AUTOMATIC DESIGN OF FUZZY CONTROL SYSTEMS FOR
AUTONOMOUS MOBILE ROBOTS
I. Ba u one1, F. J. Mo eno-Velo1, S. Sánchez-Solano1, R. Ma ín de Aga 2, A. Olle o2
1 Ins i u o de Mic oelec ónica de Se illa (IMSE). A d. Reina Me cedes, s/n.
Edi . CICA, 41012, Se illa, SPAIN
2 Dep. Ingenie ía de Sis emas y Au omá ica. E.S.Ingenie os. Camino
Descub imien os s/n. 41092, Se illa, SPAIN
[email p o ec ed]
P oc. 28 h Annual Con e ence o he IEEE Indus ial Elec onics Socie y (IECON’2002),
pp. 2457-2462, Se illa, No embe 5-8, 2002.
© 2002 IEEE. Pe sonal use o his ma e ial is pe mi ed. Howe e , pe mission o ep in / epublish his ma e ial o ad e -
ising o p omo ional pu poses o o c ea ing new collec i e wo ks o esale o edis ibu ion o se e s o lis s, o o euse
any copy igh ed componen o his wo k in o he wo ks mus be ob ained om he IEEE.
This ma e ial is p esen ed o ensu e imely dissemina ion o schola ly and echnical wo k. Copy igh and all igh s he ein
a e e ained by au ho s o by o he copy igh holde s. All pe sons copying his in o ma ion a e expec ed o adhe e o he e ms
and cons ain s in oked by each au ho ’s copy igh . In mos cases, hese wo ks may no be epos ed wi hou he explici pe -
mission o he copy igh holde .
Abs ac - This pape desc ibes he design and implemen-
a ion o a uzzy con olle o au onomous mobile
obo s. The ool X uzzy 3.0, de eloped a he IMSE
(Ins i u o de Mic oelec ónica de Se illa) has been used
o design a con olle o he Romeo 4R au onomous
ehicle designed and buil a he “Escuela Supe io de
Ingenie os”, Uni e si y o Se ille. The pape p esen s he
design o he con olle and eal expe imen s wi h Romeo
4R demons a ing he e iciency o he con olle .
I.INTRODUCTION
Some o he maneu e s ha should be pe o med by
an au onomous mobile obo , such as pa king in a
gi en place, a e easily pe o med by any human d i e
wi h a bi o p ac ice. The way in which human d i e s
usually exp ess hei con ol ac ua ion o pe o m a
maneu e (b ake, s ee ing wheel, e c.) is no qui e p e-
cise bu a he uzzy. We nei he need exac in o ma-
ion om he en i onmen o om ou ehicle o ca y
ou a success ul maneu e . Mos o he imes, we apply
a heu is ic knowledge which can be exp essed linguis-
ically by mo e o less chained i - hen ules. Fuzzy
logic p o ides a ma hema ical amewo k o ansla e
hese linguis ic and symbolic concep s in o nume ical
da a which can be handled by elec onic ci cui s. Many
wo ks ha e been epo ed in he li e a u e which show
he e ficiency o uzzy con olle s implemen ed in so -
wa e (gene al-pu pose p ocesso s) o ha dwa e (appli-
ca ion specific p ocesso s) [1-2].
As happens o any design p ocess, i is e y in e es -
ing o employ CAD ools when designing a uzzy con-
olle . This is pa icula ly ue nowadays when
educing he cos and he ime- o-ma ke o a p oduc
a e d i ing o ces o he indus y.
