OPTIMO / PRECISION - A POWERFUL 3-D PRECISION ANALYSIS TOOL                         

© Dr. Boaz Eidelberg, April 2005, Optinet Inc.

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Table of Content

Objectives

The purpose of this reference note is to familiarize the reader with  OPTIMO / PRECISION tool.  This analysis tool is quick to operate, robust and user friendly. It is intended to be used for optimizing selection of XYZR positioning stages including their mounting configuration, location of their work area and tools (point of interest ) to meet specified 3-D precision requirements. It is also intended to be used for an optimal sizing of the individual precision specifications of each stage, towards meeting the 3-Dimentional precision requirements of the complete assembled system.  The note highlights various other precision analysis tools, which are commonly used in the motion control marketplace. It then describes the benefits of OPTIMO / PRECISION and concludes with a Walkthrough in a typical sizing application example. For further support on using this tool please contact  webmaster@optinetinc.com

General Purpose Tools

Not surprisingly in a web search of precision analysis of positioning systems ,very little was found. In fact none was available from the major players in motion control positioning system manufacturing. The typical sample which was found below is analytical but does not include practical tools to work with. The second one shown was published by the author of this note.      http://www.findarticles.com/p/articles/mi_qa3685/is_200001/ai_n8900274       http://www.designnews.com/article/CA192446.html

In fact the only known tool that accommodates 3-D precision of XYZ positioning systems is BIMO of Bayside Motion Group, which is now Parker Bayside. BIMO tool was developed by the author of this note and it is limited to XYZ stages without rotating tables. The new OPTIMO / PRECISION tool is expanded here to include XYZ and Rotary stage and be available for interactive operation by the user over the internet. It is in fact based on 4 transformation matrices from one coordinate system to the other, but it is all transparent to the user. All the user sees is a set of input precision parameters of each individual axis, taken out of the selected manufacturer catalog, then assembled together into a system and finally defining the location of the point of interest ( POI ) in the coordinate system of each axis. This takes a little visualization capability but a simple sketch can greatly assist in the process.

Basic Principles

    Coordinate Systems

The importance of analyzing positioning system precision in 3 dimensions is rooted with the fact that even very high precision stages, when compounded together in various configurations, can result dramatically different precision.  The reason is that the overall system precision, at the point of interest ( POI )where work or inspecting is being done, depends not only on the individual precision specification of each stage, but more so on the offset of the point of interest from the feedback device as well as on the 3-D rotation of the slide during motion. E.g. 1 arc sec rotation generates 5 micron error at a distance of 1 m' away from the slide.

Lets start by looking at the basic of a rigid body position. If we define XYZ as an Inertial coordinate system, meaning fixed to earth without moving, then each rigid body position can be defined by 6 degrees of freedom ( DOF ). Three DOF are translational  in X,Y and Z directions respectively and three are rotational Pitch, Yaw, and Roll.

 

 

 

 

 

 

 

 

 

 

To define Pitch, Yaw and Roll, lets define another Cartesian coordinate system 1,2,3 which is body fixed, meaning it is attached to the body and moving with it where ever it goes. Lets further assume that the origin of the axes 1,2,3  is at the center of gravity of the body, where axis 1 is in the direction of motion, Axis 2 is perpendicular to axis 1 in the plane of the tracks on which the body moves, and axis 3 is perpendicular to both 1 and 2.  Now we can define Pitch as the rotation of the body about axis 2, Yaw is the rotation of the body about axis 3 and roll is the rotation about axis 1.

Typically, slides of positioning systems are designed to move in a constraint motion along a track in a single direction measured with a feedback device such as an encoder or laser interferometer. Ideally, there is no motion in any other direction other than that of the track and the absolute position of body in XYZ coordinate system is identical to the position which the feedback device reads. However, due to many factors such as manufacturing tolerances, friction, surface roughness of the track , structural deformations, vibrations and assembly misalignments,  it turns out that on a coarse scale the body is indeed moving only along its track , but on a microscopic scale the body is moving in all of its 6 DOF and its actual position along the track may not be the position which the feedback device says it is. This phenomena leads to the following definitions which constitute the precision of motion of the body:

accuracy - the linear deviation between actual position of the body and the position read by the feedback device along axis 1

