Method and apparatus for determining calibration options in a motion control system

[00002] This application claims the benefit of U.S. Provisional Application No. 60/286,834, filed Apr. 26, 2001, the entire disclosure of which is hereby incorporated by reference. This application is a continuation of prior application Ser. No. 09/293,096, filed Apr. 16, 1999, now U.S. Pat. No. 6,470,225.

[00003] The present invention relates generally to motion control systems, and more particularly, to a method and system for tuning compensation parameters.

[00004] The performance of a motion control system is of primary importance. In particular, a control system should be stable, result in acceptable responses to input commands, result in a minimum steady-state error for input commands, and be able to eliminate the effect of undesirable disturbances. Often-times, for optimum performance, adjustments are made to a control system. The alteration or adjustment of a control system in order to provide a desired performance is often referred to as compensation (i.e., compensation is the adjustment of a system in order to make up for deficiencies or inadequacies).

[00005] In some cases, an additional component (e.g., a compensator) is included within a motion control system to equalize or compensate for such deficiencies/inadequacies. For example, such compensators can be implemented as one or more parameters whose values are used by a controller to determine how to control a system. Accordingly, to provide optimum performance, appropriate values are determined for such parameters.

[00006] The process of adjusting these parameters (e.g., to compensate for system changes) is often referred to as tuning. Recently, techniques for tuning such parameters that involve minimal or no operator involvement have been developed. These techniques are generally referred to as autotuning. For example, autotuning can refer to a process in which a controller automatically determines appropriate values for such parameters.

[00007] Currently, in motion control systems such as those utilized with machine tools, relative displacement sensors, such as a telescoping ballbar, are used to provide an electrical output representing a measurement of incremental displacement caused by the motion control system. The electrical output can then be used to “tune” parameters of the motion control systems that account for errors in the motion of the machine(s) it controls. For example, a ballbar can be mounted between the two members of a machine tool that move relative to each other and create feed motion between a tool and a workpiece. More particularly, with respect to a machining center, for example, the ballbar is mounted between a spindle and a work-holding table.

[00008] Typically, the range of motion of a commercially available ballbar is limited to just a few millimeters. For example, the machine tool motion during a ballbar measurement is conventionally along a substantially constant radius arc. Measurement of deviation of the arc radius from the ideal constant radius can be used to provide information about the errors associated with the machine tool. This information can, for example, be used to determine certain aspects of positioning (static and dynamic) errors associated with the machine.

[00009] With reference to FIG. 1, a conventional use involving a motion control system and a machining center is herein described (where a user of the motion control system manually performs most steps). The center fixture of a ballbar is placed at a desired location on the machine with the setscrew loose. The spindle is then moved such that a magnetic cup in the tool holder of the spindle picks up the steel circle center ball of the ballbar and the setscrew is tightened. The positions of the axes are then recorded by performing a machine position set to redefine the coordinate system, such that the current position is the program system origin (e.g. G92X0Y0Z0).

[00010] A user of the motion control system then manually feeds the axes to move the magnetic cup off of the center ball, and creates a part program causing the machine to perform a circular move, an exemplary one of the part programs having a number of characteristics, including the following:

• • [00011] 1. The circle center corresponds to the location of the center ball.
  • [00012] 2. The circle radius is equal to or slightly less than the nominal length of the ballbar.
  • [00013] 3. The circular motion is constrained to lie within a major plane of the machine (e.g., XY, XZ, YZ).
  • [00014] 4. The circular arc includes some additional (overshoot) distance at the beginning and end of the move (allows the feedrate to accelerate from zero to full speed and to decelerate from full speed to zero), with the portion of the circle whose data is considered by the analysis being referred to as the data-arc (data collected during the feedrate transients at the beginning and end of the move is not considered during data analysis).
  • [00015] 5. The circular motion is preceded by a short (feed-in) move that travels along a circle radial line from a radius greater than the circle radius to the circle start point. Similarly, a short (feed-out) move following the circular motion travels along a circle radial line from the circle end point to a radius greater than the circle radius. These moves can be used to signal conventional software associated with the ballbar when the test begins and ends, and can be used by the software to provide an estimate of the instantaneous circle angle based on the elapsed time since the test start.
  • [00016] 6. Two successive sequences with overshoot arcs, a feed-in move, a data-arc, and a feed-out move are present for both clockwise and counter-clockwise directions within the plane.
  • [00017] 7. A program stop (M0) block is present prior to the beginning of each (clockwise and counter clockwise) sequence of moves (the program stop providing the operator with an opportunity to arm the trigger in the ballbar software).

[00018] The ballbar can then be plugged into a serial port of the computer running the ballbar software, and the ballbar length optionally calibrated (e.g., by placing the ballbar into a calibration fixture of known length and prompting the software to measure the ballbar's output). If the ballbar length is calibrated, the actual ballbar length is computed and stored by the ballbar software. One advantage of a length-calibrated ballbar is that it enables average circle diameter error to be estimated.

[00019] The ballbar software is then informed of the specific conditions for the current test by entering values for the following parameters: length of overshoot arcs, data arc start angle, data arc length, feedrate, and circle radius. The trigger in the ballbar software is armed on the machine (e.g., a personal computer) on which it is running, and the test program on the control is executed.

[00020] When the program reaches the stop (M0) after the first circular move, and after the data collection is completed, the ballbar software trigger is re-armed, and the machine is cycle started to perform the second circular move. At test completion, the test data is saved to a file and analysis software is run to diagnose error sources.

[00021] Accordingly, a method and system to automate and simplify this and other similar procedures is desired. In addition, it would be desirable to apply parametric compensation based on measurements from a ballbar.

[00022] It is an object of at least one embodiment of the present invention to provide an improved method and/or system for tuning compensation parameters of a motion control system.

[00023] Additional objects, advantages, and novel features of various embodiments of the invention will be set forth in part in the description of the exemplary embodiments that follows and, in part, will become apparent to those skilled in the art upon examining or practicing the invention.

[00024] One embodiment of the present invention is directed towards a method of and apparatus for determining calibration options in a motion control system. The method includes the steps of measuring error in a commanded motion, modeling the error, identifying sources of the error; and predicting a degree to which applying compensation for one of the sources of the error might affect the error.

[00025] Still other advantages of various embodiments will become apparent to those skilled in this art from the following description wherein there is shown and described exemplary embodiments of this invention simply for the purposes of illustration. As will be realized, the invention is capable of other different aspects and embodiments without departing from the scope of the invention. Accordingly, the advantages, drawings, and descriptions are illustrative in nature and not restrictive in nature.

[00059] In general, the present invention relates to a motion control system capable of tuning compensation parameters. For purposes of illustration, exemplary embodiments discussed primarily herein relate to test software that runs on the control of a machine tool and, in one embodiment, to such software that allows a machine tool control to self-calibrate its parametric compensation based on measurements from an instrument called a telescoping ballbar. Moreover, in one embodiment, disclosed exemplary software provides the capability to: 1) specify test conditions; 2) automatically create the motion for measurements related to the test condition; 3) initiate the motion; 4) collect the data; 5) analyze the data; and 6) update the compensation of the control based on the data.

[00060] Further exemplary embodiments can, for example, be used to:

• • [00061] 1. Validate the appropriateness of test setup data.
  • [00062] 2. Validate the integrity of collected data.
  • [00063] 3. Aid a user in fixture setup by providing prompts of what to do next based, for example, on control state and axis positions.
  • [00064] 4. Monitor a control state to coordinate a test sequence.
  • [00065] 5. Send messages to control the execution of part programs.
  • [00066] 6. Robustly handle error conditions.
  • [00067] 7. Process measured data to account for effects such as servo gains and feedforward.
  • [00068] 8. When computing proposed changes to compensation, consider the values for compensation that were active during the test execution.
  • [00069] 9. Copy updated compensation values into, for example, a database of a control.
  • [00070] 10. Activate new compensation (e.g., by issuing a system event).
    For example, certain exemplary embodiments of the present invention make use of access to controller internal data, program buffer, and system event services of an associated control to provide at least some of the aforementioned features.

[00072] According to one embodiment of the present invention, configuration of certain types of compensation can be automated. Referring specifically by example to an exemplary embodiment involving ballbars, values for the following types of compensation for axes in a machine tool that define a primary machine plane in which a ballbar circle test is performed can be automatically proposed: scale error; squareness error; straightness error; backlash; windup, and/or friction.

[00073] Ballbar measurements of radius deviations that occur during major plane circular motion of a machine are the combined result of the simultaneous motion of two axes. In order to attribute the measurements to error sources from a single axis, the analysis can be restricted based on a series of limiting assumptions about the nature of the errors. For example, the errors can be assumed to originate from a limited number of sources with a fixed mathematical relationship between an error source, the axes positions, axes velocities, and the radial error produced.

[00074] The particular mathematical relationships included within the model can be chosen based on some physical understanding of the machine structure and servo dynamics, and/or on previous experience with machine tool errors. Accordingly, each error source can be described by a single parameter that quantifies the amplitude and direction of the error. Therefore, data representing such error can be curve fit to extract optimal estimates for each parameter associated with each error type.

[00075] An alternative to the parametric model could include a data-based approach. If, for example, the desired goal of the ballbar measurement was the generation of data tables for bidirectional compensation (compensates for backlash and error due to variation in the pitch of a ballscrew) and cross-axis compensation (compensating for axis position as a function of one or more axis positions), then there might be an infinite number of data set combinations for these tables that might exactly account for the errors measured by the ballbar. However, only one of these combinations would be the “correct” one. Meanwhile, an incorrect selection could easily result in larger errors than what might occur on an uncompensated machine (e.g., for any axis motion that differs from the motion experienced during the ballbar measurement).

[00076] Although a parametric description of the errors might not exactly describe the measurement, it can allow for the derivation of an optimal combination of parameters that best describes the measurement. Accordingly, association of measured machine errors to a parametric model implies that the compensation for the machine errors can be based on the same parametric model.

[00077] In, for example, an A2100 control from Siemens Energy and Automation, Inc., the following types of parametric compensations are available: 1) backlash compensation; 2) windup compensation; 3) pitch (or scale) compensation; 4) squareness compensation; and 5) straightness compensation. Each of these compensations describes the motion of an individual axis or combined effect of two axes within a plane. Each of these error sources can also be described by a single parameter.

[00078] A mathematical model can be based on an understanding of the physical effects that often cause machine errors. If an error source exists that is outside of the scope of the mathematical model, then the error might be falsely attributed to one of the error types that is included in the model structure. To help avoid this situation, the scope of the mathematical model can be made larger than the scope of parametric compensations provided.

[00079] For example, servo mismatch error can be included within the scope of one of the exemplary models described herein for use with the A2100 control, even though the A2100 control does not have an explicit servo mismatch compensation. Such servo mismatch errors can instead be corrected in other ways based on other types of measurements. In any case, situations might arise where the mathematical model cannot properly describe the data. For example, this could occur if a machine has an error that is outside of the scope of the particular mathematical model, if there are problems with the measurement, such as the ballbar separating from the magnetic cup, or if there are high levels of noise in the measurement. To identify such a situation, statistical measures of the coherence between the model and the measurement can be computed and, for example, displayed to a user of the system (e.g., where a low value for coherence indicates to the user that a closer look at the data might be required).

[00080] According to one embodiment of the present invention, parameter estimates are used to describe machine errors within the range of motion that took place during a ballbar test. One potential concern associated with extrapolation of the model to regions outside of the range of motion of the test involves the uncertainty of the parameters in these regions. Therefore, one embodiment of the present invention includes a limitation that it reliably compensates for the region of a plane that is within the range of motion of the ballbar test. For example, for machine planes that have a rectangular (rather than square) range of motion envelope, a single circle test might not produce compensations that are reliable over the full range of motion.

[00081] Furthermore, certain error types appear to be more reliably extrapolated than others. For example, backlash, windup, and scale errors appear to be much more uniform over the full range of the axis than a parametric squareness and straightness error. Moreover, parametric compensation of a machine may be inappropriate in situations where the range of motion of the ballbar test used to develop the compensation is significantly less than the range of motion of one or more of the machine axes. In such a situation, parametric compensation based on a ballbar test may be most useful as a supplement to data-based bidirectional-axis-(pitch)-compensation and cross-axis-compensation that covers the full range of axis motion.

[00082] In other embodiments involving a ballbar test, the test may be used by the end user to fine-tune the compensation over a limited range of axis motion that represents the range of motion to be employed in an immediately upcoming machining operation. Accordingly, in one embodiment of the present invention, it is recommended that the diameter of the circle test encompass as much of the machining envelope as possible. As can be understood, the ability of the parametric model to exactly describe the motion within the limited region increases as the range of motion (circle diameter) decreases, (the error types are more likely within the scope of the model).

[00083] Another consideration that might affect the reliability of the parameters can be the nature of the circular motion itself. For example, sufficiently exciting each error source such that it produces a unique effect on the measurement allows different error sources contained within a model to be distinguished. For an exemplary model described herein, sufficient excitation involves collecting data for both clockwise and counter-clockwise moves. Additionally, in a further exemplary embodiment, the length of the data arc is as large as possible.

[00084] In an exemplary embodiment, both clockwise and counter-clockwise data arcs cover 360 degrees. In some cases, however, the mechanical limitations of the machine might not allow for a test that simultaneously covers a 360-degree arc and the maximum machining envelope of one or both of the axes. In such a situation, one embodiment of the present invention might involve using a data arc that is less than 360 degrees to enable the full axis envelope to be covered. In an exemplary embodiment described herein, the data arc is, however, at least 180 degrees, and the clockwise arc covers the same angular range as the counter-clockwise arc.

[00085] A statistical value referred to herein as the excitation may also be computed to provide the user with an indication of the sufficiency of excitation for distinguishing effects of different parameters. Two of the exemplary parametric errors, backlash and windup, are typically indistinguishable from each other during a standard (continuous motion) ballbar test, which provides an indication of the total lost motion. In one embodiment of the present invention, a separate backlash test is performed to determine the contribution of the backlash error to the total lost motion indicated by the measurement data from continuous motion test.

[00086] According to an exemplary embodiment of the present invention, a user interface, such as one containing four primary modes, can be provided for the system. In one embodiment, these modes include a setup test mode, a run test mode, an analyze data mode, and an adjust compensation mode. For example, each mode can be displayed as a separate page on a user interface.

[00087] Referring now to FIG. 2, in one embodiment of the present invention, a test-setup page 10 can be provided that acts as an interface for a user to, for example, either load a set of test condition parameters from a file or to specify such parameters manually. In an exemplary embodiment, the test conditions specified on this page can be used (e.g., by the control) to automatically generate a part program during a test execution stage (such as under the run-test mode that will be later described herein). Exemplary test conditions include information regarding the test location such as circle center data 16.

[00088] According to one exemplary embodiment, the test condition parameters may be specified for one test in each major plane of an associated machine (e.g., XY, XZ, YZ). For example, the user may modify the test condition parameters by entering an edit mode. While in edit mode, fields 12 that contain the test condition parameters could become modifiable.

[00089] Once any desired changes have been made, the user can then press an “apply” button 14. In such an embodiment, pressing an apply button 14 can be used to initiate a series of validation checks for the entered test condition parameters. For example, validation checks could be used to ensure that a reasonable test may be developed from an entered combination of test condition parameters. In one exemplary embodiment, if a validation test fails, an alert describing the problem is displayed, the user interface remains in edit mode, and the user is expected to change the test parameters until they represent a valid combination.

