Machine Design
Mathematical Model Works In Six Dimensions for Accurate Large Machine Positioning Design and Test

Mathematical Model Works In Six Dimensions for Accurate Large Machine Positioning Design and Test

It takes more than a scaled-up model to maintain accuracy when dimensions range into meters

Authored by:
Dr. Boaz Eidelberg
Parker Daedel
Irwin, Pa.

Edited by Robert Repas
[email protected]

Key points:
• When the size of the work exceeds 1 m, angular errors become amplified in three dimensions at a rate of 5 μm/arc-sec/m.
• In a typical situation, the global XYZ machine-coordinate system is defined at the center of the working envelope.
• Metrology errors in the local-stage coordinate system include the encoder, straightness, flatness, pitch, yaw, and roll.

Parker Daedel,

Correcting errors that never happen,

It’s a challenge to design a positioning system that’s highly accurate. But the task is even more problematic if there are big distances involved. When the size of the work exceeds 1 m, angular errors become amplified in three dimensions at a rate of 5 μm/arc-sec/m. These angular errors combine with other linear errors to create an overall 3D error. In addition, the accumulation of 3D errors expands with the number of positioning axes, whether they are stages or tables.

Consider that each stage has six degrees of freedom: linear motion in the X, Y, and Z axes, and rotational motion in pitch, yaw, and roll. Each degree of freedom contributes to overall machine error. Examples of large-frame equipment where this is the case include eighth-generation ink-jet printers used in flat-panel-display (FPD) manufacturing, laser scribers for photovoltaic solar-panel manufacturing, and eddy-current inspection systems in jet-engine turbine manufacturing. These large machines need a high level of accuracy and high repeatability in three dimensions.

For example, an ink-jet printer encompassing 6 × 4 × 2 m and weighing 15 tons may need to position the plotting head within a few microns over a 2 × 2-m glass-substrate surface to print RGB LEDs.

Such was the case when Parker Daedel Engineered Systems, Rohnert Park, Calif., developed a methodology and system-analysis tool to create a positioning system for a nondestructive 10-ton eddy-current inspection system for large envelopes (ECIS-LE) for Wyle Laboratories, El Segundo, Calif. The task forced Parker Daedal to meet tight 3D accuracy and repeatability specifications.

Non-Destructive Inspection (NDI) systems are commonly used for assessing structural integrity of aircraft critical systems and components such as turbine blades. Eddy-current techniques not only detect cracks, but can also discern their size. The size of the crack determines whether the component has reached its life limit or can remain in service.

Eddy currents can further define the type of crack and whether it is open or closed based on signals the sensor provides. Eddy-current tools can detect cracks on the order of 10-μm deep, 10-μm wide, and 100-μm long.

During the inspection process the eddy-current sensor travels along a 3D track close to the inspected surface while moving at a constant velocity of 30 mm/sec. It may take several days to inspect a complex part. The machine must accurately report the crack position in absolute machine coordinates to determine if the part is still suitable for use.

Wyle Laboratories developed its ECIS-LE system to serve military, commercial aerospace, and power industry customers for structural NDI applications. It has four small servostages that include three rotations and one translation. The turbine blade mounts in a chuck which then mounts on a rotary stage. Compounded Y and Z stages move the eddy-current sensor over the turbine blade. A split-X stage positions the rotary table and the blade. The positioning stages have specifications of 3D accuracy within ±75 μm (without mapping); 3D repeatability of ±6 μm; an X stroke of 1,100 mm; a Y stroke of 1,550 mm; and a Z stroke of 940 mm. In addition, the system must have a following error of ±25 μm and a constant velocity of ±1%.

For the sake of this article, we included no thermal or dynamic errors. Instead we’ll focus on the static 3D accuracy and repeatability needed with large travel and large Abbé offsets: the distance between the desired point of measurement and the reference line of the measuring system.

In a typical situation, the global X, Y, Z machine-coordinate system is defined at the center of the working envelope of the inspection system. In addition, it’s possible to define three more local, stage-fixed, coordinate systems we’ll call 1,2,3 for stages X, Y, and Z.

The origin of each local coordinate system sits at the center of the stage travel and at the bearing surface. For each stage the local coordinate system is defined as follows: Axis 1 is in the direction of stage travel; Axis 2 is perpendicular to Axis 1 in the plane of motion; and Axis 3 supplements Axes 1 and 2 in a right-hand coordinate system.