straightness - linear deviation of body position from the desired track constraint along axis 2

flatness - linear deviation of body position from the desired track constraint along axis 3

pitch - rotation of the body about axis 2

yaw - rotation of the body about axis 3

pitch -  rotation of the body about axis 1

It should be realized that the above 6 DOF deviations from desired position of the body are random variables sometimes referred to as position error. Each of the 6 errors has a deterministic value called accuracy and each has a statistical error content called repeatability. The deterministic error can be mapped out and accounted for using software tools such as calibration and mapping. The statistical portion can not be accounted for and it can not be mapped out. For this reason the repeatability of the body, which in the motion control applications is called  stage, axis, table, slide, or actuator, is probably the most important precision characteristic to be considered in all of its 6 degrees of freedom.

    Abbe Offset

 

 

 

 

 

 

 

 

 

 

 

 

Typically positioning stages are specified by their manufacturers on a single basis, when the stage is placed on a granite base and measured with laser interferometer to determine all 6 precision specifications. However, when compounded on each other, or when tools and work are mounted on the stage to do useful work, there is always an offset,  large or small, between the level of the feedback device and the point of interest ( POI ).  This offset is call Abb'e Offset. The Abb'e Offset in combination with angular error can result in an Abb'e Error as shown in the diagram above. Here the equation of a simple Abb'e Error ( mm ) = Abb'e Offset ( mm) * Angular error ( radians ).

In a more general case, the components of the Body position Error vector in directions 1,2,3 is the cross product of the Rotational Vector {R}  of the body about axes 1,2,3 and the Abb'e offset vector {A}of the point of interest expressed in body coordinate 1,2,3 plus the linear error vector {L}. 

{E} = {R} x {A}+{L}

Where,

{L}= <B,.S,,F>T = Linear error Vector consisting of feedback error ,B, straightness, S, flatness, F, components

{R) = <i,,j,,k >T Rotation Vector

i,j,k = unit vectors along axes 1,2,3

T = Notation for transposed vector  ( i.e. Columns are Transposed to rows )

x - Vector cross product as described below.

Vector Cross Product

If we have two vectors {A} and {B} in a cross product such as . {A} x {B} = {C} then the result is a new vector {C} where the magnitude of {C} equals as follows:

|{C}| = |{A}| * |{B}| * SIN (α )

where,

|  | designates the length of the vector

* designates scalar multiplication

(α ) designates the spatial angle between the two vectors

The direction of {C} is perpendicular to both vectors with direction determined according to the right hand rule as shown in the diagram below.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The components of the error vector can be expressed as follows:

E1 = P * A3 + Y * A 2  + B

E2 = Y * A1 + R * A3 + S

E3 = P* A1 +  R * A2 + F

Where,

E1,E2,E3 - The components of the Position error vector resolved in body fixed coordinate system 1,2,3

A1,A2,A3 - The components of the Point of Interest vector resolved in body fixed coordinate system 1,2,3

P = Body Pitch ( rotation about axis 2 )

Y = Body Yaw ( rotation about axis 3 )

R = Body Roll ( rotation about axis 1 )

B = Feedback Error ( Position error in axis 1 )

S = Straightness ( Position error in axis 2 )

F = Flatness ( position error in axis 3 )

If we know the orientation of axes 1,2,3 with respect to XYZ we can transform the error vector from coordinate 1,2,3 to XYZ.

{E}xyz = [T] * {E}123

Where,

{E}123 = { E1, E2,. E3 }T   = Error vector in body fixed coordinate system

{E}xyz = { Ex, Ey. Ez }T   = Error vector in inertial coordinate system

[T] = Matrix of transformation of coordinates from Body fixed system 1,2,3 to Inertial system X,Y,Z

In addition, if we compound more stages on top of the first stage we need to perform similar calculations of the errors of the compounded stage with respect to the stage it is mounted to. Then we need to transform the errors to the inertial coordinate system XYZ.

{E}xyz = [T]1 * [T]2 *[ T]3 *...{E}123

Where,

[T]1 * [T]2 * [T]3 = successive matrices of transformation of compounded bodies ( e.g. stages ) from one body fixed coordinate to the one it is riding on.

