ROAD FRICTION ESTIMATION USING A SLIDING MODE BASED OBSERVER

TYRE/ROAD FRICTION ESTIMATION USING A SLIDING MODE BASED OBSERVER Nitin Patel ∗ Christopher Edwards ∗ Sarah K. Spurgeon ∗ ∗

Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK. Emails:{np74,ce14,eon}@leicester.ac.uk Fax:+44 (0)116 252 2619

Abstract: This paper proposes a sliding mode based scheme for tyre-road friction coefficient µ estimation during a braking manoeuvre in an automotive vehicle. The scheme is model based and assumes only wheel angular velocity is measured. The proposed scheme uses a sliding mode observer to reconstruct the friction coefficient µ using equivalent injection ideas. The observer is based on a quarter vehicle representation and is independent of the model used to represent the tyre/road friction. The paper considers a dynamic LuGre friction model, a pseudo static LuGre friction model and a parameter based friction model as a basis for a comparative study. Copyright © 2007 IFAC Keywords: Friction Estimation, Sliding Mode Observer

1. INTRODUCTION Increasingly, commercial vehicles are being fitted with micro-processor based systems to enhance the safety, improve driving comfort, increase traffic circulation, and reduce environmental pollution. Examples of such products are anti-lock brake systems (ABS), traction control systems (TCS), adaptive cruise control (ACC), active yaw control, active suspension systems, and engine management systems (EMS). Many of these systems rely on the physical parameters of the vehicle and the conditions in which it is required to operate. Some of these vehicle related parameters are fixed (or are at least subject to negligible variation); some depend on the way in which the vehicle is being used (e.g. loading) and may be thought of as constant unknown parameters (which may be estimated or accounted for in terms of the robustness of the control system); but others – particularly the tyre/road friction coefficient –

are subject to severe and short term variations. Various methods have been developed to predict tyre/road friction (Kiencke, 1993; Gustafsson, 1997; Ray, 1997; Canudas de Wit et al., 2003; Yi et al., 2003; Alvarez et al., 2005; Patel et al., 2006) using observers designed around mathematical models of friction and simple vehicle models. Yi et al. (2002) proposed an adaptive method which estimates the parameters associated with a modification of the pseudo-static ‘magic formula’ of Pacejka (Bakker et al., 1987) for tyre road friction. The algorithm is such that by choosing suitable initial conditions and parameter adaptation gains, it underestimates the maximum coefficient of friction and slip in the situation when there is a lack of persistency of excitation. A second method proposed by Yi et al. (2003) uses the LuGre friction model from Canudas de Wit et al. (2003) and an adaptive observer based on passivity methods. The approach also assumes

that only information about angular velocity is known. Patel et al. (2006) uses a sliding mode observer based on the same friction and vehicle model. Recently Alvarez et al. (2005) proposed a friction estimation method which uses measurements of wheel angular velocity and vehicle longitudinal acceleration. From the measurement of wheel angular velocity, they propose to numerically compute wheel angular acceleration. The authors assume that the parameters of the friction model, the internal friction state and vehicle velocity are unknown. In this paper an observer is developed from a purely sliding mode perspective. The observer is designed around a quarter vehicle model and is completely independent of the model used to simulate the effects of friction. This distinguishes the work in this paper from the existing literature (Kiencke, 1993; Gustafsson, 1997; Ray, 1997; Canudas de Wit et al., 2003; Yi et al., 2003; Alvarez et al., 2005; Patel et al., 2006). Explicit analytical expression for the gains for the sliding mode observer are given – parameterized by a single scalar which reflects the rate at which sliding is obtained in the observer. During sliding, the equivalent output estimation error is used to estimate the friction coefficient.

2. TYRE/ROAD FRICTION AND VEHICLE MODELLING The tyre/road coefficient of friction is defined as Friction force Fx = (1) µ= Fn Normal force The quantity µ is a nonlinear function of many physical variables including the velocity of the vehicle, the road surface conditions and so-called slip. Longitudinal slip, s, can be defined for a braking scenario as v − rω where v > rω and v 6= 0 (2) s= v where r is the effective rolling radius of the tyre, v is the linear speed of the tyre centre and ω is the angular speed of the tyre. Relationship between µ and s when θ varies from 1 to 4 θ=1 1

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Fig. 1. Typical µ − slip behaviour under different road conditions.

