Mechatronic Servo System Control - M. Nakamura S. Goto and N. Kyura Part 7 potx

4.1E
nco
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81
avelocitydetector is generated as the detection noise of rotation velocity,
ripple-type velocitycan be preventedbysmoothing this detection noise with
alow pass filter. However, aripple-typevelocityfluctuation of asoftware servo
system cannotbesmoothed by alow pass filter because variedfrequencyin-
troducedlaterisrelated with the objectivevelocity. Therefore, it is necessary
to determine the encoder resolution for forcingthe velocityfluctuation within
theallowance region.
(2) PresentCondition of Encoder ResolutionDetermination
Thedeterminationofpresentencoder resolution in theindustrialmechatronic
software servesystem is carried out according to the necessityofpositioning
precisionofamechatronic servosystem
[4]
.When performing contourcontrol,
theencoder resolution calculated frompositioningprecisionisused without
change. When required, controlperformance cannot be obtained, the encoder
resolution with test error will be regulated. The determination of encoder res-
olution cannotberealized theoretically for the required control performance.
Therefore, in thischapter, the theoretical determination method forencoder
resolution forcontrol performance,especially about contour control issue, con-
sidering the relationship between ripple-typevelocityfluctuation andencoder
resolution,isproposed.
4.1.2A
Mathematical
Mo
del
and

ResolutionJ
udgement
for
Enco
der
Resolution
(1)
AM
athematical
Mo
del
of
aS
oft
wa
re
Serv
oS
ystem
An industrial mechatronic servosystem is alwaysunder the velocitycondition
of motion of theoperatedmotor at 1 / 20 ∼ 1 / 5ofmaximumvelocity. Its
dynamics is expressed by the 2nd order system as (refer to the 2.2.4)
Y ( s )=
K
p
K
v
s
2
+ K

v
s + K
p
K
v
U ( s )(4.1)
where Y ( s )isthe position outputofthe servosystem, U ( s )isthe position in-
putofthe servosystem. K
p
, K
v
have the meaning of K
p 2
, K
v 2
in the equation
(2.29)
of
the
middle
sp
eed
2nd
order
mo
del
in
the
item
2.2.4,

resp
ectiv
ely
.
The control system of the mechatronic servosystem expressed by (4.1)
is pickedout fromthe software servosystem shown in figure4.1 for encoder
resolution analysis. The model of software servosystem for simplifying the
analysis is shown in figure4.2. From thestructureofthe software servosystem
(Fig. 4.1), the velocityfeedbackcalculatedbasedondifference computation
is easily obtained from the external input. However, thisexternal input, as
asimple external input, is the same as the velocitysignal in Fig. 4.1.This
externalinput is the continuous feedbackofthe velocityoutput in an analogue
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Error
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ec
hatronic
Serv
oS
ystem
-
-
K
p
K
+

+
uy
v
d y/d t d y/d t
1
s
-
1
s
-
22
V eloc i ty feedback signa l
D i s c r e t i z a t ion a nd q u a n t i z a t ion
Fig. 4.2. Software servosystem model for encoder resolution analysis
servo system. But in asoftware servosystem, it is adiscrete feedback. The
basic
unit
of
the
po
sition
signal
is
1[pulse].
The
ve
lo
cit
ys
ignal

is
calculated
with the difference computation of the position signal. The basic unit of the
velocitysignal according to difference computation is 1 /∆t
p
[pulse/s], where
∆t
p
[s] is the sampling time.
(2)Relationship between Control Performanceand Encoder
Resolution
The relativeequation between velocityfluctuation,occurred according to en-
coder resolution, and servoparameters is derived. In thispart, thevelocity
fluctuationisanalyzedwhen the motion of theservomotor is under the con-
stan
tv
elo
cit
y,
whic
hi
sa
lw
ay
sa
doptedi
nt
he
industrial
field(

refert
oi
tem
8
of 1.1.2). The flowofsignal is as Fig. 1.1.2.
1.
The
difference
divided
according
to
ve
lo
cit
yr
esolution
1
/∆
t
p
,d
etermined
by difference computation of theposition signal, is accumulated. When
the
accum
ulatedv
aluei
so
ve
rt

he
ve
lo
cit
yr
esolution,
the
ve
lo
cit
yf
eedbac
k
signali
sa
dded
with
1
/∆
t
p
.This added velocityfeedb acksignal is the
reason for the velocityfluctuation.
2. According to the velocityloopgain K
v
added into the velocityfeedback
signal,
the
input
of

