Disse a ion
Deg ee in Indus ial Technologies Enginee ing (GETI)
Vehicle i ual sensing: es ima ion o
he longi udinal eloci y and y e o ces
Repo
Au o : Ma ia Ga aba os Saboya
Di ec o : A nau Do ia Ce ezo
Co-Supe iso : S e ano de Pin o
Summons: June 2021
Escola Tècnica Supe io
d’Enginye ia Indus ial de Ba celona
Acknowledgemen s
Fi s o all I would like o hank S e ano De Pin o and McLa en Au omo i e LTD o hei p edis-
posi ion on aking pa on a inal deg ee p ojec as his one. App ecia e S e ano o all he wo k
he has done since No embe 2020 by p opo ioning me all so o in o ma ion and knowledge
o a so impo an opic as i ual sensing. Fo in oducing me o ehicle con ol wo ld, o such
dedica ion du ing all hese mon hs and o accep ing o co-supe ise my p ojec . I am eally
pleased o all he suppo ecei ed and o eaching me ex a knowledge on his opic, ecom-
mending me eally in e es ing books and being on he lookou o me unde s anding e e y hing.
Fu he mo e, I would like o hank A nau Do ia o p oposing me he inc edible and unique
oppo uni y o aking pa in such a p ojec oge he wi h McLa en Au omo i e LTD. Fo hese
eigh mon hs o comple e dedica ion and suppo , gi ing me ad ice and sugges ions based on
his knowledge and expe ience. Fo all he mee s we ha e had discussing abou how e e y hing
should be scheduled and all he heo e ical classes he has gi en o me o help me unde s and
such a opic I was no amilia wi h be o e.
Finally, I would like o hank my amily and iends o all he suppo du ing he ealiza ion o
he p ojec and all he yea s o uni e si y s udies.
1
Abs ac
In his p ojec , a i s con ac wi h i ual sensing is explained. Deno ing he impo ance o
senso s in o ma ion, o unde s anding ehicle dynamics and he in luence he es ima o s ha e
when modeling a ehicle.
Two di e en ehicle models a e p esen ed, Bicycle and Fou -wheel model, oge he wi h a
linea and a nonlinea es ima o s, Kalman Fil e and Ex ended Kalman Fil e , espec i ely. A
he end o he p ojec , wo app oaches ha e been ca ied ou in o de o de e mine he co ec
unc ionali y o hose es ima o s.
Finally, a mo e de ailed app oach has been done in which he Ex ended Kalman Fil e is e alu-
a ed. In his case, he ehicle is no modelled, and hose es ima ed IMU pa ame e s ha e been
eplaced by eal da a p o ided by McLa en Au omo i e LTD.
2
Con en s
Nomencla u e 6
Ac onyms 8
1 In oduc ion 9
1.1 Mo i a ion......................................... 9
1.2 S a eo hea ....................................... 10
1.3 Objec i es ......................................... 11
1.3.1 Long- e mobjec i es............................... 11
1.4 Ou lineo hep ojec ................................... 12
2 Kalman Fil e 14
2.1 Kalman il e ........................................ 14
2.2 Ex endedKalmanFil e ................................. 15
3 Vehicle modelling 17
3.1 Vehicledynamics ..................................... 17
3.2 Ty ebeha iou ....................................... 18
3.3 MagicFo mula y emodel................................ 19
3.4 Fou -wheelmodel..................................... 21
3.5 Bicyclemodel ....................................... 22
3.6 Summa y.......................................... 23
4 Longi udinal speed es ima ion 25
4.1 Kinema icmodel ..................................... 25
4.2 Kinema ic ou -wheel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Summa y.......................................... 33
5 Es ima o e i ica ion 35
5.1 S ee ing ep esen a ion.................................. 35
5.2 Longi udinal wheel eloci y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Summa y.......................................... 41
6 Economical analysis 42
7 Conclusions 44
Bibliog aphy 46
3

