Early Detection of Tire-Road Friction Coefficient based on Pneumatic Trail Stiffness

Transcription

1 2016 American Control Conerence (ACC) Boston Marriott Copley Place July 6-8, Boston, MA, USA Early Detection o Tire-Road Friction Coeicient based on Pneumatic Trail Stiness Kyoungseok Han, Eunjae Lee, and Seibum Choi Abstract This paper presents a method or estimating the maximum lateral tire-road riction coeicient and wheel side slip angle based on the pneumatic trail inormation that exhibits unique characteristics according to the road surace conditions. The high sensitivity o the pneumatic trail or the wheel side slip angle enables the proposed observer to detect the peak tire-road riction coeicient in low slip regions. The conventional method that is highly dependent on the tire model has drawbacks due to model uncertainty. In order to overcome these shortcomings, the proposed method minimizes the use o existing tire models. In addition, traction orce is also considered in this paper using a correction actor. The estimation results are obtained recursively under the persistent excitation condition. A simulation is conducted irst in order to veriy the perormance o the proposed method using a combination o the Carsim and Matlab & Simulink. Then, vehicle experiments are conducted on a proving ground in order to veriy the easibility o the proposed method. The veriication results reveal that the early detection o the maximum tire-road riction coeicient is possible with less excitation signals than the conventional methods. I. INTRODUCTION The demands or vehicle active saety control systems have increased recently and the mandatory installation o saety systems or newly released vehicles is becoming more common in automotive manuacturing. Some notable systems among these include anti-lock braking (ABS) and electronic stability control (ESC) systems. The ormer system is the most well-known longitudinal vehicle saety system and the latter is concerned with vehicle lateral dynamic stabilization, which is the ocus o this paper. In order to realize the abovementioned systems, accurate vehicle state inormation such as the side slip angle and tireroad riction coeicient are required in real time. Thus ar, numerous approaches have been developed [1-3], but they have drawbacks or immediate use in real-time control areas. Among the proposed methods, this paper is motivated by the previous literature [4-6] that uses sel-aligning torque as a basis o estimation. It is well known that the sel-aligning torque increases as the slip angle increases, which results in a drop o as the lateral tire orce begins to saturate [7]. In electronic power steering (EPS) systems, sel-aligning torque is a readily measurable signal rom the assist motor torque sensor. Providing that accurate sel-aligning torque is extracted or estimated rom the steering mechanism, a pneumatic trail is calculated through dividing the aligning torque by the lateral tire orce. The pneumatic trail exhibits unique characteristics according to the road surace conditions, and previously published papers [4, 8] have revealed that early detection o the lateral tire-road riction coeicient and side slip angle is possible using this hallmark. However, the previous studies depended on a model-based estimation approach that is vulnerable to model uncertainty, including parametric error. Furthermore, the key element o the algorithm, i.e. the pneumatic trail, was too small to be accurately modeled in passenger vehicles. Although the literature has analytically demonstrated how the pneumatic trail responds to a parabolic pressure distribution, it is not suiciently accurate or practical use [7]. In order to overcome this shortcoming, this paper minimizes the use o conventional tire models such as the magic ormula, brushed model, and Dugo model [7]. Instead o the analytical pneumatic trail model, the simple linear representation that is proposed in [4] is selected or use in this study. The stiness o the pneumatic trail corresponding to the side slip angle is estimated using the recursive least square algorithm with orgetting actors [9]. The speciic tire model, except the linearized pneumatic trail, is not used; however, the six degree-o-reedom (6DoF) acceleration measurements are used instead to identiy the individual tire orces. Another contribution that distinguishes this paper rom others is that it allows some longitudinal dynamics coupled with lateral dynamics. Previous works have primarily concentrated on lateral dynamics [4, 6], which has resulted in the longitudinal dynamic being ignored. However, the experimental data demonstrates that the longitudinal tire acceleration or orce due to tire longitudinal slip partly contributes to the