the
motori
sv
ariedw
ith
the
step
of
K
v
/∆t
p
[pulse/s
2
].
3. The change of velocityoutput of themotor basedonthe added velocity
feedbacksignal is as ( K
v
/∆t
p
) × ∆t
p
= K
v
[pulse/s], according to the
integral of the input of the motorbasedonthe sampling time interval
∆t
p
.
That is to say, the size of velocityfluctuation,occurred by the signal added into

velocityfeedbackaccordingtothe effectofvelocityresolution, is consistent
with the value of velocityloopgain K
v
.This relation can be expressed, if
considering the unit, as
∆N =
60K
v
R
E
(4.2)
where ∆N[rev/min] denotesthe velocityfluctuation amplitude with the
ripple-type shape,
R
E
[pulse/rev] denotes the encoder resolution defined by
the pulse number of the encoder when the motorrotates throughone cycle.
4.1E
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Resolution
83
This derivedequation (4.2)isthe fundamental equation fordetermining the
following encoder resolution.
Next, the relationship between the velocityfluctuation periodwith the
ripple-typeshape andvelocityofthe objective trajectory is der ived. If the ve-
locityofthe objective trajectory is as V
ref
[pulse/s], the velocityfeedback, ob-
tained from the difference computation, is changed as (  V

ref
∆t
p
 ) /∆t
p
when
the velocityresolutionis1/∆t
p
,where  x  is the maximalinteger below x .
From 1, this errorisaccumulated in eachsampling time interval. Sincethe
velocityfluctuation with theripple-typeshape occurred when the error is over
1 /∆t
p
,The sampling time n at themoment of over1/∆t
p
is as
n

V
ref
−
 V
ref
∆t
p

∆t
p

=

1
∆t
p
. (4.3)
From (4.3), thevelocityfluctuation frequency f
r
[Hz] is calculated by
f
r
=
1
n∆t
p
=
V
ref
∆t
p
−V
ref
∆t
p

∆t
p
. (4.4)
From (4.4), thevelocityfluctuation frequency f
r
is depended on the velocity
of objectivetrajectory V

ref
.Inorder thatthe velocityfluctuation frequency f
r
is not changed into amonotonic functionabout V
ref
,alowpass filter cannot
be adoptedfor smoothing.
(3)Determination of Encoder Resolution
By using(4.2),the relation equation between velocityfluctuation andencoder
resolution derivedby4.1.2(2), the determination equation of the encoder res-
olution can be obtained. Whenthe motorisrotated with aconstantvelocity,
the ratio between the scale of the velocityfluctuation andthe maximalve-
locity, called velocityfluctuation ratio R
N
,isadoptedasaspecification of
am
ec
hatronic
serv
os
ystem,
in
order
to
express
clearly
the
motionl
ev
el

of
ve
lo
cit
yo
ft
he
motor.
Fr
om
this
po
in
to
fv
iew,
in
the
soft
wa
re
serv
os
ystem,
the velocityfluctuation ratio R
N
generatedi
nt
he
encod

er
resolution
can
be
expressed by
R
N
=
∆N
N
max
(4.5)
where, N
max
denotesthe maximalvelocity[rev/min] of theservomotor.Ifwe
put (4.5)into(4.2),basedonthe solutionofthe encoder resolution R
E
,the
encoder resolution can be determinedby
R
E
=
60K
v
R
N
N
max
. (4.6)
The equation (4.6)isthe finalderived result in this section. Accordingto

this equation, pr oper encoder resolution R
E
can be decidedfor satisfying the
velocityfluctuation ratio
R
N
,d
etermined
according
to
the
application
of
the
servomotor fromthe maximalvelocity N
max
andvelocityloopgain K
v
.
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Serv
oS

ystem
4.1.3Experimental Verificationofthe Encoder Resolution
Determination
(1)Experimental Verificationofthe Relationship between the
Encoder Resolution and Control Performance
From theexperiment, the relationship between the enco der resolution and the
velocityfluctuation is verified. In the experiment, DEC-1(refer to the exper-
imentdeviceE.1) wasadopted. Actually, DEC-1 wasoriginallyconstructed
with an analogueservosystem. However, in thisexperiment, asoftware servo
system using acomputerwas used. Thatistosay,the pulseoutput of the
servomotor is accumulatedbyacounter equippedinthe computer. The com-
puterprogram implementsthe servocontroller.Its outputisput into servo
(a) Experimentresults of software servosystem
(b) Simulation results of software servosystem
(c)Experimental results of analogue servosystem
Fig. 4.3. Verification of velocityripple in software servosystem
4.1E
nco
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Resolution
85
amplifier by using aD/A converterfor constructing the software servosystem.
The resolut ion of D/A conversion is adoptedwith reduction by a1/100 am-
plifier fromthe D/Aconverter, which can permit ± 5[V]with aresolution of
12[bit]. Since 1[bit] is about 2.44 × 10
− 5
[V], the effect of resolution to control
performance canbeneglected. In addition, the velocityofthe servomotor is
measured using digital data storage providing velocitydetector (tachogener-
ator) outputequippedwith aload generator.This tachogenerator output is