[00090] In a further exemplary embodiment, a separate group of test condition parameters can be stored for each of the three major planes. The test condition parameters could include the following:

• • [00091] 1) circle radius (less than or equal to the ballbar length)
  • [00092] 2) circle center (location in X, Y, Z where circle should be centered)
  • [00093] 3) data arc start angle (counter clockwise angle from horizontal axis at which to begin data collection)
  • [00094] 4) data arc length or test range (arc angle over which data is collected);
  • [00095] 5) overshoot arc length (arc angle where data is not collected to allow for transients during acceleration and deceleration; in one embodiment the acceleration overshoot arc and deceleration overshoot arc can both be specified by this parameter)
  • [00096] 6) ballbar length (this selection might be limited based on available extension bars);
  • [00097] 7) feedrate (programmed feedrate for data-arc, overshoot-arcs, feed-in, and feed-out); and
  • [00098] 8) process or feed mode (specifies the group of path parameters (e.g., G45.n) to be used for motion during the test; this may affect path acceleration, path jerk, step velocities, and feedforward status)
    In one embodiment of the present invention, a combination of test conditions parameters are validated, such as against the following: 1) Is circle radius appropriate for the ballbar radius; 2) Is axis motion within the machine range?; 3) Is circle center within the machine range?; 4) Are overshoot arcs sufficiently long to enable constant feedrate during data arc?; and 5) Are the data arcs long enough to provide sufficient excitation to distinguish between error types?

[00100] In an exemplary embodiment of the present invention, a circle radius is required to be less than or equal to a ballbar radius. According to such an embodiment, if the circle radius is less than the ballbar radius, the length of the ballbar traces a conical surface. In such an embodiment, the circle radius is slightly less than the ballbar radius because the measurements might become overly sensitive to motion in the direction of a third axis (e.g., the axis perpendicular to the major plane). Motion along the direction of the third axis should be substantially indistinguishable from motion of the axes in the plane and can often distort the parameter estimation.

[00101] For example, as the angle of the cone increases from 0° (circle radius equal to the ballbar radius) to 45°, the relative contribution to ballbar deflection from motion in the third axis increases from 0% to 100% of the deflection caused by motion in the plane. A respective exemplary validation check requires that the cone angle not exceed 26°. In such an embodiment, at 26°, the relative contribution to ballbar deflection from motion of the third axis is 50% of that of motion within the plane.

[00102] The selected values for ballbar length, circle radius, circle center, overshoot arc, data arc start angle, and data arc length can also be evaluated to ensure that the motion produced is everywhere within the range limits of the axes. For example, if it is detected that an axis range limit might be exceeded, a validation check can provide proposals for changing, for example, the circle center to allow the circle to fit within the envelope of the machine (if the motion can be brought to within the machine envelope by shifting the center).

[00103] Even if the circular arc is within the machine's range, it can still be possible that the circle center is not within the range of the machine. For example, this can occur when the circle radius is less than the ballbar length. Accordingly, in one embodiment, as a tool tip (e.g., a magnetic cup) should be brought to a circle center ball to locate a centering fixture, the circle center should be within the range of the machine.

[00104] Another exemplary embodiment can include requiring collection of data during a period of time when the circle is traversed at a constant feedrate. The frequency response of an axis typically involves distortion of the amplitude and phase as a function of frequency (for a circle, the frequency is proportional to feedrate). At a single feedrate, the amplitude and phase distortion should be constant, the shape of the circle should not be distorted (from individual axis frequency response), and it should be possible to compensate for the effects due to constant amplitude attenuation and phase lag when post processing the data.

[00105] An acceleration overshoot arc can be included to help ensure that the machine axes have an opportunity to accelerate to full feedrate before data collection begins. Meanwhile, a deceleration overshoot arc can be included to avoid collecting data during the deceleration from full feedrate to a full stop. While, in one embodiment, the overshoot arcs should be long enough to ensure constant feedrate during the data-arc, it can also be desirable to keep the length of the overshoot arc to be only slightly above its minimum possible value to minimize the total test time.

[00106] Length of the acceleration overshoot arc can depend on several factors, including: 1) an end of span speed constraint of a feed-in span preceding the acceleration overshoot arc, 2) a span maximum acceleration and jerk for an overshoot arc span, 3) programmed feedrate, 4) axis Kv gains, 5) axis feedforward levels, and/or 6) whether rate feedforward is activated. Similarly, the length of the deceleration overshoot arc can depend on similar factors. Furthermore, the span data's acceleration, jerk, end-of-span-speed-constraint, and feedforward status can vary depending, for example, on the axis configuration data, path configuration data, feed mode selection, data-arc start angle, and data arc length.

[00107] Typically, lengths for the acceleration and deceleration overshoot arcs can be substantially identical. In one embodiment, whether an overshoot arc is acceptably long can be determined by running a sequence of spans in a test move through motion algorithms of the control without directly controlling the machine. In an exemplary embodiment, appropriate subsystems of motion algorithms of the control (e.g., look-ahead and velocity management) are linked to form a limited scope emulation that can run as a callable function in a non real-time environment. The motion can be emulated in much less time than the actual move by, for example, limiting the emulation to small portions of the movement with the correct initial conditions (velocity and acceleration).

[00108] The emulated path distance signal can be passed through a filtering function that provides the same phase and amplitude distortion created by the axis Kv and feedforward level. According to one embodiment of the invention, the results of the emulation can be used to indicate whether or not the overshoot arcs are long enough. If the arcs are not long enough, then a conservative approximation can be used to develop a proposed value for overshoot arc length to provide to the user. This new estimate can also be validated against the actual motion.

[00109] With respect to one exemplary embodiment, an estimate of total lost motion for both axes in a plane can be obtained by including at least one reversal for each axis during the motion. This implies a minimum range of motion of somewhat greater than 90° if the data is aligned with one of the circle quadrants. In order to accommodate any data arc start angle, data arc length can be set somewhat greater than 180° to help ensure the occurrence of at least one reversal for each axis. Additionally, 180° of motion can help afford the ability to reasonably distinguish between squareness and straightness errors. Thus, in such an exemplary embodiment, a minimum allowable data arc is 180° and an excitation parameter exceeds a minimum defined threshold.

[00110] Setup test page 10 can also add additional content to increase overall usability. For example, a setup test page 10 can include a “Get Current” button 18 that, when pressed, copies current machine coordinates of the axes to the circle center coordinates for the current plane. It can also contain an “Auto Select” button 20 that sets the circle center as the middle of the machine's maximum range envelope and sets the circle radius equal to the ballbar length. In one embodiment, the “Get Current” and “Auto Select” buttons 18, 20 are enabled when in edit mode.

[00111] Setup test page 10 can further contain a graphical and/or animated representation 22 of the data arc and/or overshoot arc. Additional capabilities could include an ability to revert to the original test condition parameters that were present prior to any changes that were made during the current edit session (e.g., such as from within an edit mode). It can also include the ability to save the current test condition parameters (e.g., for each of three planes) to a file and/or to load new test condition parameters (e.g., for three planes) from a file. Still further, setup test page 10 can include the ability to assign the currently active test condition parameters as the default setup that is present when the user interface is opened.

[00112] In certain embodiments, setup test page 10 can contain content to assign data associated with, for example, a ballbar device itself. For example, a serial number of the ballbar can be copied to the header of a measurement data file for the purpose of creating a permanent record of the particular ballbar used to generate that data set. A ballbar scale factor can be used to adjust the measurements to account for the unique characteristics of the specific ballbar (typically printed on the side of the ballbar). Still further, a COM port to which the ballbar is attached can be specified on this page. There can also be button 24 that is used to debug ballbar problems.

[00113] Referring now to FIG. 3A, a run-test page 30 can be included, such as one that provides an interface to aid a user in setting up a fixture (e.g., a ballbar) and executing a test (e.g., a ballbar test). For example, such a run-test page 30 can comprise a drop-down list 32 for selecting a test (e.g., of three major plane tests) to perform. In one embodiment involving a ballbar test, for each of the major planes, five sequential tasks (or visible states) in the overall process of obtaining and saving a measurement are provided.

[00114] According to such an embodiment, when a major plane is selected, test setup data associated with that major plane is employed in the execution of the test. If a new major plane is selected, the state returns to its starting point in the test sequence. A finite state machine (FSM) that recurrently polls control and acquired data can coordinate the progression through these states. For example, such a test coordination FSM can contain more states than those that are made visible as a task. A detailed discussion of one embodiment of such a FSM involving a ballbar test is further discussed later herein.

[00115] According to an exemplary embodiment of the present invention, progress towards the completion of the tasks is indicated with a display on a screen. For example, each task can have an associated checkbox 34 that becomes checked when a corresponding task has been completed. A user can also be provided with the ability to, at any point in the process, go back any number of tasks by, for example, pressing a button associated with that task. Additionally, if a task fails to complete properly, the ability to automatically back up to the nearest previous state or task possible can also be provided.

[00116] In one embodiment, throughout a sequential process, a user can be provided with a text description of what to do next to progress through the sequence. For example, control data can be used to determine when the requirements for a state transition to the next state in the sequence (of tasks) have been fulfilled. In an exemplary embodiment, a user can avoid the manual intervention involved in completing the process by performing all operations using, for example, a pendant or panel of the control. One advantage associated with such an exemplary embodiment could include allowing the user to accomplish tasks at a machine without having to return to the control.

[00117] In one embodiment of the present invention involving a ballbar test, primary tasks in the process of completing the ballbar test include 1) MoveToCenter, 2) ConfirmCenter, 3) MoveToStart, 4) perform the test, and 5) save data associated with the test. A high level description of each of these tasks according to an exemplary embodiment is given below.

[00118] The state associated with the MoveToCenter task can be entered by, for example, pressing a Move to Center button 36 on run-test page 30. According to one embodiment, when the MoveToCenter state is entered, the recurrent polling for transition stimuli begins (if it has not already begun). Accordingly, when the MoveToCenter state is entered, a sequence of part program blocks to cause the axes to move to the circle center is created and loaded into, for example, a part program Manual Data Interface (MDI) buffer. A prompt text 38 can then be updated to inform the user to, for example, ensure that the path to the circle center is free of obstructions and then hit cycle start to begin the move. When it is detected that the final block in the move-to-center program has been completed and the control-state has transitioned from in-cycle to at end of program (eop) (described later herein), the FSM can transition to states associated with a ConfirmCenter task.

[00119] At the ConfirmCenter state, the FSM can update prompt 38 to ask the user to, for example, locate the center ball in the magnetic cup of the tool holder, tighten the set screw, and hit cycle start. Upon entering ConfirmCenter, the MDI program buffer can be loaded with some non-motion blocks and dwell blocks whose purpose is to message the FSM as the blocks are executed. Accordingly, when the FSM detects that the user has hit cycle start to confirm the center, the FSM can capture the current axis positions and store them as the circle center for the current test.

[00120] One advantage of such an embodiment can include facilitating the use of a slightly different circle center than the one defined in the test setup data. This can be useful because it can allow the test setup data to contain values for the circle center coordinates that are slightly offset from the plane of the center ball. Accordingly, the move to center can be performed with a high degree of confidence that the tool tip will not collide with the center ball. When the center has been confirmed, prompt 38 can be updated to, for example, suggest that the magnetic cup be power-fed or handwheeled off of the center ball.

[00121] The state associated with the MoveToStart task can be entered (e.g., from the ConfirmCenter task) when the FSM detects that the axes have been moved off of the center ball. As the state associated with MoveToStart is entered, the MDI buffer can be loaded with program blocks to move the axes to the start position for the test move and prompt 38 can be updated to, for example, suggest ensuring a safe path to the circle start point and hitting cycle start. The program blocks can move the axes to the position from which the initial feed-in move originates.

[00122] In one embodiment, when the FSM detects that the move to the start position has completed, it can generate a continuous motion ballbar test program and load it into, for example, the MDI buffer. If the user had indicated that a backlash test was also to be performed, the FSM can also include backlash test blocks in the program loaded into the MDI buffer. The display prompt can be updated to suggest that the user place the ballbar into the machine and hit cycle start to begin the test.

[00123] When the FSM detects that the ballbar test program execution has reached a certain point (e.g., prior to any motion), the FSM can cause program execution to temporarily suspend (e.g., using a technique to be described in a later section) and attempt to initialize the ballbar device. If the ballbar is successfully initialized, a button (e.g., “Cancel Test” button 40 of FIG. 3B) to enable the user to manually abort the test can be created and displayed, program execution can be allowed to continue, and the FSM can begin polling the ballbar position to detect the initial feed in the first (e.g., counter-clockwise) move.

[00124] After detecting initial feed-in, the FSM can begin, for example, storing ballbar measurements, checking for error conditions, measuring elapsed test time (e.g., based on the number of periodic ballbar measurements obtained), and testing for a feed-out move. In an exemplary embodiment, the program becomes suspended after the feed-out move. When the feed out move occurs, the data arc can be extracted from the total data set and validity checks (to be described later) can be performed. Assuming that the validity checks indicate no problems, the FSM can release the program from its suspended state and prepare to detect the initial feed-in from the second (e.g., clockwise) move.

[00125] In an exemplary embodiment, the sequence of events that occur during the second move is analogous to that of the first move. In one embodiment, once the second move completes and the validity check indicates a successful test, the program can begin to execute a backlash test (described later). Once testing is complete and validity checks are successful, the data can be extracted and stored in, for example, a temporary file.

[00126] A save data task can be initiated by, for example, the user pressing a save data button 42 after a valid test has been completed. In an exemplary embodiment, save data button 42 is enabled when a test has been completed and validated. For example, pressing save data button 42 can bring up an interactive dialog box (e.g., a window) used to assign a name for the file and to provide the user the opportunity to enter other information pertinent to the test (such as the machine type, serial number, etc). The data file can contain raw data and a header to describe the test conditions. In one embodiment, the save dialog initially contains a default file name that is created based on, for example, the test plane, machine serial number, calendar date, and number of data files with the same properties already present on the disk.

[00127] From the perspective of a user, the performance of a ballbar test according to an exemplary embodiment of the present invention could have the following sequence. The user selects a test plane from a drop down list 32, indicates a move to center (e.g., via button 36) on a user interface and indicates cycle start (e.g., via a pendant). An associated machine moves to a circle center.

[00128] The user manipulates a center fixture and uses a handwheel to make small adjustments to axis positions until a magnetic cup “picks up” a center ball. At that point, the user tightens a setscrew of the center fixture and hits cycle start. The user power feeds the axes off of the center ball, hits cycle start and the axes move to a start position. The user places the ballbar in the machine, hits cycle start, and the machine performs the test. When the test is complete, the user indicates that the data should be saved (e.g., via a save-data button 42) and the data is stored into a file.

[00129] In addition to its ability to perform the aforementioned steps, run-test page 30 can also provide some additional capabilities. For example, run-test page 30 can include an interface 44 for performing calibration of a test device (e.g., a length of a ballbar). Run-test page 30 can also provide an ability to select whether, for example, a backlash test is performed in addition to a continuous motion test. In still further embodiments, run-test page 30 can provide the ability to perform multiple tests at, for example, multiple feedrates without having to repeat the move-to-center, confirm-center and move-to-start tasks, and without having to return to, for example, setup-test page 10.

[00130] As previously alluded to, a length calibrated ballbar can enable a determination of an absolute axis scale error (a scale mismatch between axes can also be determined, even if the ballbar length is not calibrated). Accordingly, in one embodiment of the present invention, run test page 30 contains an interface 44 that enables device calibration, such as ballbar calibration. For example, such an interface 44 can include a numeric field 46 where the user can enter the length of a calibration fixture, an indicator (e.g., display 48) of whether the ballbar is currently length-calibrated, and a button 50 to request that calibration begin.