Lis o Figu es
1.1 Schema iza ion o he p ojec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Obse abili y scheme. [8] ................................. 15
3.1 Rep esen a ion o ehicle coo dina e ame. Cou esy om McLa en Au omo i e LTD. . . . 17
3.2 Ty e coo dina e ame.[1] ................................. 18
3.3 Pu e longi udinal slip cha ac e is ics acco ding o longi udinal load. . . . . . . . 20
3.4 Pu e la e al slip cha ac e is ics acco ding o la e al load. . . . . . . . . . . . . . . . 20
3.5 The ou wheel model. Cou esy om McLa en Au omo i e LTD. ................ 21
3.6 Thebicyclemodel..................................... 23
4.1 Longi udinal speed es ima ion. Compa ison wi h KF. . . . . . . . . . . . . . . . . 26
4.2 La e al speed es ima ion. Compa ison wi h KF. . . . . . . . . . . . . . . . . . . . . 27
4.3 Longi udinal eloci y o he ou wheels. . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Longi udinal eloci y o he ehicle. Compa ison wi h he EKF. . . . . . . . . . . 30
4.5 Longi udinal eloci y o on le wheel. Compa ison wi h EFK. . . . . . . . . . . 31
4.6 Longi udinal eloci y o on igh wheel. Compa ison wi h EFK. . . . . . . . . . 32
4.7 Longi udinal eloci y o ea le wheel. Compa ison wi h EFK. . . . . . . . . . . 32
4.8 Longi udinal eloci y o ea igh wheel. Compa ison wi h EFK. . . . . . . . . . 33
5.1 S ee ing ep esen a ion o bo h on wheels. . . . . . . . . . . . . . . . . . . . . . 36
5.2 Wheels longi udinal eloci y es ima ion o a high gamma ehicle o McLa en Au-
omo i eLTD........................................ 37
5.3 Wheels longi udinal eloci y o a high gamma ehicle o McLa en Au omo i e
LTD.............................................. 37
5.4 F on le wheel eloci y. Compa ison wi h EKF. . . . . . . . . . . . . . . . . . . . 38
5.5 F on igh wheel eloci y. Compa ison wi h EKF. . . . . . . . . . . . . . . . . . . 39
5.6 Rea le wheel eloci y. Compa ison wi h EKF. . . . . . . . . . . . . . . . . . . . . 40
5.7 Rea igh wheel eloci y. Compa ison wi h EKF. . . . . . . . . . . . . . . . . . . . 40
4
5
page 6 Repo
Nomencla u e
Symbols Uni s Meaning
A - Sys em ma ix
B - Inpu ma ix
C - Ou pu ma ix
D - Feed h ough ma ix
x- Inpu ec o
y- Measu emen ec o
Kk- Kalman gain
Pk- E o co a iance ma ix
Q - P ocess noise co a iance ma ix
R - Measu emen noise co a iance ma ix
d- S a e upda e unc ion
h- Measu emen unc ion
˙
ψ ad/s Yaw a e
¨
ψ ad/s2Yaw accele a ion
FxN Longi udinal o ce
FyN La e al o ce
FzN Ve ical o ce
MxNm O e u ning momen
MyNm Rolling esis ance momen
MzNm Aligning momen
mkg Vehicle mass
xm/s Longi udinal eloci y
ym/s La e al eloci y
γ ad Cambe angle
αij ad Ty e sideslip angle ( on o ea , igh o le )
β ad Vehicle sideslip angle
Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 7
Symbols Uni s Meaning
CxN/ ad Longi udinal co ne ing s i ness
CyN/ ad La e al co ne ing s i ness
CiN/ ad Co ne ing s i ness o each wheel ( on o ea )
λ- Longi udinal slip
Lim Leng h o he ehicle ( on o ea )
im Wid h o he ehicle ( on o ea )
δ ad S ee ing angle
x,wh m/s Longi udinal eloci y o he wheels
y,wh m/s La e al eloci y o he wheels
axm/s2Longi udinal accele a ion
˙ xm/s2Longi udinal accele a ion
aym/s2La e al accele a ion
˙ ym/s2La e al accele a ion
Chap e 2
Kalman Fil e
2.1 Kalman il e
The Kalman Fil e consis s o he es ima ion o a model alue, he s a e ec o , o he p e ious in-
s an which is ob ained by he measu ed alue in he ac ual ins an . This alue can be es ima ed
acco ding o he sys em dynamics and he noise measu emen s o he model [7].
This il e is linea , meaning ha he unce ain y accoun s on a Gaussian noise and he co-
a iance e o s. In his linea con igu a ion, his il e is he op imal o use, bu when non-
linea i y appea s, o he Kalman il e s should be s udied, such as he Unscen ed and he Ex-
ended Kalman il e s.
This epo will ocus on he linea Kalman il e and he Ex ended Kalman il e o nonlinea
sys ems [1]. S a ing wi h he simples , i is a linea and au onomous sys em desc ibed by he
equa ions (2.2).
˙x=Ax +Bu (2.1)
y=Cx +Du (2.2)
Being a linea sys em means ha ma ices A, B and C abo e a e cons an , and D is non-exis en .
The il e is di ided in wo pa s, he p edic ion ep esen s he es ima ion o he s a e and he e -
o co a iance acco ding o he pos e io i s a e and he sys em’s knowledge and i is ep esen ed
by he equa ions (2.3)-(2.5):
ˆxk=Adˆxk−1+Bduk−1(2.3)
ˆyk=Cˆxk+Duk(2.4)
ˆ
Pk=Adˆ
Pk−1AT
d+Q(2.5)
While he co ec ion pa uses he ou pu o he sys em o es ima e he pos e io y s a e, and i
is p esen ed by he equa ions (2.6)-(2.8):
Kk=PkCT(CP −
kCT+R)−1(2.6)
ˆxk=ˆx−
k+Kk(yk−ˆyk)(2.7)
Pk=(I−KkC)Pk(2.8)
14

Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 15
I is impo an no icing ha Q and R a e ma ices which depend on he dimensions o he s a e
and he ou pu s o he sys em, and a e he co a iance ma ices o he model and measu emen
noise, espec i ely.
Fo his il e o s a ope a ing, an ini ial alue o he s a e and he e o co a iance mus be
de ined. Only hen, he Kalman Fil e will be ini ialized and will p o ide all he necessa y
measu emen s o ob ain he minimum e o o he es ima ed s a es.
The Kalman Fil e can be seen as an op imal obse e . The la es occu s when he cu en s a e o
he p ocess is es ima ed, x, and his happens hanks o he measu emen and he inpu ec o s,
yand u, espec i ely.
Figu e 2.1: Obse abili y scheme. [8]
Fu he mo e, obse abili y can be de ined as a p ope y o he sys em o bo h model and mea-
su emen equa ions, which is independen o he ype o es ima o used [1]. The e o e, a sys em
is obse able when he ini ial s a e can be de e mined. As a consequence, he whole sys em can
be de e mined. Fo a Kalman Fil e o be co ec ly applied, he sys em has o be obse able.
2.2 Ex ended Kalman Fil e
The Ex ended Kalman Fil e is a nonlinea sys em which depends on he s a e, and i s linea iza-
ion is accomplished by means o a successi e app oxima ion o each poin om he change o
ma ices A, B and C alues.
The mos gene al o m o he disc e ized s a e space ep esen a ion is:
ˆxk= d(ˆxk−1, uk−1)(2.9)
ˆyk=h(ˆxk, uk)(2.10)
As he ini ial il e is desc ibed by a linea sys em, i s o all a linea iza ion o he sys em mus
occu [1]. This will happen by he ans o ma ion o he ma ices A and C owa ds F and H. This
ans o ma ion is al eady conside ed in he ollowing equa ions (2.11)-(2.12) o he Ex ended
Kalman Fil e .
Fk−1=∂ d(x, u)
∂x |ˆxk−1,uk−1(2.11)
Hk=∂h(x, u)
∂x |ˆxk,uk(2.12)
page 16 Repo
This il e is also di ided in wo pa s. The i s one, which co esponds o he p edic ion pa
o he linea sys em and is based on he eplica o he nonlinea sys em. The disc e iza ion o
his il e is deno ed by (2.13)-(2.15).
ˆxk= ˆxk−1+4 (ˆxk−1, uk−1)(2.13)
ˆyk=h(ˆxk−1, uk−1)(2.14)
Pk=Fk−1Pk−1+Q(2.15)
The equa ions co esponding o he co ec ion pa a e desc ibed in (2.16)-(2.18):
Kk=P−
kHT
k(HkP−1
kHT
k+R)−1(2.16)
ˆx+
k=ˆx−
k+Kk(yk−ˆy−
k)(2.17)
P+
k=(I−KkHk)P−
k(2.18)
The old A ma ix now co esponds o Fk−1while he old C ma ix o Hk. Subsequen ly, hose
ma ices alue will be ob ained acco ding o he Kalman il e equi ed in each case.
Chap e 3
Vehicle modelling
3.1 Vehicle dynamics
Be o e s a ing wi h he accu a e de ini ion o each y e and i s e ec owa ds he ehicle be-
ha iou , i is necessa y o de ine he mo e impo an dynamic cha ac e is ics in o de o model
co ec ly he ehicle.
This is why he nex coo dina e ames mus be in oduced. Those coo dina es make e e ence
on h ee o a ional-deg ees-o - eedom, bu also h ee di e en eloci ies acco ding o he sys-
em, as seen in Figu e 3.1.
To see mo e in o ma ion abou he ansla ional and o a ional equa ions o mo ion o his ame
see [1].
Figu e 3.1: Rep esen a ion o ehicle coo dina e ame. Cou esy om McLa en Au omo i e LTD.
17
page 18 Repo
In his epo , as only he longi udinal mo ion o he ehicle will be s udied, he only momen
ha will be ele an will be he yaw, as will be seen in he ollowing chap e s. Also longi udinal
and la e al eloci ies mus be conside ed.
3.2 Ty e beha iou
This sec ion is based on s udying he di e en p ope ies and cha ac e is ics o a ehicle and i s
pneuma ic y es. P o iding he mos common y e models, and making emphasis on he simple
ones. Addi ionally, his sec ion goes in o de ail abou he dis inc i e ehicle models.
Ty e o ces and momen s a e esponsible o he ehicle mo ion, and a e ex emely impo an
when i comes o ep esen he dynamic beha iou o he ehicle. Fo he co ec modeling o
he ehicle i is impo an o unde s and he ehicle dynamics o i , bu also he e ec o he
o ces o he ehicle’s beha iou .
Desc ibing he y e oad in e ac ion begins wi h desc ibing a coo dina e ame a ached o he
cen e o he con ac pa ch as seen in Figu e 3.2.
Figu e 3.2: Ty e coo dina e ame.[1]
Acco ding o he ep esen a ion o Figu e 3.2, he x-axis is he in e sec ion be ween he y e
plane line and he g ound; he z-axis is pe pendicula o he g ound plane and will always
poin upwa ds; inally, he y-axis is on he g ound plane and poin s acco ding o he igh -
handed ag eemen .
The y e o ien a ion is ep esen ed by he cambe angle γand he sideslip angle α. The i s one
de ines he inclina ion o he y e plane acco ding o he x-axis, while he second one, he z-axis
o a ion o he eloci y ec o and he x-axis. I can be measu ed acco ding o:
an(α) = y
x
(3.1)
being x, y he x and he y coo dina es o he eloci y ec o .
Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 19
S aigh away, o ces and momen s ep esen ed in Figu e 3.2 will be b ie ly explained, jus mak-
ing emphasis in hose pa ame e s ha will be ele an o he models ha will be s udied u he
on.
Those o ces a e ep esen ed as:
•Fx−Longi udinal o ce :is applied along he x-axis. When he ca is accele a ing, a
posi i e o que is applied, his o ce is de ined as posi i e. And when he ca is b aking,
ice e sa.
•Fy−La e al o ce :is applied along he y-axis. When le -co ne ing i is de ined posi i e,
he wheel o a es in an i-clockwise, and ice e sa when igh -co ne ing.
Those momen s a e ep esen ed as:
•Mz−Aligning momen :abou he z-axis. On accoun o he la e al y e o ce displaced
backwa ds om he cen e o he con ac pa ch.
As a plana mo ion is conside ed, he e ical o ces and he o e u ning and olling esis ance
momen will no be conside ed. To he con a y, longi udinal and la e al o ces a e undamen al
when i comes o descib ing ehicle dynamics because a e he ones esponsible o he appea -
ance o a shea mechanism, gene a ing a y e- oad ic ion coe icien ha will p e en slip om
happening.
3.3 Magic Fo mula y e model
The Magic Fo mula y e model was i s c ea ed by Hans Baas ian Pacejka [6], a Du ch p o esso
expe in ehicle dynamics, and consis s o he modelling o he y e model.
I is based in an empi ical model ha le us s udy he y e o ces beha iou acco ding o he slip.
The main objec i e is o ep esen he y e and ehicle beha iou o small slip alues acco ding
o a linea app oxima ion o he slip.
Y(x) = Dsin(Ca c an(Bx −E(Bx −a c an(Bx)))) + SV(3.2)
being, B (S i ness ac o ), C (Shape ac o ), D(Peak alue), E (Cu a u e ac o ) and SV(Ve -
ical shi ).
The linea app oxima ion men ioned abo e ep esen s he linea pa o bo h igu es 3.3 and
3.4, espec i ely, and i is desc ibed by he ollowing equa ions:
Fx=Cxλ(3.3)
Fy=−Cyα(3.4)
To make Figu e 3.3 easie o unde s and, ABS is in oduced. ABS is he B aking Con ol Sys em,
and i s main objec i e is o p e en he wheel om blocking when b aking. The wheel blocks
when he slip a io is no ope a ing in he op imal alues o he igu e below, in he pic o i . Bu
when his happens, ehicle s abili y and s ee abili y a e e ained so he slip s a s ope a ing in
he op imal alues and he wheel will no block. In conclusion, ope a ing in he op imal alues
allows ha ing a be e pe o mance o he wheel.

