Research Article Development and Verification of the Tire/Road Friction Estimation Algorithm for Antilock Braking System

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1 Mathematical Problems in Engineering Volume 24, Article ID , 5 pages Research Article Development and Verification of the Tire/Road Friction Estimation Algorithm for Antilock Braking System Jian Zhao, Jin Zhang, and Bing Zhu,2 State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 322, China 2 Key Laboratory of Bionic Engineering of Ministry of Education, Jilin University, Changchun 322, China Correspondence should be addressed to Bing Zhu; zhubing@jlu.edu.cn Received 8 June 24; Revised 6 August 24; Accepted 6 August 24; Published 28 September 24 Academic Editor: Ebrahim Momoniat Copyright 24 Jian Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Road friction information is very important for vehicle active braking control systems such as ABS, ASR, or ESP. It is not easy to estimate the tire/road friction forces and coefficient accurately because of the nonlinear system, parameters uncertainties, and signal noises. In this paper, a robust and effective tire/road friction estimation algorithm for ABS is proposed, and its performance is further discussed by simulation and experiment. The tire forces were observed by the discrete Kalman filter, and the road friction coefficient was estimated by the recursive least square method consequently. Then, the proposed algorithm was analysed and verified by simulation and road test. A sliding mode based ABS with smooth wheel slip ratio control and a threshold based ABS by pulse pressure control with significant fluctuations were used for the simulation. Finally, road tests were carried out in both winter and summer by the car equipped with the same threshold based ABS, and the algorithm was evaluated on different road surfaces. The results show that the proposed algorithm can identify the variation of road conditions with considerable accuracy and response speed.. Introduction The adhesion information between road and tire is very important for vehicle active safety control systems such as ABS (antilock braking system), ASR (acceleration slip regulation), and ESP (electronic stability program). Therefore, plenty of researches on tire-road friction coefficient estimationandidentificationhavebeenandarebeingfocused. For instance, it is well known that ABS is an active safety control system by utilizing the potential road-tire adhesion to improve vehicle deceleration, handling, and stability during braking ]. In order to utilize the available tire forces on variableroadsasmuchaspossible,theidentificationofpeak tire-road friction coefficient and its optimal slip ratio is a critical task for guaranteeing the performance of ABS 2]. Unlike normal vehicle dynamic information, it is hard to measure tire-road friction directly. A cutting-edge approach is the intelligent tire which monitors the tire adhesion information in real time by the sensors such as ultrasonic approach 3], optical position detection 4], and piezoelectric film sensor 5] mounted inside of tires. However, these researches arestillintheirearlystageandunsuitableforwideapplication duetothenewlyequippedsensorsandtheincreasedcost. Therefore, more researches are focused on the methods and algorithms for tire-road adhesion information estimation. As the friction phenomenon between a tire and road is affected complexly by different factors and it has significant nonlinear and dynamic properties such as viscousness and hysteresis, it isnoteasytoanalyzetheinteractionbetweenapneumatictire androad.variouskindsofdynamicmodelbasedmethodsare utilized to solve this problem 6]. For these methods, available sensorsoncarsareusedoptimallytoestimatetire/road information based on the wheel dynamic or vehicle dynamic state variables. The wheel dynamic based method described in 7], the vehicle dynamic based method in 8], and the selfaligningtorquebasedmethodin9] are three typical kinds of these solutions. The vehicle dynamic based method can be used to estimate road/tire friction forces by observers based on vehicle dynamic information. In practice, the mathematical