[00131] In an exemplary embodiment involving a ballbar calibration, to calibrate a ballbar length, a user first enters a calibrator length (e.g., as printed on the calibration fixture). The entered value is checked (e.g., by software) to ensure that it is reasonable given the configured ballbar nominal length. Whenever a valid new calibration length is entered, an internal state can transition from calibrated to uncalibrated. At that point, display 48 can indicate that the ballbar is not calibrated and the user can press calibration button 50.

[00132] According to one embodiment, pressing button 50 initiates a sequence where the ballbar initializes, powers on, collects data for a short period, and powers off. The data can then be checked for constancy and the reading used to determine an offset to add to the ballbar measurement to obtain an absolute ballbar length reading. This value can be stored and used when processing later measurements. Length calibration can be performed, for example, during the primary tasks described above, with the exception of the perform-test task. In an exemplary embodiment, the test coordinator FSM disables the calibration controls when the state is inappropriate for calibration.

[00133] With respect to ballbar testing, a backlash test can be an additional test. For example, a backlash test might be used to distinguish between windup and backlash components of the total lost motion measured by a continuous motion test. The term backlash is used herein to describe a type of lost motion that continues to persist after an axis has come to a stop. The term windup is used to describe a type of lost motion that occurs while an axis is in motion.

[00134] In an exemplary embodiment implementing a backlash test, the user can indicate that the performance of a backlash test is desired by checking a “backlash test” check box 52 at some point prior to the move to start task. Such a backlash test can be used, for example, to measure the persistent lost motion of each axis in a plane by performing ballbar measurements before and after single axis moves that end in a full stop. One exemplary backlash test part program is operative to move the axes along arcs to each of four circle quadrant boundaries (that are within the range specified in setup data). At each quadrant boundary, the part program and FSM can effect a sequence of events by incorporating schemes for messaging between the program and FSM.

[00135] In an exemplary embodiment, at each quadrant boundary, the following sequence takes place:

• • [00136] 1. Perform a single axis move toward the circle center along a radial.
  • [00137] 2. Stop axis motion, pause, and collect a first ballbar measurement.
  • [00138] 3. Feed the axis further toward the circle center along a radial.
  • [00139] 4. Stop axis motion, pause, and collect a second ballbar measurement.
  • [00140] 5. Feed the axis back towards the previous point along the radial.
  • [00141] 6. Stop axis motion, pause, and collect a third measurement.

[00142] According to one embodiment of the present invention, for every ballbar measurement, a validity check is performed to ensure that the ballbar is at a constant deflection for the duration of the measurement. Accordingly, the difference between the first and third measurement is the axis backlash. The second measurement can be used to check validity. The average measured backlash for each measurement point for an axis can be stored as the axis backlash. Meanwhile, axis windup can be computed by, for example, subtracting the axis backlash from the estimate of the axis's total lost motion from the continuous test's data.

[00143] Run test page 30 can also contain a numeric field 54 that enables the user to modify a feedrate. Such an embodiment might be useful in that it could allow the user to run multiple tests without having to return to, for example, test setup page 10. For example, each time the feedrate is changed, a validity test on the length of the overshoot angles can be performed. If the overshoot angles are not sufficiently long, an alert message can be displayed with a suggested value. In an exemplary embodiment, to perform a new test after having just successfully completed a test, the user can simply enter a new feedrate (if desired) and press a perform-test button 56. In a further exemplary embodiment, each successive test is stored in its own data file, such that every data file contains, for example, exactly one counter-clockwise result and one clockwise result.

[00144] With respect to the performance of a validity test on data acquired using a ballbar, such a validity test can be used to compare a number of data points collected through, for example, a data pipe of the ballbar, between a feed-in move and feed-out move to an expected value. If the number of data points deviates from the expected number of data points by more than a tolerated amount, the total test time is different than expected and the data can be considered invalid. An advantage associated with such a validity check can be that, if there is a significant deviation between the expected time and actual time, the estimate of circle angle based on elapsed time might not be reliable and the reliability of the data analysis might be questionable.

[00145] Among the many conceivable situations that can cause the actual test time to differ from the expected time, are: 1) incorrect feedrate, 2) circularity or center-offset errors that are so large that the deviations cause a stop-trigger to occur in the middle of the circle, and 3) lightly damped servo dynamics that result in overshoot or undershoot during the feed-in and feed-out moves. The expected total time between feed-in and feed-out triggers can be determined in advance by, for example, using motion control subsystem emulation (e.g., similar to the look-ahead/velocity manager/Kv emulation previously mentioned). The tolerance for comparing the measured time to the estimated time can be based on the uncertainty about the initial ballbar deflection prior to the feed-in move and the final ballbar deflection following the feed-out move. In one embodiment, the time to execute the emulation may be minimized, as it measures the elapsed time and distance covered during acceleration transients. The expected total time can also be used by, for example, the test coordinator FSM during test execution to detect an error condition where a feed-out trigger failed to occur when expected.

[00146] Run-test page 30 can also include numeric displays that indicate axis positions and, for example, an instantaneous ballbar measurement. It can also contain a graphical display 58 capable of indicating, for example, an instantaneous ballbar location. Such a display 58 might be useful for tracking the progress during the motion. Still further, another embodiment of run-test page 30 can include a display 60 of the part program blocks (e.g., those currently in a MDI buffer).

[00147] Referring now to FIG. 4, an analyze-data page 70 can be used to provide an interface for a user to view and manipulate an analyzed measurement. For example, a raw measurement can be stored in a data file that contains a header that describes the test conditions and relevant machine configuration at the time of the test. Analyze-data page 70 can have a file load/browse interface 72 that enables the user to load and perform analysis on a previously developed data file.

[00148] In an exemplary embodiment, when a test run is successfully completed, a data file developed during the run is automatically loaded, so that when the user switches from, for example, run-test page 30 to analyze-data page 70, the recently collected data is already loaded and analyzed. In one embodiment, whenever a raw data file is loaded, the data undergoes analysis processing. Data manipulation tasks performed for a file load according to one embodiment of the present invention involving ballbar testing are shown in the following list, and described further herein:

• • [00149] 1. Convert ballbar deflection to circle radius deflection (for cases where the circle radius differs from the nominal ballbar length).
  • [00150] 2. Estimate the circle angle for each data point.
  • [00151] 3. Estimate the circle center offset and circularity error (center offset errors could have been created when the center set screw was tightened, or by a servo drift).
  • [00152] 4. Estimate the error parameters and their statistical significance.
  • [00153] 5. Compute the model coherence statistic.
  • [00154] 6. Compute the relative contribution of each error type to the overall circularity error.

[00155] In an exemplary embodiment, some of the contents of analyze data page 70 associated with ballbar testing include: 1) displays 74 to reflect the test conditions reported in the file header, 2) an indication 76 of the center offset error, 3) an indication 78 of the circularity error, 4) a polar-error plot 80 of a center-corrected measurement, and 5) a legend 82 to identify traces on the polar-error plot 80. Polar error plot 80 can display, for example, the radius deviation from a reference circle as a function of circle angle. In one embodiment, error amplification is specified as the plot scaling (e.g., the number of microns per radial division on the polar plot's axes). The trace on polar plot 80 can be offset from the polar origin by some amount to make the plot readable. In an exemplary embodiment, the offset is specified as the percent of plot full range that defines the radius on the axes that should correspond to the average radius of the center-corrected measurement.

[00156] In still further exemplary embodiments, analyze-data page 70 contains slider controls 84, 86 and numeric-field edit boxes 88, 90 to enable a user to manipulate the scaling and offset used to generate the plot. The page can also contain an auto-scale button 92 that, for example, applies a standard offset and computes (and applies) a maximum amplification (minimum scaling) that results in a plot that is within the range of the display. According to one embodiment of the present invention, when a file is initially loaded, plot 80 is auto-scaled.

[00157] Analyze-data page 70 can also include the capability to simultaneously display plots of the results from two separate measurements (e.g., two separate data files) for comparison purposes. For example, two data sets can be classified as primary and reference data. A separate group of controls 94 for data file browse and load capability can be provided for the reference file, while statistics and error diagnostics can be displayed for the primary data. A swap button 96 can also be provided to allow for exchanging the primary and reference files. Furthermore, a button 98 to clear the reference file from the display can also be provided.

[00158] Referring now to FIG. 4A, in an exemplary embodiment, analyze-data page 70 contains a diagnostics button 100 that, when pressed, causes a new (e.g., diagnostics) window 102 to appear. Such a window 102 can contain, for example, a listing of error sources, their estimated values, and a checkbox for each error source. For example, checkboxes (e.g., checkbox 104) can be used to indicate whether polar error plot 80 should contain a contribution from the corresponding error source. In one embodiment of the present invention, the checkboxes are initially checked. Accordingly, unchecking such a checkbox (or several checkboxes) causes the estimated effect(s) of the error(s) to be removed from the measurement, and polar error plot 80 can be updated to reflect the adjusted measurement data. Diagnostics window 102 can also contain a checkbox 106 capable of enabling a user to select (or deselect) all the individual error components with a single click.

[00159] The described capability can, for example, allow a user to predict the degree to which applying compensation for one or more error types might improve the performance of the associated machine. One associated advantage could include determining which optional compensation capabilities to activate in a control.

[00160] Referring now to FIG. 4, in a further exemplary embodiment, analyze-data page 70 contains a cursor facility (such as one activated by pressing button 108) that aids a user in the task of manually assessing various aspects of the radial errors. The capability can be used to allow the user to place a polar cursor at any point on a trace on polar-error plot 80. Displays that indicate the angle and the deviation from the mean radius for the cursor point can also be provided.

[00161] Such a cursor may be controlled by, for example, a slider bar that increases or decreases the angular location, entering values into a numeric field for circle angle, touching (or clicking) a location on plot 80, or by pressing arrow keys on a keypad to increase or decrease the angular location. For example, when the cursor is moved by a slider control or arrow keys, it may be moved at either a slow speed (where the cursor-jump interval is a single measurement point) or at a high speed (where the cursor jump interval is a fixed amount of degrees (e.g., 5 degrees)).

[00162] Such a cursor facility might be useful when used in combination with a diagnostics capability for manually diagnosing cyclic errors that appear on, for example, ballbar plot 80. For example, cyclic errors can be isolated by using diagnostics window 102 to remove the effects of all the estimated error types. The amplitude and period of the remaining errors can then be manually identified with aid of a cursor.

[00163] Once again referring to FIG. 4A, in certain embodiments of the present invention, diagnostic window 102 can also be provided with the following functionality: 1) the ability to choose the order in which errors are displayed (e.g., either ranked based on relative contribution to overall error or grouped logically based on category), and/or 2) the ability to specify whether a numeric error value display should indicate either an error parameter or a contribution of that error to a total circularity error. Diagnostics window 102 can also contain readouts 110 for a measured circularity (e.g., a circularity error of the original (center-corrected) data), a corrected circularity (e.g., a circularity error of adjusted data that has error sources of the types whose checkboxes are checked), and/or a model coherence (e.g., a measurement of the ability of the error model to account for the content of the measurement data).

[00164] Referring now to FIG. 5A, an adjust-compensation page 112 can also be included that provides an interface for a user to change configuration data, such as that associated with, for example, certain types of axis compensation that have an effect on a ballbar measurement. For example, such a page 112 can contain a pull-down menu 114 for selecting a major plane (e.g., XY, XZ, YZ). Within such an adjust-compensation page 112, there can also be sub-pages associated with a particular type of compensation (e.g., six categories of compensation are described below).

[00165] In an exemplary embodiment of the present invention, within each category of error, a configuration data display 116 is provided that is limited to the axes involved in the major-plane circular move. The configuration data can be applied uniformly to all physical connections to the indicated logical axis (a detailed description of how cross-axis associations can be created for multiple physical connections to a logical axis is further provided herein). In the descriptions, the horizontal (H) and vertical (V) axes are defined differently depending on the major plane: {XY: H=X, V=Y}; {XZ: H=X, V=Z}; {YZ: H=Y, V=Z}). The displays can be used to indicate which axis is horizontal and which is vertical.

[00166] A horizontal axis cross-comp sub-page 118 such as that depicted in FIG. 5A can be included that contains configuration data to specify, for example, a straightness and squareness compensation that compensates a V-axis position as a function of a H-axis position. In one embodiment, the specific logical axes that form the association are indicated on the page header (e.g. Y=f(X)). Configuration items that can be on this page include: 1) squareness coefficient, 2) straightness coefficient, 3) H-coordinate of compensation origin, and 4) straightness compensation's range of applicability expressed in terms of ±displacement from the compensation origin.

[00167] Similarly, a vertical axis cross-comp sub-page can be included that contains configuration data to specify a straightness and squareness compensation that compensates, for example, an H-axis position as a function of the V-axis position. In an exemplary embodiment, the configuration items on this page mirror those on the horizontal axis cross-comp page discussed above and shown in FIG. 5A, except that they apply to the V-axis rather than the H-axis.

[00168] Referring now to FIG. 5B, a horizontal axis scale-error-comp sub-page 120 can be included that contains configuration data that is used to specify, for example, pitch-error or scale-error compensation of the H-axis. For example, such a page can indicate which axis is the H-axis for the particular major plane selected. The pitch or scale compensation can be achieved by creating an association for squareness/straightness compensation where the axis is associated to itself, the squareness compensation coefficient is the pitch-error coefficient, and the straightness compensation is disabled. In an exemplary embodiment, such a page also contains fields for pitch-error-comp coefficient and a compensation origin for the H-axis.

[00169] Meanwhile, referring now to FIG. 5C, a vertical axis scale-error-comp sub-page 122 can be included that contains configuration data that is used to specify scale-error-compensation for a V-axis. In an exemplary embodiment, such a page mirrors that of the horizontal axis-scale-error-comp page as described above, except that it compensates for scale errors in the V-axis instead of the H-axis.

[00170] Referring now to FIG. 5D, a lost motion compensation sub-page 126 can be further included that contains configuration data items for backlash compensation and windup compensation for the H and V-axes involved in the major plane circular move. In an exemplary embodiment, the specific logical axes associated with the H and V axes are indicated in a display.

[00171] Furthermore, and referring now to FIG. 5E, a friction comp adjustments sub-page 128 can be included that contains configuration data items for Coulomb-friction-feedforward parameters for the H and V-axes involved in the major plane circular move, for example. In an exemplary embodiment, specific logical axes associated with the H and V axes are indicated in a display.

[00172] In further exemplary embodiments, adjust-compensation page 112 can also contain a commit button 130 that copies the values from the (current-values) numeric fields to, for example, a database associated with the control and then, for example, issues a configuration-change system event to update the configuration data.

[00173] Adjust-compensation page 112 can also provide suggested values for changing existing compensation parameters. For example, each configuration parameter can be represented by three fields on the page: an original value filed 132, a current value filed 134, and a proposed value field 136. In an exemplary embodiment, each configuration item's current-value field is the source of that configuration item's value that is copied to a database when commit button 130 is pressed.

[00174] In one embodiment, the current-value field 134 is typically displayed. Meanwhile, in such an embodiment, the original-value and proposed-value fields 132, 136 can be displayed when parameter estimates for a primary file have been obtained from the most recent measurement and when the current database values for the compensation parameters are the same as those present during the most recent test. One associated advantage could be that this ensures that the compensation is adjusted using the appropriate data. One approach to accomplishing this involves displaying the original-value and proposed-value fields 132, 136 when each of the following two conditions are satisfied: 1) a primary file actively loaded (e.g., on analyze-data page 70) has a file-reference-number that matches a data-reference-number (a unique numeric string constructed at the beginning of test execution based on the test's major plane, date, and time) associated with the most recent test performed on the machine, and 2) the major plane selected on adjust-comp page 112 is the same as the major plane of the primary file.

