The Predictive Nature of Pneumatic Trail: Tire Slip Angle and Peak Force Estimation using Steering Torque

Transcription

1 AEC 8 The Predictive Nature of Pneumatic Trail: Tire Slip Angle and Peak Force Estimation using Steering Torque Yung-Hsiang Judy Hsu Stanford University J. Christian Gerdes Stanford University 38 Panama Mall Stanford, California , USA Phone: Fax: yjhsu@stanford.edu This paper presents a model-based estimation method that utilizes pneumatic trail information in steering torque to identify a vehicle s lateral handling limits, which are defined by the tire slip angle and peak achievable lateral force. This method takes advantage of the early friction information encoded in the tire pneumatic trail. Pneumatic trail decreases as a function of the tire parameters even in the linear handling region, enabling early detection of the limits before they are reached. Using readily available measurements on production vehicles, this method is unique by its estimation of peak lateral force rather than the friction coefficient alone. From successful experimental results, this method is a promising way to accurately identify tire slip angle in real-time onboard a vehicle with steer-by-wire or electric power steering (EPS without knowledge of tire normal load dynamics. Topics / Tire Property, ehicle Control, and ehicle Dynamics. INTRODUCTION Experts estimate that electronic stability control (ESC prevents 27% of loss of control accidents by intervening when emergency situations are detected []. While ESC is undoubtedly a life-saving technology, it is limited by the fact that there is no direct method of measuring sideslip angle. ESC systems currently available on production cars only have access to sideslip rate, not sideslip angle. This is due to the fact that sideslip rate integration is prone to uncertainty and errors from sensor biases, road grade and bank angle [2,3]. Some approaches have designed observers to estimate sideslip. These methods often depend on accurate tire parametrization, which is problematic since tire parameters vary based on the road surface. If onboard systems could estimate lateral tire force characteristics, control systems could further enhance vehicle handling and increase passenger safety [3]. Motivated by this necessity, researchers have looked at using steering torque information as a source of information. Steering torque measurements are readily available in vehicles with steer-by-wire or EPS systems. From steering torque, total aligning moment (the product of the total trail and lateral tire force can be extracted easily. Because aligning moment decreases well before tire force saturation (due to its dependence on pneumatic trail, several approaches have been taken to improve vehicle stabil-.5 normalized force resulting total aligning moment normalized total trail Front Slip Angle (rad Fig. : Total Aligning Moment Curve (with t m = ity by limiting driver steering input once algorithms detect a noticeable decrease in aligning torque [4 6]. By modeling total aligning moment, previous work has shown that tire slip angle and friction coefficient can be identified in real-time [7]. Other researchers developed an online neural-network based technique that uses lateral force and aligning torque measurements to estimate friction coefficient [8]. However, these methods relied on sensors that are unavailable in production vehicles in order to measure quantities such as tire vertical, longitudinal or lateral forces.

