Comparison Between Passive and Active Flexible Anti-Roll Bars Modelled by Finite Element Method on Prevention of Vehicle Rollovers

COMPARISON BETWEEN PASSIVE AND ACTIVE FLEXIBLE ANTI-ROLL BARS MODELLED BY FINITE ELEMENT METHOD ON PREVENTION OF VEHICLE ROLLOVERS

Joed Henrique Paes∗, Diego Colon´ ∗ ∗Av. Prof. Luciano Gualberto, travessa 3, n 158, CEP 05508-900 USP / Escola Politécnica - Laboratório de Automa¸cãoe Controle (LAC) SãoPaulo, SãoPaulo, Brazil

Emails: [email protected], [email protected]

Abstract— Anti-roll bars aims to increase the driving comfort by reducing the roll angle during cornering and also assist the self-steering behaviour of the vehicle chassis. This paper presents a comparison between a passive anti-roll bar and an active anti-roll bar. The bars are modelled by FEM (Finite Element Method) and a controller LQG (Linear Quadratic Gaussian) is designed to the active system. In addition, the vehicle lateral dynamic is hybridized to the flexible model in order to obtain an unique mathematical model. It reviews the vehicle roll dynamics and describes the FEM applied to dynamic modelling of flexible structures. The influence of the active anti-roll system is illustrated through numerical simulation, and a comparison among the passive anti-roll system and the system without anti-roll is also presented. To the best of the authors’ knowledge, the modelling approach presented in Sections 3.1 and 3.2 have not been reported in the literature.

Keywords— Anti-roll system, ﬂexible structures, control, ﬁnite element method.

Resumo— Barras de anti-rolagem visam aumentar o conforto atravésda redu¸cãodo ângulo de rolagem em curvas e também auxiliam na auto-dirigibilidade do chassis do ve´ıculo. Este trabalho apresenta uma compara¸cão entre um sistema com barra de anti-rolagem passiva e outro com barra. As barras são modeladas pelo MEF (Método dos Elementos Finitos) e um controlador LQG (Linear QuadráticoGaussiano) éprojetado para o sistema ativo. Além disso, àdinâmica lateral do ve´ıculo, éadicionada a parte flex´ıvel para obter um único modelo matemático. O trabalho revisa a dinâmica de rolagem ve´ıcular e descreve o MEF aplicado àmodelagem dinâmica de estruturas flex´ıveis. A influência do sistema ativo éilustrada atravésde simula¸cão, e uma compara¸cão entre os sistemas passivo e sem barra de anti-rolagem tambéméapresentada. Atéonde vai o conhecimento dos autores, a abordagem dada àmodelagem nas Se¸cões 3.1 e 3.2 nãofoi relatada na literatura.

Palavras-chave— Sistema anti-rolagem, estruturas flex´ıveis, controle, método dos elementos finitos.

Nomenclature ρ mass density per unit length. ˙ l lateral distance between left and right sus- () time derivative. pensions. ()n n-order spacial derivative. h height of the vehicle’s c.g..

ay lateral acceleration of the vehicle. 1 Introduction

ms vehicle sprung mass. The composition of the current automotive fleet mu unsprung mass on vehicle’s left/right side. consists of 36% of light trucks, minivans and sport Ixx roll moment of inertia about the c.g.. utility vehicles (SUVs). Unfortunately, the rate of fatal rollovers for pickups is twice the rate for pas- g acceleration due to gravity. senger cars, and the rate for SUVs is almost three Flat total lateral tire force. times the rate for passenger cars. While rollover φ roll angle. affects about 31% of passenger cars involved in crashes with occupant fatalities, it accounts for zs vertical deflection of the vehicle’s c.g.. 51% of passenger cars occupant fatalities in sin- zul vertical deflection of the left unsprung mass. gle crashes. These statistics are from 2016 and has motivated several studies to improve passen- zur vertical deflection of the right unsprung mass. ger cars’ safety by preventing rollover via active zrl left road displacement input. stability control (NHTSA, 2016).