2 2 Mathematical Problems in Engineering models of vehicle system are simplified for analysis and calculation; the differences between actual vehicles and these models might lead to errors of the estimation results ]. The transient characteristics and ever-changing parameters of the vehicle in motion might be another reason that affects the estimation results ]. At the same time, as signals fromthesensorsarevulnerabletothenoisepollution,there will be deviations on the observed results 2]. Therefore, the observer must have good robustness for tolerating differences between models and vehicles, variations of model parameters, and the signal errors. For this reason, Kalman filter is widely used for states observer design because of its robustness performance. It has been proven to be effective in extracting the useful information from the noised signals 3 5]. With regard to the road friction coefficient, the observer method is not a good choice. The coefficient is not the state variable of a vehicle or a wheel. The nonlinear tire model should be written into Jacobian matrix expressions, and the extended Kalman filter is necessary. Using this method, the dimension of the state variable will be very large which results in heavy computation task. Thus, this way is seldom applied. The road friction coefficient is often treated as a parameter of different kinds of tire models and detected by some kinds of parameter identification methods 6]. The relatively simple theoretical tire models such as tire brush model and dugoff model are suitable for this application 7, 8]. However, the road friction coefficient is determined bytheinteractionoftireandtheroad,andthetire/road friction coefficient changes along with the variations of tire parameters, such as inflation pressure, tread material, and wear 9]. At the same time, vehicle dynamic such as load transfer caused by the motion of vehicle can also affect the tire/road adhesion coefficient, combined with the influence of dynamic characteristics of pneumatic tire, such as relaxation and hysteresis. Thus, it is always difficult to achieve a precise tire/road friction coefficient estimation result even when the accurate tire forces are obtained 2]. In order to obtain better estimation results of tire-road friction coefficient, identification methods with good robustness are also necessary to reduce the interference of disturbance signals and converge to the actual value quickly. Recursive unbiased estimation algorithms can perform a good unbiased estimation result 2], and the recursive least square (RLS) method is one of them. Not only is the RLS method convenient for computer calculation and storage but it also has a fast response speed and strong antijamming capability by adding the forgetting factor 22, 23]. It also could deal with the nonlinear model such as tire model after linearization processing. Thus, based on the previous observation of tire forces by Kalman filter, it is a good approach by combining the RLS method to estimate road friction coefficient. The characteristics of the practical actuators are another problem in the road friction estimation for active brake control. Nowadays, almost all of the hydraulic active brake control systems are equipped with high-speed switching valves (HSV). ABS is a typical example that applies the pulse control mode by switching the HSV valves. Thus, considerablefluctuationcausedbyabscontrolwillbeobservedon T bi /p i γ a a ω i x y x y Calculator Kalman filter observer (friction force) F yf,f yr F xi F zi κ i α i Vehicle Tire/road condition estimator Recursive least square estimator (friction coefficient) Figure : The framework of the system. wheels dynamic state variables such as wheel slip ratio and brake pressure, which are important for friction estimation. In this case, the negative influences on the estimation from the nonlinear characteristics of the vehicles and tires will be amplified. Therefore, the effects on the estimation performance caused by the fluctuation from actuators should be also considered, which are not been fully discussed yet. In this paper, a tire-road friction estimation algorithm for ABS was proposed with the combination of a Kalman filter and a linearized RLS estimator. The tire forces are observed by the discrete Kalman filter using the available vehicle signals based on a planar vehicle model, and the tire-road friction coefficient, which is a parameter of tire brush model, is identified based on linearized RLS method. The proposed estimation algorithm was evaluated by both CarSim and Matlab/Simulink cosimulation and road tests. First, it was simulated using a sliding-mode-control (SMC) based ABS control algorithm that controlled wheel speeds and brake pressures more smoothly with almost no noise beingshown.then,atypicalthresholdbasedabsalgorithm was used and the performance estimation was evaluated by fluctuant input signals. Finally, road tests were carried out in both winter and summer by the car equipped with the same threshold based ABS, and the performance of the algorithm was discussed on different road surfaces. 2. The Tire/Road Friction Estimation Algorithm The sketch of the proposed system is shown in Figure. The system consists of a calculator, a Kalman filter based observer, and a recursive least square (RLS) based estimator. First, the tire longitudinal and lateral forces are estimated by the observer. Then, the road adhesion coefficient is estimated by these forces together with the vertical wheel loads, the slip ratios, and the side slip angles calculated by the calculator. All of the descriptions of the arguments in this paper can be referred to in Notations. The signals of individual wheel speed ω i and brake torque T bi or brake pressure p i,theyaw rate γ, the longitudinal and lateral accelerations a x and a y, and velocities V x and V y of target vehicle are necessary for the estimation algorithm. Based on these signals, the wheel vertical load F zi,slipangleα i,slipratioκ, longitudinaland μ F xi

3 Mathematical Problems in Engineering 3 F xfl F xfr The dynamic equations of the planar vehicle model shown in Figure 2 are F yfl F yfr ma x = H x F, F xrl a y a x γ l f ma y = H y F, I z γ=h γ F, () F xrr l r where F yrl F yrr F =F xfl F xfr F xrl F xrr F yf F yr ] T, 2t Figure 2: The planar vehicle model. F yf =F yfl +F yfr, F yr =F yrl +F yrr, H x =cos δ cos δ sin δ ], x H y =sin δ sin δ cos δ ], H γ = l f sin δ t cos δ l f sin δ+t cos δ t t l f cos δ l r]. (2) T b F p G F z ω F x F xi is the longitudinal force of each wheel, F yi is the lateral force of each wheel (where the i=fl, fr, rl,andrr represent the front left, front right, rear left, and rear right wheel, resp., hereinafter inclusive), and F yf and F yr are the lateral forces of the front and rear axle, respectively. δ is the steering angle of the front wheels, m is the mass of the vehicle, a x and a y are the longitudinal and lateral accelerations of the vehicle, respectively, γ is the yaw rate of the vehicle, I z is the moment of inertia of the vehicle, l f and l r are the distances from the center of mass of the vehicle to the front axle and rear axle, respectively, and 2t is the wheel base. The moment equilibrium equation for the single wheel model shown in Figure 3 is Figure 3: The single wheel model. I W ω i = T bi F xi r. (3) lateral forces F x and F y, and road adhesion coefficient μ can be estimated by the algorithm proposed in this paper. 2.. The Tire Force Observation Based on Kalman Filter. The schematic diagram of the vehicle model used for the tire forces estimation is shown in Figure 2, whose aerodynamic resistance and tire rolling resistance are neglected and only tire adhesion forces are considered. The dynamic analysis of a single wheel is shown in Figure 3. Inthesemodels, the longitudinal forces F x are positive for driving, whose directions are the same as shown in the figures, and negative for braking otherwise. I W is the moment of inertia of the wheel, ω i is the rotation speed of the wheel, T bi isthebrakingtorque,andr is the effective rolling radius. Furthermore, in Figure 3, F zi is the vertical force of each wheel. G is the vertical load from the mass and F p is the longitudinal force acts on the wheel from the axle. Basedon() to(3), the following states space equations can be achieved: x (t) = Α (t) x (t) + Β (t) +w(t), z (t) = Η (t) x (t) + V (t), (4)