2 AEC 8 Fig. 2: Lateral Tire Deformation This paper presents a model-based estimation method that utilizes pneumatic trail information in steering torque to identify a vehicle s lateral handling limits, which are defined by peak achievable lateral force and tire slip angle (and thus vehicle sideslip angle. The method is based on simple models. It uses sensors readily available on production vehicles, and does not require knowledge of tire normal forces. Most importantly, the algorithm utilizes the sensitivity of pneumatic trail to tire parameters even in the linear handling region, enabling early detection of peak lateral force before the limits are reached. The first section of the paper gives an overview of the concept behind the estimation method. Then, the experimental vehicle used for validation is introduced and the estimation approach is described in detail. The final section presents the experimental results and conclusions. 2. ESTIMATION CONCEPT The estimation method is based on measurements of steering torque, which is the total torque about a vehicle s steering axis. Steering torque can be modeled to include: jacking torque, which is the resulting moment produced by vertical tire forces; actuated motor torque, which is applied by either a SBW or EPS steering motor; and total aligning moment, which is the moment resulting from lateral tire force [9]. The contribution of longitudinal tire forces through scrub radius is neglected for this study. As total aligning moment τ a depends on lateral tire forces, it can be extracted from steering torque measurements and used for tire parameter estimation. Shown in Fig. 2, τ a accounts for both the selfaligning moment due to the pneumatic trail t p and the reaction torque due to the mechanical trail t m. In this simple model of τ a, lateral tire force F yf is modeled using the brush tire Fiala model and pneumatic trail is modeled as a linear function of the tire parameters (assuming the tires are not fully sliding: τ a = (t m + t p F yf ( = t m + c + c tan α µf z ( c 2 tan α + c 3 tan α tan α µf z + c 4 tan 3 α (µf z 2 ( where α is tire slip angle, µ is the tire-road friction coefficient, F z is vertical tire force, and c,..., c 4 are known values presented in Section 4.. Both pneumatic trail and lateral force in the above formulation are functions of the vehicle s lateral limits of handling, defined by the peak lateral force µf z. Illustrated in Figure, which plots normalized total aligning moment, lateral force and trail as functions of slip angle, the utility of pneumatic trail is its decrease as a function of µf z even while the lateral force is in the linear region and unaffected by friction or normal force. This makes the pneumatic trail information encoded in total aligning moment a valuable source of detecting the limits prior to reaching them. The final aspect of the estimation method lies in the following observation: in both of the pneumatic trail and lateral force models used in the observer, µ and F z always appear together (assuming cornering stiffness variation due to normal load is negligible. Thus Eq. ( can be rewritten as: τ a = (t m + c + c tan α I f (c 2 tan α + c 3 tan α tan αi f + c 4 tan 3 αif 2 (2 where I f = µf z is the inverse of the peak achievable lateral force of the tire. Thus by estimating I f, rather than µ and F z separately, this approach can identify tire characteristics without knowledge of tire normal load as the vehicle maneuvers, simplifying the sensor hardware required. Furthermore, µf z is a physical quantity of interest for control systems since either a decrease in friction or normal force of a tire results in a reduction of tire grip. 3. RESEARCH EHICLE The vehicle considered in this study is P, a steer-by-wire vehicle built in-house by the Dynamic Design Lab and Product Realization Lab at Stanford University (Fig. 3. P has independent left/right steer-by-wire steering capability. The vehicle is equipped with a sensor suite that includes steering encoders to measure steer angle, gyroscopes for yaw rate, accelerometers for lateral acceleration, and a GPS/INS system that measures vehicle sideslip and roll angles. Total aligning moment can be measured directly using load cells which are mounted on the left and right steering tierods of P. Alternatively, in the absence of load cells, total aligning moment can be determined from the motor current of a steer-by-wire or electric power steering system via a disturbance observer structure [7].

3 AEC 8 Load Cells -or- Disturbance Observer Algebraic Left Peak Force Estimator Right Peak Force Estimator Dynamic Slip Angle Observer Fig. 4: Observer Block Diagram Fig. 3: Experimental ehicle P 4. ESTIMATION METHOD Shown in Fig. 4, the estimation method comprises two main blocks: the peak force estimator and the slip angle observer. The peak force estimator utilizes the early friction information contained in the pneumatic trail to output estimates of inverted peak lateral tire force I f. Given the independent steering capability of the front tires of P, peak force is estimated separately for the left and right front tires. The second main block, the slip angle observer, uses estimated peak force and inertial vehicle measurements to update front and rear slip angle estimates. While the estimation approach outlined here is tailored for the independent left/right steering system of P, the algorithm is easily adaptable to conventionally linked steering systems. 4. Model Descriptions The proposed estimation method is based on simple models of pneumatic trail, lateral tire force, and a single-track bicycle model of vehicle motion. All models are described in detail below. 4.. Pneumatic Trail Tire pneumatic trail t p depends on slip angle, cornering stiffness C α, and inverted peak lateral force I f. At zero slip angle, t p starts at an initial length t p and vanishes to zero as α increases []. A reasonable model for t p is that it linearly decreases as a function of tan α, with a slope based on C α and I f : where t p = c + c I f tan(α f if α α sl c = t p else c = t pc αf 3 ( 3 α sl = tan C α I f (3 α f δ F yf β u x u y a r α r F yr Fig. 5: BICYCLE MODEL t po is empirically determined to be 6l (where l is the length of the tire contact patch, which is consistent with values found in previous work [] Tire Force Model Lateral tire force is modeled using the brush tire Fiala model []. This model assumes no longitudinal forces, a parabolic pressure distribution, a rigid tire carcass, and a constant coefficient of friction of sliding rubber: F y = where c 2 tan α +c 3 tan α tan αi f +c 4 tan 3 αif 2 if α α sl I f sgnα b else α sl = ( 3 tan C α I f c 2 = C α c 3 = C2 α 3 c 4 = C3 α 27 (4 and α sl is the slip angle at which full tire slip occurs. The Fiala model is used for its simplicity and qualitative correspondence with experimental tire behavior ehicle Model The model for vehicle motion is a two-wheel planar bicycle model (see Fig. 5 with nonlinear front and

