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Adaptive Differential Operator for Servo-System Applications

IP.com Disclosure Number: IPCOM000113927D
Original Publication Date: 1994-Oct-01
Included in the Prior Art Database: 2005-Mar-27
Document File: 4 page(s) / 117K

Publishing Venue

IBM

Related People

Winarski, DJ: AUTHOR

Abstract

Digital position data can be used to calculate velocity. For example, the crossing rate of uniformly spaced tracks on an optical disk can be used to calculate the radial velocity of the seek mechanism. Also, the crossing rate of uniformly spaced radial lines on an optical encoder can be used to calculate the angular velocity of a spinning optical disk or reel of tape.

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Adaptive Differential Operator for Servo-System Applications

      Digital position data can be used to calculate velocity.  For
example, the crossing rate of uniformly spaced tracks on an optical
disk can be used to calculate the radial velocity of the seek
mechanism.  Also, the crossing rate of uniformly spaced radial lines
on an optical encoder can be used to calculate the angular velocity
of a spinning optical disk or reel of tape.

      However, every means of numerically differentiating the digital
position data to best ascertain the current velocity is open to
truncation error.  Furthermore, techniques to reduce truncation error
may introduce dated information which could cause more error than it
alleviates.

      This article describes parallel and competing algorithms for
the numerical differentiation of data.  Each competing algorithm
numerically differentiates known position data via end-point
analyses.  Two sample periods later, these end-point analyses are
scored against the more accurate central-difference assessment of
velocity.  The most accurate means of numerically differentiating the
data is chosen for use in the servo-system.

Competing End-point Differentiators - Four algorithms are considered
for end-point differentiators.  These end-point differentiators
calculate the current velocity based on current and previous history.
There is a tradeoff to be considered.  The more the known history,
the less the truncation error, provided that the previously known
history is still relevant.  Hence, the goal of this disclosure is to
determine which end-point differentiator is currently the best to
use.

      The first of these is the standard two-point formula, eqn.
(1).  Eqn.(1) has a truncation error on the order of the time between
samples, h.
 V2(n) = ( Y(n) - Y(n-1) ) / h                             (1)
 Where:  Y(i)  = vector to be differentiated
         V2(n) = two-point velocity
         h     = sample period
         n     = latest time index
         n*h   = current time

      The next end-point numerical differentiator is the three-point
formula, eqn.  (2).  Eqn.(2) has a truncation error on the order of
h-squared, (h**2)/3.  Thus, eqn.(2) has better truncation error
characteristics than eqn.(1).
 V3(n) = V2(n) + (V2(n) - V2(n-1))/2                       (2)
 Where:  V3(n) = three-point velocity

      The four-point formula, eqn.  (3), has a truncation error on
the order of h-cubed.  Thus, eqn.(3) has better truncation error
characteristics than either eqns.(1-2).
                    V2(n) - V2(n-1)   V2(n-1) - V2(n-2)
 V4(n) = V2(n) + 5* --------------- - -----------------    (3)
                          6                   3
 Where:  V4(n) = four-point velocity

      The last end-point numerical differe...