Figure 1 shows curves representing the relationship between µ and s for different road conditions. When s = 0, free rolling of the wheel takes place, whilst s = 1 represents a locked wheel condition. It can be seen from Figure 1 that there is a unique value, sˆ associated with the maximum value of µ. During a braking manoeuvre, maintaining the slip at s = sˆ provides an optimal stopping distance. Less favourable (and potentially more dangerous) road conditions tend to decrease the peak value of the µ/slip curve (by up to 75% in icy conditions) and alters the value of sˆ. In many friction models (Yi et al., 2002; Canudas de Wit et al., 2003; Yi et al., 2003; Patel et al., 2006) this effect is taken into account by means of a ‘road surface condition’ parameter θ . For the purpose of developing a friction coefficient estimator, consider the vehicle dynamics: J ω˙ = −rFx − σω ω − kb Pb (3) mv˙ = 4Fx − Fav (4) where Fx is the friction force from (1), J is the moment of inertia of the wheel, σω represents the viscous rotational friction coefficient, Fav represents the aerodynamic force which can be modelled as Fav = σv gmv (Yi et al., 2003), kb the brake system gain and Pb is the actual applied braking pressure. A quarter vehicle model will be used and the assumption is made that the vehicle is travelling on a flat road and hence the load on each wheel is equally distributed. Based on this assumption, the normal force Fn is given by Fn = mg 4 , where m is the total mass of the vehicle and g is the gravitational constant. The values of the parameters used in this paper are r = 0.323m, J = 2.603kg/m2 , m = 1701kg, σv = 0.005, σw = 1 and kb = 0.9. There are many different friction models in the literature which give µ from (1) as a function of velocity v and slip s (see for example (Bakker et al., 1987; Gustafsson, 1997; Ray, 1997; Yi et al., 2002; Canudas de Wit et al., 2003)). Three different models from the literature will be described in detail in the following section. 2.1 Dynamic LuGre Friction Model The dynamic LuGre friction model is described by the following equations (Canudas de Wit et al., 2003). An internal dynamic state is assumed to satisfy the differential equation σ0 |vr | z (5) z˙ = −vr − θ h(vr ) where vr := v − rω is the relative velocity. The parameter θ in (5) captures changes in the road characteristics: typically, θ =1 represents dry, θ = 2.5 represents wet and θ = 4 represents icy road conditions. The scalar function vr α (6) h(vr ) := µc + (µs − µc )e−| vs | where vs is the Stribeck relative velocity, µs is the normalized static friction coefficient and µc is normalized Coulomb friction. The friction coefficient

produced by the tyre/road contact is then given by µ = (σ0 z + σ1 z˙ − σ2 vr ) (7) where σ0 is the stiffness coefficient, σ1 is the damping coefficient, and σ2 is the viscous relative damping coefficient. The values of these parameters used in this paper are vs = 10m/s, σ0 = 100m−1 , σ1 = 0.70s/m, σ2 = 0.011s/m, µc = 0.35, µs = 0.5 and α = 0.5. 2.2 Pseudo static LuGre Friction Model In the so–called static LuGre friction model by (Canudas de Wit et al., 2003), the value of µ is determined by the (static) algebraic relationship µ ¶ 0 θL|η| 2h(vr ) − σ2h(v h(vr ) ) r 1+γ −1) +Fn σ2 vr (e µ= θ σ0 θL|η| (8) where s σ1 θ|vr | and γ := 1 − (9) η := − 1−s h(vr ) and h(vr ) is given in (6). The quantity L is a scalar parameter which represents the dynamic patch length i.e. the contact area between the road and the tyre. In the simulations L = 0.125m is used. Again the parameter θ captures changes in the road characteristics. The formula for η in (9) converts the braking slip s as defined in (2) into driving slip on which formula (8) is based (Canudas de Wit et al., 2003).