In he las ew yea s se e al CAD ools ailo ed o
he uzzy sys em design ha e been c ea ed [3-5]. The
CAD en i onmen employed in his pape is X uzzy
3.0, which has been de eloped a he IMSE (Ins i u o
de Mic oelec ónica de Se illa) wi h he objec i e o
being an open en i onmen wi h he leas possible lim-
i a ions [6]. Wi h his gene al objec i e, X uzzy 3.0 is
based on an specifica ion language (XFL3) ha eases
he desc ip ion and manipula ion o complex uzzy
sys ems hanks o he use o use -defined membe ship
unc ions, uzzy ope a o s (including linguis ic
hedges), and ule bases (admi ing hie a chical s uc-
u es) [7]. This objec i e has also mo i a ed he use o
Ja a as he p og amming language o X uzzy 3.0. This
means he use o an ad an ageous objec -o ien ed
me hodology and he flexibili y o execu ing X uzzy
3.0 in any pla o m wi h JRE (Ja a Run ime En i on-
men ) ins alled.
This pape desc ibes how X uzzy 3.0 can help he
use o iendly design a uzzy sys em o con olling
he pa king maneu e s o Romeo 4R, an au onomous
mobile obo de eloped a he Escuela Supe io de In-
genie os (ESI) o he Uni e si y o Se ille [8]. Sec ion
II shows he desc ip ion p ocess o he uzzy sys em.
Sec ion III explains how he beha io o he con olle
can be e ified by moni o ing he in e ence p ocess as
well as simula ing he con olle in a closed loop wi h a
model o he obo . Once he sys em has been designed
and alida ed, X uzzy 3.0 allows i s syn hesis in o se -
e al p og amming languages. Sec ion IV shows how
he con olle is syn hesized as a C code and in eg a ed
in o he so wa e execu ed by he compu e ha con-
ols Romeo 4R. Se e al expe imen al esul s o diago-
nal pa king maneu e s a e included o illus a e he
e ficiency and obus ness o he designed con olle .
II.FIRST STEP:DESCRIPTION PROCESS OF THE
FUZZY CONTROLLER
Pa king a ehicle a a gi en place has been a p ob-
lem usually add essed in he li e a u e o illus a e he
capabili ies o neu al and uzzy con olle s. A ypical
goal is o back up a ehicle so as o a i e a a desi ed
loading dock a a igh angle wi h he ho izon al [9].
The inpu a iables conside ed in hese epo ed con-
olle s a e he x posi ion o he ehicle and he ehi-
cles´s o ien a ion angle wi h he ho izon al. The ou pu
con ol a iable is he equi ed s ee ing angle (see Fig-
Au oma ic Design o Fuzzy Con ol Sys ems o Au onomous Mobile Robo s
I. Ba u one1, F. J. Mo eno-Velo1, S. Sánchez-Solano1, R. Ma ín de Aga 2, A. Olle o2
1 Ins i u o de Mic oelec ónica de Se illa (IMSE-CNM). A d. Reina Me cedes, s/n. Edi . CICA, 41012, Se illa, SPAIN
2 Dep. Ingenie ía de Sis emas y Au omá ica. E.S.Ingenie os. Camino Descub imien os s/n. 41092, Se illa, SPAIN
[email p o ec ed]
This wo k has been pa ially suppo ed by he Spanish CICYT P ojec s TAP99-0926-C04-01 and TIC2001-1726.
u e 1). The speed magni ude as well as he backwa d
di ec ion o d i ing a e cons an . These epo ed con-
olle s a e e ficien whene e he ehicle is a he a
om he loading dock bu ail i i is nea he dock and
wi h a bad o ien a ion angle, like ha in Figu e 1.
The pa king p oblem ha we add ess in his pape is
mo e complex and ealis ic: ou au onomous obo has
o pa k a a desi ed place, a i ing backwa d and a a
igh angle, bu i could d i e backwa d and o wa d o
achie e success om any s a ing posi ion and o ien a-
ion.
The app oxima ion we ha e aken o design his
con olle is o di ec ly emula e wha we would do as
d i e s. In his sense, ou fi s con ol ac ion is o
decide he di ec ion o d i ing ( he sign o he speed):
backwa d o o wa d, and he magni ude o he speed.
This decision is dynamic because i akes in o accoun
no only he cu en posi ion and o ien a ion o he
ehicle bu also i s p e ious speed. This knowledge is
included in o a ule base ha we call “di ec ion”.
In addi ion, he cons ain s imposed by Romeo 4R
ha e o be conside ed when deciding he new speed.
