Robust Vehicle Roll Dynamics Identification based on Roll Rate Measurements

2012 Workshop on Engine and Powertrain Control, Simulation and Modeling The International Federation of Automatic Control Rueil-Malmaison, France, October 23-25, 2012

Robust Vehicle Roll Dynamics Identification based on Roll Rate Measurements Robert Tafner ∗ Markus Reichhartinger ∗ Martin Horn ∗ ∗

Control and Mechatronic Systems Group, Institute of Smart-System-Technologies, Alpen-Adria-Universit¨ at Klagenfurt, AUSTRIA (e-mail: [email protected])

Abstract: The modern vehicle development process is required to provide short product update cycles and high cost efficiency. Nonetheless, the demands on vehicle safety and handling quality remain high as ever. Driving characteristics evaluation relies on extensive subjective assessment as well as objective methods. Model-based objective assessment saves time and costs as most of the evaluation work can be done offline, i.e. during or after a test run the parameters of a vehicle dynamics model are identified and used for later simulation work. Referring to vehicle dynamics assessment the roll movement of the vehicle is of great importance (aside from longitudinal and lateral vehicle dynamics). This paper presents two methods for the online parameter identification of a simple roll dynamics model. These parameters can then be used for computation of the roll angle. The first method employs a recursive least-squares algorithm whereas the second method consists of a slidingmode observer combined with an uncertainty reconstruction technique. Compared to other existing methods for vehicle roll dynamics identification both estimation techniques only require the cost-efficiently measurable roll rate and lateral acceleration signals. Keywords: Vehicle dynamics, roll model parameter identification, roll angle computation, online parameter identification 1. INTRODUCTION

accuracy for extrapolation of vehicle states that have not been evaluated during the test drive already. Zomotor (2002) shows the development of simple, but accurate vehicle dynamics models and their online parameter identification during simulator test drives. Kobetz (2003) focuses on real-world in-vehicle measurements but identifies the parameters of vehicle dynamics models offline. Generally, the vehicle dynamics assessment process relies on accurate and therefore cost-intensive measurement equipment (P. E. Pfeffer and Lin (2008)). Both methods of Zomotor (2002) and Kobetz (2003) consider a lateral dynamics as well as a roll dynamics model. However, the presented identification of the model parameters requires measurement of the roll angle which is only measurable by cost-intensive sensors. Roll angle estimation has been focus of numerous publications within the last years. Due to its importance to safety related driver assistance systems several authors have published results in that area. Ryu and Gerdes (2004) estimate the roll angle and road bank angle using a disturbance observer and GPS/INS measurements. Ding and Massel (2005) derive a state-space model describing the vehicle roll dynamics on banked roads as well. A linear observer estimates the roll angle and computes its confidence interval as an indicator of the estimation error. Sebsadji et al. (2008) estimate the roll angle and road bank angle with an unknown input PI observer. All these methods have in

The vehicle dynamics development process is mainly driven by two factors: safety and handling characteristics. Evaluation of vehicle safety refers to execution of standardized testing procedures. In terms of vehicle handling evaluation the assessment of driving characteristics is still an open issue. Subjective testing methods require highly-skilled and experienced test drivers performing well-defined manoeuvres. Although trained for years in vehicle dynamics evaluation there is always a human variability in providing feedback. Consequently, the objective vehicle dynamics evaluation methods play an increasingly important role in the development process. There are numerous standards that define characteristic values to be extracted from open- and closed-loop manoeuvres (see ISO7401 (1988), ISO8725 (1988) and ISO8726 (1988)). However, it is an open issue to build a bridge between these two methodologies (see e.g. Chen and Crolla (1998), Wolff et al. (2009)). The increasing number of vehicle variants enlarges the required test drives significantly. Hence, it would be benefical to simulate some of the tests offline on a computer and extract the characteristic values for objective evaluation therefrom. The idea is to identify parameters of simple vehicle dynamics models during a test drive for further offline simulation work. Ideally, these models provide high 978-3-902823-16-8/12/$20.00 © 2012 IFAC