2.3 Parameter based Friction Model A simpler model taken from Yi et al. (2002) gives 1 (10) µ = p1 e−p2 s s(p3 s+p4 ) e−p5 v θ where s is defined in equation (2). Again the parameter θ captures the changes in the road characteristics and p1 , p2 , p3 , p4 and p5 are scalar parameters chosen such that the formula in (10) produces appropriate µ−slip behaviour similar to that given in Figure 1. In this paper the parameter values are given as p1 = 23.444, p2 = 3.3, p3 = 2.69, p4 = 1.015 and p5 = 0.01. 3. OBSERVER DESIGN In this section a sliding mode observer will be designed based on the assumption that only the angular velocity ω is available, which can be measured easily in modern vehicles. Equations (3) and (4) can be re-written as 4 v˙ = −σv gv + Fx m kb r σω ω˙ = − ω − Pb − Fx J J J

(11) (12)

which can be described in state–space form as x(t) ˙ = Ax(t) + Bu(t) + DFx (13) where   # # " " 4 0 −σv g 0 σω B = A= Kb D =  mr  0 − − − J J J (14) Since it is assumed in this section that only angular wheel speed ω is measured, the output distribution matrix £ ¤ C= 01 (15) The following model-based nonlinear observer is proposed x ˆ˙ (t) = Aˆ x(t) + Bu(t) + Gl ey + Dν (16) where x ˆ = col(ˆ v, ω ˆ ), Gl = col(g1 , g2 ) where g1 , and g2 are scalar gains, ey = ω − ω ˆ and ν = −ksgn(F ey ) where k is a scalar gain, F is a scalar which will be defined later and vˆ and ω ˆ are estimates of v and ω respectively. The main objective is to synthesize an observer to generate an estimated angular velocity ω ˆ such that ey = ω− ω ˆ ≡ 0 in finite time and to provide an estimate of longitudinal velocity. Here the gain Gl is chosen as   4J 4Jσv g − α (17) Gl =  rmσω rm  +α − J where α is a positive scalar. It follows that the eigenvalues of (A − Gl C) = {−σv g, −α} which implies (A − Gl C) is stable by design. Define e = x−x ˆ as the state estimation error. The dynamics for the error system can be obtained from (13) and (16) as e˙ = (A − Gl C)e + D (Fx − k sgn(F ey )) (18) To induce sliding the gain from the switching term ν must satisfy k > |Fx |. Consider as a Lyapunov function V = eT P e where   4J P P 1 1   rm 2 (19) P =  4J  16J P1 P2 + 2 2 P1 rm r m and P1 and P2 are positive scalars. Define ¡ ¢ Q := − P (A − Gl C) + (A − Gl C)T P (20) then from (14),(15) and (19) it can be verified that   8Jgσv P1   2gσv P1 rm (21) Q =  8Jgσ  2 32J gσ v v P + P1 2αP 1 2 rm r 2 m2 Clearly the top left element is positive, and from the Schur complement 32J 2 gσv 8Jgσv −1 8Jgσv ( rm P1 ) r 2 m2 P1 +2αP2 −( rm P1 )(2gσv P1 )

= 2αP2 > 0 and so Q is symmetric positive definite. It can be verified by direct substitution that P D = F C T for the scalar F := −2 αr J P2 . Then T for V (e) = e P e it follows that

V˙ = −eT Qe − 2eT P D (k sgn(F ey ) − Fx ) ≤ −eT Qe − 2|F ey |(k − |Fx |) ≤ 0 for large enough k and so the state estimation error e is quadratically stable as claimed. As argued in Edwards and Spurgeon (1994), in a domain of the origin, a sliding motion takes place on S = {e : Ce = 0}. Now, from first principles, the reduced order motion whilst sliding will be investigated. From equations (13) and (16), e˙y = C e˙ can be written as r r σω (22) e˙y = − ey − Fx − g2 ey + ν J J J and so if a sliding mode is enforced in finite time, ey = e˙ y = 0 (Utkin, 1992) and the above equation becomes r r (23) 0 = − Fx + νeq J J where νeq represents the equivalent output error injection signal necessary to maintain a sliding motion in the state estimation error space (Edwards and Spurgeon, 1998; Utkin, 1992). Therefore from (23) an expression for νeq is νeq = Fx (24) The equivalent injection νeq is available in realtime (since for example it can be obtained by low pass filtering ν). Therefore from (1) and (24) νeq (25) µ ˆ := Fn is an estimate of µ. It is clear that the observer and the methodology to estimate µ is completely independent of the three friction models described in Section 2.