7[V]with arotational frequencyof1000[rev/min] of theservomotor.Since
thereare many factorsofnoises in thetachogenerator,the 100[Hz] lowpass
filter is adoptedtoeliminate these noise factors. The resolution of the encoder
installed in the servomotor is 2000[pulse/rev]. But fromthe tested two in-
crease and decrease signals of the encoder outputasputting themintothe
pulsecounter, the original 1[pulse] is changed into 4[pulse].Throughthe 4
timescircuit, it can be obtained as R
E
=8000[pulse/rev]. The maximalve-
locityis N
max
=1000[rev/min], the sampling time interval ∆t
p
=4[ms] (refer
to 3.1). The position loop gain and velo cityloopgain areset as K
p
=12[1/s]
and K
v
=68[1/s]sothatthere is no oscillation or overshoot in theanalogue
servosystem (refer to 2.1.2). Since velocityfluctuation is oneofthe problems
in the industrial field, forbig velocityfluctuation in lowspeed, ramp input
for DEC 1is u ( t )=40t [pulse], i.e., rotation speed of motoris0.3[rev/min]
for lowspeed. In the steady state, the experimental results and simulation
results are illustrated in Fig. 4.3.FromFig.(a), (b), in the steady state, the
amplitudeinexperimental results and in simulation results are both 0.004[V].
The frequency in both about is 40[Hz]. The shapeofthe wavesare both tri-
angular. From theabove,itcan be verifiedthatthe experimentalresults and
simulationresults are almost the same. In Fig.(a)ofexperimental results, the
size of velocityfluctuation is about0.004[V], i.e., 0.57[rev/min]. This value is

almost the same as the size
∆N =60 × 68/ 8000 =0. 51[rev/min] of velocity
fluctuation calculated by equation (4.2). In addition, the velocityfluctuation
frequencyisalso consistentwith the frequency 40[Hz] calculated by equation
(4.4).
To verify,the experimentalresults of an analogue servosystem with same
conditionsare illustratedinFig.(c). In the analogue servosystem, the velocity
fluctuation does not occur at all. The velocityfluctuation in Fig.(a) is verified
thatitisthe cause of theresolution of software servosystem by the experiment
of 4.1.2(2).
(2)Application of Encoder Resolution Determination
Usingequation (4.6)derived by 4.1.2(3),the example of determining the
encoder resolution is illustrated. In DEC-1 adopted in the previous exper-
iment, the necessary encoder resolution is R
E
=60 × 68 × 1000/ 1000 =
4080[pulse/rev]obtainedfromequation (4.6)ifthe velocityfluctuation ra-
tion is given as
R
N
=1× 10
− 3
.I
nc
on
trast,i
ft
he
installed
encod

er
res-
olution is actually R
E
=8000[pulse/rev], the velocityfluctuation ratiois
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Error
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ec
hatronic
Serv
oS
ystem
R
N
=60 × 68/ (1000 × 8000) =5. 1 × 10
− 4
.Fromthis pointofview, according
to the encoder resolution determination equation (4.6), the encoder resolution
can be easily determined from the required velocityfluctuation ratio.
4.2
To
rqueR
esolution
In
the

soft
wa
re
serv
os
ystem,
the
feedbac
ko
ft
he
motorc
urren
te
quiv
alen
tt
o
the
torque
is
carried
out
through
am
icro-computer.B
et
we
en
the

po
we
ra
mpli-
fierfor driving the motorand the micro-computeristhe A/D, D/Aconversion.
Thetheoretical relation between the A/D, D/A conversion quantization error
andcontrol performance must be clarified.
The appropriate mathematicalmodel for the relationship between the torque
resolution of the software servosystem and control performance is derived.
According to the solutionofthe mathematicalmodel, the positioningpreci-
sion by equation (4.8)and the position flu ctuation of the ramp response by
equation (4.15)∼ (4.17), with regard to the torque resolution, can be clarified.
Accordingtothe bit number proposed in the A/D, D/A converter, the con-
trolperformance of the servosystem can be clearly estimated. Additionally,
the minimal necessary bit number of the D/A, A/D conversion fortesting
out torque command and currentfeedback, in order to implementthe neces-
sary controlperformance of the software servosystem, can be determinedby
equation
(4.25).
4.2.1Mathematical Model of the Mechatronic Servo Systemfor
Torque Resolution
The conceptual graph of the discussed software servosystem in this section
is shown in Fig. 4.4.The software servosystem is shown in Fig. 4.4.Inorder
to construct the controlcircuit of the servocontroller using micro-computer
software, the torque (current) commandoutput fromthe controlcircuit is
quantized. Therefore, the currentreference input to the poweramplifier actu-
ally needsaD/A converter. The block diagram of the2nd order system of the
servosystem includingtorquequantizationisillustrated by Fig. 4.5. K
p
[1/s],