[00175] According to such an embodiment, the file-reference-number can be extracted from a data file header when the file is loaded. When the file is originally created, the data-reference-number for the test can be copied to the file's header. The existing parametric compensation for the machine plane can also be captured at the beginning of the execution.

[00176] In an exemplary embodiment, when the currently loaded primary file is the result of the most recent measurement, and the selected major plane on adjust-comp page 112 corresponds to the plane of the most recent measurement, then fields 132, 136 for original and proposed values for each configuration item are displayed. In one embodiment, the numeric contents of these fields are not modifiable by the user. According to such an embodiment, the original value represents the value for the compensation parameter that was active when the data was collected. It can be copied from the snapshot of configuration that was taken at the beginning of test execution.

[00177] Meanwhile, the proposed value can be the value that was determined to be the correct adjustment to compensation. The proposed value can be computed based on the original value and the value for the associated error parameter as estimated from the measurement data. In an exemplary embodiment, adjust-comp page 112 provides the ability to copy the contents of the proposed value fields into the current value fields, to copy the contents of the original value fields to the current value fields, and to clear and disable the compensation for the current value fields. For example, buttons 138, 140, an 142 can be provided that are capable of respectively initiating each of these tasks when selected. In one such embodiment, each of these buttons could perform its operations for the fields on the current sub-page. Three additional buttons (e.g., buttons 144, 146, and 148) that perform these operations on all sub-pages can also be provided.

[00178] In addition to the development of the proposed values, the measured data can also be used to determine a measure of statistical significance for each of the proposed compensation parameters. As used herein, significance is a measure of whether an estimated error parameter is “significant” relative to the estimate for that error parameter's variance. Mathematics behind the development of the significance measure are further discussed herein.

[00179] Referring back to FIG. 5D, the significance for each of the proposed values can be displayed graphically, such as by a meter 150. In such an embodiment, a value that causes a needle of meter 150 to be high indicates that the proposed change to the compensation parameter is significant, while a value that causes the needle to be low indicates that the proposed change to the compensation parameter is insignificant.

[00180] In one embodiment of the present invention, the proposed value for a compensation parameter is determined by adding the (negative of the) latest estimated value for the error parameter to the compensation parameter that was active when the data was collected. Detailed descriptions of how the proposed values for each compensation parameter are developed are further provided herein.

[00181] A detailed analysis of the processing involved in an exemplary embodiment of the present invention involving ballbar testing is now provided. As part of the detailed description of the same by way of example, certain aspects of exemplary embodiments disclosed herein may be specific to a particular commercially available ballbar (the QC10 ballbar sold by Renishaw) and a particular motion control system (the A2100). One skilled in the art will, however, appreciate that embodiments of the present invention can be applied to, for example, other devices (including other ballbars) and control systems. In particular, certain of the embodiments to be further described herein relate to application software currently called Ballbar Autotuning for Parametric Compensation (ATB) and currently designed for use with the A2100 control, a detailed description of which will be provided herein as an illustrative example.

[00182] As an introductory matter, the QC10 ballbar outputs its data to a standard RS232 serial port. Certain exemplary embodiments to be discussed herein involve data collection and analysis running entirely within an operating system such as, for example, one currently sold under one of Microsoft Windows trademarks. In some of these embodiments, the real time measurement data from the ballbar is acquired at a constant sample rate and piped into memory accessible through the operating system. In this way, a contiguous data set may be obtained on a non-real time operating system.

[00183] Thus, with such an approach, ballbar data is not exactly synchronized to, for example, the real time internal data of the control. The type of analysis performed by certain embodiments of an exemplary embodiment as disclosed herein does not require synchronization of the ballbar data and internal control data. However, other embodiments of auto tuning might involve such synchronization.

[00184] With respect to the analysis of measured data in an exemplary embodiment of the present invention, the raw measurement data can undergo significant processing. For example, raw data can first be corrected for offsets between the center of rotation of the ballbar and the center of the circle. The circularity-error (a measure of the combined effect of all errors) can be extracted from the center-corrected data.

[00185] Next, estimates for the error parameters can be extracted from the data. During the parameter estimation, the coherence of the data can be determined. Once estimates for error parameters are obtained, individual effects of each error can be determined and the effect of the error may be removed from the data in response to, for example, user manipulation of the checkboxes of diagnostics window 102. Similarly, the relative impact of each error type on the overall circularity may be assessed.

[00186] Much of the data analysis involved in this specific example employs “curve-fitting” methods to extract parameters from the raw data. As used herein, the term “curve fitting” describes a process where the measured data is used to determine the parameters for a mathematical model of an assumed structure or framework. Ideally, curve fitting determines the values for the set of parameters in the mathematical model that have the maximum likelihood of having created the measured data.

[00187] For the parametric machine errors detected by an exemplary embodiment of the present invention, the curve fitting process involves a least-squares optimization. A least squares estimator can be a maximum likelihood estimator if each measurement error is normally distributed and each measurement error has the same variance. This might be untrue for the present case, but the least squares estimator can be used anyway because it tends to provide a decent result and there is no readily available alternative given the available a-priori information about the data. A relatively straightforward linear least-squares approach may be employed because the assumed structure of the radial error involves a linear combination in the model parameters.

[00188] A linear least squares formulation can be based on an overdetermined system where a linear combination of basis functions scaled by the model parameters yields the measurement. In this specific case, the basis functions can be functions of the (constant) nominal circle radius and the (varying) circle angle. The measurement can be a function of the radius measurements. The formulation may be written in matrix form as follows, where {P} is the m-element parameter vector, {f(r(t))} is the n element vector where each element is a function of measured ballbar radius at time t, and [A(R,θ(t))] is the n×m matrix whose columns are basis functions developed from the circle angle and nominal circle radius.

[heading-00189]  {ƒ(r(t))}=[A(R,θ(t))]{P}  (1)

[heading-00190] If n>m, Equation (1) represents an overdetermined system where there are more knowns than unknowns. This equation can be solved using the least squares technique which minimizes the sum of the squares of the errors between the modeled f(r) and the measured f(r).
Σe²={[A]{P}−ƒ(r)}^T·{[A]{P}−ƒ(r)}  (2)
The solution for the parameter vector {P} that minimizes this quantity includes the least squares solution and can be written using the so-called normal equations.
{P}=[[A]^T·[A]]⁻¹·[A]^T·{ƒ(r)}  (3)
The solution involves inversion of the matrix [[A]^T[A]]. This matrix has the properties of being real, symmetric, and positive-definite. These properties can be taken advantage of when finding a technique for inverting the matrix. The matrix inverse can be computed using Jacobi's method for finding the eigenvalues and eigenvectors of a real symmetric positive definite matrix.

[00195] Briefly stated, Jacobi's method involves successively applying rotation transformations to portions of the matrix until it is diagonalized. A full explanation of Jacobi's method is given in “Numerical Recipes in C” by Press, Flannery, Teukolsy and Vettering, which is hereby incorporated herein by reference. The Jacobi technique should work as long as the matrix is invertible. Another advantage of using Jacobi's technique for solving the normal equations is that it provides the eigenvalues of [A]^T[A]. These eigenvalues can be used to determine the condition number of the matrix [A]—a factor related to the excitation provided by the axis motion.

[00196] The topics that follow related to the development of the equations that describe the effects of each parametric machine error on a ballbar radius measurement. Each of these equations can be combined into an individual overdetermined system formulation in such a way that the error parameters are estimated simultaneously. This approach minimizes possibility of incorrectly attributing measurement aspects to the wrong error type.

[00197] The equations can be developed in such a way to enable a description of the effect of each error type in terms of the measured ballbar radius. The following equations are developed for the case of the X-Y plane. These results are equally applicable to the XZ and YZ planes. The x and y designators used in this development may be interpreted to mean the horizontal and vertical axes of the circle-plane rather than the machine tool axes.

[00198] It is often the case that the circle center is not coincident with the center ball of the ballbar base fixture. These differences can occur if the center ball shifts when the set screw is tightened, if the servo offsets drift with time, the axis has a significant straightness error (discussed later), or any number of other reasons. In order to properly analyze the measurement, the effect of center offset can be considered along with other parametric error effects.

[00199] In an exemplary embodiment, center offset is not considered to be a machine error; it is considered to be an error associated with measurement. The formulation of the least squares equations for determining center offset can be developed by considering the equations for the length of the vector from the centering fixture location to a point on the ideal circle as a function of circle angle. The situation can be depicted graphically as shown in FIG. 6A. A polar error plot for the ballbar measurement with a center offset error contains a “kidney bean” signature of the type shown in FIG. 6B.

[00200] Examination of FIG. 6A leads to the development of the following equation for the measured radius r as a function of the circle angle θ, where R is the average circle radius that may differ from the programmed radius if the axes have scaling errors (discussed later).
(R cos(θ)−E_CTR—X)²+(R sin(θ)−E_CTR—Y)²=(r(θ))²  (4)
The equation can be expanded to the following form.
R²−2RE_CTR—X cos(θ)−2RE_CTR—Y sin(θ)+E_CTR—X²+E_CTR—Y²=(r(θ))²  (5)
For the case of multiple measurements, the previous equation can be represented in the matrix form to accommodate the least squares solution. $\begin{array}{cc}\left[\begin{array}{ccc}-2\cos \left(\theta \left(t_1\right)\right)& -2\sin \left(\theta \left(t_1\right)\right)& 1\\ -2\cos \left(\theta \left(t_2\right)\right)& -2\sin \left(\theta \left(t_2\right)\right)& 1\\ ⋮& ⋮& ⋮\\ -2\cos \left(\theta \left(t_n\right)\right)& -2\sin \left(\theta \left(t_n\right)\right)& 1\end{array}\right]\left\{\begin{array}{c}{\mathrm{RE}}_{\mathrm{CTR\_X}}\\ {\mathrm{RE}}_{\mathrm{CTR\_Y}}\\ R^2+E_{\mathrm{CTR\_X}}^2+E_{\mathrm{CTR\_Y}}^2\end{array}\right\}=\left(\begin{array}{c}{\left(r\left(t_1\right)\right)}^2\\ {\left(r\left(t_2\right)\right)}^2\\ ⋮\\ {\left(r\left(t_n\right)\right)}^2\end{array}\right)& \left(6\right)\end{array}$
The previous equation may be “solved” using equation (3) to obtain the parameter vector where the matrix to the left is [A], the 3×1 vector is P and the n×1 vector is f(r). The equation used to determine the circle center offsets also provides an average circle radius R. Three desired parameters, R, E_CTR—X, and E_CTR—Y, may be extracted from the vector P by an iterative technique.

[00206] A first approximation of setting the quantity E_CTR—X²+E_CTR—Y² to be zero allows the parameters for the first iteration to be extracted directly. Typically, the center offset is very small relative to the circle radius and iterations are not needed. However, for the sake of robustness, an iterative algorithm can be made available in case it is needed. The iterative algorithm described below converges to the correct result as long as the ballbar center is inside the circle.

• • [00207] P=[A^TA]⁻¹A^Tr²
  • [00208] R_est=(P₃)^0.5,R_est—prev=0
  • [00209] E_CTR—X=P₁/R_est, E_CTR—Y=P₂/R_est
  • [00210] Do Forever
    • [00211] R_est=(P₃−(E_CTR—X)²−(E_CTR—Y)²)^0.5
    • [00212] If |R_est−R_est—prev|>0.01 μ, Then, Break
    • [00213] E_CTR—X=P₁/R_est, E_CTR—Y=P₂/R_est
    • [00214] R_est—prev=R_est;
  • [00215] Continue

[00216] Once the values for the circle center offsets are known, they can be used in manipulation of the raw ballbar radius data r_raw to remove the effects of center offsets from the data. The center offset effects can be removed when computing the circularity statistic. An approximate equation for removing the effects of the center offsets can be obtained by examination of FIG. 6 and making the assumption that the angle of the ballbar relative to the horizontal axis is the same as the circle angle.
$\begin{array}{cc}{r}_{\mathrm{ctr\_adj}}\left(\theta \right)=\sqrt{{\left(r_{\mathrm{raw}}\left(\theta \right)·\cos \left(\theta \right)+E_{\mathrm{CTR\_X}}\right)}^2+{\left(r_{\mathrm{raw}}\left(\theta \right)·\sin \left(\theta \right)+E_{\mathrm{CTR\_Y}}\right)}^2}& \left(7\right)\end{array}$
An equation for adjusting the measured radius for center offsets can be obtained by examining FIG. 6 and employing the law of cosines where the x_ce and y_ce designate the estimates for center offsets E_CTR—X, E_CTR—Y.
[r_ctr—adj(θ)]²−(2√{square root over (x_ce²+y_ce²)} cos(tan⁻¹(y_ce, x_ce)−θ))·[r_ctr—adj(θ)]+(x_ce²+y_ce²−r_raw²(θ))  (8)

[00220] A solution for the center-adjusted radius is the positive root of the previous quadratic equation. The roots should be real as long as the center is inside of the circle. The circularity-error or circularity is a measure of the maximum deviation of the out-of-roundness of the circle. It can be defined as the maximum center-adjusted radius minus the minimum center adjusted radius.
circularity=max(r_ctr—adj(θ))−min(r_ctr—adj(θ))  (9)
In an exemplary embodiment, the computed circularity from the measurement is displayed. The circularity for data that has been mathematically manipulated in such a way that the effects of a combination of user-selected parametric errors can be removed can also be computed and/or displayed.

[00223] The most common machine tool design involves an arrangement where linear motion within a plane is effected by coordinating the motion of two individual linear displacement actuator mechanisms (axes) that are perpendicular to each other, i.e., the axes are square. The machine tool manufacturing process may result in a squareness error where the axes are not precisely perpendicular to each other as is intended. This deviation from perpendicularity is known as the squareness error or simply squareness.

[00224] The squareness error describes a relationship between the two axes in the plane with one squareness error per major plane. Individual proposed values for squareness compensation for each axis in the plane can also be developed by equally distributing the total squareness compensation among the two axes. The equation for the effect of a squareness error on the circle radius can be developed based on FIG. 7.