page 20 Repo
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
-2000
-1500
-1000
-500
0
500
1000
1500
2000
Figu e 3.3: Pu e longi udinal slip cha ac e is ics acco ding o longi udinal load.
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
-2000
-1500
-1000
-500
0
500
1000
1500
2000
Figu e 3.4: Pu e la e al slip cha ac e is ics acco ding o la e al load.
When i comes o s udy he beha iou acco ding o he la e al o ce, he objec i e o he la e al
slip is he same as he slip a io, o ope a e nea he op imal alues, bu in his case, i s beha iou
is desc ibed in he TCS sys em.
Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 21
I s unc ionali y can be app ecia ed in Figu e 3.4. The simila i y o Figu e 3.3 can be easily
app ecia ed, bu he main di e ence is, apa om he ac ha la e al y e cha ac e is ics a e
aken in o conside a ion, he la e al o ce o he y es is always deno ed as a nega i e o ce
because i is opposed o he mo emen o he y e.
3.4 Fou -wheel model
Now ha he y e model has al eady been in oduced, his p ojec will p esen he di e en
models a ehicle can be modelled by. The i s one is he ou -wheel model, his is a comple e
ep esen a ion o he ehicle conside ing he e ec o he ou wheels, hei s ee ing angle, he
longi udinal and la e al eloci ies o all o hem and he e ec o hei o ces. E en- hough i
does no conside he suspensions o any wheel.
Figu e 3.5: The ou wheel model. Cou esy om McLa en Au omo i e LTD.
Fo hese model, as all he wheels o he ehicle a e conside ed, he e is no need he slip angles
o all ou wheel a e iden ical. Will be seen la e on bu , when hese happens he e is al eady
and exis en model which will model i . The main di e ence be ween hem is ha in his case,
all slip angles can ha e and will ha e a di e en alue. This is on accoun o he ou wheel
model is a mo e ealis ic one adjus ed in eali y, and he o he one is jus and app oxima ion o
i .
The equa ions o mo ion o his model a e:
m(˙ x− y˙
ψ) =Fx, l cos(δ l) + Fx, cos(δ )−Fy, l sin(δ l)
−Fy, sin(δ ) + Fx, l +Fx, (3.5)
m(˙ y+ x˙
ψ) =Fy, l cos(δ l) + Fy, cos(δ ) + Fx, l sin(δ l)
+Fx, sin(δ ) + Fy, l +Fy, (3.6)
2Izz ¨
ψ= 2l (Fx, l sin(δ l) + Fy, l cos(δ l) + Fx, sin(δ ) + Fy, cos(δ ))
+ (Fy, l sin(δ l)−Fx, l cos(δ l)−Fy, sin(δ ) + Fx, cos(δ ))
−2l (Fy, l +Fy, ) + (Fx, −Fx, l)(3.7)
page 22 Repo
No icing ha he only pa o he ehicle which is in con ac wi h he oad a e he y es, i is easy
o ind ou ha a co ec beha iou o ha one comes om a co ec s udy and ep esen a ion
o he y es and hei pa ame e s.
This is why when modelling a ca is as impo an modeling co ec ly he equa ions o mo ion,
as i is o model co ec ly he y es. Fo his model, hey a e ep esen ed wi h he equa ion (3.8)
below:
" x,wh,ij
y,wh,ij #=