K
v
[1/s]
ha
ve
the
meanings
of
K
p 2
, K
v 2
in
the
middle
sp
eed
2nd
order
mo
del
equation
(2.29)
of
item
2.2.4.
In
addition,
the
sampling

time
in
terv
al
of
the
velocityloopis ∆t
v
[s].
The
serv
os
ystem
is
usually
constructed
with
po
sition
feedbac
k,
ve
lo
cit
yf
eedbac
ka
nd
current
feedback

.T
he
po
sition
feedbac
ka
nd
velocityfeedbackrefer to the feedbackofthe actualmotor output forthe
servocontroller.The current feedback refers to the feedback of powerampli-
fied.Itisnot changed into theactual torque. Forthe mathematicalmodel
of the servosystem in the block diagram of Fig. 4.5, theposition feedback
andvelocityfeedbackiswidely considered. The currentfeedbackissimply
assumed
as
the
outputo
ft
he
po
we
ra
mplifier.
The
con
trol
metho
do
ft
he
ve

-
locityloopisPcontrol or PI control. But the entire prop ertyofthe velo city
4.2T
orque
Resolution
87
D / A
S e rvo c ontroller
Q u a n t i z a t ion
C ompute r
P o w e r
a mplifier
I nput ( c urrent )
O utp ut
M o t o r
Fig. 4.4. Structure of software servosystem

K
p
K
1
-
++
v
U ( s )
Y ( s )
s
1
-
s

S e rvo c ontroller M o t o r
V eloc i ty loop
P o s i t ion loop
Q u a n t i z a t ion
Fig. 4.5. The2nd order model of software servosystem including torque quantiza-
tion
loop is expressedbythe 1st order system.The position control and velocity
control are combined into the 2nd order system (refer to item 2.2.4).
In this section, the torque quantizationwith A/D, D/A conversion,asa
problem, is expressedaccordingtothe quantizationterm in Fig. 4.5.Bythe
function
f ( · )for quantization of torque, themathematical model of aservo
system includingthe torque quantizationisas
d
2
y ( t )
dt
2
= f

K
p
K
v
u ( t ) − K
p
K
v
y ( t ) − K
v

dy( t )
dt

. (4.7)
Formeasuringthe rotation angle of theservomotor by apulse[pulse] ac-
cording to the encoder, the rotation angle u of motor as aposition com-
mandisexpressed by apulse. The angular velocityinput, as theveloc-
itycommand, is K
p
{ u ( t ) − y ( t ) } [pulse/s]. The angular acceleration inp ut,
regarded as the torque command to torque quantization, is K
v
[ K
p
{ u ( t ) −
y ( t ) }−dy( t ) /dt][pulse/s
2
].
In
order
to
mak
et
he
angular
acceleration
quan
ti-
zation
function

f ( x )a
st
he
step-wise
functiono
fF
ig.
4.6,t
he
input
angular
acceleration x [pulse/s
2
]i
sq
uant
ized
by
the
angular
acceleration
resolution
R
A
[pulse/s
2
].
In addition, considering the effect of torque quantizationonthe control
performance,itassumed that position and velocitywithout quantizationare
feedback with continuous values. In the actual software servosystem, the en-

coder resolution of the servomotor is infinite. Thatis, theposition andvelocity
information is continuously obtained at the desired state. Compared with the
actual software servosystem with an encoder,the controlperformance with
this
assumption
is
the
maxim
um
po
ssible.
The
condition
of
deriving
torque
resolution
is
considered
as
thep
rerequisite
condition.
In
the
soft
wa
re
serv
os

ys-
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Error
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aM
ec
hatronic
Serv
oS
ystem
0
x [ p u l s e /s
2
]
f ( x ) [ p u l s e /s
2
]
R
A
R
A
Fig.
4.6.
Quan
tization
of
angular