ormation o tire-road riction. The ratio o longitudinal acceleration to lateral acceleration at the center o gravity o a vehicle is calculated in order to relect the eects o longitudinal dynamics. The remainder o this paper is organized as ollows. Section 2 describes the overall system model with a ocus on the linearized pneumatic trail model. Section 3 introduces the estimation strategy based on the pneumatic trail stiness characteristics. In Section 4, the simulation and experiment results are provided under a speciic test scenario, and then the paper is concluded in Section 5. II. SYSTEM DESCRIPTIONS A. Vehicle Model or Lateral Dynamics The ollowing kinematic condition [10] can be developed. Using a small angle approximation, the tire side slip angles or each axle can be represented as ollows. α = β + l r δ (1) KyoungseokHan(hks8804@kaist.ac.kr),EunjaeLee( asphaltguy@kaist.ac.kr), Seibum Choi(sbchoi@kaist.ac.kr) are with the Korea Advanced Institute o Science and technology(kaist), Daejeon, Republic o Korea /$ AACC 6326

2 α r = β l r r (2) where α and α r are the ront and rear tire side slip angle, respectively, β is the body side slip angle, r is the yaw rate, δ is the ront wheel steering angle, is the body longitudinal velocity, and l and l r are the distance rom the vehicle center o gravity to ront and rear axles, respectively. Equation (3) can be derived using the time derivative o Eq. (1) as ollows, l y l r r vx vx vx 1 1 l 1 Fy Fyr rvx Fy l Fyrlr v x m vx I z 2 1 l 1 l lr Fy Fyr r mvx I zv x mvx I zvx Then, the derived ormula in Eq. (3) assumes that the longitudinal orce is negligible. That is, the vehicle travels at a constant speed without longitudinal tire slip. In the real world, however, longitudinal orces are always present. The longitudinal orce is considered in the latter hal o this paper. α = ( 1 + l 2 m I z ) F y + ( 1 m l l r ) F I z v yr r δ (3) x where m is the vehicle mass, I z is the yaw moment o inertia, and F y and F yr are the lateral tire orce at the ront/rear axle, respectively. In Eq. (3), the available measurements are the vehicle yaw rate, ront wheel steering angle and its time-derivative. The values to be estimated are the lateral orces and vehicle side slip angle. B. Pneumatic Trail The pneumatic trail is the longitudinal distance where the eective lateral orce acts on and is deined rom the center o the tire contact area. The aligning torque is generated due to this oset. Figure 1 describes the principle o aligning torque generation, and it can be summarized as ollows, M z = (t p + t m ) F y (4) where M z is the aligning torque, t p is the pneumatic trail, and t m is the mechanical trail. The mechanical trail is determined by the steering geometry and it is a unction o the wheel steering angle. The pneumatic trail has the maximum value when the tire begins to generate F y and then the lateral orce moves toward the center o the contact area as the slip angle is increased. This causes a reduction in t p, and it approaches zero as the lateral tire orce becomes closer to the tire grip margin as depicted in Figure 2. This characteristic is used in this study or the early detection o the tire peak riction coeicient. The analytical pneumatic trail model consists o various values such as riction coeicient, normal orce, and side slip angle [7]. However, it does not provide highly accurate values or small slip angles and the accuracy is degraded by atypical normal orce distributions. Figure 1. Pneumatic trail and principle o aligning torque generation The simpliied linear relationship between the slip angle and pneumatic trail is used instead o an analytical model, as ollows: t p = c 1 tanα + c 2 c 1 α + c 2 (5) where c 1 and c 2 are the coeicients o the linear model. The coeicients o the above model were proposed in [4] and they estimate the pure lateral riction margin using the ollowing model. t p = { t p0 t p0c α I 3 tanα, i α α sl (6) 0 else where α sl = tan 1 (3/C α I ); I = 1/μF z ; t po is the initial pneumatic trail, which is assumed to be l/6 where l is the contact patch length; and C α is the cornering stiness. The coeicient or the stiness (c 1 ) is estimated recursively in the next section and the lateral tire-road riction limit can be estimated simultaneously. The beneits o using a pneumatic trail have been well documented in [4], but analysis o the linearized model was omitted. This study also uses the advantages o the pneumatic trail s characteristics, but only a partial linearized model is used in order to avoid including model uncertainty. Figure 2. Chracteristics o lateral orce and sel-aligning torque according to side slip angle III. ESTIMATION METHOD A. Side Slip Angle Estimation The peak tire-road riction limit estimation is meaningul when it is predicted in a stable region where tire orce is proportional to wheel side slip angle. For this reason, the ollowing linear tire model is suicient to estimate the side slip angle. F y = C (β + l r δ ), F yr = C r (β l rr ) (7) where C and C r are the ront/rear axle s cornering stiness, respectively. Using a orce and moment balance relationship or the lateral direction, the ollowing lateral axle orce can be derived without a complex tire orce model. 6327

