OPTIMO / CONSTANT VELOCITY - A SMOOTHNESS OF MOTION ANALYSIS TOOL                         

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

START CONSTANT VELOCITY  ANALYSIS                                                                             RETURN TO NEWSLETTER

Table of Content

• Objectives

• Constant Velocity Model

• OPTIMO / CONSTANT VELOCITY Tool

• Example Problem

Objectives

The purpose of this reference note is to familiarize the reader with the OPTIMO / CONSTANT VELOCITY tool, as a quick and intuitive, user friendly analysis tool for sizing motion platforms in positioning applications which require constant velocity.  The note presents a constant velocity model, which will be used in the analysis tool.  It then describes the benefits of OPTIMO / CONSTANT VELOCITY and concludes with a Walkthrough example a typical sizing application. For further support on using this tool please contact  webmaster@optinetinc.com

Constant Velocity Model

There are many applications where positioning systems are required to produce high smoothness of motion. For example in inc jet printing the print head may be traveling at a constant velocity shooting droplets at high frequency. The droplets then hit the substrate at fixed intervals. If velocity is constant the spacing between droplets will be equal. If the velocity varies, variations will results in between the droplets. In severe cases droplets may touch each other and cause low quality printing. In other applications line width of wafer are measured with high frequency detection of the line edge. If the velocity is constant the time of detection can be translated to edge position. Variations in velocity will result in error.

These are just 2 examples. Other applications which require smoothness of motion include, DNA Assaying, Automated Optical Inspection ( AOI ), Milling,  Water Jet Cutting, Dispensing, Laser Scribing, Plotting, Printing, EDM, Grinding, Turning, Inspection.

Our model is intended to provide the user a tool,  where the most dominant parameters and variables which effect smoothness of motion can be varied in a trial and error process which will lead to a working solution.

Our model will include the following elements:

Process Variables - These variables will highlight the specifications which must be defined to properly size a positioning system for CV application. These specifications will determine the allowable jitter

Stage Variables - These variables will highlight the ones that can be selected during the design process including structure, controller, bearing, motor, encoder, amplifier, which effect the smoothness of motion

Environment - These variables highlight those which are effected by the environment such as floor vibrations, unbalance rotation, machining forces

                                                            Constant Velocity Model

Definitions

X = Stage Position (mm)

Y= Point of Interest Position ( mounted on Moving Structure )  (mm)

M = Slide Mass ( kgm)

m = Structure Mass ( kgm)

K= Structural Stiffness ( N/m)

B= Structural Damping ( N/ m/s )

Xf = Position Feedback (mm )

Kp = Proportional Gain ( N/ m )

Kd= Derivative Gain  ( N/m/s)

F= Motor Force ( N)

Xd = Position Disturbance ( mm )

Fd= Force Disturbance ( N)

Assumptions:

• Motor Force is proportional to position error and to position derivative

• Structure is forced by stage displacement

• Any Disturbance is taken at its extreme value across the entire vibration spectrum

• Total jitter is the linear sum of all disturbance forces and displacement amplified by the traqnsfer4 function

Equations of Motion

d^2X/dt^2  + 2* Zb * Wb * dX/dt + Wb^2*X  =  2* Zb * Wb * dXd/dt + Wb^2 *Xd + Fd / M

d^2Y/dt^2  + 2* Zn * Wn * dY/dt + Wb^2 * Y  =  2* Zn * Wn * dX/dt + Wn^2 *X

Where,

Wb = Position Bandwidth Sqrt ( Kp / M )     ( rad/s )         ( Typical Value = 1/3 Wn for high performance, 1/5 Wn for low performance )

Zb = Servo Damping ( Kd /( 2* M * Wb )                            ( Typical value for good tuning 0.7, poor tuning 0.4 )

Wn = Natural Frequency  Sqrt ( K/m )           ( rad /s )          ( Sqrt ( K( N/m) /M (kgm)  )

Zn = Structural Damping ( Kn /( 2* m * Wn )                       (Typical values : 0.1 composite, 0.06 Aluminum. 0.04 Steel )

Block Diagram

Where,

 f ( S ) - Laplace Transform of f (t)

Xd - Position Disturbance Sources ( micron )

    Ball Screw Jitter                             ( Typical Values:  C0: 3um /100mm, 6um/1000mm, C2: 7um/100mm, 10um/1000mm, C5: 18um/100mm, 27um/1000mm )

    Bearing Jitter                                 ( Typical Values: Air: 0.1um, Cross Roller: 0.5 um, Recirculating: 1um, Cam: 10um )

    Servo Jitter                                    ( Typical Values: +/- 3 resolution counts )

    Floor Vibration                             ( Typical Values: 10-100 micron ) Active mount below 1 Hz, Passive Mount below 5 Hz, Resilient mount below 15 hz )

Fd - Force Disturbance Sources ( N )

    Motor Cogging Force                 ( Typical Values: 10% peak force )

    Commutation Ripple Force         (Typical Values: 10% peak force )

    Machining Force                         ( Typical Values: 10-50 N)

    Rotating Unbalance                     ( Force (N)=  M ( kgm ) * Eccentricity ( m ) * W^2 ( rad/s)^2 )

Process Variables

V = Velocity (mm/s)

fs = Sampling Rate ( Hz )

CV = Constant Velocity ( % )

Sampling Interval = V / fs   ( mm )

Allowable Jitter                                     ( Sampling Interval * CV /100 * 1000   ( micron )

OPTIMO / CONSTANT VELOCITY Tool

The advantages of OPTIMO / MCONSTANT VELOCITY are as follows:

• The only known Tool for CV estimate

• Only one simple page is needed for analysis

• Most influential motion parameters are included in model

• Self explanatory

• Available over the internet 24/7

• Free to all Optinet / Sizingtools viewers

• Interactive approach based on quick trial and error runs

• Builds excellent intuitive feel through quick repetitive iterations

• Fun to use in an attempt to "beat" target specs

• Minutes to operate and get working results

• Model can be "calibrated" as performance chart for specific manufacturer's stages

Recommended Method of Operation:

Step 1- Input Process Variable

Step 2 - Input Design and Environmental variables

Step 3 - Click Calculate to view Jitter Spectrum vs Allowable Jitter

Step 4 - Change Design and Environment Variables to move Jitter Spectrum below Allowable Jitter

Example

Consider a bill board  inc jet printing application where the print head moves at 2m/s and shoots droplets at 3,000 hz. The required smoothness of motion is 2%.

Step 1- Input Process Variable

Enter the example process variables in the SMOOTHNESS REQUIREMENT text boxes.

Step 2 - Input Design and Environmental variables

Enter assumed design variables including: encoder resolution of 1 micron, recirculating bearing with 0.5 micron jitter, Floor vibration 3 microns, motor cogging 2N, commutation ripple of 3 N and rotating pump unbalance of 1N

Step 3 - Click Calculate to view Jitter Spectrum vs Allowable Jitter

As shown the allowable jitter is 13.3 micron and the predicted jitter spectrum is higher at the spectrum region up to the structural natural frequency ( 65Hz ) .

Step 4 - Change Design and Environment Variables to move Jitter Spectrum below Allowable Jitter

In this example if  increase structural stiffness, increase structural damping and reduce commutation ripple ( e.g. with linear drive ) ,  we can expect better performance across the entire spectrum.