acceleration
tem, forrealizing the required control performance,the A/D, D/Aconversion
is carriedout with torque resolution capable of satis fying the lowerlimitation.
Moreover, since introducing thisassumption, the analysis of the effect on the
control performance of torqueresolution becomes easy and it is possible to
derivethe torque resolution conditionequation by 4.2.4(1),(2). Theappro-
priation of this conditionequation in 4.2.4(4)iscompletely expressedbya
computer simulation taking into accountthe encoder of the servomotor.
4.2.2 Deterioration of PositioningPrecision Due to Torque
Quantization Error
(1)P
osition
Determination
of
the
Soft
wa
re
Serv
oS
ystem
Fo
rd
etermining
the
po
sition
of
the
soft

wa
re
serv
os
ystem,
the
effect
of
the torque quantizationerrorisconsidered. The positioningerror E
s
p
=
P
ref
− y ( ∞ )[pulse], whichisthe errorofobjectiveposition P
ref
[pulse] and
the steady-state value of the position output y ( ∞ )[pulse], is determined based
on the servoparameter K
p
, K
v
andthe angular acceleration resolution.The
relationship equation is derivedtheoretically. As illustrated in Fig. 4.7,the
servomotor is rotatedwith aconstantvelocityinput according to the objec-
tive position P
ref
.The position can be determined. If the angular acceleration
R
A

is
quan
tized,
the
ve
lo
cit
yo
ft
he
serv
om
otor
will
be
alsoq
uant
ized
in
eac
h
sampling time interval ∆t
v
of thevelocityloop. Thatis, in the servosys-
tem with the angular acceleration quantization, the velocityoutput is only
changed with theunit of R
A
∆t
v
[pulse/s]. This quantized resolution is called

the angular velocityresolution. From this case, forthe servosystem with an-
gular accelerationquantization, the velocityfeedbackiscarriedout untilthat
angular velocityoutput becomes 0[pulse/s]. When the angular velocityoutput
becomes zero, the velocityfeedbackiscut off andthe steady state is continued
until the position outputbecomes constant.
4.2T
orque
Resolution
89
(2)Relationship between PositioningError andAngular
AccelerationResolution
At the momentthatthe input is equaltothe objective position P
ref
,the
input to the quantizationterm of Fig. 4.5 is expressed by K
v
( K
p
( P
re
f
− y ) −
dy/dt). When this value largerthanthe angular acceleration resolution R
A
,
the position and velocityisfeedback. If the angular acceleration resolution is
not full, that is, dy/dt =0[pulse/s], the outputofthe quantizationterm is 0
and the position outputremains constant.
In the steady state that the position outputisconstant, thesize of the
input to the quantizationterm is expressed by | K

p
K
v
E
s
p
| with the positioning
error E
s
p
,asFig. 4.7.When thisvalueisnot full of resolution R
A
of theangular
acceleration,the position error E
s
p
can be expressed by K
p
, K
v
, R
A
as
| E
s
p
| <
R
A
K

p
K
v
. (4.8)
From (4.8), theupper limitofthe position error E
s
p
is proportional with the
angular accelerationresolution R
A
andinversely proportional to theposition,
velocityloopgain K
p
, K
v
.
4.2.3 Deterioration of Ramp Response Due to Torque
Quan
tization
Error
(1)Ramp Response of the Software ServoSystem
Next, with regard to the ramp input of the software servosystem, the effect of
torque quantizationerrorisconsidered. The objective trajectory of the servo
motorisgiven with the constantvelocity V
re
f
[pulse/s]. When the angular
accelerationisquantized in each R
A
,i

ft
he
ob
jective
angular
ve
lo
cit
yi
st
he
integer timesofthe angular velocityresolution, the angular velocityoutput
is notchanged formaking the objective angular velocityconsistentwith an-
gular
ve
lo
cit
yo
utput.H
owe
ve
r,
if
the
ob
jective
angular
ve
lo
cit

yi
sn
ot
the
0
0
T ime
P o s i t ion
E
p
s
P
r ef
Fig.
4.7.
Deteriorationo
fp
osition
con
trol
in
soft
wa
re
serv
os
ystem
90
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tization
Error
of
aM
ec
hatronic
Serv
oS
ystem
0
0
0
T ime
V eloc i ty
T
d
T
u
T
f
E
v
r
= R
A
∆ t
v
V
r ef
V

u
V
d
P o s i t ion
E
p
r
E
d
E
u
Fig. 4.8. Deterioration of ramp response in software servosystem
integer times of theangularvelocityresolution, the angular velocityoutput
is changed because of inconsistence between objectiveangularvelocityand
angular velocityoutput.
Fig. 4.8 illustratedthe variation of theangularvelocityoutput.The upper
part of Fig. 4.8 showsthe position fluctuation and the bottom part shows the
angular velocityfluctuation.FromFig. 4.8,the response is dividedintotwo
states: one is that the angular velocityoutput is belowthe objective angu lar
velocity(scale of
T
d
[s]) and another is that the angular velocityoutput is over
the objectiveangularvelocity(scale of T
u
[s]).
(2)State of Angular VelocityOutput under ObjectiveAngular
Velocity V
re
f

