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HAL Id: tel-02592741

https://hal.inrae.fr/tel-02592741

Submitted on 15 May 2020

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Stabilité dynamique des véhicules légers tout-terrain.

Nouvelles solutions : application aux véhicules légers de type quad

N. Bouton

To cite this version:

N. Bouton. Stabilité dynamique des véhicules légers tout-terrain. Nouvelles solutions : application aux véhicules légers de type quad. Sciences de l’environnement. Doctorat Vision et Robotique, Université Blaise Pascal de Clermont-Ferrand, 2009. Français. �tel-02592741�

(2)

o

CemOA: archive ouverte d'Irstea / Cemagref

(3)

CemOA : archive ouverte d'Irstea / Cemagref

(4)

(5)

(6)

TCL

(7)

TCL

(8)

(9)

LLT

(10)

(11)

(12)

(13)

h kr

TCL

(14)

TCL

(15)

(16)

(17)

(18)

•

(19)

TCL

(20)

(21)

(22)

m·ay

hT

mayhT=c

2(Fn2−Fn1)

mg

ay ≥ cg 2hT

(23)

ϕv h=OG hr=OQ hT=h+hr

may(hcos(ϕv)+hr)+mghsin(ϕv)=c

2(Fn2−Fn1)

ay ≥g^c₂−hsin(ϕv) (hcos(ϕv)+hr)

(24)

(25)

quad+pilote

CdGpilote

(26)

+

•

••

hT

SSF= c 2hT

(27)

ay

g= ^c²−hsinϕv

hcosϕv+hr

fr

P −m.ay

(28)

θ1 θ2

l1 l2 G

α

θ1 θ2

fr

α=min(θ1,θ2)·fr

fr l1 l2 fr fr

l1 l2 α

fr

(29)

Ucrit

U T

2·h m

J ϕv

(30)

U=mg(h(cos(ϕv)+sin(ϕv)))

dϕdUv=0 ϕv=π/4

ϕv π/4

T=1

2(J+2mh²)˙ϕv2

E∆=Ucrit−(U+T)

DRM =1−U+T Ucrit

TCL

(31)

TCL

TCL TCL=Fn1−Fn2

Fn1 Fn2

TCL TCL=Fn1−Fn2

Fn1+Fn2

TCL

TCL TCL |TCL|<0.8

(32)

(33)

TCL

TCL TCL

(34)

(35)

TCL

(36)

y

TCL

|TCL|<Rs Rs

(37)

0<Rs<1

δr

TCL

|TCL|> 0.9 δr=kr(|TCL|−0.9) kr δs

δs= kp·˙ϕv+kd·¨ϕv

(38)

+

TCL

(39)

TCL

(40)

+

(41)

TCL

•

(42)

•

TCL

•

••

•

(43)

km.h⁻¹ kg

cm³ cm³

•

(44)

•

• cm³

•

(45)

(46)

(47)

TCL

TCL ^Ce^m

OA: archive ouverte d'Irstea / Cemagref

(48)

cm³

kg m.s⁻¹

(49)

TCL

• ψ˙

• v

• δ

◦

GHz

(50)

m.s⁻¹

◦.s⁻¹

◦.s⁻¹ ^Ce^m

OA: archive ouverte d'Irstea / Cemagref

(51)

km.h⁻¹

Hz

TCL

(52)

TCL

v δ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0

5 10 15 2120.6 25

Temps (s) Vitesse (km.h−1 )

A

B C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0

5 10 15 20 25

Temps (s)

Angle de braquage (°)

B

C

A

(53)

t=3s 6m.s⁻¹=21.6km.h⁻¹ t=7.5s

t=14.3s

TCL

t=9st=12s t=14s TCL t=14.3s

TCL

(54)

c hT L ay

TCL TCL

|TCL|≈2

c hTay

g

TCL

m m

TCL m TCL

m

m m

TCL

m m

TCL m

m

m m

(55)

TCL

m

m TCL

TCL

ay=v·˙ψ=v²·tan(δ) L

|TCL|≈2 c

hTv²tan(δ) Lg

TCL v²tan(δ) L=1.28m L=1.35m

TCL v²tan(δ)

TCL

(56)

L=1.28m

=0.12 =

0.1205 L=1.35m =0.114

=0.112

TCL v²·tan(δ)

TCL

(57)

TCL

quad+pilote

TCL TCL

(58)

TCL

(59)

(60)

•

(61)

TCL

(62)

•C

•Cm

•Fl

•Ft

•Cf

•Ca

•Mg

•Rx

•Ry

•Rz

•ω CemOA: archive ouverte d'Irstea / Cemagref

(63)

gl

gl = ^ω−^Rdyn^Vx

max |ω|,|_Rdyn^Vx |

Rdyn

ω Vx

gl [−1,1] gl

gl=1

(64)