4 4 Mathematical Problems in Engineering Β (t) = 6 A (t) = T bfl I W r I I W, (5) H γ ] I 5 z ] T bfr I W T brl I W T T brr ], (6) I W H x m 5 H Η (t) = y m 5. (7) ] 5 6 I 5 5 ] The system state is an -dimensional vector x(t) = F xfl,f xfr,f xrl,f xrr,f yf,f yr,ω fl,ω fr,ω rl,ω rr,γ] T, and the measurement variable is a 7-dimensional vector z(t) = a x,a y,γ,ω fl,ω fr,ω rl,ω rr ] T.Thesystemprocessnoise w(t) is an -dimensional zero-mean white noise vector, the measurement noise V(t) is a 7-dimensional zero-mean white noise vector, and w(t) and V(t) are mutually unrelated. The discrete Kalman filter method based on linear discretesystemiswidelyusedinengineeringbecauseofits advantages in computer aid calculation and the less requirements of data storage. The discretization of the continuous vehicle system is necessary for the discrete Kalman filter approach. In this paper, the zero-order hold (ZOH) method is utilized to discretize the continuous vehicle model (4), and thediscretesystemstatespaceequationis x k+ = Α k x k + Β k +w k, z k = Η k x k + V k. Basedon (8), the discrete Kalman filter estimation process is as follows. Prediction: State estimation: (8) x k k = Α k k x k + Β k k. (9) x k = x k k + Κ k (z k Η k x k k ). () Filter gain matrix calculation: Ρ k k Η T k Κ k = () (Η k Ρ k k Η T k + V). Step prediction error variance matrix calculation: Ρ k k = Α k k Ρ k Α T k k + W. (2) The estimated error variance matrix calculation: Ρ k =(I Κ k Η k ) Ρ k k, (3) where the matrices W and V represent the process noise and measurement noise covariance matrix, respectively. As long as the initial values x and P aregiven,accordingto the measured values z k of the kth step, the state estimation of the kth step x k canberecursivelycalculated,andthe corresponding tire force included in x k could be achieved Information from the Calculator. The practical slip ratios κ, the side slip angles α, and the vertical tire loads F zi are calculated by the calculator which is shown in Figure : κ fl κ fr κ rl κ rr F zfl F zfr F zrl F zrr 2t 2t ] = 2t 2t l f +l r l f +l r ] ] l f +l r l f +l r ] mgt + ma y h g mgt ma y h g mgl r +ma x h g ], mgl f ma x h g ] ω fl r (V x t γ)cos δ+(v y +l f γ)sin δ ω fr r (V x +t γ)cos δ+(v y +l f γ)sin δ ] = ω rl r V ] x t γ, ω rr r V x +t γ ] ] α fl α fr α rl α rr V y +l f γ V x t γ V δ y +l f γ ] = δ ] tan V x +t γ, V ] ( y l r γ ] V x t γ ) V y l r γ] ( V x +t γ] ) (4) where g is the gravitational acceleration and h g is the height of center of mass The Combined Longitudinal and Lateral Tire Brush Model. Plenty of researches have been carried on exploring the dynamic phenomenon and transient properties of the tireroad forces, and various tire friction models, including the theoretical and empirical models, were proposed 24]. The empirical models could describe tire characteristic precisely, but their expressions are complicated and difficult