At the state of that the angular velocityoutput is belowthe objective angu-
lar velocity V
ref
,f
romt
he
angular
ve
lo
cit
yq
uant
ization,
the
outputa
ngular
velocityisas V
d
=  V
re
f
/ ( R
A
∆t
v
)  R
A
∆t
v
[pulse/s] (where  x  is expressed as

the maximalinteger below x ). The error V
re
f
− V
d
between objectiveangular
velocityand angular velocityoutput is made integral as theposition output
error. If the angular acceleration input is overhalf of theangularacceleration
resolution R
A
/ 2(refertoFig. 4.6), the positivepulseequivalenttothe angu-
lar acceleration resolution is generated. Whengenerating the pulse and the
position outputerroris E
d
[pulse], the angular acceleration input is expressed
as
K
v
( K
p
E
d
− V
d
)with the loop of Fig. 4.5.When thisvalueishalf of the
angular acceleration resolution R
A
/ 2, thefollowing relationship equation is
4.2T
orque

Resolution
91
successful
K
v
( K
p
E
d
− V
d
)=
R
A
2
. (4.9)
If solving the equation (4.9)for E
d
,itisas
E
d
=
R
A
+2K
v
V
d
2 K
p

K
v
. (4.10)
The amplitudeofthe position error is in the positivedirection in equation
(4.10). Additionally,the amplitude of this velocityerrorisas V
ref
− V
d
.
(3) State of Angular VelocityOutput overObjectiveAngular
Velocity V
ref
If generating the pulse equivalenttothe angular acceleration resolution,
theangularvelocityoutput R
A
∆t
v
is increased as V
u
=  V
ref
/ ( R
A
∆t
v
)+
1  R
A
∆t
v

[pulse/s]. The error V
u
− V
re
f
of theobjectiveangularvelocityand an-
gular velocityoutput is made integral as theposition output error. If the angu-
lar acceleration input is overhalf of theangularaccelerationresolution R
A
/ 2
(refer to Fig. 4.6), the negativepulseequivalenttothe angular acceleration
resolution is generated. If generating the pulseand the position output error
is E
u
[pulse], the angular acceleration input is expressed as − K
v
( K
p
E
u
+ V
u
).
If
this
va
lue
is
half
of

the
angular
acceleration
resolution
R
A
/ 2, thefollowing
relation equation is successful
− K
v
( K
p
E
u
+ V
u
)=−
R
A
2
. (4.11)
If solving the equation (4.11) for E
u
,itisas
E
u
=
R
A
− 2 K

v
V
u
2 K
p
K
v
. (4.12)
The amplitudeofthe position error is in the positivedirection in equation
(4.12). Moreover, the amplitude of thevelocityerrorisas V
u
− V
ref
.When
generating thisnegativepulse, the angular velocityoutput is back to the
state that the angular velo cityoutput of 4.2.3(2) is overthe objective angular
velocity. Because of these two states, afluctuation of theramp response exists.
(4) Amplitude and Cycle of PositionFluctuation
From Fig. 4.8, if theposition hasdeviation E
u
+ E
d
,when the errorbetween
the objectivevelocityand tracing velo city V
ref
− V
d
is continued at the time
T
d

,the time T
d
of that theangularvelocityoutput is continuously belowthe
ob
jective
angular
ve
lo
cit
yi
sa
s
92
4Q
uan
tization
Error
of
aM
ec
hatronic
Serv
oS
ystem
T
d
=
E
d
+ E

u
V
ref
− V
d
=
R
A
(1 − K
v
∆t
v
)
K
p
K
v
( V
ref
− V
d
)
(4.13)
by using equation (4.10) and(4.12).Similarly,the time T
u
of that is the
angular velocityoutput is continuously overthe objective angular velocity, is
as
T
u

=
E
d
+ E
u
V
u
− V
re
f
=
R
A
(1 − K
v
∆t
v
)
K
p
K
v
( V
u
− V
ref
)
. (4.14)
The fluctuation period T
f

[s] is as
T
f
= T
d
+ T
u
=
R
2
A
∆t
v
(1 − K
v
∆t
v
)
K
p
K
v
( V
ref
− V
d
)(V
u
− V
ref