4 AEC 8 [algebraic] [dynamic] Fig. 6: Algorithm Procedure rear tire forces F yf and F yr described in Eq. (4: β = m (F yf + F yr r (5 ṙ = I z (af yf bf yr, (6 where a and b are the distances of the front and rear axles from the CG, sideslip angle β is the angle between the vehicle s heading and the direction of its velocity, r is the yaw rate, I z is the moment of inertia, and m is the vehicle mass. It is assumed that the vehicle speed is constant. Using kinematics, the front and rear tire slip angles are linearized to be: α f = β + ar δ (7 α r = β br (8 where δ is the steer angle at the tire. 4.2 Estimation Algorithm Shown in Fig. 6, the estimation algorithm is as follows. Prior to beginning estimation, the initial front axle slip angle α fo is set to zero (i.e. the vehicle is driving straight and inverted front peak lateral force Îfo is set to I fnom = µ nom F zfnom (9 where the nominal friction coefficient µ nom =, and F zfnom is the nominal front axle load. Then, repeat:. Form estimates of lateral force. Once the front slip angle is known, the rear slip angle is also known from combining Eqs. (7 and (8: α r = α f + δ (a + br ( where δ is measured from onboard steering encoders and r is measured from onboard INS. Assuming that the vehicle is traveling on an even surface and longitudinal weight transfer is negligible, the friction coefficients for the front and rear axles are be assumed to be equal. Therefore, the rear axle peak force is Î r = Îf F zfnom, ( F zrnom where F zrnom is the nominal rear axle load and Îf is the front axle peak force estimate. Using the Fiala tire force model in Eq. (4, one can calculate F yf and F yr based on current estimates of Îf, Îr, α f, and α r. 2. Update estimates of front/rear slip angle. In this step, the front slip angle estimate is dynamically updated given lateral force. The update equation for the front slip angle is derived by taking the derivative Eq. (7 and substituting in Eqs. (5 and (6: α f = ( m + a2 I z ab I z ( F yf + m F yr r δ. (2 Thus, the observer update law for α f is as follows: α f = where ( ( m + a2 F yf + I z m ab F yr I z r δ + K( F yf F yfmeas (3 F yfmeas = ma y F yr (4 K is a constant feedback gain of K = 3 7 and a y is the lateral acceleration measurement. Once α f is updated, it is straightforward to update the rear slip estimate from Eq. (. 3. Form pneumatic trail estimate. Using the measured total aligning moment τ a and lateral force estimate F yf, one can construct an estimated pneumatic trail t p for each tire: t p = ( τ a F yf + t m (5 where the mechanical trail t m is determined kinematically as a function of the measured steer angle []. A five point moving average filter is included in the calculation of t p to prevent the dynamics of the estimate from changing faster than the physical system. 4. Use pneumatic trail to solve for peak lateral force. From the pneumatic trail estimate, the linear model in Eq. (3 is used to solve for the estimated peak lateral force Î f : = t poc αf tan α f. (6 Î f 3(t po t p For onboard vehicle implementation, certain considerations should be made to ensure the algorithm outputs physically reasonable estimates. First,

5 AEC 8 δ (rad α f (rad a y (g Steering Angle Front Slip Angle Lateral Acceleration δ (rad α f (rad a y (g Steering Angle 5 5 Front Slip Angle Lateral Acceleration Fig. 7: Ramp Steer Fig. 9: Slalom Maneuver Front slip angle (deg 5 5 Actual Estimate Linear Ramp Steer Results Front slip angle (deg Slalom Results Actual Estimate Linear Rear slip angle (deg Rear slip angle (deg Fig. 8: Ramp Slip Angle Estimates Fig. : Slalom: Slip Angle Comparison the algorithm should observe a decrease in pneumatic trail from its initial value ( t p < t po and a slip angle bounded away from zero ( α f > α thres before the peak force estimation can be meaningful. Secondly, because the slip angle is expected to change at a frequency of -5 Hz, the total aligning moment measurement should be low-pass filtered at the tire hop frequency (f = -5 Hz to prevent the high frequency dynamics transmitted from the road to the wheel. Other signals such as yaw rate and lateral acceleration should similarly be conditioned prior to using them in the algorithm. 5. EXPERIMENTAL RESULTS Two experimental maneuvers are presented here to validate the observer performance. First, a quasisteady-state ramp steer is performed on P, shown in Fig. 7. Driven at a constant speed of m/s on flat dry pavement, this maneuver achieves full lateral force saturation. Using the estimation algorithm outlined in the previous section, the resulting slip angle estimates are shown in Fig. 8. The estimates are compared with both GPS-based measurements, which are taken as truth, and a purely linear slip angle feedback observer which assumes F y = C α α and uses lateral acceleration and yaw rate as measurements [2]. In the linear region of handling, as expected, both observers match well with GPSbased measurements. However, after the vehicle enters nonlinear region of handling (after t = 6s, the limits of a linear estimator are clearly evident as large errors begin to develop. Meanwhile, the slip angle estimates for the pneumatic trail-based estimator are comparable to those from GPS well into the nonlinear region of handling. This demonstrates that the simple total aligning moment model presented in Eq. has a reasonable correspondence to experiment. After t = 5 s, some estimation error develops. This is chiefly due to the fact that once the tire forces have fully saturated, the system becomes unobservable a single force measurement corresponds to a range of slip angles. However, the overall goal of this work is to incorporate this observer structure into a stability control scheme, which would prevent the vehicle from entering this region of handling. The second maneuver performed was a slalom