zrr right road displacement input. An active anti-roll system allows the control of the roll angle and, thus, improves the vehicle k suspension stiﬀness. s stability, especially when turning or moving on bs suspension damping coeﬃcient. slopping ground (Jamil et al., 2017). A rollover

kt tire stiffness. is a type of vehicle crash in which a vehicle tips over onto its side or roof and have a higher fa- v nodal displacements. tality rate than other types of vehicle collisions EI bending rigidity. (Phanomchoeng and Rajamani, 2012). This present paper is focused on finding hy- troller for a vehicle, as well as an experimen- bridized dynamic models that merge the car’s dy- tal study using the electrically actuated roll con- namic roll movement with a flexible anti-roll bar trol system. Firstly, parameter sensitivity anal- that is attached to the suspension system. This ysis is performed based on the 3DOF linear ve- anti-roll bar can be passive or active. This last hicle model. The controller is designed in the model must be well suited to the design of active framework of lateral acceleration control and gain- feedback controllers, which are used to mitigate scheduled control scheme considering the varying the rollover occurrence. Only linear time invariant parameters induced by laden and running vehicle models are considered. The flexible bar’s mathe- condition (Kim and Park, 2004). matical model is obtained by FEM. A method for designing a controller which The suitability for controller synthesis is de- uses an active anti-roll bar (AARB) and an elec- termined by a number of factors, including the tronic stability program (ESP) for rollover pre- accuracy of the model for moderate steering in- vention is presented by Yim. An ESP can carry puts (e.g. linear tire behaviour and small roll out active braking to reduce vehicle’s speed and angles) and the order and simplicity of the dy- lateral acceleration to prevent rollover. The con- namic model. The finite element approach pro- troller for the AARB was designed based on linear duces models with large number of degrees of free- quadratic static output feedback control method- dom, which causes numerical difficulties in dy- ology, which attenuates the effect of lateral ac- namic analysis and control design. Thus, a model celeration on the roll angle and the roll rate, by reduction is applied aiming to glean a suitable sys- controling the suspension stroke and the tire’s de- tem with lower order to design the LQG controller. flection of the vehicle (Yim et al., 2012). The remainder of the paper is organized as follows: the section 2 presents the results of the 3 Mathematical models literature survey. The section 3 presents the models chosen for this study. The control design is Assuming that the axles and tires have a known presented in the section 4. The section 5 exhibits mass, a four degrees-of-freedom (dof) roll dynamic the results taken from three models: 1) the roll model is developed in the following, considering dynamics model without anti-roll system, 2) with vertical translation of the sprung mass, denoted by passive and 3) active anti-roll bar system as well. zs, vertical translation of the left unsprung mass, Finally, the section 6 presents preliminary conclu- denoted by zul, vertical translation of the right sions from this ongoing study. unsprung mass, denoted by zur, as well as the roll motion of the sprung mass, denoted by φ. The 2 Literature survey Fig. 1 illustrates the modelled system.