3 F y = ml ra y +I z r, l +l F yr = ml a y I z r (8) r l +l r where a y is the lateral acceleration and r is the yaw acceleration, which comes rom the yaw rate sensor. Substituting Eq. (8) into Eq. (3), the ollowing open-loop observer can be designed. α = ( 1 + l 2 m I z ) F y + ( 1 m l l r ) I z v F yr r δ (9) x The estimated α r can also be represented as ollows with an estimate o the above α. α r = α l +l r r + δ (10) The ultimate goal o the tire-road riction coeicient estimation is to provide an accurate potential riction limit to the vehicle controller unit (VCU). That is, when the current vehicle state is about to move beyond the stable area in a muslip curve, the active saety control systems such as the ESC are activated in order to recover the vehicle to its original position. Early detection o the maximum tire-road riction limit is important in determining whether the control system o the vehicle activates or not; thereore, the linear region is signiicantly more important than remainder o the mu-slip curve in Figure 2. Thus, the open-loop observer or α only considers the stable area that is suicient to achieve the goal o this study. Cornering stiness is a unction o normal tire orce, in the ollowing equation [11]. The area that exhibits nonlinear characteristics is not in the scope o this paper, so the cornering stiness adaptation [12] is not be perormed. C α = af z bf z 2 (11) where a and b are the constant coeicients. B. Tire-Road Friction Coeicient Estimation Method The primary assumption made in this section is that the vehicle travels at a constant speed, i.e. a pure side slip condition, and thus there is no longitudinal tire slip. Figure 3 describes how the proposed normalized and linearized pneumatic trail model in Eq. (6) is changed according to the wheel side slip angle with a constant C α and F z (C α = 90,000 N/rad, F z = 10,000 N/rad). The pneumatic trail has a distinguishable stiness according to the dierent road surace conditions. Figure 3. Plot o linearized pneumatic trail model. Unlike previous works [4, 13], a partial linearized pneumatic trail model is used in this study in order to avoid including model uncertainty. Dividing both sides by t p0 in Eq. (6), the stiness in Figure 3 can be written as ollows. m = C αi 3 = C α 3μF z (12) The nominal stiness or each road surace was 3 (mu = 1.0), 4.28 (mu = 0.7), 6 (mu = 0.5), and 10 (mu = 0.2). The recursive least squares (RLS) algorithm is used to estimate the stiness iteratively through minimizing the weighted linear least squares cost unction. The algebraic manipulation o Eq. (6) is perormed in order to apply the RLS algorithm as ollows. t p t p0 1 = C α 3μF z tanα m α (13) The available values are t p /t p0 and α rom the previous estimation results. C α is assumed to have a constant value because the tire stays in the stable region where it exhibits a linear property. F z can be easily estimated without considering the road slope angle as ollows. F z = mgl r ma x h, F l +l zr = mgl +ma x h (14) r l +l r where h is the vehicle s height o center o gravity, F z and F zr are the ront/rear axle s normal orces, respectively, and a x is the longitudinal acceleration. Equation (13) can be rewritten into a standard RLS algorithm ormat as ollows. y(k) = T (k)θ(k) + e(k) (15) where y(k) =(t p /t p0 1) is the system output, θ(k) = m is the unknown parameter, T (k) = α is the measured regression vector, and e(k) is the identiication error. The speciic procedures or the RLS algorithm at each time step k are as ollows: 1. Calculate the identiication error: e(k) = y(k) T (k)θ(k) (16) 2. Calculate the updated gain vector: K(k) = P(k 1) (k) T (k)p(k 1) λ+ T (k)p(k 1) (k) 3. Calculate the covariance matrix: (17) P(k) = 1 [P(k 1) P(k 1) (k) T (k)p(k 1) ] (18) λ λ+ T (k)p(k 1) (k) 4. Update the parameter estimate vector: θ (k) = θ (k 1) + K(k)e(k) (19) where λ is the orgetting actor that is used to reduce the inluence o the old data and it typically has a value in [0.9,1]. In this way, the maximum tire-road riction limit can be derived using the stiness estimated in Eq. (12). This paper only uses the stiness as a source o estimation and use o the ull pneumatic trail model is discouraged because the analysis o the ull model is insuicient or atypical road suraces. The various parameters such as normal orce, slip angle, and longitudinal orce can aect the orm o the pneumatic trail due to the tire s strong nonlinear characteristics as in Eq. (20). t p = (α, F z, F x ) (20) 6328