)
(4.15)
by combining the T
d
of equation (4.13) andthe T
u
of equation (4.14). The
amplitude of position fluctuation E
r
p
[pulse] is as
E
r
p
= E
d
+ E
u
=
R
A
+ K
v
( V
d
− V
u
)
K
p

K
v
=
R
A
(1 − K
v
∆t
v
)
K
p
K
v
(4.16)
by combining the equation(4.10) andequation (4.12). The amplitudeofthe
velocity E
r
v
[pulse/s]
is
as
E
r
v
= R
A
∆t
v
(4.17)

from the angular accelerationresolution.
From above derivedequation (4.15), the equation (4.17) is the relationship
equation expressing the relation among fluctuation period T
f
,a
mplitude
of
po
sition
fluctuation
E
r
p
,velocityfluctuation amplitude E
r
v
andangularac-
celerationresolution R
A
.B
ya
nalyzing
their
prope
rties
fromt
heser
elation
equations,
the

amplitudeo
fp
osition
fluctuation
E
r
p
andthe velo cityfluctua-
tion amplitude E
r
v
areproportional to theangularaccelerationresolution R
A
,
andthe fluctuation period T
f
proportional to twice that of the angular ac-
celeration resolution R
A
.Inaddition, the fluctuation period T
f
is dependent
on the objectivevelocity V
ref
.The amplitude of position fluctuation E
r
p
and
thevelocityfluctuation amplitude E
r

v
arenot reliant on theobjectivevelocity
V
re
f
.T
he
angular
acceleration
resolution
R
A
is
dep
enden
to
nt
he
parameter
K
p
, K
v
, ∆t
v
of theservocontroller.
4.2T
orque
Resolution
93

4.2.4Derivation of Torque ResolutionDetermination
(1) Positioning Precision
If the positioning error output E
s
p
is not full of 1[pulse] of encoder outpu t,
thereisnoeffect of torque resolution compared with the encoder resolution.
If the rightside of equation (4.8)isnot full of 1as
| E
s
p
| <
R
A
K
p
K
v
< 1(4.18)
andbysolvingthe R
A
,the conditionofangularaccelerationresolution R
A
if
the position error E
s
p
is not full of 1can be expressed as
R
A

<K
p
K
v
. (4.19)
That is, in order to makethe positioning precision E
s
p
of theservomotor
is full of 1, theangularaccelerationresolution R
A
should be determinedby
satisfying the equation (4.19).
(2)Fluctuation of Ramp Response
In the ramp response, the torque resolution is determined when the amplitude
of angular velocityoutput deterioration E
r
v
andthe ampli tude of position out-
putd
eterioration
E
r
p
arewithin the allowance E
r
limv
[pulse/s] and E
r
limp

[pulse],
respectively.The upper bound R
Ap
[pulse/s
2
]ofthe angular acceleration res-
olution satisfying the conditionofamplitude of position output det erioration
can be calculated usingequation (4.16)
R
Ap
=
K
p
K
v
E
r
limp
1 − K
v
∆t
v
. (4.20)
The upperbound R
Av
[pulse/s
2
]o
ft
he

angular
acceleration
resolution
satisfy-
ing the conditionofthe amplitude of theangularvelocityoutput deterioration
can be calculated usingequation (4.17)
R
Av
=
E
r
limv
∆t
v
. (4.21)
The angular accelerationresolution R
A
is needed from equation (4.20) and
equation (4.21)
R
A
≤ min( R
Av
,R
Ap
) . (4.22)
That is, when the angular acceleration resolution R
A
can be determinedfor
satisfying the equation (4.22), therestraintofthe deterioration of ramp re-

sponse within thedemanded allowance canberealized.
94
4Q
uan
tization
Error
of
aM
ec
hatronic
Serv
oS
ystem
(3)Calculation of Bit Numbers of torque resolution
The angular accelerationresolution R
A
should correspond to the bit number
of the D/A, A/D conversion whichisadoptedfor thecurrentreference and
feedbackofthe software servosystem. First of all, the angular acceleration
resolution is convertedintothe torque resolution R
T
[Nm] usingthe pulse
number P [pulse/rev] equivalenttothe moment of inertia J
M
[kgm
2
]ofthe
motorand the encoder of one time rotation.
R
T

=
2 πR
A
J
M
P
. (4.23)
Next, the bit number of the A/D, D/A conversion should be converted.
Thatis, the maximum of the bit number of the A/D, D/A conversion adopted
forthe software servosystem must be able to output the maximumtorque
of themotor except the symbol bit.The relation equation amongst thebit
number B [bit] of the resolution of the A/D, D/A conversion,the maximum
torque T
max
[Nm] of themaximumtorqueofmotor andthe torque resolution
R
T
[Nm] should be given as
2
B − 1
=
T
max
R
T
=
T
max
P
2 πR