Matrix  [T] is typically expressed with 3 Euler angles Yaw, Pitch and Roll of a body fixed coordinate 1,2,3 with respect to the one it is riding on, Alternatively it can be expressed  with 9 directional cosines,where each directional cosine consists of the component of a unit vector along axis 1,2 or 3 resolved into axes X,Y,Z.

Although the above procedure is general for any free body kinematics, it can be simplified in the special case of position stages compounded in XYZ orientation with typical angles of rotation of 90 degrees. OPTIMO - PRECISION is based on the above procedure.

Definition of Angular Variables and Precision Parameter

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Axes 1 and 2 in the theoretical plane of rotary motion fixed to the Rotary table Base

Axis 3 Perpendicular to the theoretical plane of rotation

Wobble - The angle between the the perpendicular axis normal to the actual Rotating Table Slide ( indicated by dotted lines in the diagram above ) and Axis 3

Radial Runout - The deviation of the Rotary table Slide from the Theoretical boundaries of rotation in directions of Axes 1 and 2

 

Statistical Sums of  Error Random Variables

As shown in the above formulation there are considerable additions of error components, each of which is a random variable. We therefore need to add these contributions statistically.

In general if two random variables A and B are of normal distribution with average values Aav and Bav respectively and standard deviation As and Bs respectively, then the sum of the two is a new random variable C with average Cav and standard deviation Cs with he following relationship to A and B

Cav= Aav+Bav

Cs= SQRT ( As^2 + Bs^2 )

In our model we will assume that each precision variable is a normally distributed random variable. Therefore the overall 3-D error in position is the statistical sum of all the contributing factors.

 

    Specifications of Position Errors with Sigma Levels

Typically precision variables are defined by their standard deviation  in +/- microns for linear error and or +/- arc sec for angular errors. It is important to specify how many sigma the +/- precision value represents.

 

Where,

+/- 1 sigma indicates that in 68.27 % of all cases the result will be within the specified limits

+/- 2 sigma indicates that in 95.45 % of all cases the result will be within the specified limits

+/- 3 sigma indicates that in 99.73 % of all cases the result will be within the specified limits

+/- 4 sigma indicates that in 99.99 % of all cases the result will be within the specified limits

Note: The familial expression 6 Sigma which was introduced in the 1980's by Motorola indicates that only 1 out of million parts or samples can fail. This has become the industry standard for quality. In positioning precision users therefore should insist on a minimum of +/-3 Sigma test results for stage precision and should compare these values among their options.

 

OPTIMO / PRECISION ANALYSIS TOOL

Title Page

 

 

 

 

The Title page consists of the following items:

Buttons

EXAMPLE - Click on this button to fills the INPUT text boxes  with sample values

RETURN - returns to www.sizingtools.com  Home web page

CALCULATE - Performs the precision Analysis and populates the RESULT text boxes

CLEAR - Clears all values in text boxes for a new analysis

Coordinate Systems

XYZ - Inertial coordinate system fixed to the Machine base and does not move.

1,2,3 - Body fixed coordinate system ( see definitions in previous sections ).  Note that the example above shows one body fixed coordinate system for Stage A which is attached to the center of gravity of the moving slide of Stage A, and second body fixed coordinate system 1,2,3 attached to the Base of the Rotary stage.

Compounded XYZR diagram

The diagram shows 4 axes compounded in XYZR configuration, where for input purpose and not to confuse with the Inertial coordinate system XYZ,  stage X is called stage A, stage Y is called stage B , stage Z is called Stage C and the Rotary stage called Stage R.

Point of Interest ( POI )

This  is a point within the Inertial coordinate system, which  in our case is the "Machine" coordinate system,  where precision is of interest. It may not necessarily be throughout the entire travel of the axes, but we must be able to specify its location or identify its range within the machine coordinates, and be able to resolve the Position Vector of the POI into any local coordinate System 1,2,3 of a participating machine stage. A side bar sketch of the specific application may help in this visualization process.

General Titles

The general titles allow the user of this tool to identify the specific analysis with a specific date, customer nam , project name, analyst name and comments.

To save the analysis please use Print Screen key on your keyboard and paste the results into your documents. You can also use Microsoft Paint or similar Tools to cut and paste portions of the analysis page.