4 The practical pneumatic trail is diicult to model accurately, but it has a dierent reduced speed according to the road surace, which can be seen rom the experimental data. This is why only the stiness according to the road surace is used in the linearized model in Eq. (6). In addition, incorrect estimates o lateral orce and side slip angle can adversely aect the estimation o the tire-road riction limits when the entire model is accepted without modiication. C. Correction Factor based on Accelerations The proposed algorithm that assumes pure side slip becomes invalid when an excessive longitudinal orce is included. Because the pneumatic trail is easily aected by the longitudinal orce and its stiness demonstrates that dierent aspect in Figure 3. Thereore, the primary assumption that a vehicle travels at a constant speed is identical to the previous works, but the correction actor that can allow some longitudinal dynamics coupled with lateral dynamics is introduced in this section. In the real world, the longitudinal orce generation in the tire is inevitable even though the vehicle travels at almost a constant speed. I the longitudinal orce is added to the proposed algorithm as depicted in Figure 4, the estimation result might be underestimated due to the coupling eect between the longitudinal orce and lateral orce. That is, the proposed algorithm assumes that the peak tire road riction is dominated by the lateral tire orce, but only partial longitudinal orce contributes the ormation o the riction coeicient. Figure 4. Coupling eect o longitudinal orce and lateral orce The ollowing heuristic rule is created in order to allow some longitudinal orce in the estimation result. k ˆ where ax k 0, 0.05 a ax k ax / ay, ay ax k undeined (combinedslip), 0.2 ay where k is the correction actor, μ is the estimated riction coeicient rom the previous section, μ is the corrected riction coeicient, a x and a y are the longitudinal/lateral acceleration at the center o gravity o vehicle, respectively, and α is the arbitrary constant value that is proportional to a x /a y, which can be determined rom the trial and error. I the ratio o the longitudinal acceleration to the lateral acceleration is less than 5%, then the longitudinal intervention level can be neglected; thus, the correction actor is assumed to be zero. However, provided that a x /a y is in the range o [0.05, 0.2], the correction actor should be determined in order y to allow some longitudinal dynamics. The heuristic correction actor is determined using a constant value ( α ) that is proportional to a x /a y. The proposed algorithm does not have an eect with a relatively large a x /a y o more than 20%. The pneumatic trail model should be modiied in this excessive longitudinal dynamics through including the longitudinal tire slip. IV. SIMULATION AND EXPERIMENT RESULTS A. Simulations In order to illustrate the perormance o the developed algorithm, a simulation using Carsim was conducted. The vehicle used in the simulation was a D-class sedan stored in Carsim and the tire was modeled using a magic ormula. A slalom maneuver was perormed in order to give an excitation signal to the proposed algorithm, as depicted in Figure 5; the vehicle speed was assumed to be constant at approximately 90 km/h. In order to evaluate the convergence rate at the transient area, the mu-transition occurred twice ater the homogenous road surace. In the linear region o the mu-slip curve, the open-loop observer or the wheel side slip angle tracked the actual values well as illustrated in Figure 5, even though abrupt mutransitions occurred at 20 s and 40 s. From this, it is determined that the axle lateral orces are not aected by the road surace conditions. The estimation result presented in Figure 6 demonstrates that the pneumatic trail stiness based estimation algorithm can identiy the peak tire-road riction coeicient in the early stages. Moreover, the current mu (induced rom the coulomb riction) is insuicient to predict the potential riction coeicient that is a crucial actor in the vehicle control system. As depicted in Figure 6, the current mu rom the Carsim signal was signiicantly lower than the actual value, but the estimated riction coeicient rom the proposed algorithm could track the true value in most areas. The current mu had a similar value on dierent road suraces because there was no signiicant curve shape change in the low slip region. That is, it was diicult distinguish between the dierent road suraces or the small wheel side slip angle that necessitates the usage o the pneumatic trail s beneits. Figure 5. Plot o simulation maneuver and wheel side slip angle eistmation result. 6329

