When evaluating the accuracy of a linear motion system, the area of focus is often the positioning accuracy and repeatability of the drive mechanism. But there are many factors that contribute to the accuracy (or inaccuracy) of a linear system, including linear errors, angular errors, and Abbé errors. Of these three types, Abbé errors are probably the most difficult to measure, quantify, and prevent, but they can be the most significant cause of undesirable results in machining, measuring, and high-precision positioning applications.

Abbé errors begin as angular errors: Abbé errors are caused by the combination of angular errors in the motion system and the offset between the point of interest — for tooling or the transport of load — and the origin of the error such as the linear-axis screw or guideway. Angular errors — commonly called roll, pitch, and yaw — are unwanted motions due to the rotation of a linear system around its three axes.

If a system is moving horizontally along the X axis, as shown below, pitch is defined as rotation around the Y axis, yaw is rotation around the Z axis, and roll is rotation around the X axis.

Errors in roll, pitch, and yaw typically result from inaccuracies in the guide system, but mounting surfaces and methods can also be sources of angular errors. For example, mounting surfaces that are not precisely machined, components that are not sufficiently fastened, or even varying rates of thermal expansion between the system and its mounting surface can all contribute to angular errors greater than those inherent in the linear guides themselves.

Note that crossed-roller linear bearings are particularly sensitive to mounting errors. Their rigidity and accuracy make them less forgiving than other options of mounting inaccuracies. That’s why many linear-bearing manufacturers recommend only mounting crossed-roller variations to honed surfaces expected to exhibit no more than a few micrometers of deflection.

Abbé errors are especially problematic because they amplify what, in most cases, are very small angular errors, increasing in magnitude as the distance from the error-causing component (called the Abbé offset) increases.

For overhung loads, the farther the load is from the cause of the angular error the higher the Abbé error will be. The cause of the angular error is typically the guideway or a point on the mounting surface. What’s more, Abbé errors for multi-axis configurations are even more complex because they’re compounded by the presence of angular errors in each axis.

The best methods for minimizing Abbé errors are to use high-precision guides and to ensure that mounting surfaces are sufficiently machined — so they don’t introduce additional inaccuracies to the system. Reducing the Abbé offset by moving the load as close as possible to the center of the system will also minimize Abbé errors.

Abbé errors are most accurately measured with a laser interferometer or other optical device that is completely independent of the system. But laser interferometers aren’t practical for most setups, so linear encoders are used in many applications where Abbé error is a concern. In this case, the most accurate measurements of Abbé error are achieved when the encoder read head is mounted on the point of interest such as the tooling or the load.

XY tables are less susceptible to Abbé errors than other types of multi-axis systems such as Cartesian robots. That’s primarily because they minimize the amount of cantilevered travel and typically operate with the load located at the center of the Y axis carriage.

In an ideal world, a linear motion system would exhibit perfectly flat, straight motion and reach the intended position with zero error every time. But even the highest precision linear guides and drives (screws, rack and pinions, belts, linear motors) have some errors due to machining tolerances, handling, mounting, and even the way they’re applied.

There are three types of errors found in linear motion systems — linear errors, angular errors, and planar errors — and each type has a different effect on the system and the application. To avoid paying for high-precision components where they’re not needed or ending up with a system that doesn’t meet the application requirements, it’s important to understand the differences between these three types of linear motion errors and their causes.

Linear errors in linear systems: Linear errors include positioning accuracy and repeatability. These errors are sometimes referred to as positioning errors because they specify the system’s ability to reach the desired position. In the context of linear systems, the term “accuracy” typically refers to positioning accuracy, which is the deviation between the target position and the position the system achieved. Repeatability refers to how well a system returns to the same position over multiple attempts. The main contributor to linear errors is the drive mechanism (screw, rack and pinion, or linear motor, for example), but the system’s tuning can also affect its ability to reach the target position accurately and repeatably.

Angular errors in linear systems

As mentioned earlier, angular errors are errors in which the point of interest rotates around an axis. These are typically referred to as roll, pitch, and yaw errors, denoting rotation around the X, Y, or Z axis, respectively. If the point of interest is the center of the table, or slide, angular errors may not have a significant effect on the application. But when the point of interest is some distance away from the table or slide, Abbé errors, which are angular errors amplified by distance, can produce undesirable results, especially in machining, measuring, and assembly applications. The primary causes of angular errors, and by extension, Abbé errors, are inaccuracies in the linear guides and poorly machined mounting surfaces.

Planar errors in linear systems

Planar errors — often called straightness and flatness issues — occur during the system’s travel, but rather than rotation around an axis, planar errors are deviations from an ideal, straight reference plane. Straightness defines the extent of motion along the Y axis as the system travels along the X axis. Similarly, flatness defines the extent of motion along the Z axis as the system travels along the X axis.

Note here that the point of reference is the axis of travel (typically the X axis), so there are only two types of planar errors, involving motion along the remaining two axes.

Planar errors are detrimental to applications such as dispensing, machining, or measuring, where the system’s behavior during motion is critical. In multi-axis systems, planar errors in one axis affect the adjacent axis (or axes) especially when the axes are stacked such as in X-Y tables, planar tables, and some Cartesian systems.