α

V

α

α = arctan _|^V_V^y_x_|

Vx Vy V

gl gt

gt = tan(α) gt = (1−|gl|)tan(α)

g= g_l²+g_t²

(65)

Fl Ft

l

L Kz

Fz δr

δr=Fz

Kz

Rnom

R²_nom= L 2

2+(Rnom−δr)²

δr δr<<Rnom

L= 8Rnom Fz

Kz

(66)

P xp yp

Pmax

P(xp)=4Pmax

L xp(1−xp

L)

Fz= ^L

0P(xp)ldxp=2PmaxLl 3

P(xp)=6Fz

L²lxp(1−xp

L)

A xa (xp,yp)

(67)

µ

σg=µP(xp),xa<xp<L

σg=6µFz

L²lxp(1−xp

L),xa<xp<L γ σgx σgy

σgx =σgcos(γ) σgy =σgsin(γ)

γ cos(γ)=^g_g^l

sin(γ)=^g_g^t

σgx =^6µF_L2l^zxp(1−^x_L^p)^g_g^l σgy =^6µF_L2l^zxp(1−^x_L^p)^g_g^t

σax

σay ∆xp= glxp

∆yp=gtxp

σax(xp,gl) = Kxglxp, 0≤xp≤xa

σay(xp,gt) = Kygtxp, 0≤xp≤xa

Kx Ky

(68)

xa

(σax)²+(σay)²=σ_g²

xa=L 1− lL²

6µFz (Kxgl)²+(Kygt)^{2 (1/2)}

xa

glc = _K^6µF_x_lL^z2

gtc = ^K_K^x_y (g_lc²−g_l²)







Fx = ^L

0 l/2

−l/2(σax+σgx)dxpdyp

= ^K^x^x₂²â^lg^l+^µg_g^l^F^z 1−3(^x_Lâ)²+2(^x_Lâ)³







Fy = ^L

0 l/2

−l/2(σay+σgy)dxpdyp

= ^K^y^lx₂²â^g^t+^µg^t_g^F^z 1−3(^x_Lâ)²+2(^x_Lâ)³

Fx =µxFz

Fy =µyFz

µx=µcos(γ) µy=µsin(γ)

(69)

TCL

y x

y(x+Sh)=Dsin[Carctan(Bx−E(Bx−arctan(Bx)))]+Sv

Ft Fl

α gl

Fl(gl+Shx) = Dxsin[Cxarctan(Bxgl−Ex(Bxgl−arctan(Bxgl)))]+Svx

Ft(α+Shy) = Dysin[Cyarctan(Byα−Ey(Byα−arctan(Byα)))]+Svy

Dx Dy

Bx By Cx Cy

Ex Ey

SvxSvyShx

Shy

(70)

γ Fz

Fl = Ds in(Carctan(BΦ))









Φ = (1−E)gl+^E_Barctan(Bgl) D = a1F_z²+a2Fz

C = a0

B = ^a³_CDe^F^z²^+a_a5Fz⁴^F^z

E = a6F_z²+a7Fz+a8

Ft = Ds in(Carctan(BΦ))+∆Sv









Φ = (1−E)(α+∆Sh)+^E_Barctan(B(α+∆Sh)) D = a1F_z²+a2Fz

C = a0

B = (1−a12|γ|)â³^sîn(a⁴ârctan(a_CD ⁵^F^z⁾⁾ E = a6F_z²+a7Fz+a8

∆Sh = a9γ

∆Sc = (a10F_z²+a11Fz)

ai

•

ai

(71)

•

Fz

α∈[−4^◦,4^◦]

(72)

gl α





Fl = Cl.gl

Ft = Ct.α Cl,Ct>0

Cl Ct

ai

Cl Ct

(73)

TCL

•

•R0(x0,y0,z0)

•R1(x1,y1,z1)

•O

•a

•b

(74)

•L

•R

•δ

•ψ R1 R0

•v ψ˙

ψ=˙ v·tan(δ) L

•u

•αfαr β

•Iz O

•m

•FlfFlrFf Fr





m ˙ucos(β)−u˙βsin(β)−u˙ψsin(β) = (Flr+Flfcos(δ)+Ffsin(δ)) m ˙usin(β)+u˙βcos(β)+u˙ψcos(β) = (−Fr+Flfsin(δ)−Ffcos(δ))

(75)

Izψ =[bF¨ r+a(Flfsin(δ)−Ffcos(δ))]