A
J
M
. (4.24)
The final conversion is basedonequation (4.23). The equation (4.24) is
changed aftersolution aboutbit number
B of theresolution of the A/D,
D/A conversion as
B =log
2
T
max
P
πR
A
J
M
. (4.25)
By usingthe bit number fr om equation (4.25), the performance of aA/D,
D/A conversion satisfying the demanded precisionofthe ramp response can
be determined.
(4) Numerical Example of Torque ResolutionDetermination
Thee
ffectiv
eness
of
usingt
he
relationship
be

twe
en
the
deriv
ed
con
trol
pe
r-
formance of thesoftware servosystem and the bit number of the A/D, D/A
con
ve
rsion
in
thes
oftwa
re
serv
os
ystem
is
ve
rifiedh
ere.
The
designed
po
sition
lo
op

gain
and
the
ve
lo
cit
yl
oo
pg
ain
of
thes
erv
o
controller are K
p
=40[1/s]and K
v
=200[1/s], respectively.The sampling
time interval forthe velocityloopis ∆t
v
=50[µ s]. The rated values of the servo
motorare J
M
=0. 13 × 10
− 4
[kgm
2
], T
max

=1. 47[Nm], P =5000[pulse/rev].
Formaking the positioning precision as (4.19), the deterioration of the ramp
resp onse as E
r
limp
=1[pulse] and E
r
limv
=1[pulse/s], the angular acceleration
resolution is determined by equation (4.22), and the calculationofthe bit
nu
mb
er
B of
thet
orquer
esolution
usinge
quation
(4.25)
for
B =1
5[bit].
4.2T
orque
Resolution
95
When existingquantizationofthe position informationinthe actualsoft-
ware servosystem, in order to investigate thedegree of the obtained the
control performance basedonthe torque resolution,the computer simulation

is made usingtorqueresolution and considering the quantizationofposition
information.The objective trajectory is u ( t )=10000t [pulse] (0 ≤ t<1[s]),
u ( t )=10000[pulse] (1 ≤ t ≤ 2[s]).The approximation of thevelocityinforma-
tion is basedonthe discrete errorofthe position information. Additionally,the
positioning precision is E
s
p
=1[pulse], the position fluctuation of the ramp re-
sponse is E
r
p
=2[pulse] and the velo cityfluctuation is E
r
v
=200[pulse/s].Even
without considering the torque resolution and only considering the quantiza-
tion of the position information, all of the positioningprecision, the position
fluctuation of the ramp response andthe velocityfluctuation have the same
values. When existing quantizationofthe position, theeffect of torque quanti-
zationinthe derivedtorqueresolution can be neglected formaking the results
consistentbetween consideringthe torque quantizationand notconsidering
the torque quantization. Moreover, in the desiredstate without considering
the quantizationofposition,ifcomparingthe designvalues andsimulation re-
sults, the expected control performance canbeobtainedbyusing thederived
torque resolution about the positioning precision and the position fluctuation
of the ramp response. The reason for deteriorationofthe designvalues caused
by velocityfluctuation is that theposition information is simply discrete when
approximating the velocityinformation.Thatis, theresolution of the velocity
approximationvalues is as 1[pulse]
/∆t

v
[s] =1/ (50 × 10
− 6
)=20000[pulse/s]
whenapproximating the velocityinformation basedonthe discrete values. If
standardizing the resolution of the velocityapproximation value, the velocity
fluctuation is very small at 1%.
Next,inorder to calculate the bit number of the torque resolution sat-
isfyingthe general requiredcontrol performance,the relationship equation
amongthe positioning error E
s
p
expressed by equation (4.8), the amplitude of
po
sition
fluctuation
E
r
p
expressed by equation (4.16), the velo cityfluctuation
amplitude E
r
v
expressed by equation (4.17), and the angular acceleration res-
olution R
A
can
be
con
ve

rtedi
nt
ot
he
bit
nu
mb
er
B of
thet
orquer
esolution
by equation (4.25). This relationship is shown in Fig. 4.9.
Accordingtothe use of Fig. 4.9, althoughthe bit number of the torque
resolution
cannot
be
wo
rk
ed
outf
romt
he
requiredc
on
trol
pe
rformance,t
he
positioning precision and the control performance of theramp response canbe

obtained fromthe bit number of the torque resolution of the actual operated
software servosystem.
(5)Relationship Among Control Performance, Torque Resolution
and ServoParameter
Therelationship amongst the control performance,torqueresolution and servo
parameter of the software servosystem is summarized as below.