5 Mathematical Problems in Engineering 5 Longitudinal tire force (N) 3 Longitudinal tire force calculated by different tire model MF model, μ= Brush model, μ= Slip ratio ( ) Figure 4: Longitudinal tire force calculated by magic formula and brush model. Speed of vehicle and front left wheel (m/s) 2 Braking pressure and speed Vehicle speed Braking pressure 5 5 Front left wheel cylinder pressure (MPa) Figure 5: The velocity and wheel pressure for SMC based ABS (Without noises). for parameters fitness, while the theoretical models have simpler expression with less parameters and are convenient for parameters fitness in real time. Therefore, the theoretical tire models are widely used in road identification generally. The tire brush model is one of theoretical tire models and it is widely used for estimation or control purposes because of its simplicity and qualitative correspondence to the experimental tire behaviors such as the friction ellipse influence and tire nonlinear characteristics, for example, tire force saturation 7]. The expression of tire brush model for the braking action is 25] F x = C x (κ/ (+κ)) F f t, where F y = C α (tan α/ (+κ)) f F t, F t = { f f 2 + { 3μF z 27μ 2 F 2 f 3, if f 3μF z, z { μf z, else, f= Cx 2( κ +κ )2 +Cα 2 α (tan +κ )2. (5) (6) C x, C α are the longitudinal and lateral stiffness of tire, respectively, and μ is the tire-road friction coefficient. Generally, the magic formula is considered the accurate model for road friction illustration. Figure 4 shows the comparison of the longitudinal tire forces calculated by the tire brush model and the magic formula model at a tire vertical load of 4 N. It can be seen that the result of tire brush model is close to that of the magic formula model within the range of zero to optimal wheel slip ratio, while its precision decreases at large wheel slip ratio comparing with magic formula. As the tire brush model is precise enough aroundtheoptimalwheelslipratio,whichistheabscontrol target, it could meet the requirements of road identification as longasthewheelslipratiocouldbecontrolledintheoptimal range by ABS. The parameters C x, C a,andμ in tire brush model are needed to be identified. According to 2], the values of C x and C a which depend on tire properties such as tire size, tread width, tread stiffness, inflation pressure, and load tend to be static for a short period; therefore, they could be assigned as the fixed value in a short time. In this paper, the value of tireroad friction coefficient μ is estimated by the usage of the RLS method, which is described in the following section Tire-Road Friction Coefficient Estimation. Recursive least squares (RLS) method is an iterative algorithm which could estimate the parameters recursively by minimizing the sum of the squares of the difference between observed data and computed data 26]. However, RLS method is only available forlinearsystems.asthetirebrushmodelisnotonlya nonlinear function but also a segmented function, it is hard to transform the tire brush model to a linear representation directly. Therefore, in order to utilize the RLS to estimate tire model parameters, an approximate linearization method was adopted. First, tire brush model in nonlinear form could be written as follows: y (k) =f(θ (k)) + V l. (7) Consider that y=f x,f y ] is the observed values of tire forces estimated by Kalman filter, f(k, θ) is the expression of tire

6 6 Mathematical Problems in Engineering Speed of vehicle and front left wheel (m/s) Braking pressure and speed Front left wheel cylinder pressure (MPa) Road friction coefficient μ Estimation of road friction coefficient Vehicle speed Braking pressure Figure 6: The velocity and wheel pressure for SMC based ABS (with noises) Actual μ Estimated μ (noised sig) Estimated μ (smooth sig) 3 Estimation of the wheel longitudinal force Figure 8: The road friction coefficient estimation result for SMC based ABS. Front left wheel longitudinal force (N) Actual F x Estimated F x (noised sig) Estimated F x (smooth sig) Figure 7: The tire force estimation result for SMC based ABS. brush model, θ=μ]is tire brush model parameter vector, and V l is the observed noise. y(k) can be written approximately as follows: where y (k) F(k) ( θ (k) θ (k ))+f(k, θ (k )), (8) F (k) = Define a variable h(k): f (k, θ).x (9) θ θ= θ(k ) h (k) =y(k) +F(k) ( θ (k )) f(k, θ (k )). (2) Speed of vehicle and front left wheel (m/s) 3 2 Braking pressure and speed Vehicle speed Braking pressure Front left wheel cylinder pressure (MPa) Figure 9: The velocity and wheel pressure for threshold based ABS. Bring the expression of y(k) into (2); h(k) can be written approximately as follows: h (k) F(k) θ (k). (2) After approximately linearizing the nonlinear tire brush model, the RLS method is applicable for the tire-road friction coefficient estimation. Equation (2)canbeusedasthelinear observationmodelandthecostfunctionj( θ, k) is as follows: J( θ, k) = k i= λ k i (h (i) F(i) θ) 2, (22)

7 Mathematical Problems in Engineering 7 Front left wheel longitudinal force (N) Estimation of the wheel longitudinal force Actual F x Estimated F x Figure : The tire force estimation result for threshold based ABS. Road friction coefficient μ Estimation of road friction coefficient Actual μ Estimated μ Figure : The road friction coefficient estimation result for threshold based ABS. where λ is a forgetting factor. The least-squares (LS) solution is classically obtained by zeroing the gradient of the cost function J( θ, k), which gives the LS estimation: θ (k) =( k i= k F T (i) λ k i F (i)) (F T (i) λ k i h (i)). (23) The parameter vector θ(k) can be updated for each new observation and thus be estimated online. For this, the error covariance matrix P (k) =( k i= i= F T (i) λ k i F (i)) (24) is computed recursively by using the Sherman-Morrison formula 23]: P (k) =P(k ) P (k ) FT (k) F (k) P (k ) λ+f(k) P (k ) F T. (25) (k) By putting (24) and(25) into(23) and setting the initial θ() and P(), the RLS algorithm is obtained: P (k ) F T (k) K (k) = λ+f(k) P (k ) F T (k), θ (k) = θ (k ) +K(k) (h(k) F(k) θ (k )), P (k) = (I K(k) F (k)) λ P (k ). (26) When the wheel slip ratio is very small, the algorithm might result in fluctuant and unexpected estimation 27]. In order to avoid incorrect estimation, a slip ratio threshold s update was set, and the estimation results are only updated when the slip ratio is larger than s update andheldasthe latest valid estimation value at a slip ratio smaller than s update. Consider θ (k ) +K(k) { θ (k) = (h(k) F(k) { θ (k )), if s>s update, (27) { θ (k ), else. According to 25], the relationship between calculative slip ratio s and practical slip ratio κ for braking maneuver is as follows: s= κ. (28) VBOX system ABS AutoBox system Sensors Figure 2: Tested vehicle and testing system. The performance of the algorithm is sensitive to the preset threshold s update.ifs update istoosmall,theremightbefluctuant and unexpected estimation when the actual wheel slip ratio is pretty low, while, if s update is set too large, low sensitivity might occur and the estimated μ will not be updated in time. As the slip ratio for is around 6% in the ABS algorithm used for the verification, with integrated consideration of reliability and sensitivity, threshold of 5% was chosen in this paper: s update =.5. (29)