Carlson and Gerdes use Model Predictive Control zs Flat (MPC) theory to develop a framework for auto- CG P = msg motive stability control. The framework is then φ h demonstrated with a roll mode controller, which ks bs ks bs seeks to actively limit the peak roll angle of the z Roll center z vehicle while simultaneously tracking the driver's ur ul yaw rate command (Carlson and Gerdes, 2003). l Cameron and Brennan present results of zrr k k zrl an initial investigation into models and control t t strategies suitable to prevent vehicle rollover due to untripped driving maneuvers. A challenging Figure 1: Four degree-of-freedom roll dynamics task is identifying suitable vehicle’s models from model (source: the authors). the literature, comparing these models with experimental results, and determining suitable parame- Rajamani (Rajamani, 2012) shows that the ters for the models (Cameron and Brennan, 2005). overall equations of motion for the model is given A pair of MPC capable of modifying the nom- by: inal roll dynamics of a vehicle through control of the planar vehicle dynamics are presented by Beal msz¨s = ks (2zs zur zul) and Gerdes. Each of these controllers is based − − − (1) bs (2z ˙s z˙ur z˙ul) , on a linear model of the vehicle. Results from − − − a nonlinear vehicle model demonstrate that the l differential-drive technique results in significant murzür = ks zs sin φ zur − 2 − lateral-longitudinal tire force coupling and satu- ration, which degrades the validity of the model l + bs z˙s φ˙ cos φ z˙ur kt (zur zrr) , used for controller design (Beal and Gerdes, 2010). − 2 − − − Kim presents the design of an active roll con- (2) l mulzül = ks zs + sin φ zul and 2 − m y˙2(0, t) m y˙2(L, t) k l L = ur + ul t (y(0, t) z )2 + bs z˙s + φ˙ cos φ z˙ul kt (zul zrl) , B rr 2 − − − 2 2 − 2 − 2 2 (3) kt 2 ksy (0, t) ksy (L, t) (y(L, t) zrl) − 2 − − 2 − 2 and k s l sin φ (y(0, t) y(L, t)) Jsφ¨ =msayh cos φ + msgh sin φ − 2 − + k z (y(0, t) + y(L, t)) . k l2 b l2 s s s sin φ s φ˙ cos φ (10) − 2 − 2 (4) ksl bsl + (zul zur) + (z ˙ul z˙ur) , The generalized Hamilton’s principle is used 2 − 2 − to derive the equations of motion, i.e.: 2 where Js = (Ixx + msh ). tf L δLB + δLˆ dx + δLB + δWnc dt = 0, 3.1 Passive anti-roll bar Zt0 " Z0 # (11) Consider a system, as shown in Fig. 2, which con- where the virtual works of a dissipative force and sists of a flexible bar connected to the unsprung external forces are given by: masses of the roll dynamic model. It is desired to 2 obtain a set of equations that represents this sys- l 2 δWnc = Flath cos φ bs φ˙ cos φ) tem so that the influence of the added bar show − 2 that the roll angle is reduced by this passive sys- l tem when compared to the roll dynamic model. If + bs cos φ y˙(L, t) y˙(0, t) δφ 2 − EI is the equivalent bending rigidity, ρ is the mass + [bs (y ˙(0, t) +y ˙(L, t)) 2bsz˙s] δzs (12) density of the beam (per unit length) and y(x, t) − the deflection of the bar, the kinetic energy T and l + bs z˙s φ˙ cos φ y˙(0, t) δy(0, t) the potential energy V formulas are: − 2 − 2 2 2 l msz˙s mury˙ (0, t) muly˙ (L, t) + bs z˙s + φ˙ cos φ y˙(L, t) δy(L, t). T = + + 2 − 2 2 2 ˙2 L (5) Jsφ 1 2 The discretization techniques, such as finite + + ρy˙ (x, t) dx, 2 2 differences, finite elements, and the Rayleigh-Ritz Z0 method, are appropriate to extract the character- kt 2 V = msgh cos φ + (y(0, t) zrr) istics of the systems. In this paper, the finite ele- 2 − ment method is applied, which is based on selec- kt 2 tion of a set of shape functions satisfying the ge- + (y(L, t) zrl) 2 − ometric boundary conditions of the problem. We 2 ks l will search for solutions of the form: + zs sin φ y(0, t) 2 − 2 − (6) n 2 ks l y(x, t) = Ni(x)vi(t), (13) + zs + sin φ y(L, t) 2 2 − i=1 X L 1 00 2 where n is the number of degree of freedom of the + EI (y (x, t)) dx. element, N(x) are known polynomial shape func- 2 0 Z tions of the spatial coordinates, which are linearly The Lagrangian’s formula is: independent over the domain 0 x L as ex- ≤ ≤ L pressed in Eq. 14, and v(t) are unknown functions L = T V = LD + Lˆ dx + LB, (7) of the time t: − Z0 2 Ni(x) = C1i +C2ix+C3ix +..., i = 1, ..., n. (14) where Lˆ is a Lagrangian density and in which: Hence the variations of the displacements can be ˙2 2 Jsφ msz˙s LD = + msgh cos φ put in a matrix form as in Eq. 15. 2 2 − 2 (8) 2 l 2 δy = Nδv. (15) ksz ks sin φ, − s − 4 The substitution of Eq. 15 into the variational and principle in Eq. 11 leads to:

1 2 1 00 2 Lˆ = ρy˙ (x, t) EI (y (x, t)) , (9) mv¨ + cv˙ + kv = fd, (16) 2 − 2 zs Flat the deﬂection of the bar B. The kinetic energy T