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common that they require a priori knowledge of the roll model parameters. Zomotor (2002), Kobetz (2003) and Abdellatif et al. (2003) identify the dominating parameters of roll models but also require measurement of the roll angle itself. Rajamani et al. (2011) present the identification of the center of gravity height and roll angle estimation by sensor fusion (low-frequency tilt angle and gyroscope) and a nonlinear dynamic observer. This paper focuses on solving the problems of roll angle estimation and model parameter identification simultaneously. The two presented methods only require the sensor signals of lateral acceleration and roll rate, that are measurable in a cost-efficient way. It demonstrates the model parameter identification and verifies the results using a complex vehicle dynamics simulation software (IPG CarMaker 1 ). The remainder of this paper is organized as follows: Section 2 presents the modelling of the roll dynamics. Section 3 introduces the first identification algorithm based on recursive least-squares (RLS). The second method consists of a higher-order sliding-mode observer with an uncertainty reconstruction technique and is presented in Section 4. As both methods require a differentiated input signal, a robust differentiator is introduced in Section 5. Section 6 demonstrates the application of both approaches to simulation data. The results are evaluated by comparing the estimated roll angle and the simulation data of CarMaker. Section 7 concludes the paper.

z

CoG

ms ay ms ay,m

hr

ms g RC hrc

ls Fz,l

Fz,r lt

Fig. 1. Schematics of simplified roll model. (6) The wheel track lt is equal to the distance ls between left and right suspension elements. The overall vehicle mass m is divided into a sprung mass ms and an unsprung mass. Considering the vehicle’s roll dynamics only the sprung mass is taken into account. The torque balance in the roll center (RC) yields a roll moment of lt lt Mr = Fz,l −Fz,r = ms ay cos(ϕ)hr +ms gsin(ϕ)hr , (1) 2 2 where ϕ is the roll angle, hr is the distance between the center of gravity and the roll center, Fz,l/r are the vertical tire forces and g is the gravitational constant. Mostly, the lateral acceleration sensor is mounted on the chassis and therefore its output signal ay,m is affected by the roll angle ϕ. The transformation between road-fixed and chassisfixed coordinate systems reads as ay,m = ay cos(ϕ) + gsin(ϕ). (2) Combining (1) and (2) the resulting roll moment Mr can be formulated as Mr = ms ay,m hr . (3) The differential equation of the roll dynamics reads as Jxs ϕ¨ + (df + dr ) ϕ˙ + (cst,f + cst,r + cf + cr ) ϕ = Mr , (4) | {z } | {z }

2. MODELLING OF VEHICLE ROLL DYNAMICS A detailed model of the roll dynamics which takes into account the nonlinear characteristics of all suspension elements can be found in Kiencke and Nielsen (2004). Shim and Chinar (2008) develop a 14-DOF full vehicle model and show its limitations for vehicle rollover detection or integrated chassis control. However, simpler 4DOF models (lateral and roll dynamics) which are more relevant to parameter estimation problems (due to their number of reduced unknown parameters) are investigated by Hamblin et al. (2006). It is found that (for this class of models) parameter fitting plays a more important role than modelling differences. Ammon (1997) claims that due to the roll movement the nonlinear characteristics of the opposed damping elements can be neglected. Fig. 1 shows the schematics of the simplified roll dynamics model considered for the presented work. Therein z represents the vertical axis of the leveled vehicle coordinate system and zv the vertical axis of the chassis-fixed coordinate system respectively. For modelling the following assumptions are made:

=:de

=:ce

where Jxs is the moment of inertia of the sprung vehicle parts w.r.t. the x-axis 2 and dϕ d2 ϕ ϕ˙ = , ϕ¨ = 2 . (5) dt dt The effective damping coefficient de comprises the damping constants of the front (df ) and rear (dr ) damping elements. Abdellatif et al. (2003) show how to compute individual damping constants from the effective damping coefficient. Similarly the effective spring stiffness ce represents the overall stiffness that consists of the front (cf ) and rear (cr ) suspension spring stiffnesses and two stabilizers stiffnesses (cst,f , cst,r ). Finally, the dynamic model of the roll dynamics can be formulated as Jxs ϕ¨ + de ϕ˙ + ce ϕ = Mr (6) In contrast to many publications neither the coefficients de and ce nor the roll angle ϕ are assumed known or measurable. Solely the vehicle parameter Jxs as well as the measurements of ay,m and ϕ˙ are required. The first parameter identification method also provides an estimate for the term

(1) The road bank angle is negligible during test manoeuvres. Hence, the identification drive has to be performed on a flat test track. (2) The pitch dynamics are negligible. (3) The vertical dynamics of the tires can be neglected, i.e. vertical tire stiffness is sufficiently high. (4) The height of the front and rear roll centers are equal. (5) The effective stiffness and damping coefficients are constants. Hence, the suspension elements show linear behaviour. 1

zv ϕ

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ms hr . These vehicle parameters have to be known a priori for the second method. The main benefit of the presented identification methods is the minimal requirement in terms of costly measurement equipment. Using the variables x1 := ϕ and x2 := ϕ, ˙ (7) a state-space representation of (6) is given by dx1 =x2 , (8a) dt dx2 = − k1 x1 − k2 x2 + ku, (8b) dt y =x2 , (8c) where u = ay,m and the coefficients k1 , k2 , k are defined as ce de ms hr k1 := , k2 := and k := . (9) Jxs Jxs Jxs The unknown parameters k1 and k2 can be split into a nominal and an uncertain part, i.e. k1 = k¯1 + ∆1 and k2 = k¯2 + ∆2 , (10) ¯ ¯ where the constants k1 and k2 represent given nominal values of k1 and k2 respectively and the constants ∆1 , ∆2 the corresponding uncertainties. The constant k is estimated by the RLS based algorithm whereas it has to be known a priori for the observer based method. Since both proposed identification algorithms, see Section 3 and 4, presume a relative degree of 2 w.r.t. output y, system (8) is rewritten as dz1 =z2 , (11a) dt dz2 = − k1 z1 − k2 z2 + kv, (11b) dt y =z1 , (11c) where the coordinates     z1 x2 := , (12) z2 −k1 x1 − k2 x2 + ku and the input v :=

du dt

Consequently, the estimation yˆk of (14) is given by yˆk = ψkT θˆk−1 .

(17) The recursive algorithm for computation of the parameter vector θˆ can be formulated as (Ljung (1999)) h i θˆk =θˆk−1 + γk−1 yk − ψkT θˆk−1 , (18a) Pk−1 ψk , (18b) λ + ψkT Pk−1 ψk   1 Pk = I − γk−1 ψkT Pk−1 . (18c) λ The vector γk is a correction term for the actual parameter estimates θˆk and the matrix Pk can be interpreted as an estimate of the estimation error covariance matrix. The forgetting factor λ > 0 ensures that recent measurements are stronger weighted than older ones. For avoidance of the blow-up phenomenon (see e.g. Saelid and Foss (1983)) the implementation of an adaptive forgetting factor is highly recommended 5 . γk =

3.2 Conversion From z- to s-Domain For computation of the unknown parameters (9) the transˆ ∗ (z) needs to be approximated by the timefer function G ˆ continuous transfer function G(s). Therefore it is assumed that the identification algorithm yields a BIBO stable transfer function (13). The transformation itself is based on the final value theorem ! ˆ Gˆ∗ (z) = G(s) (19) z=1