8 8 Mathematical Problems in Engineering Lateral accelerometer Yaw rate sensor a y Yaw rate (γ) Wheel cylinder pressure sensors Brake pedal Brake pressure (p i ) Brake trigger VBOX Node Brake trigger x a x AutoBox DS5 DS222 PC Control desk Matlab/Simulink sensors Wheel speed signals ABS ECU Node 2 (ω i ) μ-jump flag CAN bus Figure 3: Sketch of testing system. Figure 4: The summer road testing ground. timeof.s,whichactedonboththediscretekalman filtering and RLS estimator. The parameters used in this paper can bereferredtointable 2. Figure 5: The winter road testing ground. 3. Verification by Simulink and CarSim Cosimulation The algorithm was validated by the Matlab/Simulink and CarSim cosimulation platform, and a sedan vehicle model was chosen. The road was set as a μ-jump road whose friction coefficient is shown in Table. The vehicle was braked at an initial speed of 2 km/h, and the algorithm runs at a loop 3..VerificationbytheSMCBasedABSAlgorithm. The sliding mode control (SMC) is suitable for the systems with uncertainty and disturbance because of its advantages in quick response, insensitivity to parameter change or disturbance, no need of online system identification, and usability of physical implement 28]. Therefore, it is often used for ABS and realizing stable slip ratio control in simulation. Take the error e between the calculative slip ratio s and the optimal slip ratio s anditsderivativeasthestatevariables: e=s s, e= s. The switching function is set as follows: (3) δ slide =ce+ e= s+cs s ], (3) where c is a constant, which can be referred to in Table 2.

9 Mathematical Problems in Engineering 9 Speed (m/s) Road friction coefficient μ Speed (m/s) Road friction coefficient μ Estimation result of road friction coefficient Vehicle speed Estimation μ μ-jump μ-jump Figure 6: Dry concrete pavement to wet tiles (summer) Estimation result of road friction coefficient Vehicle speed Estimation μ μ-jump μ-jump Figure 7: Dry concrete pavement to ice (winter). Table : Tire-road friction coefficient for simulation. Distance travelled l/m 5 5 > Road friction coefficient μ Table 2: Parameter List. Variable Value Unit Simulated vehicle model r.335 m m 529 kg l f.39 m l r.637 m t.775 m g 9.8 m/s 2 h g.59 m I z kg m 2 I W kg m 2 Tested vehicle r.37 m m 77 kg l f.2 m l r.34 m t.735 m g 9.8 m/s 2 h g.5 m I z 4 kg m 2 I W kg m 2 Sliding mode control ε 3 Ω 2 c 2 T.5 s Tire C x 6 4 N C α 6 4 N/rad Forgetting factor λ.9 Table3:Variancesforthewhitenoisesofrealsignals. Signals Variance Unit a x.675 (m/s 2 ) 2 a y.2 (m/s 2 ) 2 γ 6.758e 5 (rad/s) 2 ω i 3 km/h.337 (rad/s) km/h.369 (rad/s) km/h (rad/s) 2 p i.75 (Mpa) 2 Adopting the constant speed reaching law of SMC, δ slide =ce+ e= ε tanh ( δ slide ), (32) Ω where ε and Ω are positive real constants.

10 Mathematical Problems in Engineering Speed (m/s) Road friction coefficient μ Estimation result of road friction coefficient Vehicle speed Estimation μ Slip ratio s Deceleration (m/s 2 ) Slip ratio s Braking pressure (MPa) Deceleration Braking pressure Figure 8: Details of dry concrete pavement to ice (winter).