cos δij sin δij 0
−sin δij cos δij 0
0 0 1







x
y
0

+


0
0
˙
ψ

×


±li
± j
2
0





(3.8)
The wheel’s eloci y is necessa y when i comes o inding he alue o he la e al o ces ep e-
sen ed in he equa ions o mo ion. This la e al o ce is ob ained by a linea app oxima ion on
he co ne ing s i ness.
Fy,ij =Ciαij (3.9)
No ice ha ideno es on and ea wheels and jdeno es igh and le wheels.
The necessa y sideslip angle comes om inding he longi udinal and la e al eloci ies o each
wheel acco ding o he equa ion (3.10).
αij = a c an y,wh,ij
x,wh,ij !(3.10)
Fo mo e in o ma ion abou hese model and i s epe cussion o he e ical o ces see sec ion
2.1.2 o [1].
3.5 Bicycle model
A e in oducing he comple e model o he ehicle, ou -wheel model, a mo e simple one
will be p esen ed. This one is he bicycle model, which is a simpli ica ion o he p e ious, by
assuming iden ical slip angles on bo h le and igh wheels and also igno ing he la e al load
ans e .
I is based in he s udy o he longi udinal and la e al ehicle dynamics. Al hough i is no a
sui able model o simula ion, i is use ul o es ima ion and con ol pu poses. As his epo
ocuses on he s udy o he longi udinal ehicle dynamics, his model will be modelled in o -
de o s udy he ehicle beha iou owa ds hese cha ac e is ics and can e lec on whe he he
simula ion ob ained is speci ic and exac o can be imp o ed.
Figu e 3.6 is a ep esen a ion o his, om whe e can be ex ac ed he longi udinal, la e al and
yaw equa ions o mo ion.
Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 23
l l
Vy
Fx,
Fy,
β
α
α
δ
Ψ
Vx
Fy,
Fx,
Figu e 3.6: The bicycle model
Those equa ions o mo ions a e:
˙ x=1
m(Fx, cos(δ )−Fy, sin(δ ) + Fx, ) + y˙
ψ(3.11)
˙ y=1
m(Fy, cos(δ ) + Fx, sin(δ ) + Fy, )− x˙
ψ(3.12)
¨
ψ=1
Izz
(l Fx, sin(δ ) + l Fy, cos(δ )−l Fy, )(3.13)
Conside ing only on ac ion and a co ne ing s i ness linea app oxima ion, equa ion (3.14)
is desc ibed.
Fy,i =Ciαi(3.14)
being i, on o ea , espec i ely. No ice ha , o ind he sideslip angle, he wheel eloci ies in
he x-axis and y-axis a e necessa y. Those equa ions a e ind below:
αi= a c an y,wh,i
x,wh,i !(3.15)
" x,wh,i
y,wh,i#=