β ψ˙ u







˙

u = _m¹[Flfcos(δ−β)+Flrcos(β)+Ffsin(δ−β)−Frsin(β)] β =˙ _um¹ [Flfsin(β−δ)−Ffcos(β−δ)−Frcos(β)−Flrsin(β)]− ˙ψ ψ =¨ _I¹_z[bFr+a(Flfsin(δ)−Ffcos(δ))]









u = ^vcos(α_cos(β)^r⁾

αr = arctan tanβ−_ucos(β)^b˙^ψ αf = arctan tanβ+_ucos(β)^a˙^ψ −δ

αr = β−^b˙_u^ψ αf = β+^a˙_u^ψ−δ











˙

u = _m¹[Flfcos(δ−β)+Flrcos(β)+Cf(.)αfsin(δ−β)−Cr(.)αrsin(β)] β =˙ _um¹[Flfsin(δ−β)−Cf(.)αfcos(β−δ)−Cr(.)αrcos(β)−Flrsin(β)]− ˙ψ ψ =¨ _I¹_z[bCr(.)αr+a(Flfsin(δ)−Cf(.)αfcos(δ))]

Cf(.) Cr(.)











β =˙ _um¹ [−Ffcos(δ−β)−Frcos(β)]− ˙ψ ψ =¨ _I¹_zz[bFr−aFfcos(δ)]

(76)











ψ =¨ _I¹_z(−aCf(.)αf+bCr(.)αr) β = −˙ _um¹(Cf(.)αf+Cr(.)αr)− ˙ψ αr = β−^b˙_u^ψ

αf = β+^a˙_u^ψ−δ u ≈ v

a b L

m Iz

Cf(.) Cr(.)

ψ˙ β

TCL

•• z2y2 x2z2

R2(x2,y2,z2)

IG/R2 =



Ix 0 0 0 Iy 0 0 0 Iz





(77)

•ϕv

•G m

•h

•c

•P=mg g

•Fn1

•Fn2

•Fa

TCL

ϕv

Fa kr br

−→ Fa=1

h(krϕv+brϕ˙v)−→y2

h kr











m−→aG·−y→1= −→ P+−→

Fa+−→

Fn1+−→

Fn2 ·−y→1

m−→aG·−z→1= −P+→ −F→a+−→Fn1+−→Fn2 ·−z→1

−−−→∆G/R2·−x→2= −−−−→MG,Fn1+−−−−→MG,Fn2 ·−x→2

(78)

−→aG

G −−−→∆G/R2 G R2(x2,y2,z2) −−−−→MG,Fn1

−−−−→

MG,Fn2

−→Fn1 −→Fn2

G ϕ¨v Fn1 Fn2

¨

ϕv= 1

hcos(ϕv)h˙ϕv2sin(ϕv)+h˙ψ²sin(ϕv)+v˙ψ+b¨ψ− krϕv+brϕ˙v

mh cos(ϕv) Fn1+Fn2=m −h¨ϕvsin(ϕv)−h˙ϕv2cos(ϕv)+g− krϕv+brϕ˙v

mh sin(ϕv) Fn1−Fn2=2

cIxϕ¨v+(Iz−Iy) ˙ψ²cos(ϕv)sin(ϕv)−hsin(ϕv)(Fn1+Fn2)

Ix Iy

Iz TCL

TCL ϕv

TCL TCLpermanent=

−2c (Iz−Iy) ˙ψ²cos(ϕv)sin(ϕv)−hsin(ϕv)(Fn1+Fn2) Fn1+Fn2

(Iz−Iy) ˙ψ²cos(ϕv)sin(ϕv) [hsin(ϕv)(Fn1+Fn2)]

TCLpermanent=2

c(hsin(ϕv)) hsin(ϕv)

TCLpermanent hsin(ϕv) =₂^c

ϕv sin(ϕv)≈ϕv cos(ϕv)≈1 h˙ψ²sin(ϕv)<<v˙ψ h˙ψ²sin(ϕv)<<^k_mh^r^ϕ^vcos(ϕv)

0=v˙ψ− krϕv

mh v˙ψ=aG,y1

|TCLpermanent|≈2

c h²aG,y1 kr

m

TCLpermanent

|TCLpermanent|≈2

c hTaG,y1

g

(79)

hT

h²

kr

m

=hT

g

hT=mh²g kr

TCL

¨

ϕv = 1

hcos(ϕv)[h˙ϕv2sin(ϕv)+h˙ψ²sin(ϕv)+u˙ψcos(β)+˙usin(β) +u˙βcos(β)− krϕv+brϕ˙v

mh cos(ϕv)]

β

TCL

(80)