CG P = msg and potential energy V expressions are: φ k b k b 2 2 2 ˙2 s s s s msz˙s mury˙A(0, t) muly˙B(L, t) Jsφ y T = + + + 2 2 2 2 zur = y(0, t) zul = y(L, t) L L x 1 2 1 2 Roll center + ρy˙A(x, t) dx + ρy˙B(x, t) dx, z 2 2 Z0 Z0 z z rr kt kt rl (26)

kt 2 V = msgh cos φ + (yA(0, t) zrr) Figure 2: Roll dynamic model with passive anti- 2 − roll bar (source: the authors). kt 2 + (yB(L, t) zrl) 2 − 2 ks l where: + zs sin φ yA(0, t) 2 − 2 − Js 0 0 2 m = 0 ms 0 , (17) ks l   + zs + sin φ yB(L, t) 0 0 m22 2 2 − L l2   l bs 0 bs (N(0) N(L)) 1 2 2 2 2 − 00 00 c = 0 2bs bs (N(0) + N(L)) , + EI (yA(x, t)) + (yB(x, t)) dx.  l T T T T T− T T  bs 2 N (0) N (L) bs N (0) + N (L) bs N (0)N(0) + N (L)N (L) 2 0  − −  Z (18) (27) l2 l ks msgh 0 ks (N(0) N(L)) 2 − 2 − k = 0 2ks ks (N(0) + N(L)) , As in Eq. 7, the Lagrangian terms are:  l T T T T −  ks N (0) N (L) ks N (0) + N (L) k22 2 − −  (19) ˙2 2 Jsφ msz˙s m h 0 0 LD = + msgh cos φ s 2 2 − (28) f = 0 0 0 , (20) 2 2 l 2  T T  ksz ks sin φ, 0 ktN (0) ktN (L) − s − 4   where ˆ 1 2 1 2 L = ρy˙A(x, t) + ρy˙B(x, t) T 2 2 m22 = Mv + N (0)murN(0) (29) (21) 1 00 2 1 00 2 T EI (y (x, t)) EI (y (x, t)) , + N (L)mulN(L), − 2 A − 2 B 2 2 T mury˙A(0, t) muly˙B(L, t) LB = + k22 = Kv + N (0) (ks + kt) N(0) 2 2 T (22) + N (L)(ks + kt) N(L), kt 2 kt 2 (yA(0, t) zrr) (yB(L, t) zrl) − 2 − − 2 − L 2 2 T ksy (0, t) ksy (L, t) Mv = ρN (x)N(x) dx, (23) A B Z0 − 2 − 2 L l 00T 00 ks sin φ (yA(0, t) yB(L, t)) Kv = EIN (x)N (x) dx, (24) − 2 − 0 Z + kszs (yA(0, t) + yB(L, t)) . and (30) T Again, the generalized Hamilton’s principle v = (φ, zs, v1, v2, ..., vn) , (Eq. 11) is used to derive the equations of motion, T (25) d = (ay, zrr, zrl) . where the virtual work of dissipative and external forces, considering the control torque provided by the motor, is given by: 3.2 Active anti-roll bar l δWnc = Flath cos φ + bs cos φ (y ˙B(L, t) y˙A(0, t)) 2 − Consider the system, as shown in Fig. 3, which 2 consists of two flexible bars, each one connected l 2 bs φ˙ cos φ) δφ uδθ + uδθ to an unsprung mass and the electric motor, which − 2 − yA(L,t) yB (0,t) provides the control torque for the active system. + [bs (y ˙A(0, t) +y ˙B(L, t)) 2bsz˙s] δzs As in the previous section, it is desired to seek a − l set of equations that represents the system, so that + bs z˙s φ˙ cos φ y˙A(0, t) δyA(0, t) − 2 − the influence of the control torque shows that the roll angle will be minimized by the active system l + bs z˙s + φ˙ cos φ y˙B(L, t) δyB(L, t), when compared to the passive system. Denote by 2 − yA(x, t) the deflection of the bar A and yB(x, t) (31) zs where u is the torque input applied by the motor. Flat

CG As a simpliﬁcation, the dynamic model of the mo- P = msg tor is neglected and the torque is directly applied φ as moments at the tips of each bar connected to ks bs u u ks bs y − the motor, as shown in Fig. 3. zur = yA(0, t) zur = yB(L, t) Likewise the previous section, the ﬁnite ele- x motor Roll Center ment method is employed in this analysis, where: z

n z z rr kt kt rl yk(x, t) = Nki(x)vki(t), (32) i=1 X Figure 3: Roll dynamic model with active anti-roll and k = A, B. Hence the variations of the dis- bar (source: the authors). placements have the form:

δyk = Nkδvk. (33) 3.3 Parameters The substitution of Eq. 33 for each bars into the variational principle in Eq. 11 leads to: Table 1 presents the parameter’s values used for simulation. These parameters were obtained mv¨ + cv˙ + kv = fd + hu, (34) by Yim from an small SUV car in software CarSim (Yim et al., 2012). where: ®

Js 0 0 0 0 ms 0 0 Table 1: Parameters values. m =   , (35) 0 0 m22 0 Variable Value Units  0 0 0 m33 ms 984.6/2 kg   mur 40 kg l2  l  l bs 0 bs NA(0) bs NB(L) 2 2 − 2 mul 40 kg 0 2bs bsNA(0) bsNB(L) c =  l T T −T − , bs N (0) bsN (0) bsN (0)NA(0) 0 h 0.45 m 2 A − A A  b l N T (L) b N T (L) 0 b N T (L)N (L) 2  s 2 B s B s B B  Ixx 439.9/2 kgm − −  (36) ks 28721 N/m l2 l l ks msgh 0 ks NA(0) ks NB(L) 2 − 2 − 2 bs 2000 n/(m/s) 0 2ks ksNA(0) ksNB(L) k =  l T T − − , kt 230000 N/m ks 2 NA (0) ksNA (L) k22 0 l T − T l 1.6 m  ks N (L) ksN (L) 0 k33  − 2 B − B  2  (37) g 9.81 m/s msh 0 0 E 210 GP a 3 0 0 0 ρ 7850 kg/m f =  T  , (38) 0 ktNA (0) 0 T  0 0 ktN (L)  B    0 Table 2: Passive bar geometry coordinates. 0 h = , (39) Spatial distribution of the nodes  N T (L) − A Description x [m] y [m] z [m]  N T (0)   B  n1 0 0 0 where:   n2 0 0 -0.1 T n3 0 0 -0.2 m22 = MvA + murNA (0)NA(0), (40) n4 0 0 -0.3 T n5 0.2 0 -0.3 m33 = MvB + mulNB (L)NB(L), (41) n6 0.4 0 -0.3 k = K + (k + k ) N T (0)N (0), (42) 22 vA s t A A n7 0.6 0 -0.3 T n8 0.8 0 -0.3 k33 = KvB + (ks + kt) NB (L)NB(L), (43) L n9 1.0 0 -0.3 T Mvi = ρNi (x)Ni(x) dx, (44) n10 1.2 0 -0.3 Z0 n11 1.4 0 -0.3 L 00T 00 n12 1.6 0 -0.3 Kvi = EIN (x)N (x) dx, (45) i i n13 1.6 0 -0.2 0 Z n 1.6 0 -0.1 and 14 n15 1.6 0 0 T v = (φ, zs, vA1 , vA2 , ..., vAn , vB1 , vB2 , ..., vBn ) , T d = (ay, zrr, zrl) . Figure 4 illustrates the spatial discretization (46) of the passive bar. Table 2 shows the coordinates xyz of each node. All models use beam element, where each node has six dof’s: displacements xyz model is expressed in following form: and their respective rotations. The cross-section x˙ = Ax + Bu + Gw, is circular with radius r = 21.8mm. (47) y = Cx + v, y n4 n5 n6 n7 n8 n9 n10 n11 n12 where w is white process noise signal and v is the n3 n2 white measurement noise signal. n1 n13 n14 x n15 z KLQG w v Figure 4: Geometry of the passive bar (source: kalman u P lant y K the authors). ﬁlter − P

Figure 5 illustrates the spatial discretization of the active bar. The Tables 3 and 4 shows the coordinates x-y-z of each node. Figure 6: LQG scheme (source: the authors).

Bar A Bar B Considering Fig. 6: the Kalman filter is y n n n n n n n n n n n 5 6 7 8 9 1 2 3 4 5 firstly determined by minimizing the steady-state 4 T n3 error covariance E[e(t), e (t)], where the error is n2 n6 n1 n7 motor n8 e(t) = x(t) xˆ(t). The process of minimization of x n9 − z the error covariance is strongly associated with the matrices Ξ and Θ, that are expressed in Eq. 48, Figure 5: Geometry of the active anti-roll bar which also represent the covariances of the process (source: the authors). and measurement noises, respectively: E[wwT ] = Ξ,E[vvT ] = Θ,E[wvT ] = Ψ. (48) Properly chosen values of the above matrices Table 3: Active bar ”A” geometric coordinates. lead to the gain L of Kalman filter, that is: Spatial distribution of the nodes Description x y z xˆ˙ = Axˆ + Bu + L[y Cxˆ]. (49) − n1 0 0 0 The state feedback gain matrix K is deter- n2 0 0 -0.1 mined by the minimization of the cost functional n3 0 0 -0.2 expressed in Eq. 50 for the particular selection of n4 0 0 -0.3 the weight matrices Q and R: n5 0.2 0 -0.3 n6 0.4 0 -0.3 ∞ 1 T T n 0.6 0 -0.3 J = x Qx + u Ru dt. (50) 7 2 n 0.8 0 -0.3 Z0 8 The control law resulting from this process has the following form:

u = Kx,ˆ (51) Table 4: Active bar ”B” geometric coordinates. − Spatial distribution of the nodes wherex ˆ is the estimated values delivered by the Description x y z Kalman ﬁlter. n1 0.8 0 -0.3 Finally, the state-space equation of LQG con- n2 1.0 0 -0.3 troller with the Kalman ﬁlter can be written in n3 1.2 0 -0.3 following form: n4 1.4 0 -0.3 n5 1.6 0 -0.3 xˆ˙ = [A LC BK]x ˆ + Ly, − − (52) n6 1.6 0 -0.2 u = Kx,ˆ n7 1.6 0 -0.1 − n8 1.6 0 0 where y is the vector of the plant’s output (sen- sors) measurements.

4 Control design 5 Results

The design of LQG control law for the active anti- For the analysis, it was adopted a known unitary roll system is performed in two steps, where the lateral acceleration input aiming to compare the ﬁnal results between the three models chosen for In order to improve steady-state response of simulation. The chosen input represents a mod- controller, an integrator was added throughout erate driver’s maneuver that starts a cornering at the designing. The controller’s transfer function 1.0 second and maintain a circular trajectory for obtained is given in Eq. 54, where 8e6 stands for 3.0 seconds. Figure 7 illustrates the lateral accel- 8 106 and so forth: eration used as input to the models. × 8e6s4 + 1e8s3 + 3e9s2 + 2e10s + 1e11 Lateral acceleration K(s) = 1.2 s6 + 56s5 + 2e3s4 + 4e4s3 + 5e5s2 + 3e6s (54) 1 Figure 9 a) shows the model eigenvalues with-

0.8 out anti-roll system, which is naturally stable with

) all complex conjugate poles. In Fig. 9 b) it is 2

0.6 presented both passive and active anti-roll system

(m/s y

a poles. It is seen the typical characteristics of flex- 0.4 ible structures, which are the complex conjugate poles with small real parts. The addition of the 0.2 controller included some real poles with large real part in order to attain the control objective. Fig- 0 0 1 2 3 4 5 6 ure 9 c) shows a zoom of b) where it is seen the Time (s) controller’s influence in the poles near the origin. The arc pattern of the poles resulted of the ap- Figure 7: Input taken for lateral acceleration plication of the Rayleigh damping on the flexible (source: the authors). portion to make the system well conditioned nu- merically. Typically, the models provided by the finite element method present large number of degree of

10 5 freedom, and the design of the controller becomes 80 1.5 4000 A a complicated task. In this paper, the order re- B 60 3000 duction through truncation was applied by using 1 C the Hankel singular values of the balanced repre- 40 2000 0.5

sentation of the system. Figure 8 shows the origi- 20 1000

) ) nal model with 196 states and the reduced model )

0 0 0

Im( Im( with the four more relevant states to the controller Im( -20 -1000 design. The obtained controller was applied to -0.5 the original system. In the ﬁrst step, the gain L -40 -2000 -1 of the Kalman ﬁlter is determined. In this case, -60 -3000 the steady-state error covariance is minimized by -80 -1.5 -4000 choosing the weighting matrices Q and R as: -40 -20 0 -2 -1 0 -40 -20 0 Re( ) Re( ) 10 9 Re( ) a) b) c) Q = CT C ,R = 1e−12 . (53) Figure 9: Eigenvalues for the systems: A(without anti-roll bar), B(passive bar), C(active bar) Magnitude plot (source: the authors). -50

-100

-150 Figure 10 shows the main result of this work. The system without anti-roll bar suﬀers a roll an- -200 gle of 0.58 degree under inﬂuence of a unitary lat-