Fo example, Romeo has no an elec onically con-
olled b ake cu en ly, and i is impo an o ensu e
ha he d i ing di ec ion changes so ly. This means
ha he con olle should ne e decide o go o wa d a
a a he high speed i p e iously, he ehicle was d i -
ing backwa d a a a he high speed. This kind o con-
s ain s a e conside ed by a ule base ha we call
“b ake”. The inpu a iables o his ule base a e he
speed decided by he ule base “di ec ion” and he p e-
ious speed. I s ou pu is he new speed ha will be
adop ed by Romeo.
The second decision is o selec he p ope angle o
he wheels once we ha e decided o d i e backwa d o
o wa d. The speed selec ed by he ule base “b ake”
oge he wi h he x posi ion and he o ien a ion o he
ehicle a e he inpu a iables o ano he ule base ha
we call “wheel”.
As a esul o ou knowledge emula ion, he uzzy
con olle ha we ha e ob ained is a hie a chical sys-
em wi h he s uc u e shown in Figu e 2. The global
inpu a iables a e he posi ion (x, y), o ien a ion
(angle), and p e ious speed (olddi ) o he obo ; and
he ou pu a iables a e he s ee ing wheel angle
(wheel) and he new speed (di ec ion).
We ha e employed X uzzy 3.0 o desc ibe his con-
olle . X uzzy 3.0 di ides he desc ip ion o a uzzy
sys em in o wo pa s. One pa is he logical defini ion
o he sys em (i s s uc u e, he membe ship unc ions
ha ep esen he uzzy se s, and he ules o each ule
base). This pa can be defined ia g aphical use in e -
aces by using he ool x edi o by edi ing di ec ly a
“.xfl” file. The o he pa is he ma hema ical defini ion
o he di e en unc ions ha appea in he logical de -
ini ion (membe ship unc ions, connec i e ope a o s,
de uzzifica ion me hods, linguis ic hedges, e c.). This
pa can be defined ia g aphical use in e aces by
using he ool x pkg o by edi ing di ec ly a “.pkg” file.
This wo old defini ion allows us o c ea e and use
ou own membe ship unc ions, de uzzifica ion me h-
ods, e c. In ou case, all he ma hema ical unc ions
employed a e desc ibed in he xfl.pkg file p o ided
wi h X uzzy 3.0. Fo example, he de uzzifica ion
me hod employed in he ule base “wheel” is he Fuzzy
Mean me hod, which calcula es he weigh ed a e age
o he consequen single on alues. This p o ides a so
in e pola ion among he 7 single on alues conside ed
o ep esen he wheel angle. On he o he side, he
de uzzifica ion me hod ha we employ in he ule
bases “di ec ion” and “b ake” is a me hod ha we call
“MaxLabel”. I selec s he single on consequen o he
ule whose ac i a ion deg ee is maximum, because he
decision made by hese ule bases has o be c isp: o -
wa d o backwa d bu no an a e age o bo h. The
desc ip ion o he MaxLabel me hod can be seen in
Figu e 3. This figu e illus a es he g aphical in e ace
o he ool x pkg whe ein he X uzzy use can define
new uzzy ope a o s (logical connec i e, linguis ic
hedges, membe ship unc ions, o de uzzifica ion
me hods).
wheel
(x,y)
Fig. 1: Example o he diagonal pa king p oblem.
he objec i e
angle
ehicle
angle
Fig. 2: S uc u e o he designed con olle .
x
angle
di ec ion
y
olddi
b ake
wheel
di ec ion
wheel
We ha e used he ool x edi o speci y he logical
defini ion o ou con olle , as can be seen in Figu e 4.