11 Mathematical Problems in Engineering Table 4: The statistic data of estimation results for threshold based ABS. Actual friction Estimation result coefficient Lag time/s Means Standard deviation.5 (mid-μ) (low-μ) (high-μ) With the consideration of the hysteresis of hydraulic brakesystem,thebrakesystemcanbemodelledasfollows: T b + T T b = U, (33) T where T b is the brake torque applied on the target wheel, U is the output from the SMC ABS controller, and T is a time constant. By substituting (3)into(33), we get I W ω= T U F x r+ T ( I W ω F xr). (34) Assuming F x =,andputting(34)into(32), we have the output of SMC controller: U=I W T { V w r ε tanh (δ slide Ω )+(c 2 V w ) e] V w + I W T ( I W ω F x r)}, (35) where V w is the speed of center of the wheel. By adjusting the braking torque on slipping wheels, the wheel slip ratio could be limit in a suitable range. During the evaluation, the influence by the sensors noises should be considered for real application. Therefore, sensor noises were treated as white noise and added onto the smooth simulated measurements. Their variances were analyzed based on the experimental data and listed in Table 3. These data are derived from raw signals of sensors used in our test system, which will be introduced in Section 4. Normally, these signals should be prefiltered before being used by controller such as ABS or ESP, and the noises will be reduced significantly. Therefore, if the proposed estimation algorithm performs well on these noises, it will be qualified for the practical usage. Furthermore, it should be noticed that the magnitude of noise of raw wheel speed changes with the valueofwheelspeeditself:thelargerthespeed,thehigherthe noise power. Therefore, a segmented variance of white noise was adopted in this paper. The simulation results for the SMC based ABS control are showninfigures5 to 8.Duetothelimitedspaceinthispaper, onlytheresultscurvesoftheleftfrontwheel,whicharetypical in all of the four wheels, are shown and analyzed. The velocity and wheel pressure without noises are shown in Figure 5. It should be noticed that using the SMC based ABS control, the wheel slip ratio was controlled at the target valueaccurately,andbothofthewheelspeedandthebrake pressure are smooth and stable. Therefore, it is possible to analyze the estimation performance by means of the step response. Then, the white noises were added onto a x, a y, γ, ω i,andp i, and the velocity and wheel pressure with noises are shown in Figure 6. The curves of longitudinal tire forces are shown in Figure 7. TheactualF x curveshowsthelongitudinalforce from the CarSim vehicle model, the estimated F x (smooth sig) and the estimated F x (noised sig) show the longitudinal forces estimated by Kalman filter based observer from signals without noises and with noises. It can be seen that the observer showed well-qualified estimation performance. When the road friction coefficient change happened, the estimatedtireforcebasedonbothsmoothsignalsandnoised signalsfollowedtheactualvaluefastlyandthelagoftherising time was only about.2 s. The errors between the actual value and the observation values were not more than 5 N when the calculations reached the steady state. Thus, it is concluded that the algorithm can respond to the change of road surface conditions effectively, and asymptotically unbiased estimation of the tire forces can be achieved. The Kalman filter also performed well on noised signals, although the estimated F x based on noised signals wobbled a little bit more than F x estimated by smooth signals. The friction coefficients from CarSim vehicle model and RLS based estimator are shown in Figure 8. The estimated friction coefficient based on smooth signals shows satisfied performance with fast response and good accuracy. The rising time of estimated value had a lag of about.4 s. As its input F x is generated bythekalmanfilterbasedobserver,thelagoftheobserver should be considered and the lag of the RLS estimator is pretty small. At steady state, the error between the actual andtheestimatedvalueisnotlargerthan5%,whichis precise enough. It is also seen from the estimated curve based on noised signals that the transient performances became a bit worse. More fluctuation, larger steady state error, and extended settling time can be seen. It is concluded that the noise of estimated F x basedonnoisedsignalsfurtheraffected the RLS estimation negatively. However, the result is still good enough at both response speed and accuracy Verification by the Threshold Based ABS Algorithm. The threshold based algorithm is widely used in practical ABS system. Although the products from various manufactures are different from each other, their principles are the same. () Wheel slip ratio thresholds and acceleration thresholdsaresetasthecontroltargets. (2) The controlled wheel speed will be fluctuating around the target value, and obvious and regular control cyclescanbeseen. (3) In the first control cycle, large pressure dump will be set to ensure that the wheel pressure decreased to a proper level as soon as possible. In following cycles, there will be pulse pressure increments and decrements.

