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# Variable-Sample-Period Digital Velocity Estimator

IP.com Disclosure Number: IPCOM000099619D
Original Publication Date: 1990-Feb-01
Included in the Prior Art Database: 2005-Mar-15
Document File: 3 page(s) / 119K

IBM

## Related People

Winarski, DJ: AUTHOR

## Abstract

This article derives a numerical differentiator for use in a digital motion-control servo-system, such as might be used in a computer peripheral. This digital velocity estimator calculates the exact velocity of the moving parts of a disk or tape drive when the servo-system operates at a constant acceleration.

This text was extracted from an ASCII text file.
This is the abbreviated version, containing approximately 52% of the total text.

Variable-Sample-Period Digital Velocity Estimator

in a digital motion-control servo-system, such as might be used in a
computer peripheral.  This digital velocity estimator calculates the
exact velocity of the moving parts of a disk or tape drive when the
servo-system operates at a constant acceleration.

First it is shown how a parabola is fit through three samples
or data points from a digital tachometer.  This digital tachometer
may be an optical encoder which can detect linear or rotary motion.
Once this parabola has been fitted through these three points, it
will be differentiated to derive the variable-sample-period digital
velocity estimator.

Use of the variable-sample-period digital velocity estimator
can add stability to the velocity loops of servo-systems in computer
peripheral equipment.  For equipment having velocity profiles with
constant acceleration, this additional stability is provided by
having no error between the numerically calculated and actual
velocities. Derivation of the Variable-Sample-Period Velocity

The generalized parabola used in this study is shown in
equation (1) with reference to the figure.  In equation (1) there are
three undetermined constants a, b, and c, which need to be
ascertained. Y(T) = a*T2 + b*T + c (1)

The data points through which the parabola is fitted are given
in Table 1.  In the table, the incremental distance, d, is the unit
displacement between pulses of the linear or rotary optical encoder.
Because the incremental linear or rotary distance between pulses is
constant on either optical encoder, the time period between samples
varies as the velocity of the computer peripheral changes. Thus, in
Table 1, T2 is not normally an integral multiple of T1.

(Image Omitted)

Fitting the parabola, equation (1), through the data points in
Table 1, gives equations (2-4).  Equation (2) shows that the constant
term in equation (1), c, now contains all prior history of
displacement.  This constant term is not referenced any further in
this analysis. c = Yo (2) d = a*T12 + b*T1 (3) 2d = a*T22
+ b*T2 (4)

Equations (3) and (4) are two simultaneous, linear equations
with two unknowns.  These two equations can be written in a matrix
form.  By use of Cramer's rule, the matrix can be solved for the
undetermined constants a and b.

By differentiating the generalized parabola in equation (1)
with respect to time T, the...