cos δisin δi0
−sin δicos δi0
0 0 1







x
y
0

+


0
0
˙
ψ

×


±li
0
0





(3.16)
3.6 Summa y
In his chap e a b ie in oduc ion o ehicle dynamics is p esen ed, oge he wi h he impo -
ance o y e beha iou and modelling. Fo his, all he necessa y pa ame e s ha will be used
o model he y e and ehicle beha iou a e desc ibed.
A e ha , he Magic Fo mula y e model is de ined. S aigh away, he y e’s beha iou acco d-
ing o he Pacejka o mula is ep esen ed in igu es 3.3 and 3.4, and i s ela ion wi h he secu i y
sys ems explained in Chap e 1 is de ined.
Fu he mo e, no only y es a e impo an o be modelled, bu so is he ehicle. This is why
he nex sec ions o his chap e ep esen wo di e en ehicle models, he i s one, conside s
page 30 Repo
As can be seen in Figu e 4.4, his p oblem is sol ed. This has occu ed because he ehicle
model conside ed o his es ima ion is mo e complex and, consequen ly, p o ides mo e and
mo e accu a e in o ma ion. In eali y, his in o ma ion will also be ob ained by he IMU senso ,
he same as wi h he o he model, bu in he case o simula ions, his in o ma ion will be mo e
simila o eali y.
The compa ison o he la e al eloci y ob ained in Figu e 4.2 wi h he one ob ained om an
Ex ended Kalman Fil e is no objec o his simula ion, bu can be ound in [5]. E en hough,
i is impo an knowing ha he con e gence p oblem ob ained in he p e ious sec ion will be
sol ed.
0 5 10 15 20 25 30 35 40 45 50
0
10
20
30
40
50
60
70
80
90
es ima ed eloci y
ou -wheel model
Figu e 4.4: Longi udinal eloci y o he ehicle. Compa ison wi h he EKF.
Es ablishing a colo e e ence acco ding o Figu e 4.3, a on le , on igh , ea le , and
ea igh eal eloci y is compa ed o hei espec i e es ima ions a e applying and Ex ended
Kalman Fil e . This can be ound in igu es 4.5, 4.6, 4.7 and 4.8.
The discussion o he eloci y es ima ion done o Figu e 4.5 will apply o he o he wheels
ep esen ed in he ollowing igu es. They will no be discussed as he explana ion is he same.

Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 31
0 5 10 15 20 25 30 35 40 45 50
66
68
70
72
74
76
78
80
82
84
86
es ima ed
eal
Figu e 4.5: Longi udinal eloci y o on le wheel. Compa ison wi h EFK.
No ice ha his es ima ion is qui e accu a e and is no much in luenced by noise, excep when
lowe ing he eloci y. Despi e ha , hose simula ions can be conside ed co ec wi h an e o
calcula ed o 0,8715%. Following, Table 4.2 will p esen he e o pe cen age o he es ima ed
eloci ies ob ained om he oo squa e mean (RSME). Fo ha , he ansien s a e has no been
conside ed.
Table 4.2: RSME calcula ion be ween he eal and he es ima ed eloci y.
Longi udinal eloci y E o (%)
Vehicle eloci y 0,8501
F on le wheel 0,8715
F on igh wheel 0,8715
Rea le wheel 0,8533
Rea igh wheel 0,8534
page 32 Repo
0 5 10 15 20 25 30 35 40 45 50
66
68
70
72
74
76
78
80
82
84
86
es ima ed
eal
Figu e 4.6: Longi udinal eloci y o on igh wheel. Compa ison wi h EFK.
0 5 10 15 20 25 30 35 40 45 50
70
72
74
76
78
80
82
84
86
es ima ed
eal
Figu e 4.7: Longi udinal eloci y o ea le wheel. Compa ison wi h EFK.
Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 33
0 5 10 15 20 25 30 35 40 45 50
70
72
74
76
78
80
82
84
86
es ima ed
eal
Figu e 4.8: Longi udinal eloci y o ea igh wheel. Compa ison wi h EFK.
Fo all o hese simula ions, he slip a io has no been conside ed when desc ibing he Ex ended
Kalman Fil e . As explains [1] in Chap e 4, some app oaches ha e been made which conside
he slip a io so small ha can be neglec ed, as can be app ecia ed in he ollowing equa ions:
ij =(λij + 1) w,ij (4.13)
ij = w,ij (4.14)
Fo his eason, each wheel eloci y is app oxima ed o he longi udinal eloci y o each wheel
in he con ac pa ch. This con ac pa ch is desc ibed in Figu e 3.2. I is he con ac poin be ween
he wheel and he g ound plane, which in no mal condi ions, i s eloci y is desc ibed acco ding
o he ollowing equa ions:
w,ij =