1 ln(zi ) = si . Ts

3. RLS BASED IDENTIFICATION METHOD 3.1 RLS Algorithm The transfer function of the time-discretized system (11) can be written as ˆb1 z −1 + ˆb0 z −2 y¯(z) Gˆ∗ (z) = = . (13) v¯(z) 1+a ˆ1 z −1 + a ˆ0 z −2 The corresponding difference equation 4 reads as yk = ˆb1 vk−1 + ˆb0 vk−2 − a ˆ1 yk−1 − a ˆ0 yk−2 . (14) The parameter vector θˆ to be identified is defined by  T θˆ = a (15) ˆ1 a ˆ0 ˆb1 ˆb0 and the data vector ψ as T

(20)

ˆ Let s1,2 be a conjugate complex pole pair of G(s). Then the parameters k1 and k2 can be computed by k1 =s1 s2 = Re(s1 )2 + Im(s1 )2 , (21a) k2 = − s1 − s2 = −2Re(s1 ), (21b) 6 ˆ respectively. Computation of the dc-gain K0 of G(s) allows the calculation of the parameter k as k = K0 k1 . (22)

are introduced. 3

ψ = [ −yk−1 −yk−2 vk−1 vk−2 ] .

s=0

and pole mapping

4. OBSERVER BASED METHOD 4.1 Observer Design and Uncertainty Reconstruction The estimation of the positive parameters k1 and k2 assumes a well-known parameter k = mJsxshr . It is based on robust state estimation of system (11), where the applied second-order sliding-mode observer reads as p dˆ z1 =ˆ z2 + κ1 |z1 − zˆ1 |sign (z1 − zˆ1 ) , (23a) dt dˆ z2 = − k¯1 z1 − k¯2 zˆ2 + kv + κ2 sign (z1 − zˆ1 ) , (23b) dt

(16)

3

Although changing the coordinates the structure of the system is unaffected. Additionally, the z-coordinates are never transferred back to x-coordinates and hence the uncertain parameters in the diffeomorphism are not critical! 4 The z-transforms of the sequences (y ) and (v ) are denoted as k k y¯(z) and v¯(z).

5

A possible algorithm is presented by the bounded-gain forgetting algorithm (Slotine and Li (1991)) but there exist numerous other solutions (e.g. So et al. (2003) and Wang (2009)) 6 K := lim G(s) ˆ 0 s→0

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time-derivatives is realized in terms of control techniques. Therefore the control loop depicted in Fig. 2 is considered, where the integrator dˆ u = rˆ (36) dt serves as system with u ˆ as the variable to be controlled.

see e.g. M’Sirdi et al. (2008). Defining the state estimation errors as e1 := z1 − zˆ1 and e2 := z2 − zˆ2 , (24) yields the error dynamics p de1 =e2 − κ1 |e1 |sign (e1 ) , (25a) dt de2 =∆ (z1 , z2 ) + k¯2 e2 − κ2 sign (e1 ) . (25b) dt Here the function ∆ is defined as ∆ (z1 , z2 ) := −∆1 z1 − ∆2 z2 . (26) Suppose that ∆ (z1 , z2 ) + k¯2 e2 < Ψ < ∞ (27) holds for any possible z1 , z2 and |ˆ z2 | ≤ sup |z2 |, where Ψ is a positive constant. Then it is sufficient to choose the observer parameters according to √ κ1 = 1.5 Ψ and κ2 = 1.1Ψ, (28) such that the solution of system (25) converges to (e1 , e2 ) = (0, 0) within finite time Te , i.e. e1 (t) = 0 and e2 (t) = 0 ∀t ≥ Te , (29) see e.g. Davila et al. (2005). Exploiting the finite time convergence of e1 and e2 the function ∆ can be approximated by filtering the so-called equivalent part of equation (25b), i.e. ˆ 1 , z2 ) = lowpass ((κ2 sign (e1 )) =: [κ2 sign (e1 )] . ∆(z lp (30)