in the observer. The discontinuous term in the observer (18) has been replaced by a sigmoidal ey approximation |ey |+δ ,where δ = 0.001 Plant ω and estimated ω

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4. SIMULATION RESULTS

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Three different friction models (the dynamic LuGre, the static LuGre and the parameter based) described in Section 2 will be used for producing the results. Equations (3)-(4) will be used together with one of the friction models from Section 2 to represent the plant. In the plant model (3)-(4) the term Fx is replaced by µFn = µ mg 4 , where µ is computed from one of the expressions (7), (8) or (10) given in Section 2. Three different input signals Pb were generated to provide braking action for three different road conditions (θ = 1, θ = 2 and θ = 4). In the observer, the gain for the nonlinear injection term has been chosen as k = 9000 whilst g1 = −0.180 and g2 = 0.6158 were used which have been obtained from (17) using α = 1. Initially the dynamic LuGre friction model from Section 2.1 will be investigated; this will then be followed by the so-called static friction model (Section 2.2) and finally the parameter based friction model (Section 2.3). In the simulations which follow the initial conditions for the plant using the dynamic LuGre model are ω = 124, v = 40 and z = 0 whilst ω ˆ = 155 and vˆ = 50 are used

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Fig. 4. Comparison of plant states {dotted line}, estimated states {solid line} and input signal when θ = 4. Figures 2, 3 and 4 show the plant states ω and v {dotted line}, the estimated states ω ˆ and vˆ {solid line}, and the error in the states (ey , ev ) using the dynamic LuGre friction model given in (5). Three different road conditions were considered (θ = 1, θ = 2 and θ = 4) to produce the above results. A Deliberate large mismatches between the plant and observer velocity at time zero has been introduced to demonstrate the observer performance

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Fig. 5. Comparison of µ using a dynamic LuGre considering θ = 1 {top}, θ = 2 {middle}and θ = 4 {bottom}. level. It can be seen from the figures that the error associated with the angular velocity converges to zero very quickly, whilst the error associated with the longitudinal velocity converges slowly. This is due to the small invariant zero (−σv g = −0.0491) associated with the triple (A, D, C). Figure 5 shows the error in µ for the three different road conditions (θ = 1 {top}, θ = 2 {middle} and θ = 4 {bottom}). From the above figure, the error between the plant µ and estimated µ ˆ is very small, which indicates an excellent estimation of the friction coefficient.

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Fig. 7. Comparison of µ using a pseudo static LuGre friction model considering θ = 1 {top}, θ = 2 {middle} and θ = 4 {bottom}. the static LuGre model are ω = 124 and v = 40 whilst ω ˆ = 155 and vˆ = 50 are used for the observer. This represents a deliberate mismatch for the purpose of demonstration. Figure 7 shows the error in µ for the three different road conditions (θ = 1, {top},θ = 2 {middle} and θ = 4 {bottom}). Again, the error between the plant µ and estimated µ ˆ is very small, which indicates excellent estimation of the friction coefficient. Plant ω and estimated ω

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Fig. 6. Comparison of plant states {dotted line}, estimated states {solid line} and input signal when θ = 1. Figure 6 shows the plant states ω and v (dotted lines), the estimated states ω ˆ and vˆ (solid lines), and the error in the states (ey , ev ) when using the so–called static friction model given in (8) in the plant computation. Here only one road condition (θ = 1) is shown. A similar level of performance is achieved for θ = 2 and θ = 4 but the associated simulations are not included here. The initial conditions for the plant using