5 In contrast, the model-based methods rom previous studies can lose eectiveness due to rapid steering maneuvers. That is, the estimation results cannot converge to a constant value because the model-based estimation depends on the linearized pneumatic trail model, which reacts sensitively to excitation signals. This shortcoming necessitates the use o the orgetting actor in order to provide more weight to the recent data than old data. The primary role o the orgetting actor is to prevent the divergence o the estimation results, but a time delay exists. Traditionally, there is a trade-o between the convergence rate perormance and sensitivity. For example, i a relatively large orgetting actor (λ = 1) is used, then the estimation result cannot converge to a constant value because all data is given the same weight. Thereore, λ = is used in this simulation in order to prevent this divergence. when the driver pressed the pedal shit. Thereore, the presence o the longitudinal slip cannot be avoided in actual driving. In addition, a perectly constant velocity such as in the Carsim simulation cannot be achieved by a human driver. Considering the practical usage o the proposed algorithm, the inclusion o longitudinal orce is necessary in order to present a more reasonable estimation result. Although the combined slip, which reers to the relatively large longitudinal slip being included, is not considered in this paper, the intervention level o the longitudinal orce due to the acceleration or deceleration is considered using a correction actor. Figure 7. Test vehicle : ull size SUV equipped with wheel orce transducer Figure 8 presents the experiment maneuver and wheel side slip angle estimation results. In general, a sine sweep was used to evaluate the vehicle s handling perormance in various requency domains, and it also generated the pneumatic trail in this experimental validation. As in the simulation, the tire remained in the linear region in the mu-slip curve; thus, the estimation o the wheel side slip angles matched well with the actual values in all areas as depicted in Figure 8. Figure 6. Plot o tire orce and riction coeicient estimation results (mutransition) B. Experiments In order to demonstrate the possibility o implementing the proposed algorithm in a commercial vehicle, an experiment was conducted on a proving ground. The test vehicle traveled on a proving ground paved with dry asphalt and the excitation signal was a sine sweep maneuver as depicted in Figure 8. The wheel orce transducer sensor that can measure the individual tire orces and moments was used to measure the aligning torque. Accurate estimation o the aligning torque is beyond the scope o this paper; thereore, direct measurement rom the sensor was used instead o the assist motor torque in the EPS. The disturbance observer used to estimate the sel-aligning torque was proposed previously and it has been well documented in [8]. Figure 7 describes the coniguration o the test vehicle equipped with the wheel orce transducer. The other necessary signals, e.g. accelerations, gyroscope measurements and wheel steering angles, were obtained using the Can Bus signal in realtime. The test vehicle was driven at an almost constant speed o approximately 80 km/h, but the longitudinal orce appeared to be signiicantly larger than that o the simulation. This resulted rom the engine throttle control occurring constantly Figure 8. Plot o experimental maneuver and wheel side slip angle eistmation result. The lumped lateral orce in both the ront and rear axle was also well estimated as seen in Figure 9. However, the aligning torque measurement rom the wheel orce transducer was not robust to sensor noise, and thereore signal iltering was perormed in order to suppress the divergent measurements. 6330

6 The selected ilter was the rate limiter that limited the slope o the raw data without phase lag. As depicted in Figure 9, the model-based estimation result cannot provide a meaningul value due to its highly sensitive responsiveness to the excitation signal. The use o the orgetting actor enabled it to converge in [0.8, 0.9]. The currently used riction coeicient remained in [0.2, 0.3], but the estimated value was signiicantly higher. This means that early detection o the maximum riction limit was enabled using the pneumatic trail stiness characteristics. As in the simulation results, the convergence rate perormance was degraded using the orgetting actor; thus, the estimation results converged to the true value ater 5 s. I a relatively large orgetting actor was