Consider when the slide of each stage moves under servocontrol from one point to another along Axis 1. There are both systematic and random errors in the three linear directions 1,2,3 and there are three rotational errors about axes 1,2,3. The terminology of these positioning errors in the local-stage coordinate systems are commonly defined as follows: linear error along Axis 1 – Encoder error 1; linear error along Axis 2 – Straightness; linear error along Axis 3 – Flatness; angular error about Axis 1 – Roll; angular error about Axis 2 – Pitch; and angular error about Axis 3 – Yaw.

Note that this model assumes the encoder mounts in the local plane 1-2 along the center of axis travel. Otherwise the measured linear error along Axis 1 contains contributions from pitch and yaw errors and will need to be subtracted from the total tested value to provide net encoder accuracy. Alternatively, this value may be determined from specifications obtained from the encoder manufacturer.

Well-established laser interferometer methods are used to determine the errors in six degrees of freedom in any local positioning system of a stage. But there is no unified method that combines these results into total machine errors in a global-coordinate system.

Laser interferometer test results are valid only for the tested stage in the tested orientation. They do not imply any information about the overall accuracy of the machine. To impute the contribution of the stage precision on the overall machine precision, we must first establish a way to transform errors from the local-stage coordinate system into global machine coordinates. Then we must combine them into a unified machine error.

Coordinate transformation
In general, the assembly configuration of the X, Y, and Z stages in the inspection machine determines the direction that each stage contributes to the global system error. A position stage having fixed, local coordinates 1,2,3 is oriented in a 3D space within the global X, Y, Z coordinate system of the machine. In the example, local Axis 1 points in the global Y direction, local Axis 2 points in the global Z direction, and local Axis 3 points in the global X direction.

The point of interest within the local stage is defined by an Abbé offset vector R in the body fixed-coordinate system. The point of interest is where process work or measurements take place. The components of vector R are further defined as R1, R2, and R3, respectively, in the local 1,2,3 directions.

Therefore, the equation that transforms the errors from the local coordinate system, as measured by the laser interferometer, to the global machine-coordinate system is:

{E} = [T] * ({L} + {A} × {R})

where {E} = <Ex, Ey, Ez>T, theEx, Ey, and Ez along the X, Y, and Z global coordinates, respectively, with T the superscript designating vector transpose; [T] = transformation matrix from body fixed-coordinate system 1,2,3 to the global-coordinate system X, Y, and Z; {L} = <Enc, Str, Flt>T, the local linear-error vector with components Enc, Str, and Flt along the local 1,2,3 coordinates, respectively. Enc is the encoder error in the direction of travel 1, Str is the straightness error in local direction 2, and Flt is the flatness error in local direction 3; {A} = <Rol, Pch, Yaw>T is the local rotational error vector with components Rol, Pch, and Yaw of small angular rotations about local axes 1, 2, and 3, respectively. × designates a vector cross product while {R} =<R1, R2, R3> T, the Abbé offset vector with components R1, R2, and R3 along local axes 1, 2, and 3, respectively.

It can be shown by observation that for the example depicted here the results are as follows:





Similarly the cross product expands as follows:



where | | designates a determinant commonly used to develop the expressions of vector cross product with unit vectors i, j, and k along local axes 1, 2, and 3, respectively.

Substituting the expressions of {A} × {R} , [T] and {L} into {E}, we get the following transformation expression for the example stage:

Ex = E3 = Flt + Rol × R2 – Pch × R1
Ey = E1 = Enc + Pch × R3 – Yaw × R2
Ez = E2 = Str + Yaw × R1 – Rol × R3

The transformation expression can be easily validated for sign notation by visualizing the rotation using the schematic diagram.

Laser-interferometer testing
As mentioned earlier, a laser interferometer measures the metrology errors in the local-stage coordinate system including the encoder, straightness, flatness, pitch, yaw, and roll. If roll optics are not available, roll may be measured or estimated by other means. Encoder accuracy may come from the encoder manufacturer or read from the laser interferometer at the stage slide elevation without an Abbé offset.

A typical laser interferometer tests for bidirectional errors along the stage travel. As shown, the curves appear to be random. Yet within any given machine they consist of a deterministic or repeatable mean error and a random or nonrepeatable error.

The nature of the mean-error curve is relatively smooth with peaks and valleys characterized by the maximum accuracy error. The nature of the repeatability curve is an oscillation about the mean with maximum error between the back and forth curves. The curves are used in calculating the standard deviation of the error within any given machine. You can get a more rigorous definition of the number of standard deviations which the laser interferometer curves represent from the printout of the laser-interferometer test results.

There are seven assumptions about the statistical nature of the observed errors:

First, the accuracy curve of each tested stage is a random process with a repeatable deterministic mean error within a given machine. This variable is obtained from laser-interferometer tests. These errors come from repeatable sources, such as bearing straightness, flatness, and structural deformation, and can be mapped out in the machine controller by an error mapping routine.

