) due to their inherent friction and direct drive characteristics. Overall, they do not contribute greatly to deflection or positioning errors in a system.
However, they are not suited for heavy loads because of their limited force range. Also, their speed is limited to approximately 300 mm/sec. They are a good choice for applications needing nanometer stability even with power off, small moves that need speed and fast settling times, and applications where magnetic fields are detrimental, such as disk drive testing and electron-beam lithography.
Differentiating terms of precision
Engineers, as well as manufacturers' specifications, interchangeably use (or confuse) the terms accuracy, repeatability, and resolution. Accuracy, however, is not the same as resolution. Plus, it is more difficult to obtain. In addition, there's no guarantee that if a component has one of these qualities, it will have another or both.
NMTBA standards define these terms. But it's up to the engineer to determine whether the linear component manufacturer measured such qualities according to those standards.
Arthur Holzknecht is manager, Engineered Systems at Anorad Corp., Hauppauge, N.Y.