u

Fig. 2. Time-differentiation realized as control loop. The control signal rˆ is an estimate of du dt . In case of RED the positive constants α and β of a supertwisting controller given by p (37a) rˆ =α |ε|sign(ε) + r1 , dr1 =βsign(ε), (37b) dt are tuned, such that the integrator output u ˆ tracks the signal u perfectly, i.e. ε := u − u ˆ≡0 (38) is satisfied. The closed-loop system consisting of integrator (36) and controller (37) reads as p dε =w − α |ε|sign(ε), (39a) dt 2 dw d u = 2 − βsign(ε), (39b) dt dt where the variable du w := − r1 (40) dt is introduced. The structure of system (39) corresponds to that of (25) and therefore ε(t) and w(t) vanish after finite time. Moreover, the control signal rˆ corresponds to an estimate of the time-derivative of the signal u, i.e. du rˆ = , (41) dt see e.g. Reichhartinger and Horn (2011).

In order to obtain estimates for the parameters k1 and k2 the time dependent function eq. (26)

=

ˆ 1 z1 (t) − ∆ ˆ 2 z2 (t) −∆

(31)

is considered, see Baev et al. (2006). Obviously for t ≥ Te and a constant time update interval Tu > 0 the delayed function ˆ 1 z1 (t − Tu ) − ∆ ˆ 2 z2 (t − Tu ) b(t − Tu ) = −∆ (32) ˆ 1 and is available. Therefore the estimated uncertainties ∆ ˆ 2 of (31) and (32) satisfy the set of equations ∆      ˆ1 b(t) −z1 (t) −z2 (t) ∆ = ˆ 2 , (33) b(t − Tu ) −z1 (t − Tu ) −z2 (t − Tu ) ∆ | {z } | {z } =:b(t)

=:A(t)

where det (A(t)) 6= 0 ∀t ≥ Te is assumed. According to (10) the parameters k1 and k2 are computed as     k¯ k1 −1 = A b + ¯1 . (34) k2 k2

6. SIMULATION AND RESULTS 6.1 Numerical Simulation For the evaluation of the presented identification methods a simulation environment is set up. The vehicle motion during a test drive is simulated with the help of vehicle dynamics simulation software CarMaker. It generates input and output signals for the roll dynamics parameter identification algorithms. R The latter are implemented in the Simulink environment, see Fig. 3. Both algorithms provide the estimated ˆ 1 (s), parameters k1 , k2 and k 8 for the transfer functions G

5. ROBUST EXACT DIFFERENTIATOR The proposed identification methods require real-time estimation of the first time-derivative of the input signal u = ay,m . It is assumed that 2 d u (35) dt2 ≤ K < ∞ holds 7 . Exploiting the ideas of robust exact differentiation (RED) (see e.g. Levant (2003, 1998)) the computation of 7

rˆ

u ˆ

4.2 Parameter Estimation

b(t) := [κ2 sign (e1 )]lp

ε

8

Note that k is only estimated in the RLS based parameter identification method.

The first time-derivative of u is assumed to be continuous.

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ˆ 2 (s) to compute the roll angle ϕ. G The quality of any identification procedure depends on a

ay,m ϕ˙

d dt

Gˆ1 (s)

k,k1,k2

RLS

ϕ ˆ˙ 2

Manoeuvre d dt

Road

k1 ( s12 ) 0

Gˆ2 (s)

60 t (s)

80

100

0

20

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60 t (s)

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20

Results

0 R Matlab/Simulink

CarMaker

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ϕ k1, k2

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ϕ−ϕ ˆ1,2

ϕ ˆ2

Observer

0

ϕ˙ −ϕ ˆ˙ 1,2 ϕ ˆ˙ 1

ϕ

Vehicle

RLS

Result ϕ ˆ1 Visual.

k2 ( 1s )

Experiment Design

500

Fig. 5. Identified model parameters k1 and k2 . Fig. 3. Schematics of the simulation environment.