12 2 Mathematical Problems in Engineering Speed (m/s) Road friction coefficient μ Estimation result of road friction coefficient Vehicle speed Estimation μ μ-jump μ-jump Figure 9: Wet tiles to dry concrete pavement (summer). (4) Road surface changes are also identified by preset wheelslipratiothresholdsandaccelerationthresholds. Generally, only simple classification of high μ and low μ road is provided. The ABS control cycles of both wheel speed and pressure can be seen clearly in Figure 9. Itisseenthatwhenbraking on low μ road, there are 3 or 4 cycles in one second and the amplitudes and periods of these cycles are regular, while, on roads with high friction coefficient, the cycles are less regular. The curves of longitudinal tire forces for the threshold based ABS are shown in Figure. It is obvious that the F x also exhibited regular cycles along with the fluctuations of wheel speeds. In spite of this, the Kalman filter based observer also showed quite precise estimation, which provided a good basis for the identification of the road friction coefficient. The friction coefficient estimation results are shown in Figure. It is clear that the errors are more significant comparing with those in Figure 8,whichmeansthefluctuationsof ABS control did affect negatively the estimation performance. The statistic data of the estimation is shown in Table 4. The lag time of the first step is not considered because the estimated value was not updated in the beginning and kept as the initial value.42 until the slip ratio exceeded 5%. In the following estimation, the values also kept constant at any time the slip ratio was smaller than 5%. The lag time in the case of mid-μ to low-μ-jump was only.36 s which is almost the same as that in the SMC case. However, before the threshold based ABS figures out the car driving from a slippery surface to a rough surface, it will apply small pressure increments, and the slip ratio will be tiny, which can be seen in Figure 9.Thus,theincreaseinestimated value in the case of low-μ-tohigh-μ-jump will be slow, whose lag time was.372 s in this simulation. When the estimation values were steady, their means and standard deviations were calculated. The maximum steady state error occurred on the mid-μ road, which was about.42%, and the minimum one occurred on the low-μ road, which was only 2.4%. Furthermore, all the standard deviations were small enough. It is concluded that the algorithm is accurate even in cases of fluctuations. 4. Verification by Road Tests The road tests were carried out by the car equipped with the same threshold based ABS. The parameters of the car are listed in Table 2. As shown in Figures 2 and 3, the testing system consists of an ABS, a VBOX system, a dspace Autobox system, and a series of sensors. The VBOX captures the brake trigger signal and sends out vehicle speed and acceleration captured by its GPS. The wheel speeds and μ-jump flag can be achieved from the ABS ECU internal signals. Both of them are configured as nodes on CAN bus and send their information to the Autobox. The Autobox also captures the signals from lateral accelerometer, yaw rate sensor, and the pressure sensors on wheel cylinders. Finally, all of the signals are transferred and saved in a notepad computer. It can be seen in Figure 3 that all of the necessary signals for the estimation, except for the lateral velocity, were captured directly, while the lateral velocity was calculated by the following equation: V y = (a y V x γ) dt. (36) Furthermore, the μ-jump flag is not necessary for the road identification. It is only used as the flags of road changes. Differing from simulation, the sampling time of the variable devices is different in road test. The lateral acceleration, the yaw rate, and the brake pressures were captured by AutoBox directly at sampling time of ms, the ABS ECU control ran at a loop time of 5 ms, and the sampling time of CAN signals from VBOX was ms. On data processing, linear interpolations were used to achieve ms interval samples for the signals of ABS ECU and VBOX system. By this way, the algorithm also runs at a sampling time of ms. The tests were carried out in both summer and winter. The summer tests were implied on a road test ground located at the south of China. The car was equipped with all-season tires. The high-μ road is dry concrete pavement, whose adhesion coefficient is about.8 to.9, and the low-μ road is wet basalt tiles road, whose adhesion coefficient is smaller than.2. The test road is shown in Figure 4. The winter tests were implied on a lake located at the north-east of China. The snow tires were equipped on the car. The high-μ road is dry concrete pavement and the low-μ road is the ice surface, which is shown in Figure 5.Astheconcrete pavement is not as rough as the one of summer test ground,

13 Mathematical Problems in Engineering 3 25 Estimation result of road friction coefficient 2 Speed (m/s) Road friction coefficient μ Vehicle speed μ-jump Estimation μ μ-jump Figure 2: Ice to dry concrete pavement (winter). its friction coefficient is about.7 generally. And the friction coefficient of the icy surface is about.2. () The Braking Test from High-μ to Low-μ Road. According to the ABS test regulation we used, the car was tested at an initial speed of 8 km/h, and the length of high-μ road is 5 m. The results of summer and winter tests are shown in Figures 6 and 7, respectively. It is seen that the curves of estimated friction coefficients are quite similar to those in simulation. When braking on the low-μ roads, the estimation values are very steady. The friction coefficient is about. for the tiles roadandabout.2fortheicysurface. However, there are fluctuations for the estimation results on the concrete pavement in Figure 7, and its detailed information is shown in Figure 8. Itisseenthattheestimated value varied from.35 to.83. Its valley occurred at.522 s which followed the maximum slip ratio s of 23%. Furthermore, the absolute value of the car deceleration kept on increasing and reached the peak of.78 g at.798 s, while the peak of the estimated μ appeared at.794 s. As the brake efficiency cannot reach %, it is reasonable to achieve apeakdecelerationof.78gonasurfacewiththepeak adhesion coefficient of.83. The precision of the estimation is acceptable, but it took the algorithm about.25 s to reach the correct level. The fluctuation might be caused by the relaxation characteristics of the snow tires. Furthermore, it was discovered that some snow was taken by the tires from the compacted snowfield in front of the pavement during the test, which might also affect the results. (2) The Braking Test from Low-μ to High-μ Road. Inthese cases,thecarwasbrakedataninitialspeedof6km/h. The length of the low-μ road is m in summer test and 2 m in winter test. The results of summer and winter tests are shown in Figures 9 and 2, respectively.again,the estimation showed perfect performance on the low-μ roads. And larger lags were shown when the μ-jump happened, which is similar to simulation. Furthermore, the estimated value on concrete pavement in winter is located between.45 and.7. Considering the average deceleration of.49 g of the car, the estimation is acceptable. 5. Conclusion In this paper, a road friction estimation algorithm based on discrete Kalman filter and RLS method is proposed. The Kalmanfilterisusedtoobservethelongitudinaltireforces andtherlsisusedtoestimatetheroadfrictioncoefficient. The algorithm was verified by simulation as well as road test. A SMC based ABS and a threshold based ABS were used for the verification. The SMC based ABS can control the wheel slipratioatthetargetvaluesmoothly,andthethresholdbased