Analysis Page

 

 

The Analysis page consists of Input text Boxes and Output of Results. The following is a recommended Analysis Procedure -

INPUT VARIABLES

Individual Stage Specification

The input text boxes must be entered for each Stage that participates in the Analysis. The parameters for the Linear stages include the following:

Encoder Accuracy, Straightness, Flatness, Pitch, Yaw, and Roll. Most of these parameters are specified by the vendors within their catalogs. If you do not have these values call and ask for them before making a precision sizing choice. Most manufacturers do not list roll as a parameter but you can assume the larger value of Pitch and Yaw. Make sure that all values are +/- 3 Sigma. If they are not specified by the manufacturer - ask for them. Comparing cost performance of stage precision without knowing the sigma value is like comparing apples and oranges.

The parameters for the rotary stage include Encoder accuracy, wobble and radial runout

Stage Assembly Configuration

In this section we assemble the individual stage to the machine. We do it by specifying the direction in which we mount and align the body fixed, local, coordinate of each stage 1,2,3 with the machine, Inertial or global coordinate system X,Y,Z. For example in the above title page diagram, local axis 1 of stage A coincides with global X axis. local axis 2 of stage A coincides with global machine Y axis and axis 3 lines up with Z. Similarly for Stage C, for example, axis 1 ( not shown but as defined previously it is in the direction of motion of axis C ) coincides with global axis Z , axis 2 lines up with Y and axis 3 coincides with axis Y. We place these letters in their corresponding text boxes of the analysis page.

Abb'e Offset of Point of Interest (POI )

This is probably the most complicated portion of the analysis. We must resolve the global position of the point of interest as specified by the customer of the positioning system X,Y,Z into each local coordinate system of each stage 1,2,3. It is recommended that the analyst will draw a representing sketch of the system configuration and use it for this determination. Alternatively consult with the machine designer for these values. Make sure you always take the maximum Abb'e offset by moving the stages to their extreme positions. Finally, it is recommended that the Abb'e offset within any stage will be with respect to the origin of the slide when placed at the center of travel on the stage it is riding on. Note: make sure you are consistent with the specified INPUT units. The input can define repeatability or accuracy. The values can all be +/- or TIR ( total indicated readings ).

RESULTS

The results are provided in a Table that combines the statistical errors in X, Y and Z directions with their respective contributions from each individual stage. The results also provides the Total radius of a 3D sphere which contains  all errors with the probability corresponding to the specified Sigma Value of the input variables..

 

Walkthrough EXAMPLE

 

case1- Lets start with a simple case

put 1 micron error in stage A, as shown in the above diagram, and calculate. We get as expected 1 micron 3D error because the error is in direction 1 of Stage A

case2- Lets  add now an Abb'e offset of 1000 mm in the direction of axis 3 of stage A and a pitch of 1 arc sec

What we see is that we added an error of about 5 microns in the direction of X.

case 3-  Lets add now 10 micron straightness and 5 micron flatness to stage A.We get ,as expected, the following results. Notice that the Results table indicates that the total 3-D  error is 12.2 micron, which is the statistical sum of 4.9, 10. and 5. Notice also that the 1 micron encoder error was masked by the 4.9 micron Abb'e error. The table shows us which element of the stage contributes the most to the total  3-D total and in which direction it is the worse. We can use this information to improve the stage selection, assembly configuration or the location of POI with respect to teach individual stage.

Case 4 - Suppose now that we change the mounting configuration of Stage A and we mount it sideways without changing the location of the point of interest or the specification of the stage. The interesting results are as follows:

The total error was reduced. The Abb'e Error disappeared and the encoder error showed up. It is recommended at this point to do some visualization exercises to get more comfortable with the transformations and confidence with the results.

What this example demonstrates is that the common practice of trying to improve system precision by paying higher price for a more accurate feedback device could not be further away from a good sizing practice with a reliable sizing tool.

Case 5 - Click on the EXAMPLE button to input the Tool example variables, then click CALCULATE and try to rationalize the RESULTS

Support

If you have any question or comment about this tool, or if you discover any error ,please contact us at webmaster@optinetinc.com. We offer limited free advertising bonus to comment's contributions, as shown in http://www.sizingtools.com/Comments.htm. Optinet is dedicated to excellence in positioning system analysis and would like to thank you for your interest in our work.