The membe ship unc ions employed a e: 5 uzzy se s
o co e he x posi ion; 1 single on alue and 3 uzzy
se s o co e he y posi ion; 7 uzzy se s o co e he
o ien a ion; 5 single on alues o co e he speed (i s
sign eflec s he d i ing di ec ion); and 7 single on al-
ues o co e he s ee ing wheel angle. The speed alues
a e a he slow as co esponds o pa king maneu e s
(be ween -1m/s and 1m/s). The wheel angle alues a e
limi ed by he maximum cu a u e ha Romeo can
apply. Figu e 5 shows he window o x edi whe ein we
ha e defined he membe ship unc ions o he o ien a-
ion a iable.
The ule bases can employ di e en ma hema ical
unc ions o ep esen he uzzy ope a o s. Fo exam-
ple, he ule bases “di ec ion” and “wheel” use di e -
en de uzzifica ion me hods, as commen ed abo e.
Figu e 6 illus a es he window o x edi whe ein we
ha e selec ed he ma hema ical unc ions o he uzzy
ope a o s in he ule base “di ec ion”. An ad an age o
X uzzy 3.0 is ha he use can eely modi y he ma h-
ema ical unc ions ha desc ibe hese linguis ic ope a-
o s (wi h he ool x pkg men ioned abo e).
We ha e also used x edi o define he ules o each
ule base. The XFL3 language employed by X uzzy
3.0 eases he ansla ion o linguis ically exp essed
ules because admi s linguis ic hedges like “mo e o
less equal o”, “sligh ly equal o”, e c., and ela ions
like “g ea e han” o smalle han”. Fo ins ance, one
o he ules in he ule base “di ec ion” is:
‘i (y is “equal o” nea and x is “s ongly equal o”
cen e and angle is “equal o g ea e han” le small
and angle is “equal o smalle han” igh small) hen
di is backwa d’.
III.SECOND STEP:OFF-LINE VERIFICATION
PROCESS OF THE FUZZY CONTROLLER
Al hough he defini ion o he uzzy con olle
ansla es ou expe knowledge, we migh ha e o go -
en o conside some si ua ions o no conside p op-
e ly o he ones. This is why pe o mance o he
con olle has o be e ified p io o expe imen wi h
Romeo 4R. Fo his pu pose, we ha e employed h ee
e ifica ion ools o X uzzy 3.0: x 3dplo ,x m and
x sim.
The ool x 3dplo allows us o isualize he beha io
o one o he con ol a iables e sus wo o he ones.
Fig. 3: G aphical use in e ace o he ool x pkg.
Fig. 4: Main window o x edi .
Fig. 5: Membe ship unc ions o he ehicle o ien a ion.
Fig. 6: Window o x edi o selec he uzzy ope a o s.
This is e y use ul o s udy, o ins ance, i ou con ol-
le is sa e enough o a oid c ashes wi h a possible
pa emen a y=0. Figu e 7 shows he su ace co e-
sponding o he new speed decided by he con olle
agains he y posi ion and he o ien a ion o Romeo,
when he x-posi ion coo dina e is ze o and he p e ious
speed was -1m/s. We can see ha i he y posi ion is
nea ze o and he angle is no qui e ze o, he con olle
decides o s op o be e s aigh en he ca by d i ing
o wa d in subsequen s eps.
The ool x m is e y use ul o moni o how is wo k-
ing he in e ence p ocess. Fo ins ance, i we wan o
know why he new speed is -1m/s when he x posi ion
is ze o, he p e ious speed was -1m/s, he y posi ion is
2 m and he angle is 180º, we can use x m as shown in
Figu e 8 o disco e ha he ule esponsible o his
decision is he ule 30 o he ule base “di ec ion”:
‘i (y is “equal o” nea and x is “equal o” cen e
and (angle is “smalle han” le o “g ea e han”
igh )) hen di is backwa d’.
Al hough wi h he p e iously men ioned ools we
can analyze he con olle i sel , a e y impo an s ep
in any con ol design is o simula e he con olle wo k-
ing in a closed loop wi h he plan . Fo his simula ion,
we ha e employed he ool x sim o X uzzy 3.0. The
beha io o ou plan , Romeo 4R, has been desc ibed
by he bicycle kinema ic model [10], conside ing a
fi s -o de dynamic esponse in he se ling o he
speed and he wheel angle imposed by he con olle .