Gq

Gp

G quad+pilote

Mp

Mq

hp Gq Gp

h1 Gq

h2 Gq G

h2= Mp

Mp+Mq hp

h3 G

ϕx quad+pilote

ϕv quad+pilote

z z=ϕx−ϕv

αp

TCL quad+pilote

αp

h3 z TCL

(81)

hidentifie

h h1

h1=hidentifie−h2=hidentifie− Mp

Mp+Mq hp

ϕv

αp=0 h3=hidentifie

αp h3

quad+pilote

h3= h²₁+h²₂−2h1h2cos(π−αp)

z=cos⁻¹ h²₃+h²₁−h²₂ 2h3h1





ϕx = ϕv+sign(αp)z

¨

ϕx = ¨ϕv

˙

ϕx = ˙ϕv

Fn1 Fn2

ϕx ϕv

quad+pilote h3

TCL

αp

(82)

d

d= Mp

Mp+Mq hpsin(αp) αp

αp=sin⁻¹(NKδ)

K N =+1

N =−1 N =0

d= Mp

Mp+Mq hpNKδ

TCL

v ψ˙

δ h kr br

h kr TCL

br

TCLTCL

h kr

TCL

(83)

TCL h kr

TCL

+

Y = Φ(X,ξ)

X Y ξ

Y_p^m m

J(ξ) ξ

J(ξ)=1 2

M m=1

(Y_p^m−Φ(X^m,ξ))²

X^m m M

J ∇J(ξ) J

(84)

ξ0 ξ1 ξ2

ξn+1=ξn−γn∇J(ξn), n≥0

γn ξn

ξ ξ0

J J

J ξ0

ξ

r^m r^m(ξ)=Y_p^m−Φ(X^m,ξ)

r^m ξ

r^m(ξ)≈r^m(ξn)+∇^m_r(ξ)δξ

δξ=ξ−ξ ∇^m_r(ξ) r^m(ξ) ξ

r^m(ξ)=0 δξ

δξ=−∇^m_r(ξ)⁺r^m(ξ)

∇^m_r(ξ)⁺ ∇^m_r(ξ)

ξ0 ξn

ξn+1=ξn−∇^m_r(ξn)⁺r^m(ξn)

(85)

ξ

δξ λ

(∇^m_r(ξn)^t∇^m_r(ξn)+λI)δξ=∇^m_r(ξn)^tr^m(ξn)

∇^m_r(ξ) λ

λ

h k

r

h kr TCL

Y_p^m TCL

Y =Φ(X^m,ξ) TCL X

ξ=(h,kr)

TCL

sin(ϕv)≈ϕv,cos(ϕv)≈1 TCL TCL











TCLpermanent = −2 c

(Iz−Iy)˙ψ²ϕv−h(Fn1+Fn2)ϕv

Fn1+Fn2

ϕv = mhv˙ψ kr−mh²ψ˙² Fn1+Fn2 = mg−krϕ²_v

h

ξ ξ

r^m(ξ)=TCLpermanent(X^m,ξ)−TCLmesure(X^m) TCLpermanent(X^m,ξ)

h kr

(86)

ξ=(1000h,kr)

λ

ξn+1=ξn+λ·δξ, λ<1

h kr

TCL

250kg Ix Iy Iz 45110130kg.m²

ab 0.580.7m c 0.95m

TCL

(87)

h kr TCL δ=4^◦ v m.s⁻¹

h=1.2475m kr=5900N.m.rad⁻¹ TCL

TCL

h kr hT

m m

cm

(88)

h kr

◦ ◦ ◦

h kr TCL

δ=4.8^◦ h=1.32m kr=8600N.m.rad⁻¹ TCL

quad+pilote 0.65m

TCL TCL TCL

km.h⁻¹ ^◦ ¹ ² ³ TCL¹ TCL²

(89)

• TCL TCL

•

TCL TCL

h kr

h kr hT

m m

◦ ◦ ◦ ◦

h kr TCL

δ=6^◦ h=1.14m kr=8100N.m.rad⁻¹

TCL quad+pilote 0.64m

h kr hT

m m

(90)

TCL TCL TCL

km.h⁻¹ ^◦ ¹ ² ³ TCL¹ TCL²

TCL

(91)

(92)

TCL

(93)

X Y u f(X,u)

X = AX+Bu˙ Y = CX+Du

X n×1 Y m×1

u (l×1 A B C

D n×n n×lm×n m×l

(94)

X Y u

Oobs

Oobs=





 CAC

. . CAⁿ⁻¹







X = AX+Bu+L(Y−Y)˙ Y = CX+Du

Y−Y L n×m

X

X =AX+Bu+L(Y−CX−Du)˙ X =(A−LC)X+(B−LD)u+LY˙

X =X−X X =(A−LC)X˙ X(0)=X(0)−X(0)

X(t)=exp[(A−LC)(t)]X(0) A−LC

L

(95)

TCL

(96)

ψ˙ v δ

ψ˙ β

Ce

Ce=C0

Ce<C0

ψ v δ˙