RLS

20

◦

eϕ˙ 1 ( s )

◦

40

60 t (s)

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80

100

80

100

◦

−1 20

40

(3) 40

60 t (s)

0 0

(2)

20

1

100

60 t (s)

Fig. 6. Estimation errors eϕ˙ 1 and eϕ˙ 2 . RLS

0

20

40

60 t (s)

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100

0.1 0 −0.1 0

ϕ˙ ( s )

10 0 −10

0 −1

Observer

eϕ1 (◦ )

10 0 −10

1

0

eϕ˙ 2 ( s )

50 0 (1) −50 0

ay,m ( sm2 )

δw (◦ )

sufficiently high excitation of the system to be identified. Therefore, accurate planning of the experiment needs to be guaranteed. The ISO standards ISO7401 (1988), ISO8725 (1988) and ISO8726 (1988) provide an overview of relevant driving manoeuvres for transient vehicle behaviour testing. For the identification of the roll dynamics a driving maneuovre consisting of different phases is performed. Fig. 4 shows the steering wheel angle δw and the resulting lateral acceleration ay,m and roll rate ϕ. ˙

0

20

40

60 t (s)

80

40

60 t (s)

80

100

80

100

100 Observer eϕ2 (◦ )

Fig. 4. Excitation signal (steering wheel angle) of simulated driving manoeuvre. The steering wheel actuation phases are of ramp (1) and step form (2) followed by a sinusoidal sweep with f = 0.1− 4.0Hz (3). Hence, steady-state as well as transient phases are included in the excitation signal. The estimated model parameters k1 and k2 of both identification methods are depicted in Fig. 5. The errors eϕ˙ 1 and eϕ˙ 2 defined as eϕ˙ := ϕ˙ − ϕˆ˙ 1 and eϕ˙ := ϕ˙ − ϕˆ˙ 2 (42) 1

20

0.1 0 −0.1 0

20

40

60 t (s)

Fig. 7. Estimation errors eϕ1 and eϕ2 . The maximum absolute value of eϕ1 is 0.08◦ (RLS) and eϕ2 0.04◦ (observer based) and can be seen in Fig.7. Related to the maximum value of the roll angle arising during the driving manoeuvre this means a percentual error of 4% (RLS) and 2% (SMO) respectively.

2

are shown in Fig. 6. After a settling time of the parameter identification algorithms the maximum absolute values of |eϕ˙ 1 | and |eϕ˙ 2 | are 0.99◦ /s and 0.42◦ /s respectively. As depicted in Fig. 3 the roll angle computation uses the individually identified values of the parameters k1 , k2 and k as well as the input signal ay,m . The output (ϕˆ1 , ϕˆ2 ) of ˆ 1 (s), G ˆ 2 (s) is finally compared the two transfer functions G to the roll angle ϕ of the CarMaker simulations. Similar to (42), the difference between real and estimated values is defined by eϕ1 := ϕ − ϕˆ1 and eϕ2 :=ϕ − ϕˆ2 . (43)

6.2 Roll Dynamics Model Evaluation So far, it has been shown, that the identification algorithms deliver good estimations of the vehicle’s roll angle ϕ in the time domain. Referring to Decker (2009) the frequency response also provides important characteristic values for the vehicle dynamics evaluation 9 . Therefore, it is essential, that the identified roll model’s frequency response matches the reality well. In this case the frequency 9