[00225] The coordinates of a point on the actual (non-circular) curve as a function of circle angle θ can be described by the following equations where R is the average radius which may differ from the programmed radius if the machine also has a scale error.
x(θ)=R cos(θ)+R sin(θ)sin(½φ)−R cos(θ)sin(½φ)tan(½φ)  (10)
y(θ)=R sin(θ)+R cos(θ)sin(½φ)−R sin(θ)sin(½φ)tan(½φ)  (11)
In these equations, the final terms on the right that involve the quantity sin(½φ)tan(½φ) are insignificantly small for small values of φ. This is a valid assumption for the amplitude of squareness errors found on most machine tools. The ballbar radius as a function of circle angle is the desired result and can be developed by computing the sum of the squares of the x and y components of the point on the curve and employing the aforementioned simplifying assumption.
r²(θ)=x²(θ)+y²(θ)  (12)
r²(θ)=R²(cos²(θ)+sin²(θ))(1+sin²(½φ))+2R² sin(θ)cos(θ)sin(½φ)  (13)
For small φ, the term sin²(½φ) becomes insignificant and the previous equation may be simplified to the following
r²(θ)=R²(1+2 sin(θ)cos(θ)sin(½φ))  (14)
The parameter that describes the squareness error in the X-Y plane can be defined as the unitless proportionality factor sin(½φ). Analogous parameters can be obtained for the X-Z and Y-Z planes.
E_SQR—XY=sin(½φ)  (15)
The following matrix equation is for an overdetermined system of equations represented as a linear combination of terms in the squareness parameter. $\begin{array}{cc}\left(\begin{array}{c}{\left(r\left(t_1\right)\right)}^2\\ {\left(r\left(t_2\right)\right)}^2\\ ⋮\\ {\left(r\left(t_n\right)\right)}^2\end{array}\right)=\left[\begin{array}{cc}2\sin \left(\theta \left(t_1\right)\right)\cos \left(\theta \left(t_1\right)\right)& 1\\ 2\sin \left(\theta \left(t_2\right)\right)\cos \left(\theta \left(t_2\right)\right)& 1\\ ⋮& ⋮\\ 2\sin \left(\theta \left(t_n\right)\right)\cos \left(\theta \left(t_n\right)\right)& 1\end{array}\right]\left\{\begin{array}{c}{R}^2{E}_{\mathrm{SQR}}\\ R^2\end{array}\right\}& \left(16\right)\end{array}$

[00236] The axis squareness error is most often described as the perpendicular deviation in units of parts-per-million or equivalently microns/meter or inches/mil. This is the way the squareness compensation is represented and the way the squareness error can be displayed, such as on a screen. The conversion from the unitless squareness parameter to parts-per-million squareness can be obtained by multiplying the unitless parameter by 1 million.
squareness=1,000,000·E_SQR(μ/m)  (17)

[00238] A conventional linear motion machine tool axis includes an accurate feedback measurement to move to the correct position. Axis scale errors can be created by an inaccurate feedback mechanism or an inaccurate parameter for converting a feedback measurement from, for example, a rotary device to linear displacement. One common linear axis mechanism involves a rotary motor that drives a ballscrew. A ball-nut converts the rotary motion to linear motion. The axis position may be estimated by measuring the motor angle and computing the axis position based on an assumed value for screw pitch. Alternatively, the axis position may be measured using a linear displacement sensor such as a glass scale, for example.

[00239] In the first case, differences between the assumed and actual pitch of the ballscrew can cause the axis to travel an incorrect distance due to this pitch error. In the second case, errors in the scale itself, due to factors such as thermal growth, for example, can cause the axis to travel to an incorrect position due to the scale error. Pitch error and scale error can cause an axis to travel a distance that is different than desired.

[00240] This type of error also appears on other axis mechanisms, such as rack-and-pinion and linear motor driven axes. The fundamental type of error experienced with these types of mechanisms is a position error that is linearly proportional to the distance traveled. The constant of proportionality may be represented using units such as parts-per-million (equivalent to microns/meter or mils/inch).

[00241] Effects of axis linear-scale-errors on circular motion in a plane are shown in FIG. 8. The figure shows the actual circle (not a polar error plot) for a case where the scale errors are very large relative to the radius. The figure shows how the two scale errors from each axis act together to create two effects: 1) a relative scale mismatch between the axes that causes the circle to be stretched or compressed in a direction parallel to the axis, and 2) an overall enlargement or reduction in the average (best-fit) radius of the (distorted) circle. The specific example shown in FIG. 8 is for a case where the scale error on the X axis is positive (the axis moves further than desired) and the scale error on the Y axis is negative (the axis moves a smaller distance than is desired). Since the (positive) X axis scale error is larger than the (negative) Y axis error, the average circle radius (best-fit-circle) is larger than the ideal circle radius.

[00242] The equations for the effect of each scaling error on the measured circle radius can be developed by first considering the x and y axes separately.
x(θ)=R_prog cos(θ)(1+E_SCL—X), y(θ)=R_prog sin(θ)(1+E_SCL—Y)  (18)
The square of the instantaneous radius is the sum of the squares of the x and y vector components. With the (valid) assumption that the square of the error parameters are insignificant, the following relation can be obtained.
r²(θ)=R_prog²+2R_prog²E_SCL—X cos²(θ)+2_prog²E_SCL—Y sin²(θ)  (19)
The above equation can be reconstructed as a n^x2 matrix equation. This can be done if the ballbar is length calibrated such that the difference between r²(θ) and R_prog is meaningful.

[00247] In an exemplary embodiment, differences between x and y scaling errors are accounted for even if the ballbar is not length-calibrated. One approach involves transforming the individual X and Y scale errors into a scale mismatch error and an average radius. If the ballbar is not length-calibrated, then the estimate for average radius can be discarded. If the ballbar is length-calibrated, then the estimate for average radius can be used to extract the individual axis scaling errors from the scale mismatch error. The parameters in the new formulation have the following definitions: $\begin{array}{cc}{E}_{\mathrm{SCL\_XY}}=\frac{E_{\mathrm{SCL\_X}}-E_{\mathrm{SCL\_Y}}}{2},R_{\mathrm{avg}}=R_{\mathrm{prog}}\left(1+\frac{E_{\mathrm{SCL\_X}}+E_{\mathrm{SCL\_Y}}}{2}\right)& \left(20\right)\end{array}$

[00248] If the scaling error is small enough to make the square of the error terms insignificant, then the following equations for the x and y locations on the curve may be used. These equations assume that the circle is stretched in the x-direction; if the circle is actually stretched in the y-direction, then the scale mismatch parameter E_Smismatch will be a negative number.
x(θ)=R_avg cos(θ)(1+E_SCL—XY), y(θ)=R_avg sin(θ)(1−E_SCL—XY)  (21)
The sum of each component squared in the previous equation yields the expression for the instantaneous ballbar radius (where the error term squared can be once again ignored).
r²(θ)=R_avg²+2_avg²(cos²(θ)−sin²(θ))E_SCL—XY  (22)
The previous equation can be represented in the following matrix form for measurements at corresponding times. $\begin{array}{cc}\left(\begin{array}{c}{\left(r\left(t_1\right)\right)}^2\\ {\left(r\left(t_2\right)\right)}^2\\ ⋮\\ {\left(r\left(t_n\right)\right)}^2\end{array}\right)=\left[\begin{array}{cc}2\left({\cos }^2\left(\theta \left(t_1\right)\right)-{\sin }^2\left(\theta \left(t_1\right)\right)\right)& 1\\ 2\left({\cos }^2\left(\theta \left(t_2\right)\right)-{\sin }^2\left(\theta \left(t_2\right)\right)\right)& 1\\ ⋮& ⋮\\ 2\left({\cos }^2\left(\theta \left(t_n\right)\right)-{\sin }^2\left(\theta \left(t_n\right)\right)\right)& 1\end{array}\right]\left\{\begin{array}{c}{R}_{\mathrm{avg}}^2{E}_{\mathrm{SCL\_XY}}\\ R_{\mathrm{avg}}^2\end{array}\right\}& \left(23\right)\end{array}$

[00253] If the ballbar is not length calibrated, then the estimate for R_avg is meaningless and the individual axis scale errors cannot be extracted from the scale mismatch errors. In this case, values for the individual axes' scale errors can be computed by assuming that each axis contributes equally toward the observed scale mismatch.

[heading-00254]  E_SCL—X=E_SCL—XY, E_SCL—Y=−E_SCL—XY  (24)

[00255] If the ballbar is length calibrated, then the individual axis scale errors may be extracted using the following equations. $\begin{array}{cc}{E}_{\mathrm{SCL\_X}}=\frac{R_{\mathrm{avg}}}{R_{\mathrm{prog}}}-1+E_{\mathrm{SCL\_XY}},E_{\mathrm{SCL\_Y}}=\frac{R_{\mathrm{avg}}}{R_{\mathrm{prog}}}-1-E_{\mathrm{SCL\_XY}}& \left(25\right)\end{array}$
The individual axis scale errors and the scale mismatch errors in the previous equation are unitless factors of proportionality. It is common, however, to represent these factors in parts-per-million units, also described as microns/meter or mils/inch. The conversion of a scale error (E_SCL—X, E_SCL—Y, E_SCL—XY) from a unitless factor to a parts-per-million factor can be simply a matter of multiplying the scale error term by a factor of 10⁶.

[00257] In the preceding discussions, it has been assumed that an axis' linear scale error is the phenomenon that can cause the average circle radius to differ from the ideal radius. The other parametric error types considered do not appear to affect the average circle radius. Specifically, the average circle radius does not appear to be affected by squareness errors, straightness errors, lost-motion errors, and servo mismatch errors.

[00258] There is, however, one phenomenon in addition to scale errors that does appear to affect the average circle radius. The dynamic characteristics of a position controller can result in amplitude attenuation (or amplification) that varies according to circle radius and feedrate. Position controllers have a low-pass frequency characteristic, and some may exhibit amplification at certain frequencies. Since the circle frequency is the per-second-feedrate divided by the radius, the frequency increases with increases in feedrate or reductions in nominal radius. For cases where the ballbar is length calibrated, an adjustment can be made to the estimate of average radius to account for the expected behavior of the position loop's dynamic response.

[00259] The expected behavior of the position loop response can be based on a number of approximating assumptions. The assumptions are expected to be reasonable for circle frequencies that are below the cutoff frequency of the position loop. Even the highest feedrate ballbar tests are expected to correspond to frequencies that are within the bandwidth of the axes for even the lowest gain controllers. For example, an axis with a Kv gain of 1.0 and no feedforward has a cutoff frequency of approximately 2.7 Hertz; for a circle radius equal to the minimum ballbar length of 100 mm, the feedrate associated with the cutoff frequency is 100,000 mm/min—a value beyond the capability of most machines and higher than what would typically be used during a ballbar test. The assumptions about the dynamic response of the axes can be as follows:

• • [00260] 1. Both axes that actuate motion in the circle plane have identical values for Kv and VFF.
  • [00261] 2. Both axes have identical amplitude response up to the cutoff frequency.
  • [00262] In 3. The effective Kv is identical to the nominal Kv adjusted by the Kv_adjust.
  • [00263] 4. The axis dynamics are approximated as a continuous time system with an infinite bandwidth velocity loop.

[00264] Based on these assumptions, the effect of axis dynamic response on the average radius can be developed from the system shown in FIG. 9. A transfer function of this simplified model for a position controller is shown below. $\begin{array}{cc}\frac{x_{\mathrm{fdbk}}}{x_{\mathrm{cmd}}}\left(s\right)=\frac{\left(\mathrm{VFF}/100\right)s+\left(1000\mathrm{Kv}/60\right)}{s+\left(1000\mathrm{Kv}/60\right)}& \left(26\right)\end{array}$
The amplitude scaling for a sinusoidal input of angular frequency ω can be determined by evaluating the magnitude of the transfer function at s=jω. $\begin{array}{cc}\frac{x_{\mathrm{fdbk}}}{x_{\mathrm{cmd}}}\left(s=j\omega \right)=\sqrt{\frac{{\left({\mathrm{VFF}}_v/100\right)}^2{\omega }^2+{\left(1000{\mathrm{Kv}}_v/60\right)}^2}{{\omega }^2+{\left(1000{\mathrm{Kv}}_v/60\right)}^2}}& \left(27\right)\end{array}$

[00266] For a constant feedrate circular move, the axis motion can be sinusoidal where ω=Feedrate/(60R). Since both axes are assumed to have identical responses, the radius amplitude scaling is identical to the axis amplitude scaling. Performing the substitution for ω, yields the final result for the radius scaling due to the position loop dynamic response. $\begin{array}{cc}\begin{array}{c}{R}_{\mathrm{avg}\mathrm{dyn}}=R_{\mathrm{prog}}·\\ \sqrt{\frac{{\left(\mathrm{VFF}/100\right)}^2{\left(\mathrm{Feedrate}/R_{\mathrm{prog}}\right)}^2+{\left(1000\mathrm{Kv}/60\right)}^2}{{\left(\mathrm{Feedrate}/R_{\mathrm{prog}}\right)}^2+{\left(1000\mathrm{Kv}/60\right)}^2}}\end{array}& \left(28\right)\end{array}$

[00267] The dynamic response can be factored into the estimate of the axis scale errors by measuring the overall scaling relative to the expected radius, given the programmed circle radius and the model for dynamic response. $\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{SCL\_X}}=\frac{R_{\mathrm{avg}}}{R_{\mathrm{prog}}}\\ \sqrt{\frac{{\left(\mathrm{Feedrate}/R_{\mathrm{prog}}\right)}^2+{\left(1000\mathrm{Kv}\right)}^2}{{\left(\mathrm{VFF}/100\right)}^2{\left(\mathrm{Feedrate}/R_{\mathrm{prog}}\right)}^2+{\left(1000\mathrm{Kv}\right)}^2}}-\\ 1+E_{\mathrm{SCL\_XY}}\end{array}& \left(29\right)\end{array}$ $\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{SCL\_Y}}=\frac{R_{\mathrm{avg}}}{R_{\mathrm{prog}}}\\ \sqrt{\frac{{\left(\mathrm{Feedrate}/R_{\mathrm{prog}}\right)}^2+{\left(1000\mathrm{Kv}\right)}^2}{{\left(\mathrm{VFF}/100\right)}^2{\left(\mathrm{Feedrate}/R_{\mathrm{prog}}\right)}^2+{\left(1000\mathrm{Kv}\right)}^2}}-\\ 1-E_{\mathrm{SCL\_XY}}\end{array}& \left(30\right)\end{array}$

[00268] An axis straightness error is a linear displacement error in a direction perpendicular to the axis containing the straightness error. Straightness errors can be treated in a similar way as squareness errors such that the linear displacement errors that have a direction that is parallel to the direction of one of the machine's linear axes (e.g., x, y, and z) are considered. In other words, the straightness errors can be considered for one machine major-plane at a time.

[00269] Straightness errors can be distinguished from squareness errors in that the relationship between the error in the direction of the perpendicular-axis relative to the position of the axis containing the error is non-linear. In the present discussion, the non-linear relationship is restricted to the case of a symmetric parabolic curve where the axis of symmetry passes through the location of the center of the circle in the ballbar test. The assumed structure for the straightness error of the X axis in the direction of the Y axis can be defined by the following equation, where the deviation caused by straightness errors is ΔY, X₀ is the x coordinate of the center of the test circle, and E_STR—X—Y is the X-axis straightness error factor for the Y directed component of straightness error.

[heading-00270]  Δy_STR(x)=E_STR—X—Y(x−x₀)²  (31)

[00271] The assumption that a straightness error is reasonably approximated by a parabolic curve can be based on experience with machine tools that has shown that axis guideways often have a fundamental straightness error that involves a bowing shape. Limiting the straightness error to include the fundamental component prevents the analysis from accounting for errors where the guideway has multiple bows or wiggles. Straightness errors that are more complex than a fundamental component can be handled using a data-based approach, such as developing tables based on laser measurements and employing the cross-axis error compensation.

[00272] The assumption that the parabola is symmetric about a line parallel to the perpendicular axis can be a reasonable one because the squareness error should account for the presence of any linear trend. The assumption that the parabola's axis of symmetry passes though the center of the test circle x₀ is not based on any physical insight about the machine, since the location of the circle center is arbitrary and may change from test to test. This restriction can be included because it greatly simplifies the analysis, enabling the error parameters to be estimated from a single evaluation of the least-squares normal equations. This restriction does not limit the ability to identify error sources within the range of the ballbar test because an expansion of equation (31), shown below in equation (32), reveals that the choice of x₀ may be arbitrary since it is a factor in: 1) the linear trend accounted for by the estimate for squareness error and 2) a y-component of center offset accounted for by the estimate of center offset error.
E_STR—X—Y(x−x₀)²=(E_STR—X—Y)·x²−(2E_STR—X—Y)·x²+(E_STR—Xx₀²)  (32)

[00274] The choice for x₀ may be arbitrary in the sense that the least squares approach should provide an optimal result by providing an estimate for squareness and y-center offset that accommodates the particular choice of x₀ for straightness. However, the arbitrary choice for x₀ makes extrapolation of the error model outside the range of the test invalid. Application of compensation based on the estimate to locations outside of the range of the test can easily cause larger errors to occur in these regions than those that might have occurred without any compensation.