6 AEC 8 at a constant speed of 5 m/s that enters the nonlinear operating region of the tires Fig. 9. The slip angle estimates for this maneuver are shown in Fig.. Again, both estimators match well with GPSbased measurements in the linear handling region. However, once the vehicle enters the nonlinear handling region (near t = 4s and 2s, the benefit of the pneumatic trail-based estimation approach over the linear estimator is clear. The former tracks the actual slip angle better and is within the accuracy of the GPS measurement. The small estimation error seen in its rear slip angle estimate is likely due to the effects of tire relaxation length. During high frequency maneuvers such as this, non-instantaneous slip angle generation introduces unmodeled effects in the kinematic equation relating the front and rear slip angles in Eq. (. Thus, while the front slip angle matches well with GPS, the rear slip angle deviates slightly. It is worthwhile to note that both of these experimental results were obtained without including a lateral load transfer model in the estimation process. From successful experimental results, this method is a promising way to provide accurate estimates of slip angle up to the limits in real-time using readily available measurements on production vehicles. 6. CONCLUSIONS Tire pneumatic trail is a valuable source of information for lateral tire characterization by enabling early detection of the limits before they are reached. This paper presented an estimation method that utilizes pneumatic trail information contained in steering torque measurements to estimate a tire slip angle and peak lateral force. By estimating peak lateral force rather than the friction coefficient alone, this method is able to accurately identify slip angle without knowledge of tire normal load dynamics. One avenue of ongoing work is integrating the estimation method with a control scheme utilizing the independent steering capability of P. Independent peak force sensing would enable using the inside tire as a sensor for early peak lateral force detection and the outside tire for optimal lateral force generation. Additionally, from a stability control perspective, independent steer angle control can optimize tire adhesion of the front tires separately. Finally, future work aims to develop an observer stability proof to guarantee estimation convergence. REFERENCES []. Electronic stability control coalition fact sheet. ESC Coalition and DEKRA Automotive Research 23. [2]. Fukada, Y., 998. Estimation of vehicle slipangle with combination method of model observer and direct integration. Proceedings of the International Symposium on Advanced ehicle Control (AEC, Nagoya, Japan. [3]. van Zanten, A. T., 22. Evolution of electronic control systems for improving the vehicle dynamic behavior. Proceedings of the International Symposium on Advanced ehicle Control (AEC, Hiroshima, Japan. [4]. Ono, E., Asano, K., and Koibuchi, K., 23. Estimation of tire grip margin using electric power steering system. Proceedings of the 8th International Association for ehicle System Dynamics (IASD Symposium, Kanagawa, Japan. [5]. Yasui, Y., Tanaka, W., Muragishi, Y., Ono, E., Momiyama, M., Katoh, H., Aizawa, H., and Imoto, Y., 24. Estimation of lateral grip margin based on self-aligning torque for vehicle dynamics enhancement. SAE Paper No [6]. Endo, M., Ogawa, K., and Kurishige, M., 26. Cooperative control of active front steering and electric power steering based on self-aligning torque. Proceedings of the International Symposium on Advanced ehicle Control (AEC, Taipei, Taiwan. [7]. Hsu, Y. H. J., and Gerdes, J. C., 26. A feel for the road: A method to estimate tire parameters using steering torque. Proceedings of the International Symposium on Advanced ehicle Control (AEC, Taipei, Taiwan. [8]. Pasterkamp, W. R., and Pacejka, H. B., 997. Application of neural networks in the estimation of tire/road friction using the tire as sensor. SAE Paper No [9]. Hsu, Y. H. J., Laws, S., Gadda, C. D., and Gerdes, J. C., 26. A method to estimate the friction coefficient and tire slip angle using steering torque. Proceedings of ASME International Mechanical Engineering Congress and Exposition (IMECE. []. Pacejka, H. B., 22. Tire and ehicle Dynamics. Society of Automotive Engineers, Inc, 4 Commonwealth Dr. Warrendale, PA []. Laws, S., Gadda, C. D., Kohn, S., Yih, P., Gerdes, J. C., and Milroy, J. C., 25. Steerby-wire suspension and steering design for controllability and observability. Proceedings of IFAC World Congress, Prague. [2]. Rock, K. L., Beiker, S. A., Laws, S., and Gerdes, J. C., 25. alidating gps based measurements for vehicle control. Proceedings of ASME International Mechanical Engineering Congress and Exposition (IMECE.