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Baev, S., Shkolnikov, I., Shtessel, Y., and Poznyak, A. (2006). Parameter identification of non-linear system using traditional and high order sliding modes. In Proceedings of American Control Conference. doi: 10.1109/ACC.2006.1656620. Chen, D. and Crolla, A. (1998). Subjective and objective measures of vehicle handling: Drivers & experiments. Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 29, 576–597. doi: http://dx.doi.org/10.1080/00423119808969588. Davila, J., Fridman, L., and Levant, A. (2005). Secondorder sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control, 50(11), 1785 – 1789. doi:10.1109/TAC.2005.858636. Decker, M. (2009). Zur Beurteilung der Querdynamik von Personenkraftwagen. Ph.D. thesis, Fakult¨ at f¨ ur Maschinenwesen der TU M¨ unchen. Ding, E. and Massel, T. (2005). Estimation of vehicle roll angle. In Proceedings of the 16th IFAC World Congress. International Federation of Automatic Control, Elsevier, Czech Republic. doi:10.3182/20050703-6CZ-1902.01908. Hamblin, B., Martini, R., Cameron, J., and Brennan, S. (2006). Low-order modeling of vehicle roll dynamics. In Proceedings of American Control Conference. doi: 10.1109/ACC.2006.1657345. ISO7401 (1988). Road vehicles - lateral transient response test methods. ISO8725 (1988). Road vehicles - transient open-loop response test method with one period of sinusoidal input. ISO8726 (1988). Road vehicles - transient open-loop response test method with pseudo-random steering input. Kiencke, U. and Nielsen, L. (2004). Automotive Control Systems. Springer Berlin Heidelberg New York, 2nd edition. Kobetz, C. (2003). Modellbasierte Fahrdynamikanalyse durch ein an Fahrman¨ overn parameteridentifiziertes querdynamisches Simulationsmodell. Ph.D. thesis, Fakult¨at f¨ ur Maschinenbau / TU Wien. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34(3), 379–384. Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, 76, 924941. Ljung, L. (1999). System Identification: Theory for the User. Prentice Hall Ptr, 2nd edition. M’Sirdi, N., Rabhi, A., Fridman, L., Davila, J., and Delanne, Y. (2008). Second order sliding-mode observer for estimation of vehicle dynamic parameters. International Journal of Vehicle Design, 48(3/4), 190 – 207. doi: 10.1504/IJVD.2008.022576. P. E. Pfeffer, M.H. and Lin, J. (2008). Vehicle dynamics measurements. In Proceedings of the Institution of Mechanical Engineers,Part D: Journal of Automobile Engineering, volume 222, 801–813. doi: 10.1243/09544070JAUTO413. Rajamani, R., Piyabongkarn, D., Tsourapas, V., and Lew, J. (2011). Parameter and state estimation in vehicle roll dynamics. IEEE Transactions on Intelligent Transportation Systems, 12(4), 1558 –1567. doi: 10.1109/TITS.2011.2164246.

∠G(jω) (◦ )

|G(jω)|dB (−)

response of the CarMaker data has been approximated by using Fast-Fourier Transform (FFT). Fig. 8 reveals that the frequency responses of the identified roll models and an approximation of the reference system based on CarMaker signals match satisfyingly. −40 −50 −60 10−1

100 ω ( rad ) s

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0 −100 −200 10−1

CarMaker RLS Observer

Fig. 8. Bode plots of identified frequency responses. 7. CONCLUSION The roll angle of automotive vehicles is an important indicator for vehicle safety-related systems as well as vehicle dynamics assessment. However, its in-vehicle measurement is costly and therefore omitted for standard production vehicles. There exist several estimation methods that rely on simple linear roll models. Practically, these estimation techniques require exact knowledge of the plant parameters of the roll model. This work has presented two different methods that aim to identify a linear roll model using the measurement signals of lateral acceleration and roll rate. Once the identification is performed, the roll angle can be calculated e.g. by a trivial observer. The presented methods have been evaluated in simulation. They show high accuracy and can therefore be used instead of a roll angle sensor for vehicle dynamics assessment work. Aside from estimation of the roll angle the methods also provide parameters of a simple roll dynamics model that can be further used for offline simulation work. It is intended to implement them on a vehicle rapid prototyping system and determine their robustness to measurement noise on both input and output signals. As the algorithms are efficient in terms of required computational power their online implementation is straightforward. ACKNOWLEDGEMENTS This work was supported by the Austrian Research Promotion Agency (FFG), project number 834176. REFERENCES Abdellatif, H., Heimann, B., and Hoffmann, J. (2003). Nonlinear identification of vehicle’s coupled lateral and roll dynamics. In Proceedings of the 11th Mediterranean Conference on Control and Automation. IEEE, Rhodes,Greece. Ammon, D. (1997). Modellbildung und Systementwicklung in der Fahrzeugdynamik. B.G. Teubner Stuttgart.

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