14 4 Mathematical Problems in Engineering one generally shows significant fluctuations in pulse pressure control. Simulations for the SMC based and threshold based ABS showed that the Kalman filter based observer can detect the longitudinal tire forces accurately and quickly, even if there are noises of sensor signals or wheel slip ratio and pressure fluctuations caused by braking control. AsthewheelspeedisquitesmoothinthecaseofSMC based ABS control, the algorithm shows perfect performance on road friction coefficient estimation. While the fluctuations caused by the threshold based ABS control will affect negatively its precision as well as response speed, its performance is still satisfactory. The experiments were carried out in both winter and summer. When braking on more slippery roads, the wheel speed control cycles will be more regular in both amplitude and period. Thus, the estimation performance is better on roads with lower friction coefficient, and estimation errors are larger on roads with higher friction coefficient. Furthermore, when low-μ to high-μ-jump happens, there will be larger lags than those of high-μ- to low-μ-jump cases because of the characteristics of the tested ABS algorithm. Still and all, the performance of the proposed algorithm is satisfactory. In conclusion, the Kalman filter based observer is robust enough and affected slightly by the fluctuations of wheel dynamics, while the RLS based estimator is more likely affected by the pulse pressure control. The smoother and more regular the wheel controlled by the ABS is, the better the estimation performance is. Notations ω i : Rotation speed of wheel, i=fl, fr, rl, rr represent the front left, front right, rear left, and rear right wheel, respectively. V x : Longitudinal speed of vehicle V y : Lateral speed of vehicle a x : Longitudinal acceleration of vehicle a y : Lateral acceleration of vehicle γ: Yawrateofvehicle T b : Brake moment of wheel p: Brake wheel cylinder pressure F x : Tire longitudinal force F y : Tirelateralforce F z : Tireverticalforce F yf : Lateral forces of the front axle F yr : Lateral forces of the rear axle δ: Steer angle of the front wheels m: Massofthetargetvehicle I z : Momentofinertiaofthevehicle l f : Distances from the center of mass of the vehicle to the front axle l r : Distances from the center of mass of the vehicle to the rear axle 2t: Wheelbase I W : Momentofinertiaofthewheel r: Tireeffectiverollingradius G: Wheel vertical load from the mass F p : Longitudinal force acts on the wheel from the axle x: System state vector z: Systemmeasurementvariable w: Systemprocessnoise V: System measurement noise W: Process noise covariance matrix V: Measurement noise covariance matrix κ: Practical slip ratio α: Side slip angle g: Gravitational acceleration h g : Height of center of mass C x : Longitudinal stiffness of tire C α : Lateral stiffness of tire μ: Tire-road friction coefficient y: Observed values of tire forces estimated by Kalman filter θ: Tire brush model parameter vector f(k, θ): Expression of tire brush model V l : Observednoise s: Calculative slip ratio s : Optimal slip ratio e: Difference between s and s δ slide : Switchingfunction U: OutputfromtheSMCABScontroller V w : Speed of center of wheel. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments ThisworkispartiallysupportedbyNationalNaturalScience Foundation (5569, 57525, and 52556), Jilin Province Science and Technology Development Plan Projects (228 and 2424GX), Science Foundation for Chinese Postdoctoral (2M553 and 22T5292), and Fundamental Research Funds of Jilin University. References ] D.Park,D.Hwang,K.Leeetal., DevelopmentofHILSsystem for ABS ECU of commercial vehicles, SAE Technical Paper , 2. 2] M. Choi, J. J. Oh, and S. B. Choi, Linearized recursive least squares methods for real-time identification of tire-road friction coefficient, IEEE Transactions on Vehicular Technology, vol.62,no.7,pp ,23. 3]M.Ohori,T.Ishizuka,T.Fujita,N.Masaki,andY.Suizu, Fundamental study of smart tire system, in Proceedings of the IEEE Intelligent Transportation Systems Conference, pp , Toronto, Canada, September 26. 4] A. J. Tuononen, Optical position detection to measure tyre carcass deflections, Vehicle System Dynamics,vol.46,no.6,pp , 28. 5] Y. Zhuang, K. Guo, and Y. Chen, Research on the in-tire sensing technology applied in tire-road friction status recognition, Automobile Technology,no.6,pp. 2,2.

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