cos δij sin δij 0
−sin δij cos δij 0
0 0 1







x
y
0

+


0
0
˙
ψ

×


±li
± j
2
0





(4.15)
4.3 Summa y
A e ha ing desc ibed he di e en Kalman il e s and he di e en ways o model a ehicle,
his chap e is in oduced o de e mine he unc ionali y o his es ima o s acco ding o each
ehicle model.
page 34 Repo
The i s sec ion es ima es, acco ding o a linea sys em, he eloci ies o he cen e o g a i y o
a ehicle, jus conside ing one on and one ea wheel (Bicycle Model in Sec ion 3.5).
Figu es 4.1 and 4.2, ep esen he longi udinal and la e al eloci y o a ehicle compa ed o he
es ima ed eloci ies acco ding o a linea Kalman Fil e , espec i ely. I can be seen how he
la e al eloci y is no es ima ed co ec ly. In his sec ion he mo i e om his is explained and a
solu ion is p oposed.
The solu ion is o implemen a nonlinea es ima o which will conside he ou wheels o he
ehicle (Fou -Wheel Model in Sec ion 3.4). These es ima ions s a s wi h he linea iza ion o
he sys em and hen i s disc e iza ion.
As can be seen in igu es om 4.5 o 4.8, and wi h he calcula ed e o o Table 4.2, he ou
wheels a e es ima ed co ec ly. So his il e is he sui able o be implemen ed.
Chap e 5
Es ima o e i ica ion
As has been explained du ing he i s chap e o his epo , one o he main objec i es was
o design an es ima o able o compa e he eal eloci y o a McLa en Au omo i e LTD high
gamma ehicle wi h he es ima ed eloci y owa ds an Ex ended Kalman Fil e . In his chap e ,
he simula ions ob ained will be p esen ed and explained in de ail.
Be o e s a ing explaining he esul s ob ained, i is impo an no icing a big di e ence in com-
pa ison wi h he simula ions explained in Chap e 4. P e iously, he ehicle was modelled
acco ding wo di e en models, Bicycle o Fou -Wheel model, depending on he in o ma ion
wan ed. Remembe he la es allows o ob ain mo e in o ma ion han he o me because o
ha ing conside ed he ou wheels o he ehicle and bo h s ee ing angles o he on wheels,
he e is no hing simpli ied in he e. On he con a y, o s a his simula ion, he model o he
ehicle was no conside ed. This is because he in o ma ion ex ac ed om he model o en e as
an inpu in he il e o Kalman implemen ed (in his case, Ex ended Kalman Fil e ) is di ec ly
p o ided by McLa en Au omo i e LTD as inpu signals.
Ha ing now cla i ied why he p e iously explained models will no be conside ed he e, he
esul s ob ained in he simula ions will be discussed in he ollowing sec ions.
5.1 S ee ing ep esen a ion
Figu e 5.1 ep esen s he s ee ing angle o bo h on wheels. This igu e helps unde s anding
he upcoming igu es o he longi udinal eloci ies o he wheels.
The in o ma ion o he s ee ing angle desc ibed when he ehicle is u ning igh o le , ol-
lowing he signal c i e ia deno ed by Figu e 3.5. In he es ima o designed, i en e s as an inpu
which helps es ima e he di e en wheels eloci ies and he longi udinal eloci y o he ehicle.
I can be seen how his ajec o y is eally complex, bu , wha is mo e impo an , is ha bo h
on wheels will be submi ed o he same s ee ing. This may no happen, and le s o a mo e
complex s udy which is no objec o his p ojec , bu mo e in o ma ion abou his di e en
con igu a ions can be ound in [9].
35