[00275] The presence of the linear trend constant terms in equation (32) implies estimating the squareness error and center offset errors at the same time as the straightness error. Accordingly, in an exemplary embodiment, the center correction is not applied to the raw data prior to performing the diagnostics. This can be potentially important for data arcs that cover less than 360 degrees.

[00276] An example of a situation in the X-Y plane where the Y-axis has no straightness-error and the X-axis has a straightness error that complies with the assumptions is shown in FIG. 10. The plot shows how a straightness error might affect the shape of a curve on a polar error plot with a high degree of amplification. The equation for the distorted curve can be developed by applying the error from equation (31) to the equation for a circle where R is the actual best-fit circle radius that may differ from the nominal radius due to a scale error.
x=R·cos(θ), y=R·sin(θ)+E_STR—X—Y·(R·cos(θ))²  (33)

[00278] The square of the radius is the sum of the squares of the x and y coordinates of the curve. If terms involving the square of the straightness coefficient are ignored by assuming that the straightness error is small, then the following equation for circle radius as a function of angle can be derived.
r²(θ)=R²+2E_STR—X—YR³ sin(θ)cos²(θ)  (34)
An analogous equation for the impact of a Y-axis straightness error on the instantaneous circle radius may be derived in a similar manner. An equation for the effect of the Y-axis straightness error E_STR—Y—X is shown below.
r²(θ)=R²+2E_STR—Y—XR³ cos(θ)sin²(θ)  (35)
The combined effect of both planar straightness errors is shown below with the equations written for multiple measurements in matrix form. $\hspace{1em}\begin{array}{cc}\begin{array}{c}\left\{\begin{array}{c}{r}^2\left(t_1\right)\\ r^2\left(t_2\right)\\ ⋮\\ r^2\left(t_n\right)\end{array}\right\}=\left[\begin{array}{ccc}2\sin \left(\theta \left(t_1\right)\right){\cos }^2\left(\theta \left(t_1\right)\right)& 2\cos \left(\theta \left(t_1\right)\right){\sin }^2\left(\theta \left(t_1\right)\right)& 1\\ 2\sin \left(\theta \left(t_2\right)\right){\cos }^2\left(\theta \left(t_2\right)\right)& 2\cos \left(\theta \left(t_2\right)\right){\sin }^2\left(\theta \left(t_2\right)\right)& 1\\ ⋮& ⋮& ⋮\\ 2\sin \left(\theta \left(t_n\right)\right){\cos }^2\left(\theta \left(t_n\right)\right)& 2\cos \left(\theta \left(t_n\right)\right){\sin }^2\left(\theta \left(t_n\right)\right)& 1\end{array}\right]·\\ \left\{\hspace{1em}\begin{array}{c}{R}^3{E}_{\mathrm{STR\_X}\mathrm{\_Y}}\\ R^3{E}_{\mathrm{STR\_Y}\mathrm{\_X}}\\ R^2\end{array}\right\}\end{array}& \left(36\right)\end{array}$

[00283] As mentioned previously, the estimation of straightness errors can be performed simultaneously with the estimation of squareness and center offset; thus, in an exemplary embodiment, the previous matrix equation is not used directly. A global matrix equation that simultaneously estimates the parameters is further provided herein.

[00284] A servo mismatch error is an error created due to a difference in the dynamic response between the two axes of the circle plane. The dominant effect of servo mismatch is a phase mismatch. A previous discussion herein described how the amplitude response of an axis is expected to be flat beyond the highest conceivable circle frequency, even for low Kv gains. This means that the Kv gains may be mismatched without creating a mismatch in the amplitude response.

[00285] However, this is not the case for the phase response which, for a position loop without feedforward, begins to roll off immediately. For any given circle-frequency, i.e., feedrate/radius, the phase difference between the sinusoids for the x and y axes is desired to be 90 degrees, but may differ from this ideal amount by some constant deviation. The constant deviation should vary with circle frequency, and, in most cases, the deviation is more likely to be significant as the circle frequency is increased (but the relationship is not necessarily monotonic).

[00286] The phase deviation can be caused by mismatches in the effective position loop gain, velocity loop response, or both. It is not conventionally possible to diagnose the source of the phase mismatch with a single ballbar measurement—which can be accomplished by performing frequency response measurements for each axis individually. Even though no attempt to propose a compensation for an observed servo mismatch error is made, a servo-mismatch error can be computed because it informs the user that such an error is present, prevents servo mismatch errors from being falsely attributed to other error types, and allows the user to gauge the servo-mismatch error's impact on the circularity and to predict the improvement that might be accomplished if the error were removed.

[00287] The combined phase error between the two axes during a constant feedrate circle may be represented by the following equation where the total phase error φ is equally divided among the two axes by assuming a half Q lead on the X axis and a half φ lag on the Y-axis. For a counter-clockwise move, the circle angle θ increases with time (i.e., the frequency dθ/dt=ω is positive). For clockwise moves, the reverse is true—the frequency is negative. Equations for these two cases can be shown below. $\begin{array}{cc}{X}_{\mathrm{CCW}}\left(t\right)=R·\cos \left(\omega t+\frac{1}{2}\varphi \left(\omega \right)\right),Y_{\mathrm{CCW}}\left(t\right)=R·\sin \left(\omega t-\frac{1}{2}\varphi \left(\omega \right)\right)& \left(37\right)\\ X_{\mathrm{CLW}}\left(t\right)=R·\cos \left(-\omega t+\frac{1}{2}\varphi \left(\omega \right)\right),Y_{\mathrm{CLW}}\left(t\right)=R·\sin \left(-\omega t-\frac{1}{2}\varphi \left(\omega \right)\right)& \left(38\right)\end{array}$

[00288] In these equations, ω is the per-second feedrate divided by the programmed radius and, as mentioned, φ is expected to be constant for a given fixed value of |ω|. The analysis will proceed first for the counter-clockwise (positive-frequency) case. The results for the clockwise case can be derived in an analogous manner. The square of the radius for the counter-clockwise case is the sum of the squares of the axis locations. In the following equation, the term cot is replaced by the circle angle θ. $\begin{array}{cc}{r}_{\mathrm{CCW}}^2\left(\theta \right)=R^2·\left({\cos }^2\left(\theta +\frac{1}{2}\varphi \left(\omega \right)\right)+{\sin }^2\left(\theta -\frac{1}{2}\varphi \left(\omega \right)\right)\right)& \left(39\right)\end{array}$
The previous equation can be simplified by employing some trigonometric identities to the following form.
r_CCW²(θ)=R²+R²·2 sin(θ)cos(θ)sin(φ)  (40)

[00291] Now returning to the clockwise case, the equation (38) can be reformulated by changing the sign of the arguments of the sine and cosine functions. $\begin{array}{cc}{X}_{\mathrm{CLW}}\left(t\right)=R·\cos \left(\omega t-\frac{1}{2}\varphi \left(\omega \right)\right),Y_{\mathrm{CLW}}\left(t\right)=-R·\sin \left(\omega t+\frac{1}{2}\varphi \left(\omega \right)\right)& \left(41\right)\end{array}$
These equations can be used to obtain the following result for the instantaneous radius of the clockwise circle.
r_CLW²(θ)=R²−R²·2 sin(θ)cos(θ)sin(θ)  (42)
The impact of a servo mismatch error on a polar error plot is shown in FIG. 11.

[00295] Comparison of equations (40) and (42) to equation (14) indicates that the formulation for servo mismatch error can be the same as the formulation for squareness error. This means that the two errors can be indistinguishable unless data for both clockwise and counterclockwise measurements are present in the matrix formulation. Thus, in an exemplary ballbar test, each measurement contains both a clockwise and counter-clockwise run, and for the sake of equal weighting, both runs cover the same data-range. This can be accomplished by “stacking” the matrix with rows developed from the counter-clockwise measurement on top of rows developed from the clockwise measurement. In the following equation, the superscripts clw and ccw on the i'th measurement indicator t_i are used to identify whether a measurement is from a clockwise or counter-clockwise test. $\begin{array}{cc}\left\{\begin{array}{c}{r}^2\left(t_1^{\mathrm{ccw}}\right)\\ r^2\left(t_2^{\mathrm{ccw}}\right)\\ ⋮\\ r^2\left(t_n^{\mathrm{ccw}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{clw}}\right)\\ r^2\left(t_2^{\mathrm{clw}}\right)\\ ⋮\\ r^2\left(t_n^{\mathrm{clw}}\right)\end{array}\right\}=\left[\begin{array}{cc}2\sin \theta \left(t_1^{\mathrm{ccw}}\right)·\cos \theta \left(t_1^{\mathrm{ccw}}\right)& 1\\ 2\sin \theta \left(t_2^{\mathrm{ccw}}\right)·\cos \theta \left(t_2^{\mathrm{ccw}}\right)& 1\\ ⋮& ⋮\\ 2\sin \theta \left(t_n^{\mathrm{ccw}}\right)·\cos \theta \left(t_n^{\mathrm{ccw}}\right)& 1\\ \dots & \dots \\ -2\sin \theta \left(t_1^{\mathrm{clw}}\right)·\cos \theta \left(t_1^{\mathrm{clw}}\right)& 1\\ -2\sin \theta \left(t_2^{\mathrm{clw}}\right)·\cos \theta \left(t_2^{\mathrm{clw}}\right)& 1\\ ⋮& ⋮\\ -2\sin \theta \left(t_n^{\mathrm{clw}}\right)·\cos \theta \left(t_n^{\mathrm{clw}}\right)& 1\end{array}\right]\left\{\begin{array}{c}{R}^2\sin \left(\varphi \right)\\ R^2\end{array}\right\}& \left(43\right)\end{array}$

[00296] Lost motion describes an error that commonly exists in an individual axis mechanism that is caused by looseness in the mechanical components and linkages. When an axis with a lost motion error reverses its direction of motion (sign of velocity), the position of the end-point lags the position of the motor. The A2100 control identifies and compensates for two classes of lost motion, backlash and windup.

[00297] Backlash is a type of lost motion that is present regardless of whether the axis is in motion or at rest. That is, it persists after the axis has come to a stop. Windup is a type of lost motion that exists when the axis is in motion where the axis appears to creep to the correct position after the motor has stopped. An explanation of the differences between windup and backlash and how they can be compensated is given in U.S. Pat. No. 6,060,854, entitled “Method and Apparatus for Compensating for Windup in a Machine,” issued to Stephen J. Yutkowitz, the entire disclosure of which is incorporated herein by reference.

[00298] Both types of lost motion exhibit nearly indistinguishable signatures on a ballbar plot and cannot conventionally be separately identified based on a single ballbar circle-test measurement. An additional test that uses the ballbar to allow the two types of lost-motion to be distinguished from each other has been developed and can be called the backlash-test. In an exemplary embodiment, the backlash test is a separate type of test from the circle-test whose measured data has been the topic of discussion up to this point. An explanation of how to estimate the total lost motion error from the circle-test data will be presented first, then an explanation for the backlash test will be given.

[00299] The parametric description of total-lost motion makes the assumption that the lost motion parameter does not vary with axis position or velocity (as long as there are no full stops). Axis lost motion of the backlash type that does vary with position can be handled (e.g., in the A2100 by the bidirectional compensation capability whose data-tables are developed from laser measurements).

[00300] An effect of lost-motion on circular motion is shown in FIG. 12. The specific example shown in FIG. 12 is for a case with a very high amount of backlash on the Y-axis and no backlash on the X-axis. Unlike many of the previous plots, FIG. 12 is not a polar-error plot; it is a plot of the actual circular contour.

[00301] The plot of FIG. 12 shows the contour for clockwise motion. The case for counter-clockwise motion should look like the current plot reflected about the Y-axis, as can be imagined. The center-corrected curve is simply the actual curve shifted downward such that its geometric center corresponds to the center of the ideal circle. The actual y-coordinate of the curve center depends on the direction from which the ballbar center fixture was approached when the circle center was defined during the fixturing phase of the ballbar test.

[00302] Examination of FIG. 12 indicates that the total lost motion error E_TLM causes a flat area following the axis reversal where the motion of the Y-motor does not result in any Y-axis motion while the X-axis moves with no error. This flat area can cause the best-fit circle to have a radius that is somewhat less than the radius of the programmed circle. It will be shown that for the type of data generated by an exemplary embodiment of the present invention, the amount of “shrinkage” caused by lost motion should be insignificantly small compared to the total lost motion error and need not be considered when determining the absolute scale errors. The command circle angle that defines the distance from the reversal where the (Y-axis) motion begins can be well approximated by the following equation when the ratio of E_TLM to R is fairly small. $\cos \left(\alpha \right)=\frac{R-\frac{1}{2}{E}_{\mathrm{TLM\_Y}}}{R+\frac{1}{2}{E}_{\mathrm{TLM\_Y}}}$

[00303] For actual measurements on conventional machine tools, a minimum circle radius can be about 100 mm and the largest expected lost-motion error should be well below about 0.2 mm. Thus, the maximum conceivable value for the angle α should be less than about 2 degrees. This confirms that the amount of shrinkage of the best-fit-circle radius due to lost motion should be insignificant relative to the lost motion error itself and need not be considered.

[00304] The observation that the flat area, S_TLM, covers a small percentage of the total circle also suggests that the exact shape of the curve just after the reversal (henceforth referred to as the transient portion) need not be considered by the technique for estimating the total lost motion from a ballbar measurement (since, e.g., a built-in requirement could be that the data arc is relatively large). Nevertheless, a good approximate representation of the transient portion can be useful when attempting to predict how backlash or windup compensation might improve circularity. The predicted effect of compensation can be obtained by subtracting a curve developed from the model parameter from the measurement curve, wherein it is desired to closely match the model to the actual transient. Thus, for a clockwise circle with a y-axis lost motion error, the following set of equations applies.