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Fig. 8. Comparison of plant states {dotted line}, estimated states {solid line} and input signal when θ = 1. Figure 8 shows the plant states ω and v (dotted lines), estimated states ω ˆ and vˆ (solid lines), and the error in the states (ey , ev ) using the parameter based friction model given in (10). Here again only one road condition (θ = 1) is shown. A similar level of performance is achieved for θ = 2 and θ = 4 but the associated simulations are not included here. Figure 9 shows the error in µ for the three different road conditions (θ = 1 {top}, θ = 2 {middle} and θ = 4 {bottom}). Again, the error between the plant µ and estimated µ ˆ is small, which indicates excellent estimation.

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with the brake input). The potential advantage offered by the method is that the underlying observer is friction model independent. Almost all other methods in the literature are built around an assumed model structure for the tyre/road friction coefficient. The approach has been demonstrated by using three different friction models from the literature.

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REFERENCES

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Fig. 9. Comparison of µ using a parameter based (bottom) friction model considering θ = 1 {top}, θ = 2 {middle} and θ = 4 {bottom}. Plant ω and estimated ω

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Fig. 10. Comparison of plant states and friction coefficient µ {dotted line}, estimated states and estimated friction coefficient µ {solid line} when θ = 1. Figure 10 shows simulations when around 3% measurement noise was introduced to the angular velocity ω. Figure 10 shows the plant states ω, v and friction coefficient µ {dotted line}, the estimated states ω ˆ , vˆ and estimated friction coefficient µ {solid line} and the errors using the dynamic LuGre friction model given in (5). A low–pass filter was used for the estimated friction coefficient µ. It is clear from Figure 10 that even in the presence of measurement noise associated with the angular velocity ω, the estimation of the friction coefficient µ is maintained reasonably well.

5. CONCLUSIONS This paper has considered a unique sliding mode based scheme for estimating tyre friction µ in an automotive braking manoeuvre. Only angular wheel speed is assumed to be available (together

Alvarez, L., J. Yi, R. Horowitz and L. Olmos (2005). Dynamic friction model-based tyre/road friction estimation and emergency braking control. Trans. of the ASME on Journal of Dynamic Systems, Measurement and Control. Bakker, E., L. Nyborg and H. Pacejka (1987). Tyre modelling for use in vehicle dynamics studies. Soc. Automotive Eng. Paper # 870 421. Canudas de Wit, C., P. Tsiotras, X. Claeys, J. Yi and R. Horowitzs (2003). Nonlinear and Hybrid Systems in Automotive Control edited by R. Johansson and A. Rantzer. pp. 147–210. Springer-Verlag, London, UK. Edwards, C. and S.K. Spurgeon (1994). On the development of discontinuous observers. International Journal of Nonlinear Control 59, 1211–1229. Edwards, C. and S.K. Spurgeon (1998). Sliding mode control: theory and applications. London: Taylor & Francis Ltd. Gustafsson, F. (1997). Slip-based tyre-road friction estimation. Automatica pp. 1087–1099. Kiencke, U. (1993). Real-time estimation of adhesion characteristic between tyre and road. In: Proceedings of the IFAC World Congress. Patel, N., C. Edwards and S.K. Spurgeon (2006). A sliding mode observer for tyre friction estimation during braking. In: Proceeding of the American Control Conference, Minneapolis, USA. pp. 5867–5872. Ray, L.R. (1997). Nonlinear tyre force estimation and road friction identification - simulation and experiments. Automatica 33, 1819–1833. Utkin, V. I. (1992). Sliding Modes in Control Optimization. Springer-Verlag, Berlin. Yi, J., L. Alvarez and R. Horowitz (2002). Adaptive emergency braking control with understanding of friction coefficient. IEEE Trans. on Control System Technology 10, 381–392. Yi, J., L. Alvarez, X. Claeys and R. Horowitz (2003). Emergency braking control with an observer-based dynamic tyre/road friction model and wheel angular velocity measurement. Vehicle System Dynamics 39, 81–97.