page 36 Repo
0 0.5 1 1.5 2 2.5 3 3.5 4
104
-6
-4
-2
0
2
4
6
Figu e 5.1: S ee ing ep esen a ion o bo h on wheels.
5.2 Longi udinal wheel eloci y
Figu e 5.2 ep esen s he es ima ed eloci y o he ou wheels. To ob ain hem, was c ucial
unde s anding he measu es p o ided by McLa en Au omo i e LTD, and he uni s wi h which
hey whe e measu ed. A e ha , he ou wheels in o ma ion en e he es ima ion as an inpu
wi hou conside ing whi e Gaussian noise. This is because as his da a is eal da a ob ained
om he senso s o he ehicle when going es ing, he noise is al eady included. In ela ion
wi h Chap e 2, his da a is he measu ed da a yko he disc e ized s a e space ep esen a ion
o he Ex ended Kalman Fil e (equa ions (2.9) - (2.10)).
Meanwhile, Figu e 5.3 ep esen s he exac same g aphic as be o e bu wi hou ha ing es ima ed
he eloci y, meaning, his is done di ec ly om he da a p o ided. A i s sigh , an impo an
simili ude be ween bo h o hen can be app ecia ed. This is why, wi hou u he delay, hose
eloci ies will be compa ed in de ail.
Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 37
0 0.5 1 1.5 2 2.5 3 3.5 4
104
-20
0
20
40
60
80
100
120
140
160
180
on le
on igh
ea le
ea igh
Figu e 5.2: Wheels longi udinal eloci y es ima ion o a high gamma ehicle o McLa en Au o-
mo i e LTD.
0 0.5 1 1.5 2 2.5 3 3.5 4
104
0
20
40
60
80
100
120
140
160
180
on le
on igh
ea le
ea igh
Figu e 5.3: Wheels longi udinal eloci y o a high gamma ehicle o McLa en Au omo i e LTD.
page 38 Repo
To keep in mind he eloci ies de ined ollowing, emembe Figu e 3.5. Figu es 5.4 and 5.5 com-
pa es on eloci ies om bo h, le and igh , wheels, while Figu es 5.6 and 5.7 compa es he
ea eloci ies om bo h wheels. As was p e iously app ecia ed, he es ima ion o each eloc-
i y is done co ec ly, meaning, he Ex ended Kalman Fil e design is a good il e ha complies
wi h his unc ionali y independen ly o he ype o ehicle s udied un il now.
The ma hema ical ex ac ion o hese eloci ies is he same as explained in Chap e 4, and a e
deno ed by he equa ion (4.15). In his case, he slip is no s ill being conside ed, and because o
he du a ion o he p ojec , his will be one o he u he objec i es o achie e, as been explained
in Sec ion 1.3.1.
0 0.5 1 1.5 2 2.5 3 3.5 4
104
-20
0
20
40
60
80
100
120
140
160
180
es ima ed
eal
Figu e 5.4: F on le wheel eloci y. Compa ison wi h EKF.
One o he main objec i es o es ing McLa en Au omo i e LTD da a wi h he es ima o designed
in Chap e 4 was o de e mine whe he ha es ima o was good calib a ed o any pa ame e
change should be done.
Vehicle i ual sensing: es ima ion o he longi udinal eloci y and y e o ces page 39
The conclusion o ha objec i e is ha he e is a li le pa ame e modi ica ion which will be
objec o he u u e objec i es explained be o e (Sec ion 1.3.1) in o de o educe he e o pe -
cen age o he ea le wheel, which inc eases a lo in compa ison wi h he o he wheels. Those
e o pe cen ages a e shown in Table 5.1.
As has been explained in he p e ious chap e , his e o has been calcula ed acco ding o he
oo mean squa e p ocedu e (RSME). Bu , as eal da a is used in his simula ions, unc ion
nanmeadian o Ma lab has been necessa y o dele e he NaN alues o he look-up able.
Table 5.1: RMSE calcula ion o he longi udinal eloci y o he wheels.
Longi udinal eloci y E o (%)
F on le wheel 1,8833
F on igh wheel 1,9007
Rea le wheel 6,1109
Rea igh wheel 1,8767
0 0.5 1 1.5 2 2.5 3 3.5 4
104
-20
0
20
40
60
80
100
120
140
160
180
es ima ed
eal
Figu e 5.5: F on igh wheel eloci y. Compa ison wi h EKF.
Bibliog aphy
[1] Sebas iaan Van Aals . “Vi ual Sensing o Vehicle Dynamics.A model-based app oach o
indi ec meadu emen o he ehicle mo ion s a es and y e o ces.” PhD hesis.
[2] Wenchang Li e al. “Adap i e Ca -Following Con ol o In elligen Elec ic Vehicles”. In:
2018 IEEE 4 h In e na ional Con e ence on Con ol Science and Sys ems Enginee ing (ICCSSE).
2018, pp. 86–89.
[3] A. Alloum, A. Cha a a, and M. Rombau . “Vehicle dynamic sa e y sys em by nonlinea
con ol”. In: P oceedings o Ten h In e na ional Symposium on In elligen Con ol. 1995, pp. 525–
530.
[4] Manuel Acos a, S a is Kana achos, and Michael E. Fi zpa ick. “Accu a e i ual sensing
o e ical i e o ces o enhanced handling dynamics”. In: IECON 2017 - 43 d Annual
Con e ence o he IEEE Indus ial Elec onics Socie y. 2017, pp. 4009–4014.
[5] Da id Gomez. Obse e design o la e al ehicle dynamics con ol. 2021.
[6] H. Pacejka. Ti e and Vehicle Dynamics. Jan. 2012.
[7] Xiao Chao e al. “Vehicle Longi udinal Speed Es ima ion Based on Kalman Fil e ”. In: 2020
IEEE In e na ional Con e ence on A i icial In elligence and Compu e Applica ions (ICAICA).
2020, pp. 718–721.
[8] Domingo Biel e al. “Lec u e 6a: Con ol using s a e obse e s”. In: Con ol Theo y. Bachelo ’s
deg ee in Enginee ing Physics. 2017.
[9] Na haniel William T oupe. “Op imiza ion o s ee ing geome y o ul a-high-mileage e-
hicles.” PhD hesis. 2011.
46