The ou pu s o he simula ion pe o med by x sim can
be sa ed o a log file o pos e io g aphical ep esen a-
ion. As an example, Figu e 9 illus a es he simula ed
beha io o Romeo 4R when i s a s (wi h speed 0) a
x=3.9m, and y=9m, wi h an angle o 90º. The a ows
indica e he d i ing di ec ion and he shaded one ma ks
he s a ing poin .
Wi h his o -line simula ion, we can analyze he
obus ness o ou con olle agains pe u ba ions. Fo
ins ance, i he ue speed aken by Romeo 4R is 40%
g ea e han ha imposed by he con olle and we
epea he simula ion o Figu e 9, he esul s ob ained
a e shown in Figu e 10.
IV.THIRD STEP:ON-LINE VERIFICATION PROCESS
OF THE FUZZY CONTROLLER
Once checked ha ou sys em is obus enough, we
ha e e ified i s con ol beha io wi h he ue plan ,
Romeo 4R. This obo is an elec ical ehicle p o ided
wi h a se o senso s and ac ua o s ha make i capable
o au onomous na iga ion (Figu e 11). The in o ma-
ion collec ed by he senso s and ha equi ed by he
ac ua o s is cen alized by a compu e placed a he
back o he obo and which also implemen s he con-
ol algo i hms. In ou pa king applica ion, he com-
pu e has o go e n a mo o con ol ca d which in u n
go e ns, independen ly, he s ee ing and ac ion elec-
ical mo o s o Romeo. These elec ical mo o s has o
ecei e, espec i ely, he wheel angle and new speed
commands om ou con olle . In addi ion, he mo o
Fig. 7: S udying he beha io o he a iable “di ec ion”.
Fig. 8: Moni o ing he in e ence p ocess.
F
ig. 9: Simula ing he con olle wi h a model o Romeo 4R.
0
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con ol ca d eads he di ec ion and ac ion encode s
o he engines. These measu es, oge he wi h he
in o ma ion p o ided by a gy oscope ha e o be p o-
cessed by he compu e o es ima e he cu en posi-
ion, o ien a ion and speed o he obo , which a e he
inpu a iables equi ed by ou con olle .
The compu e ope a es wi h Linux and all he d i -
e s o he senso s and ac ua o s ha e been w i en in C
code. In addi ion, an in e ace has been de eloped ha
include a wide se o unc ions (p og ammed in C++)
o wo k easily wi h senso s and ac ua o s. Ha ing his
in e ace, he inclusion o ou con olle is as simple as
gene a ing i s C code. Fo his ask, we ha e employed
he syn hesis ool x c o X uzzy.
Despi e execu ing he con ol code and all he o he
equi ed ou ines, he compu e ope a es a eal ime
wi hou p oblems because a con ol cycle pe iod o
100 ms is enough o ou applica ion.
Figu e 12 shows wo examples o expe imen al a-
jec o ies ollowed by Romeo when s a ing a di e en
posi ions (ma ked by he shaded a ows) and wi h di -
e en o ien a ions. Compa ing Figu e 12a wi h Figu e
9 we can see ha expe imen al esul s do no di e
e y much om simula ed esul s.
Figu e 13a illus a es he e olu ion in ime o he
wheel angle e e ence gi en by he uzzy con olle (in
solid line) and he eal angle aken by Romeo (in
dashed line), o he expe imen in Figu e 12a. We can
see how he angle e e ence changes so ly (as a conse-
quence o he Fuzzy Mean de uzzifica ion me hod
applied by he ule base “wheel”) and how i is ol-
lowed apidly by he eal angle.
Figu e 13b compa es he e olu ion in ime o bo h
he speed e e ence (in solid line) and he eal speed (in
dashed line) co esponding o he expe imen in Figu e
12a. In his case, he e e ence changes ab up ly
(because o he MaxLabel de uzzifica ion me hod
applied) and he dynamic o he eal speed is slowe
Fig. 10: Simula ing he con olle wi h pe u ba ions.