[heading-00305]  x(θ)=R cos(θ)  (44)

[00306] $\begin{array}{cc}{y}_{\mathrm{clw}}\left(\theta \right)=(\begin{array}{ccc}R\sin \left(\theta \right)+\frac{1}{2}{E}_{\mathrm{TLM\_Y}},& -\frac{1}{2}\pi \le \theta <\frac{1}{2}\pi -\alpha & \left(a\right)\\ R-\frac{1}{2}{E}_{\mathrm{TLM\_Y}},& \frac{1}{2}\pi -\alpha \le \theta <\frac{1}{2}\pi & \left(b\right)\\ R\sin \left(\theta \right)-\frac{1}{2}{E}_{\mathrm{TLM\_Y}},& \frac{1}{2}\pi \le \theta <\frac{3}{2}\pi -\alpha & \left(c\right)\\ -R+\frac{1}{2}{E}_{\mathrm{TLM\_Y}},& \frac{3}{2}\pi -\alpha \le \theta <\frac{3}{2}\pi & \left(d\right)\end{array}& \left(45\right)\end{array}$

[00307] The equation for the radius squared as a function of circle angle can be obtained by summing the square of the x and y components. The estimation of the total lost motion error may be based on regions (a) and (c) in equation (45) because regions (b) and (d) are very small and should not significantly bias the result if their effects are ignored. As a practical matter, the angle ox is not known until E_TLM is known, so the transient effects cannot be considered until a first approximation in an iterative method is obtained. The approximate equation for the radius squared can thus be obtained by applying the equation for region (a) to both regions (a) and (b), and by applying the equation for region (c) to both regions (c) and (d). The following equation can be developed by assuming that the lost-motion squared terms are insignificantly small. $\begin{array}{cc}{r}_{\mathrm{clw}}^2\left(\theta \right)=(\begin{array}{cc}{R}^2+R\sin \left(\theta \right)E_{\mathrm{TLM\_Y}},& -\frac{1}{2}\pi \le \theta <\frac{1}{2}\pi \\ R^2-R\sin \left(\theta \right)E_{\mathrm{TLM\_Y}},& \frac{1}{2}\pi \le \theta <\frac{3}{2}\pi \end{array}& \left(46\right)\end{array}$
The results for the counter-clockwise circle may be analogously obtained and can be shown below. $\begin{array}{cc}{r}_{\mathrm{ccw}}^2\left(\theta \right)=(\begin{array}{cc}{R}^2-R\sin \left(\theta \right)E_{\mathrm{TLM\_Y}},& -\frac{1}{2}\pi \le \theta <\frac{1}{2}\pi \\ R^2+R\sin \left(\theta \right)E_{\mathrm{TLM\_Y}},& \frac{1}{2}\pi \le \theta <\frac{3}{2}\pi \end{array}& \left(47\right)\end{array}$
The equations for estimating the lost motion errors of the X-axis may also be obtained using similar techniques or by replacing θ with (θ+½π) in the previous equations. $\begin{array}{cc}{r}_{\mathrm{clw}}^2\left(\theta \right)=(\begin{array}{cc}{R}^2-R\cos \left(\theta \right)E_{\mathrm{TLM\_X}},& 0\le \theta <\pi \\ R^2+R\cos \left(\theta \right)E_{\mathrm{TLM\_X}},& \pi \le \theta <2\pi \end{array}& \left(48\right)\\ r_{\mathrm{ccw}}^2\left(\theta \right)=(\begin{array}{cc}{R}^2+R\cos \left(\theta \right)E_{\mathrm{TLM\_X}},& 0\le \theta <\pi \\ R^2-R\cos \left(\theta \right)E_{\mathrm{TLM\_X}},& \pi \le \theta <2\pi \end{array}& \left(49\right)\end{array}$
Equations that are analogous to (45) for the counter-clockwise y-axis case and for both x-axis cases can also be developed, but are not shown here. The matrix equation for total lost motion presented below includes factors for lost motion on both axes. In the equation, the notation QI, QII, QIII, QIV are used to represent the circle quadrants (where the measurement data is segregated into quadrants). $\begin{array}{cc}\begin{array}{c}\left\{\begin{array}{c}{r}^2\left(t_1^{\mathrm{ccwQI}}\right)\\ ⋮\\ r^2\left(t_{n_{\mathrm{ccwl}}}^{\mathrm{ccwQI}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{ccwQII}}\right)\\ ⋮\\ r^2\left(t_{n_{\mathrm{ccwII}}}^{\mathrm{ccwQII}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{ccwQIII}}\right)\\ ⋮\\ r^2\left(t_{n_{\mathrm{ccwIII}}}^{\mathrm{ccwQIII}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{ccwQIV}}\right)\\ ⋮\\ r^2\left(t_{n_{\mathrm{ccwIV}}}^{\mathrm{ccwQIV}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{clwQI}}\right)\\ ⋮\\ r^2\left(t_{n_{\mathrm{clwl}}}^{\mathrm{clwQI}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{clwQII}}\right)\\ ⋮\\ r^2\left(t_{n_{\mathrm{clwII}}}^{\mathrm{clwQII}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{clwQIII}}\right)\\ ⋮\\ r^2\left(t_{n_{\mathrm{clwIII}}}^{\mathrm{clwQIII}}\right)\\ \dots \\ r^2\left(t_1^{\mathrm{clwQIV}}\right)\\ ⋮\\ r^2(t_{n_{\mathrm{clwIV}}}^{\mathrm{clwQIV}}\end{array}\right\}=[\begin{array}{ccc}+\cos \left(\theta \left(t_1^{\mathrm{ccwQI}}\right)\right)& -\sin \left(\theta \left(t_1^{\mathrm{ccwQI}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ +\cos (\theta \left(t_{n_{\mathrm{ccwI}}}^{\mathrm{ccwQI}}\right)& -\sin \left(\theta \left(t_{n_{\mathrm{ccwl}}}^{\mathrm{ccwQI}}\right)\right)& 1\\ \dots & \dots & \dots \\ +\cos \left(\theta \left(t_1^{\mathrm{ccwQII}}\right)\right)& +\sin \left(\theta \left(t_1^{\mathrm{ccwQII}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ +\cos \left(\theta \left(t_{n_{\mathrm{ccwII}}}^{\mathrm{ccwQII}}\right)\right)& +\sin \left(\theta \left(t_{n_{\mathrm{ccwII}}}^{\mathrm{ccwQII}}\right)\right)& 1\\ \dots & \dots & \dots \\ -\cos \left(\theta \left(t_1^{\mathrm{ccwQIII}}\right)\right)& +\sin \left(\theta \left(t_1^{\mathrm{ccwQIII}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ -\cos \left(\theta \left(t_{n_{\mathrm{ccwIII}}}^{\mathrm{ccwQIII}}\right)\right)& +\sin \left(\theta \left(t_{n_{\mathrm{ccwIII}}}^{\mathrm{ccwQIII}}\right)\right)& 1\\ \dots & \dots & \dots \\ -\cos \left(\theta \left(t_1^{\mathrm{ccwQIV}}\right)\right)& -\sin \left(\theta \left(t_1^{\mathrm{ccwQIV}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ -\cos \left(\theta \left(t_{n_{\mathrm{ccwIV}}}^{\mathrm{ccwQIV}}\right)\right)& -\sin \left(\theta \left(t_{n_{\mathrm{ccwIV}}}^{\mathrm{ccwQIV}}\right)\right)& 1\\ \dots & \dots & \dots \\ -\cos \left(\theta \left(t_1^{\mathrm{clwQI}}\right)\right)& +\sin \left(\theta \left(t_1^{\mathrm{clwQI}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ -\cos \left(\theta \left(t_{n_{\mathrm{clwI}}}^{\mathrm{clwQI}}\right)\right)& +\sin \left(\theta \left(t_{n_{\mathrm{clwI}}}^{\mathrm{clwQI}}\right)\right)& 1\\ \dots & \dots & \dots \\ -\cos \left(\theta \left(t_1^{\mathrm{clwQII}}\right)\right)& -\sin \left(\theta \left(t_1^{\mathrm{clwQII}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ -\cos \left(\theta \left(t_{n_{\mathrm{clwII}}}^{\mathrm{clwQII}}\right)\right)& -\sin \left(\theta \left(t_{n_{\mathrm{clwII}}}^{\mathrm{clwQII}}\right)\right)& 1\\ \dots & \dots & \dots \\ +\cos \left(\theta \left(t_1^{\mathrm{clwQIII}}\right)\right)& -\sin \left(\theta \left(t_1^{\mathrm{clwQIII}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ +\cos \left(\theta \left(t_{n_{\mathrm{clwIII}}}^{\mathrm{clwQIII}}\right)\right)& -\sin \left(\theta \left(t_{n_{\mathrm{clwIII}}}^{\mathrm{clwQIII}}\right)\right)& 1\\ \dots & \dots & \dots \\ +\cos \left(\theta \left(t_1^{\mathrm{clwQIV}}\right)\right)& +\sin \left(\theta \left(t_1^{\mathrm{clwQIV}}\right)\right)& 1\\ ⋮& ⋮& ⋮\\ +\cos \left(\theta \left(t_{n_{\mathrm{clwIV}}}^{\mathrm{clwQIV}}\right)\right)& +\sin \left(\theta \left(t_{n_{\mathrm{clwIV}}}^{\mathrm{clwQIV}}\right)\right)& 1\end{array}]\\ \left\{\begin{array}{c}R·E_{\mathrm{TLM\_X}}\\ R·E_{\mathrm{TLM\_Y}}\\ R^2\end{array}\right\}\end{array}& \left(50\right)\end{array}$

[00311] An example for a polar-error plot for clockwise and counterclockwise moves on a machine with lost motion on both X and Y axes is shown in FIG. 13. The post-reversal transient portions for this synthesized example are very similar to the post-reversal portions of measurements on actual machines. In the example, the best-fit circle and the ideal circle are so-similar that they are substantially indistinguishable, as was expected based on the argument of insignificantly small circle shrinkage.

[00312] Up to this point, the discussion has been limited to the identification of the total lost motion from a continuous motion circle-test measurement. Experience with these measurements indicates that there is no substantially detectable difference between the effect of the backlash and windup components of total lost motion. In order to separately identify each of these errors, an additional backlash test can be employed.

[00313] An exemplary backlash test involves a sequence where the ballbar is used to detect the lost motion that occurs when the axis moves to position stops and dwells (as previously described). The windup error for an axis can then be computed by subtracting the measured backlash from the estimate for total lost motion. If a backlash test is not performed, backlash can be assumed zero and the measured lost motion from the circle-test attributed to a windup error. For example, this could be based on the assumption that the machine is properly compensated for backlash using bidirectional compensation.
E_windup—X=E_TLM—X−E_{backslash—X}

[00315] Friction on a machine tool axis can be the source of a transient error following axis reversals. Currently, for example, the A2100 employs friction compensation algorithms as part of its integrated feedforward capability. The A2100 friction compensation assumes that the friction forces and torques acting on the axis are reasonably represented by a Coulomb+viscous friction model.

[00316] A torque discontinuity due to Coulomb friction occurs after an axis reversal and results in a transient error where the axis lags the command for some period of time until the integrator in the velocity control loop builds up to contain a torque that counters the friction torque. In general, the size of the transient error increases with the friction level and decreases with the velocity loop's proportional and integral gains. A description of how to iteratively tune the friction compensation based on a ballbar measurement is described in the A2100 Installation Service & Support Manual (referred to hereinafter as the “ISS manual”), which is hereby incorporated by reference.

[00317] An effect of Coulomb friction errors on a ballbar plot is an increase in the circle radius following the reversal for some angle until the servo detects the error and the radius returns to the desired value. A signature of a friction reversal error on a polar-error plot is shown in FIG. 14.

[00318] In contrast to the effect of lost motion, friction errors can cause the best-fit radius to be slightly larger than the ideal radius. The amplification of the best fit radius over the ideal radius relative to the magnitude of the friction reversal error depends mostly on the angular duration of the friction error, which itself is related (primarily) to the velocity loop's integrator time constant and the circle frequency. To assess the degree to which friction errors may bias the best fit radius, a worst case scenario could be considered to get an approximate value for the largest expected angle of the reversal spike.

[00319] For example, the anticipated highest circle frequency should occur at a minimum ballbar radius of about 100 mm and a maximum anticipated feedrate of about 15,000 mm/min—making the frequency about 2.5 rad/sec. Experience shows that the integrator time constant is typically less than about 0.020 sec. The duration of the reversal error is considered to be significant over two integrator time constants. The angular distance traveled in two time constants can thus be about 0.040 sec*2.5 rad/sec=0.10 rad or 5.7 degrees. Even for the worst case scenario, the friction error should not significantly amplify the best-fit radius relative to the amplitude of the friction error itself.

[00320] In another embodiment of the present invention, algorithms for estimating the parameters associated with the friction errors can also be included. The results for friction compensation parameters obtained via feedforward auto-tuning can often be improved upon by performing some manual adjustments based on ballbar measurements. The manual adjustments can be typically performed based, for example, on the procedure outlined in the ISS manual. An explanation for why the results of the feedforward auto tuning may be improved by further manual tuning is that the transient friction forces/torques that occur during an axis reversal are more complex than the assumed step discontinuity at zero velocity. The exact nature of these transient friction torques may be very complex and may be a function of a number of state variables including the positions, velocities, and accelerations of a number of bodies in a multi-body system.

[00321] This complex behavior may be partially approximated by adjusting the amplitude and time constant of friction compensation (feedforward) to differ from the ideal values obtained from feedforward auto-tuning's torque measurements and measurements of the response of the velocity loop compensator. This observation has led to the addition of a Coulomb-Width-Adjust parameter to the set of feedforward parameters to enable adjustments of the friction compensation time constant without affecting the other types of feedforward that operate directly on the integrator time constant (acceleration and viscous friction feedforward). Thus, the ballbar measurement may be used to determine an optimal value for the Coulomb-Width-Adjust parameter, given that the feedforward-auto-tuning process has already been performed and has resulted in a value for the estimate of the integrator time constant.

[00322] In one embodiment, the ability to estimate friction parameters (e.g., amplitude and time constant) directly from a measurement of a machine without any active friction compensation can be included (e.g., if a precise estimate of the circle angle for every measurement can be obtained). For example, such an algorithm could first remove known error types from the data using the techniques described below. Next, the radial error in the immediate region of the axis reversal could be converted to deviation of the reversing axis and the corresponding angles could be converted to the time since reversal. Then, an iterative process can begin by selecting a candidate value for the time constant τ (inverse of integrator gain) and performing a simulation based on this time constant. A model used to develop the simulation is shown in FIG. 15.

[00323] The transfer function for the position error in response to a friction force signal can be derived from the block diagram in FIG. 15 and can be shown in equation (51) below. The DC amplitude of this transfer function is arbitrary. The transfer function formulation may be used to develop a numerical solution to the associated linear differential equation. $\begin{array}{cc}\frac{e\left(s\right)}{f\left(s\right)}=\frac{s}{s^3+K_P{s}^2+K_P\left(\frac{1}{\tau }+K_V\right)s+\frac{K_V{K}_P}{\tau }}& \left(51\right)\end{array}$

[00324] The known axis Kv, an assumption that Je≈J (where Je is an inertia estimate from feedforward auto tuning and J is actual total axis inertia), and an estimate for the axis Kp (velocity loop proportional gain) can be used to develop an estimate for the error-transient versus time function of the data sequence for the axis error transient following a reversal. The scaling of such a time sequence might be arbitrary since the friction torque is initially unknown. The value for Kp need not be very precise as the pulse shape is quite insensitive to this parameter (assuming the velocity servo is reasonably well tuned with a phase margin of more than 40°). The simulation parameter for Kp can be the same as the iff_kp_estimate.

[00325] The amplitude of the estimated transient can be determined using least squares to obtain an optimal scaling between the arbitrary-amplitude simulated transient and the y-axis deviation sequence derived from the measured radius. The sum of the residual errors squared can be computed and retained. Residual errors can be computed for a number of selections for the time constant as part of an iterative optimization process. The time constant and associated amplitude that results in the minimum sum-of-residual-error-squared can be chosen as the estimated friction parameters. The process can be constructed to simultaneously include available measurements of reversals of the axis under consideration.

[00326] A potential problem with the above-described procedure for estimating the friction reversal error can be that it might not be valid for cases where, for example, the friction compensation is actively applied during the ballbar measurement. The ballbar measurement is expected to have the most value in determining an adjustment to nominal values for friction compensation parameters obtained via feedforward auto-tuning. Thus, in an exemplary embodiment, a ballbar measurement will be performed under circumstances where friction compensation is actively applied during the measurement.

