Resolution, accuracy and repeatability play crucial roles in determining the precision of a position sensor and ultimately the performance of the overall system. This is why it's crucial to understand how resolution, accuracy and repeatability relate in the context of encoders. The previous entry in our blog series reviewed resolution. Now, we finish up by reviewing accuracy and repeatability and how they relate to encoder precision. We will also discuss how calibration can correct for errors caused by eccentricity or poor accuracy.

Accuracy

Accuracy represents how much a measured position deviates from a calibrated standard. Figure 1 depicts Encoder A and Encoder B. A blue dot shows the true position to be measured. Encoder A is perfectly accurate and has counts representing 1 degree with identical, uniformly sized bins. In contrast, Encoder B's bins are differently sized, with bin 0 ranging from 0 to 1.1 degrees and bin 1 ranging from 1.1 to 2 degrees.

Encoder A's blue dot is in bin 1 and will produce a count of 1. However, in Encoder B, the blue dot's position is incorrectly reported as count 0 due to the unevenly spaced bins. This measurement has an error of 0.1°. If we assume this error is the largest error throughout a revolution, the encoder will have an accuracy rating of ± 0.1°.

Figure 1

Practical Accuracy

With an optical encoder and assuming a concentrically mounted scale, scale pitch variation is often the largest source of error. In other encoder technologies, there are analogous geometric non-uniformities that also contribute to error.

For example, gain, offset and phase variations between the sin/cos signals contribute to sub-divisional error (SDE). Harmonic distortion in the sin/cos signals can further contribute to SDE.

Calibration can account for scale pitch errors, but it is more difficult to correct high-frequency SDE through calibration. Figure 2 shows high-frequency SDE superimposed on pitch error over one revolution.

Figure 2

Repeatability

Repeatability is the degree that successive position measurements vary at a specific location, and it is tied to resolution because the measurement can only be as repeatable as the smallest measurable distance. Typically, repeatability is specified as one least significant bit (LSB), which, in terms of angle, is the nominal size of one bin. Sometimes, repeatability isn't specified because it is assumed to be identical to the resolution.

Some manufacturers specify repeatability as < 1 LSB. This, however, does not consider a potential boundary condition as shown in Figure 3. Here, the blue dot depicts a position which could fall into either bin.

Figure 3

The magnetic encoder may give a different reading depending on the direction of approach to the position to be measured. This is due to magnetic hysteresis, and repeatability may be 2 LSBs or more.

Repeatability and Accuracy

Although accuracy is critical to achieving precise positioning control, it is possible to achieve superior position control with low-accuracy measurements because they can be corrected through calibration. An example of this can be seen in Figure 4, which depicts a shooting target.

When trying to hit the target's center, the cluster of green shots are the most precise because they are accurate and repeatable. Because of the blue cluster's repeatability, it would also be precise if its accuracy were effectively compensated for through calibration. In contrast to the blue, the yellow shots are highly accurate but have poor repeatability.

Figure 4

Eccentricity Error

If the readhead on an optical encoder is mounted at a specific location on the scale, it can generate significant angular error if the scale is not precisely centered on the axis of rotation. Bearing runout also contributes to eccentricity errors. Because eccentricity error is repeatable, except when generated by non-repeatable bearing runout, it can be corrected through calibration.

Figure 5

The angular error attributed to eccentricity can be so great that it dominates the entire system's accuracy. To determine the magnitude of angular error, use the following formula:

$$\text { Angular Error }=\frac{e^* 1.296^* 10^6}{\pi^* D} \operatorname{arcsec}$$

Where e is the distance between the scale's center and the axis of rotation. As an example, a 50-millimeter scale with 100-micrometer eccentricity results in an angular error of 0.23 degrees. Capacitive and inductive encoders measure position over 360 degrees and are much more tolerant to eccentricity than optical encoders.

Calibration

Encoder calibration requires a high-precision reference, typically, an interferential optical encoder. After capturing one revolution of position data from the reference encoder and the encoder to be calibrated, the variations between the data sets are stored in a lookup table of offsets. Capturing more data points increases the calibration effectiveness, and many points are required for high-frequency errors.

Some encoders have calibration routines and the offset table is stored in the encoder memory. The drive or controller may also feature a calibration function. In either case, the offset is added to the position reading. Interpolation provides an offset value when the position reading falls between values on the offset table. The process of adding offsets introduces a delay which may affect servo loop performance.

Figure 6

The Gearbox Effect

A gearbox or leadscrew can increase an encoder's effective resolution by decreasing the smallest movement possible. For example, the output of a 10:1 gearbox moves 0.1 degrees when the input moves 1 degree. Using this gearbox would reduce the smallest possible move by a factor of ten. However, keep in mind that gearboxes and leadscrews are subject to backlash.

Consider the gearbox shown in Figure 7 When the green input gear changes direction, the blue output gear is physically disconnected until the mechanical play is eliminated. Even though the gearbox increased the effective resolution, it delivers no value during the backlash because the output gear doesn't move. Because backlash compromises accuracy, calibration is complex and only partially effective. Backlash typically reduces system accuracy by 1 to 2 orders of magnitude.

Figure 7

Conclusion

An encoder's resolution and repeatability are key to achieving high-performance position control, and it is important to understand that they are distinct parameters. Encoder accuracy is also critical to system performance, but should not be considered in isolation because mechanical errors can dominate. Also, keep in mind that if errors are repeatable, they can be accounted for with calibration which will increase encoder latency and add time to system commissioning.

Read our tech note to learn more.

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How Resolution, Accuracy and Repeatability Relate to System Performance: Part II