Vehicle virtual sensing : estimation of the longitudinal velocity and tyre force

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. 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