used, then the estimation result might exhibit oscillation as in the model-based method. model was used. The simulation and experiment results revealed that the pneumatic trail stiness could be used to identiy the riction limit in the linear region and it is more robust to model uncertainty than the conventional method. Unlike previous works, the intervention level o the longitudinal orce was also relected using the accelerations at the vehicle s center o gravity. The easibility o the proposed method is increased when considering actual human driver characteristics. However, highly accurate estimations o aligning the torque are required in order to maintain the proposed method s robustness against signal noise. Thereore, uture work should include a method o accurately estimating the aligning torque and more tests are required in order to evaluate the robustness o the algorithm against various road suraces. ACKNOWLEDGMENT This research was supported by the MSIP (Ministry o Science, ICT and Future Planning), Korea, under the C-ITRC (Convergence Inormation Technology Research Center) (IITP-2015-H ) supervised by the IITP (Institute or Inormation & Communications Technology Promotion) and this work was supported by the National Research Foundation o Korea (NRF) grant unded by the Korea government ( ). Figure 9. Plot o tire orce and riction coeicient estimation results A key eature o this experiment is that highly accurate extraction o aligning torque is essential in this algorithm. Because the pneumatic trail is deined as in Eq. (4), it can be easily aected by estimation errors in the aligning torque. The wheel orce transducer used in this study provided an error rate o less than 1%, but when it was converted to a physical value, the possible error was in the range o [ 45 N, +45 N], which cannot be neglected when computing the accurate pneumatic trail. For this reason, a more accurate method to measure or estimate the aligning torque is required in order to maintain the algorithm s robustness against signal noise. However, this is not within the scope o the paper, so uture work should include a method or estimating the aligning torque rom the assist motor torque in the EPS. In addition, more analytic linearized pneumatic trail models should be proposed through considering the physical aspects. Because the practical linearized model suggested in [8] is not ully veriied or various road surace conditions, uture work should also consider the design o a pneumatic trail model that includes the primary parameters such as C α, F z, and α. V. CONCLUSION This paper presented a method that estimates the tire-road riction limit based on the pneumatic trail stiness. In order to avoid model uncertainty, part o the linearized pneumatic trail REFERENCES [1] M. Choi, J. J. Oh, and S. B. Choi, "Linearized recursive least squares methods or real-time identiication o tire road riction coeicient," Vehicular Technology, IEEE Transactions on, vol. 62, pp , [2] J. J. Oh and S. B. Choi, "Vehicle velocity observer design using 6-d imu and multiple-observer approach," Intelligent Transportation Systems, IEEE Transactions on, vol. 13, pp , [3] R. Rajamani, G. Phanomchoeng, D. Piyabongkarn, and J. Y. Lew, "Algorithms or real-time estimation o individual wheel tire-road riction coeicients," Mechatronics, IEEE/ASME Transactions on, vol. 17, pp , [4] Y.-H. J. Hsu, S. M. Laws, and J. C. Gerdes, "Estimation o tire slip angle and riction limits using steering torque," Control Systems Technology, IEEE Transactions on, vol. 18, pp , [5] K. Nakajima, M. Kurishige, M. Endo, and T. Kiuku, "A vehicle state detection method based on estimated aligning torque using EPS," SAE Technical Paper , [6] C. Ahn, H. Peng, and H. E. Tseng, "Robust estimation o road rictional coeicient," Control Systems Technology, IEEE Transactions on, vol. 21, pp. 1-13, [7] H. Pacejka, Tire and vehicle dynamics: Elsevier, [8] Y.-H. J. Hsu, Estimation and control o lateral tire orces using steering torque: ProQuest, [9] D. Simon, Optimal state estimation: Kalman, H ininity, and nonlinear approaches: John Wiley & Sons, [10] R. Rajamani, Vehicle dynamics and control: Springer Science & Business Media, [11] M. Doumiati, A. C. Victorino, A. Charara, and D. Lechner, "Onboard real-time estimation o vehicle lateral tire road orces and sideslip angle," Mechatronics, IEEE/ASME Transactions on, vol. 16, pp , [12] C. Sierra, E. Tseng, A. Jain, and H. Peng, "Cornering stiness estimation based on vehicle lateral dynamics," Vehicle System Dynamics, vol. 44, pp , [13] S. Song, M. C. K. Chun, J. Huissoon, and S. L. Waslander, "Pneumatic trail based slip angle observer with Dugo tire model," in Intelligent Vehicles Symposium Proceedings, 2014 IEEE, 2014, pp