Second, the accuracy curve of each tested stage has a statistical error that is nonrepeatable within a given machine. This repeatability variable comes from the laser-interferometer test. These errors are the result of nonrepeatable sources such as friction and bearing jitter and cannot be mapped out by software.

Third, the mean error in an accuracy curve of each tested stage is random with a nonrepeatable shape. Thus it will vary when comparing one machine to another machine of the same type.

Fourth, the mean error at any position along a stage travel is a normal distribution random variable with a mean of zero since there is no bias in the assembly process one way or another.

Fifth, the standard deviation of the mean value in the accuracy curve is proportional to the maximum accuracy error. The constant of proportionality (meaning whether the maximum accuracy error represents ±1, 2, or 3 sigma) depends on the level of consistency in the assembly process, the quality of the components used, and the consistency of machining tolerances.

Sixth, all random-error variables in any local and global direction have a normal distribution centered about a certain error point and can be defined as a ± error value about this point. The location of this point can be found from metrology charts. A correction of any error bias can be made by software within the controller. For example, if the maximum accuracy error is from –4 to 8 μm for a total error of 12 μm, we can refer to it as ±6 μm and introduce a 2-μm correction at any position along the travel.

And finally, seventh, global-system errors in the X, Y, and Z directions are noncorrelated. This means that an error in one direction is not a known function of an error in another direction. This assumption implies that the maximum 3D machine error is the largest error among the global X, Y, Z directions.

Under these assumptions it’s possible to estimate the total error in any global direction as the statistical sum of all local errors in that direction. As an example, remember that the transformation of local-stage errors to global-machine errors are given by:

Ex = Flt + Rol × R2 – Pch × R1;
Ey = Enc + Pch × R3 – Yaw × R2; and
Ez = Str + Yaw × R1 – Rol × R3.

Now, if each error variable in this expression is a random number, it’s possible to determine the total accuracy error in global coordinates as follows:

Esx = Sqrt (Eflt2 + R2 × Erol2 + R1 × Epch2)
Esy = Sqrt (Eenc2 + R3 × Epch2+ R2 × Eyaw2)
Esz = Sqrt (Estr2+ R1 × Eyaw2 + R3 × Erol2)

where Esx, Esy, and Esz = the total statistical-system error in global X, Y, and Z directions, respectively, and Eflt, Eenc, Erol, Epch, Eyaw, and Estr = the maximum accuracy errors of the contributing errors in local coordinates as measured by the laser interferometer.

Similarly, total repeatability error in global coordinates is determined as follows:

Rsx = Sqrt (Rflt2 + R2× Rrol2 + R1 × Rpch2)
Rsy = Sqrt (Renc2 + R3 × Rpch2 + R2 × Ryaw2)
Rsz = Sqrt (Rstr2 + R1× Ryaw2+ R3 × Rrol2

where Rsx, Rsy, and Rsz = the system-repeatability errors in global X, Y, and Z directions, respectively, and Rflt, Renc, Rrol, Rpch, Ryaw, and Rstr = the values of the repeatability of the contributing errors in local coordinates as measured by the laser interferometer

The case of 3D error in a multiaxis machine is simply an extension of a single stage. First, formulate the 3D errors of each stage as described above and take the statistical sum of all contributions from all stages in the global X, Y, and Z direction as follows:

TEi = Sqrt ( Σ ( Esi2) ) over all stages
TRi = Sqrt ( Σ (Rsi2) ) over all stages

where TEi = total maximum accuracy error in the global i direction and TRi is the total repeatability error in global i direction.

The final results
In this example, the basic mathematical model went into a spreadsheet to create a tool for future use. The model compared favorably with test results. For example, the X accuracy at 0.4 Abbé offset was ±6.4 μm for the test result while the model calculated an offset of ±6.28 μm, Y accuracy at full Abbé offset was measured at ±18.4 μm, with the model predicting ±19.9 μm.

The use of this modeling tool provides detailed insight to the contribution of each stage-error element on the total accuracy and repeatability of the machine. These contributions are directly related in the model to machine-design parameters, part specifications, manufacturing tolerances, and assembly variables. The model and its supporting analysis tool provide a robust guideline, throughout the entire machine-development process, for error budgeting, setting design objectives and manufacturing tolerances, purchasing specifications, and assembly guidelines. It can also be used as a quick troubleshooting tool in case deviations from budget are found or engineering design changes occur.

© 2010 Penton Media, Inc.

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