[00327] If the friction compensation that is applied is not optimal, then some error at the reversal might occur, and such an error might be more complex than the type produced when friction compensation is turned off. One approach to take in this case could involve attempting to remove the effects of friction compensation. However, the effects of friction compensation are not easily removed from the data because the DC scaling of the transfer function may no longer be arbitrary. This means that a 1-dimensional optimization problem could become a 2-dimensional optimization problem.

[00328] One solution could include automatically applying the iterative manual tuning procedure outlined in the ISS manual. Such an approach could iteratively execute the following sequence: 1) perform a ballbar measurement, 2) isolate the reversal errors, 3) use heuristic rules to determine whether to increase or decrease the Coulomb-Width-Adjust and whether to increase or decrease the Coulomb-Friction-Level, 4) perform a model-based curve fitting to determine candidate values of these parameters for the next iteration, 5) configure the friction feedforward with these new values, and 6) return to step 1. The iterative sequence can be exited once the errors are below a defined threshold or the improvements obtained between iterations become insignificant.

[00329] Estimation of the friction-related parameters is a capability that could be useful for the purpose of data analysis in ranking the impact of different errors on the overall circularity. It could be used to auto-tune friction compensation for systems where feedforward-auto-tuning is unavailable. For systems where feedforward-auto-tuning is available, it could, for example, be used to propose modifications to the friction compensation parameters obtained via a previously run feedforward-auto-tuning test.

[00330] The preceding discussion developed the equations for the individual error types by assuming that each of these error types was the one existing in a measurement. In real applications, the error types occur simultaneously and are present in some proportion. In order to get an optimal estimate for each error types, the effect of all known errors can be considered simultaneously. This can be accomplished by formulating a single matrix equation that includes the effects of all known errors.

[00331] The matrix of basis functions can contain, for example, ten columns, with each column containing a basis function for one of the following exemplary parameters:

• • [00332] 1. X axis center offset.
  • [00333] 2. Y axis center offset
  • [00334] 3. XY Squareness Error
  • [00335] 4. XY Scale-mismatch error
  • [00336] 5. XY Servo mismatch
  • [00337] 6. X Straightness Error
  • [00338] 7. Y Straightness Error
  • [00339] 8. X Total Lost Motion
  • [00340] 9. Y Total Lost Motion
  • [00341] 10. Best Fit Radius

[00342] The matrix equation can be formulated by simply combining each of the matrix equations previously discussed. Since the servo-mismatch and lost-motion estimate includes separate treatment of data based on circle direction, the data can be segregated into clockwise and counter clockwise segments. Additionally, the total lost motion estimate involves separate treatment of data based on quadrant, with the data being further partitioned based on circle quadrant (estimate angle). Partitioning of this type has already been described during the discussion of lost motion estimation as seen in equation (50).

[00343] A combined matrix equation for simultaneous estimation of all parameters is shown below. To enable the equation to fit on the page, the data partitioning is not explicitly shown (the±symbols are used to designate matrix columns whose basis function depends on direction and/or quadrant). $\begin{array}{cc}\left\{r^2\left(t_i\right)\right\}={\left[\begin{array}{c}{\left\{-2\cos \left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{-2\sin \left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{\cos \left(\theta \left(t_i\right)\right)\sin \left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{2\left({\cos }^2\left(\theta \left(t_i\right)\right)-{\sin }^2\left(\theta \left(t_i\right)\right)\right)\right\}}^T\\ {\left\{\pm 2\cos \left(\theta \left(t_i\right)\right)\sin \left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{2\sin \left(\theta \left(t_i\right)\right){\cos }^2\left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{2\cos \left(\theta \left(t_i\right)\right){\sin }^2\left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{\pm \cos \left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{\pm \sin \left(\theta \left(t_i\right)\right)\right\}}^T\\ {\left\{11\dots 1\right\}}^T\end{array}\right]}^T·\left\{\begin{array}{c}R·E_{\mathrm{CTR\_X}}\\ R·E_{\mathrm{CTR\_Y}}\\ R^2·E_{\mathrm{SQR\_XY}}\\ R^2·E_{\mathrm{SCL\_XY}}\\ R^2·\sin \left({\varphi }_{\mathrm{svo}}\right)\\ R^3·E_{\mathrm{STR\_X}\mathrm{\_Y}}\\ R^3·E_{\mathrm{STR\_Y}\mathrm{\_X}}\\ R·E_{\mathrm{TLM\_X}}\\ R·E_{\mathrm{TLM\_Y}}\\ R^2\end{array}\right\}& \left(52\right)\end{array}$
The following symbolic matrix equation can be used to describe the previous specific equation.
{r²(t)_n×1)=[A]_n×10)·{P}_(10×1)  (53)

[00346] The estimation of the above ten parameters can be accomplished by applying the least squares solution of equation (3) to equation (52) above. In the following equations the ^ symbol is used to designate a parameter as being an estimated value.
{{circumflex over (P)}}=[[A]^T[A]]⁻¹[A]^T{r²}  (54)

[00348] The inversion of the matrix A^TA may be accomplished using eigenvalues and eigenvectors obtained by Jacobi's technique. Using the Jacobi technique has the advantage that it provides the eigenvalues of [A^TA], which are used in the computation of the condition number as described below.

[00349] Once the matrix is inverted, the scaling of the parameter by powers of the best fit radius can be removed by multiplying each parameter by the appropriate power of the inverse of the best-fit radius obtained by square-rooting the final element of the parameter vector. $\begin{array}{cc}\begin{array}{ccc}\hat{R}=\sqrt{{\hat{P}}_{10}},& E_{\mathrm{CTR\_X}}=\frac{{\hat{P}}_1}{\hat{R}},& E_{\mathrm{CTR\_Y}}=\frac{{\hat{P}}_2}{\hat{R}},\\ E_{\mathrm{SQR\_XY}}=\frac{{\hat{P}}_3}{{\hat{R}}^2},& E_{\mathrm{SCL\_XY}}=\frac{{\hat{P}}_4}{{\hat{R}}^2}& \\ {\varphi }_{\mathrm{svo}}={\sin }^{-1}\left(\frac{{\hat{P}}_5}{{\hat{R}}^2}\right),& E_{\mathrm{STR\_XY}}=\frac{{\hat{P}}_6}{{\hat{R}}^3},& E_{\mathrm{STR\_YX}}=\frac{{\hat{P}}_7}{{\hat{R}}^3},\\ E_{\mathrm{TLM\_X}}=\frac{{\hat{P}}_8}{\hat{R}},& E_{\mathrm{TLM\_Y}}=\frac{{\hat{P}}_9}{\hat{R}}& \end{array}& \left(55\right)\end{array}$

[00350] Examination of the matrix [A] in equation (52) reveals that there are opportunities in the formulation for some columns to be redundant with other columns. If a column is not unique, then the parameter associated with that column cannot be uniquely identified and the least squares solution to the matrix equation fails to exist. The columns can be made unique by ensuring that the data arc is sufficiently large and that both clockwise and counter-clockwise data is present.

[00351] An indicator of the overall uniqueness of the columns in the matrix formulation is the matrix condition number (the ratio of the maximum singular value to the minimum singular value). The singular values may be obtained by computing the square root of the eigenvalues of the [A]^T[A]] matrix (from equation (3)). The condition number for a given set of test conditions can be normalized based on the (best case) condition number for a full 360 degree data arc in both directions.

[00352] A statistic called the “excitation” is defined as the inverse of the normalized condition number. A test for the sufficiency of excitation provided by a collection of test setup parameters might be to compare excitation statistic to a threshold. When an inappropriate set of test conditions is proposed by the user, the excitation falls below a predefined threshold and the test conditions can be rejected. $\begin{array}{cc}\left\{\lambda \right\}=\mathrm{eig}\left[{\left(\left[A\right]\right)}^T\left[A\right]\right],C_{\mathrm{NUM}}=\sqrt{\frac{{\lambda }_{\max }}{{\lambda }_{\min }}},\mathrm{excitation}=\frac{C_{\mathrm{NUM}}}{C_{\mathrm{NUM}}\left({360}^{°}\mathrm{clw}\&\mathrm{ccw}\right)}& \left(56\right)\end{array}$

[00353] A plot of the excitation as a function of data arc length for a data arc starting at 0.0 is shown in FIG. 16. Similar plots for other (non-zero) data start angles are very similar to the one shown in FIG. 16. The plot indicates that the best excitation is obtained for a data arc whose length is greater than about 280 degrees. The minimum acceptable excitation allowed for by the test setup validity test can be chosen to be about 0.2 or approximately 220 degrees of data arc.

[00354] Some data analysis tasks provided as described by example herein involve computing a synthesized value for the ballbar radius for each angle at which a ballbar radius measurement has been taken. The sequence of synthesized radii can be referred to as r_synth(t). The synthesized radius sequence can be used by the following analyses: 1) modeling coherence, 2) ranking the relative impact of each error type on the overall circularity error, 3) predicting how the circularity error would change if a selected combination of axis compensations were employed, and/or 4) estimating the measurement variance used to determine the significance of each estimated parameter.

[00355] The synthesized radius vector can be obtained by evaluating equation (52), where the parameter vector is populated with the estimates, and performing the square-root of the result. The synthetic data can then be modified at the axis reversals to account for the lost motion transients of the type described by equation (45)b and (45)c. The data following the reversal can be modified by computing the radius for a constant axis deviation over the region defined by the angle a. The following equations describe how the synthetic data can be generated. $\begin{array}{cc}\left\{r_{\mathrm{synth}}\left(\theta \right)\right\}=\sqrt{\left[A\right]·\left\{\stackrel{̑}{P}\right\}}& \left(57\right)\\ {\hat{\alpha }}_X={\cos }^{-1}\left(\frac{\hat{R}-\frac{1}{2}{\hat{E}}_{\mathrm{TLM\_X}}}{\hat{R}+\frac{1}{2}{\hat{E}}_{\mathrm{TLM\_X}}}\right),{\hat{\alpha }}_Y={\cos }^{-1}\left(\frac{\hat{R}-\frac{1}{2}{\hat{E}}_{\mathrm{TLM\_Y}}}{\hat{R}+\frac{1}{2}{\hat{E}}_{\mathrm{TLM\_Y}}}\right)& \left(58\right)\\ r_{\mathrm{synth}}\left(\theta \right)=\{\begin{array}{c}\sqrt{{\hat{R}}^2{\sin }^2\left(\theta \right)+{\left(\hat{R}-\frac{1}{2}{\hat{E}}_{{\mathrm{TLM}}_X}\right)}^2}\\ \mathrm{\text{'}\text{'}}\end{array}\begin{array}{c}\forall \mathrm{clw}\text{:}-{\hat{\alpha }}_X\le \theta <0\mathrm{or}\pi -{\hat{\alpha }}_X\le \theta <\pi \\ \forall \mathrm{ccw}\text{:}0\le \theta <{\hat{\alpha }}_X\mathrm{or}\pi \le \theta <\pi +{\hat{\alpha }}_X\end{array}& \left(59\right)\\ r_{\mathrm{synth}}\left(\theta \right)=\{\begin{array}{c}\sqrt{{\hat{R}}^2{\cos }^2\left(\theta \right)+{\left(\hat{R}-\frac{1}{2}{\hat{E}}_{{\mathrm{TLM}}_Y}\right)}^2}\\ \mathrm{\text{'}\text{'}}\end{array}\begin{array}{c}\forall \mathrm{clw}\text{:}\frac{\pi }{2}-{\hat{\alpha }}_Y\le \theta <\frac{\pi }{2}\mathrm{or}\frac{3\pi }{2}-{\hat{\alpha }}_Y\le \theta <\frac{3\pi }{2}\\ \forall \mathrm{ccw}\text{:}\frac{\pi }{2}\le \theta <\frac{\pi }{2}+{\hat{\alpha }}_Y\mathrm{or}\frac{3\pi }{2}\le \theta <\frac{3\pi }{2}+{\hat{\alpha }}_Y\end{array}& \left(60\right)\end{array}$
In the previous equations, the value for mean radius was for the case where all errors, except for the lost motion under consideration, are present in the data set.

[00357] The sequence of synthesized radii can be used to demonstrate what is expected to occur in the absence of certain error types or combinations of error types. As described earlier, for example, analyze-data page 70 can allow a user to select a combination of error types to be removed from the measurement (e.g., by checking or unchecking checkboxes to designate the desired combination). For example, when a checkbox is checked, the associated error type can be included in the predicted-radius polar-error plot. When it is unchecked, the effect of the corresponding error can be removed. In this way, the user can get a realistic idea of how application of compensation can improve performance by removing the known error effects from measured data that may contain error types for which no compensation is available.

[00358] The first step in constructing the predicted radius sequence can be to compute the residual error by subtracting elements of the synthesized radius sequence from corresponding elements in the measured radius sequence. The residual error can then be added to a second synthesized radius sequence r_{synth—partial} that can be created by zeroing all elements of the {P} vector except for those whose error types remain (e.g., those checked on diagnostics window 102). If an axis lost motion error is included, then modification of the synthesized sequence can be performed by employing equations (59) and (60) for the axis or axes whose lost motion error should be included in the prediction sequence.

[00359] In an exemplary embodiment, checkboxes for the center offset errors are not provided because the center offsets are not machine errors and the predicted radius sequence should have the effect of center offsets removed. This means that, for the computation of r_{synth—partial}, the first two elements of {P} can be set to zero. A procedure for computing the predicted radius r_predicted is now explicitly described.

[00360] First, the residual error between the synthesized radius and the measurement can be computed.

[heading-00361]  e_r=r−r_synth  (61)

[heading-00362] Next, the synthesized radius including only the error types whose checkboxes are checked can be computed.

[00363] In the following equation, each element of the vector {c} can be assigned a value of 0.0 when the checkbox associated with the parameter is checked, and a value of 1.0 when the checkbox associated with the parameter is not checked.
$\begin{array}{cc}\left\{r_{\mathrm{synth\_part}}\left(\theta ,\left\{c\right\}\right)\right\}=\sqrt{\left[A\left(\theta \right)\right]·{\left\{❘0❘0❘c_3{\hat{P}}_3❘c_4{\hat{P}}_4❘c_5{\hat{P}}_5❘c_6{\hat{P}}_6❘c_7{\hat{P}}_7❘c_8{\hat{P}}_8❘c_9{\hat{P}}_9❘{\hat{P}}_{10}❘\right\}}^T}& \left(62\right)\end{array}$
If c₇=1, use equation (59) to apply the x-axis reversal transient modification to r_synth—part. Likewise, if c₈=1, use equation (60) to apply the y-axis reversal transient modification to r_synth—part. Finally, the predicted radius for the selected combination of error types can be computed.
r_predicted(θ,{c})=r_synth—part(θ,{c})+e_r(θ)  (63)

[00367] The predicted radius sequence from the previous equation has no center offset error. It can be used directly in the computation of the corrected circularity displayed, for example, in the diagnostics window 102.

[00368] A capability that can be provided by diagnostics window 102 can include displaying and ranking error types in order of their relative contribution to the circularity error. Each error type's contribution can be determined by evaluating equation (62) for r_synth—part(θ,{c}) where the elements in the vector {c} are set to zero, except the one associated with the particular error type under consideration (set to 1.0). The circularity error for the resulting synthesized radius can be computed and retained.

[00369] The process can be performed for each of the, in this embodiment, seven error parameters. The circularity errors associated with each error type can be sorted and displayed when the user requests. The process can be summarized by the following equation, which makes use of the Kronicker delta δ(k): a 10 element vector whose elements are zero, except for the k'th element which has a value of one.
E_CTRC(k)=max(r_synth—part(θ,{c}=δ(k)))−min(r_synth—part(θ,{c}=δ(k))), k=3,4, . . . ,9  (64)