0
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Fig. 11: Romeo 4R success ully pa ked.
Fig. 12: Expe imen al esul s.
0
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✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧
(a)
(b)
han ha o he eal angle. The wo hwhile ac shown
by his figu e is ha he obo is con olled e ficien ly
o commu e so ly be ween d i ing backwa d and o -
wa d, hus mee ing he imposed equi emen s.
V.CONCLUSIONS
The au oma ic design and implemen a ion o uzzy
con ol sys ems in ol es a numbe o ac i i ies ela ed
o he defini ion o he con olle , including i s logical
s uc u e and he ma hema ical defini ion o unc ions,
he o -line e ifica ion o he con olle by means o
simula ion and he on-line e ifica ion and es ing wi h
he eal p ocess.
This pape p esen s he applica ion o he X uzzy
3.0 ool o he design o he uzzy con olle o au ono-
mous ehicles. Pa icula ly he design and implemen-
a ion o a con olle o he Romeo 4R is desc ibed.
The esul s ob ained wi h he eal ehicle a e simila o
he simula ion esul s and Romeo 4R is able o pe o m
success ully a pa king maneu e e en when he ehicle
is ini ially close o he pa king posi ion, wi h a bad o i-
en a ion, and ha ing o maneu e au onomously o
pa king.
Fu u e wo k will include a compa ison o he p e-
sen ed uzzy logic me hod wi h o he echniques based
on he consecu i e execu ion o pa h planning, gene a-
ion and con ol echniques in di e en p ac ical cases.
VI REFERENCES
[1] Munaka a, T., Jani, Y., Fuzzy sys ems: an o e iew,
Communica ions o he ACM, Vol. 37, N. 3, 1994.
[2] Sugeno, M., Ed., Indus ial Applica ion o Fuzzy
Con ol, No h-Holland, pp. 19-40, 1985.
[3] Home page o FuzzyTech: h p://www. uzz-
y ech.com
[4] Home page o FIDE: h p://www.ap onix.com/
ide/
[5] Home page o TILShell: h p://www.o ech
eng .com/ uzzy/TilShell.h ml
[6] F. J. Mo eno-Velo, I. Ba u one, S. Sánchez-Solano,
A. Ba iga, "X uzzy 3.0: A De elopmen En i on-
men o Fuzzy Sys ems", P oc. 2nd IEEE In .
Con . on Fuzzy Logic and Technology (EUS-
FLAT’2001), pp. 93-96, Leices e , 2001.
[7] F. J. Mo eno-Velo, S. Sánchez-Solano, A. Ba iga,
I. Ba u one, D. R. López, “An Speci ica ion Lan-
guage o Fuzzy Sys ems”, Ma hwa e & So Com-
pu ing, Vol. 8, No. 3, pp. 239-253, 2001.
[8] A. Olle o, B. C. A ue, J. Fe uz, G. He edia, F.
Cues a, F. López-Pichaco and C. Nogales. “Con ol
and pe cep ion componen s o au onomous ehi-
cles guidance. Applica ion o he Romeo Vehicles".
Con ol Enginee ing P ac ice, Vol. 7, No. 10, pp
1291-1299, Oc obe 1999.
[9] S.-G. Kong, B. Kosko, “Compa ison o Fuzzy and
Neu al T uck Backe -Uppe Con ol Sys ems”,
Chap e 9 in Neu al Ne wo ks and Fuzzy Sys ems,
B. Kosko, P en ice Hall, 1992.
[10] Y. Zhao, S. L. Bemen , “Kinema ics, Dynamics and
Con ol o Wheeled Mobile Robo s”, P oc. IEEE
In . Con . on Robo ics and Au oma ion, pp. 91-96,
Nice, 1992.
Fig. 13: E olu ion o : (a) wheel angle e e ence (solid line)
and eal angle (dashed line), (b) speed e e ence (solid line)
and eal speed (dashed line).
515253545
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
ime (s)
cu a u e (m-1)
ime (s)
speed (m/s)
5 15253545
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
(a)
(b)