-250 eral acceleration. When the passive bar is added, Gain (dB) Gain the roll angle is reduced to 0.41, that represents a -300 Original model (196 states) decrease of approximately 30%. The passive bar Reduced model (4 sates) -350 also improves the roll occurrence during the ramp acceleration between 1.0 and 3.0 seconds. The -400 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 third model, with active bar, reduces the roll angle Frequency (rad/s) to zero under inﬂuence of the unitary lateral acceleration at steady-state. During the ramp accel- Figure 8: Magnitude plot of the original and re- eration, the active model is not able to reduce the duced models for active anti-roll bar (source: the roll angle to zero. However, the controller tends authors). to maintain the roll angle below 0.1 degree. Comparison roll angles 0.7 out it. On the other hand, the active bar reduced the roll angle to zero under constant lateral accel- 0.6 eration and showed a signiﬁcant reduction during varying lateral acceleration. 0.5

(deg) 0.4 Acknowledgements

0.3

Without anti-roll bar The authors would like to thank the reviewers for Roll angle angle Roll 0.2 Passive anti-roll bar Active anti-roll bar their thoughtful comments on the ﬁrst draft of this paper. 0.1

0 0 1 2 3 4 5 6 References Time (s) Beal, C. E. and Gerdes, J. C. (2010). Predic- Figure 10: Roll angle for system: 1) without bar, tive control of vehicle roll dynamics with rear 2) with passive and 3) with active anti-roll bar wheel steering, American Control Conference (source: the authors). (ACC), 2010, IEEE, pp. 1489–1494. Bharane, P., Tanpure, K., Patil, A. and Kerkal, The active system presented a dc gain of G. (2014). Design, analysis and optimization 299Nm/degree. It means that the motor provides of anti-roll bar, Journal of Engineering Re- 299Nm of torque to compensate one degree of roll search and Applications 4(9): 137–140. angle. This value is similar to others presented in the literature, on real active anti-roll system Cameron, J. T. and Brennan, S. (2005). A com- (Bharane et al., 2014). Figure 11 shows the con- parative, experimental study of model suit- trol torque applied by the motor whereby the roll ability to describe vehicle rollover dynamics angle was reduced. While the vehicle was cor- for control design, ASME, American Society nering at a circular trajectory, the motor kept a of Mechanical Engineers, pp. 405–414. torque of 122Nm to compensate the roll angle. Carlson, C. R. and Gerdes, J. C. (2003). Optimal rollover prevention with steer by wire and dif- Torque of the motor 140 ferential braking, ASME, American Society of Mechanical Engineers, pp. 345–354. 120 Jamil, F. M., Abdullah, M., Ibrahim, M., Harun, 100 M. and Abdullah, W. W. (2017). The dy- 80 namic veriﬁcation of vehicle roll dynamic models using diﬀerent softwares, Proceedings 60

Torque (N.m) Torque of Innovative Research and Industrial Dia- 40 logue 16: 135–136.

20 Kim, H.-J. and Park, Y.-P. (2004). Investiga- tion of robust roll motion control consider- 0 0 1 2 3 4 5 6 ing varying speed and actuator dynamics, time (s) Mechatronics 14(1): 35–54. Figure 11: Control signal: torque provided by the NHTSA (2016). Passenger vehicle occupant fa- motor (source: the authors). talities, https://www.nhtsa.gov/. Accessed: 2018-02-30. 6 Conclusion Phanomchoeng, G. and Rajamani, R. (2012). Pre- diction and prevention of tripped rollovers. There is a significant opportunity to improve the safety of the current fleet by preventing vehicle Rajamani, R. (2012). Vehicle Dynamics and Con- rollover. In this work, it was proposed a model trol, Mechanical Engineering Series, 2 edn, of the flexible anti-roll bar system with FEM Springer US. in both passive and active version. The model Yim, S., Jeon, K. and Yi, K. (2012). An in- proposed was hybrid, in the sense that it inte- vestigation into vehicle rollover prevention grated the lumped parameter part (roll vehicle’s by coordinated control of active anti-roll bar dynamic) and the flexible part. The active model and electronic stability program, Interna- was used to design a LQG controller to reduce the tional Journal of Control, Automation and rolling. The passive anti-roll bar reduced the roll Systems 10(2): 275–287. behaviour in 30% compared to the system with-