- Email: [email protected]
Slip Angle Estimation: Development and Experimental Evaluation
2013 IFAC Intelligent Autonomous Vehicles Symposium The International Federation of Automatic Control June 26-28, 2013. Gold Coast, Australia
Slip Angle Estimation: Development and Experimental Evaluation Seung-Hi Lee ∗ Youngseop Son ∗∗ Chang Mook Kang ∗∗∗ Chung Choo Chung ∗∗∗∗ Div. of Electrical and Computer Engineering, Hanyang University, Korea (e-mail: [email protected]). ∗∗ Global R&D Center, MANDO Corporation, Korea (email: [email protected]). ∗∗∗ Div. of Electrical and Computer Engineering, Hanyang University, Korea (e-mail: [email protected]). ∗∗∗∗ Div. of Electrical and Biomedical Engineering, Hanyang University, Korea (e-mail: [email protected]). ∗
Abstract: This paper develops a slip angle estimation method using measurements of lateral acceleration and yaw rate. A unique model for slip angle estimation is developed by combining velocity kinematics, yaw dynamics, and lateral translational dynamics, so as to avoid sensor drift and parameter variation effect. The model is used in designing a state estimator to estimate slip angle, which compensates slip angle prediction error using the measurements of lateral acceleration and yaw rate. Experimental performance verification of the proposed slip angle estimation scheme is presented. 1. INTRODUCTION Slip angle estimation is an important technical component in the vehicle stability control system development. Thus, the problems of slip angle estimation has received considerable attention over the decades. In this regard, there has been some work on slip angle estimation (see e.g. Bevly et al. (2006, 2001); Hac and Simpson (2000); Hiemer et al. (2005); Nishio et al. (2001); Piyabongkarn et al. (2009); Ryu et al. (2002); Tseng et al. (1999a); Ungoren et al. (2002); Van Zanten (2000)). In an aspect of models to use, slip angle estimation methods can be classified into two categories: velocitykinematics-based and dynamics-model-based approaches. The velocity-kinematics-based approach make use of kinematics only to describe the slip angle rate in terms of lateral acceleration, yaw rate, longitudinal velocity, it is thus independent of parameter variation. Thus it is robust against variations in the vehicle parameters that change according to tire-road conditions and driving operations, because it involves no vehicle parameters (see e.g. Ungoren et al. (2002)). However, an integration is required to obtain the slip angle. Thus it is prone to sensor drift. For example, direct integral method Van Zanten (2000) estimates the slip angle based on integration of the kinematics model. In this method, estimated slip angle contains steady state error, therefore it cant estimate slip angle exactly. Meanwhile, the dynamics-model-based approach make use of the vehicle lateral dynamics to describe the slip angle rate. The slip angle can be represented by either a differential equation form (in terms of slip angle, yaw rate, steering angle, longitudinal velocity) or an algebraic form (in terms of lateral acceleration, yaw rate, longitudinal velocity). In this case, no integration is required to obtain an expression for the slip angle. This dynamics model based method is 978-3-902823-36-6 © 2013 IFAC
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known to be relatively robust against road bank angle and sensor bias (see e.g. Bevly et al. (2001); Nishio et al. (2001); Van Zanten (2000)). However, the dynamics model is a function of the vehicle parameters. The slip angle estimate thus tends to deviate from the actual values as a result of mismatch between the vehicle actual parameters and those used by the model, or disturbances, such as unmodeled lateral forces and moments due to side wind. Moreover, this estimate will be reasonably good only under somewhat restrictive conditions such as nominal longitudinal tire forces. By dynamically combining the two approaches, one can utilize advantages of each approach. In Piyabongkarn et al. (2009), a method to dynamically combine the velocity kinematics model and dynamics model approaches. They used a combination of model-based estimation and kinematics-based estimation. At low frequencies, the signal from the model-based estimation plays an important role in the estimator. Higher frequency behavior is obtained using the kinematics-based estimation. Applying however a state estimator to the dynamics model based approach is known to be helpful in reducing to some extent the effect of parameter variation, which compensates slip angle prediction utilizing the differences between the model output and the measured output signals. Applying approximate linear tire models (with a small slip angle assumption) causes poor slip angle estimation in the case of large slip angle motion. Nonlinear observers (see e.g. Bevly et al. (2006, 2001); Daily and Bevly (2004); Grip et al. (2009, 2008); Phanomchoeng et al. (2011); Rodic and Vukobratovic (1999); Ryu and Gerdes (2004)) aim to design an accurate model based on actual vehicles dynamics and to estimate the slip angle. In Grip et al. (2009, 2008), they presented nonlinear slip angle observers with slip friction adaptation. However, these methods suffer from complexity of the models, and thus are difficult 10.3182/20130626-3-AU-2035.00071
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia in estimating slip angle in real-time. It is worthwhile to note that GPS measurements from a two-antenna system combined with inertia sensors can also be used to estimate vehicle sideslip angle (see e.g. Ryu et al. (2002)), though such sensors are expensive. The aim is to develop a slip angle estimation scheme to overcome the aforementioned problems of sensor drift in the velocity-kinematics-based approach as well as parameter variation effect in the dynamics-model-based approach. Fully utilizing advantages of each approach, we are to first develop a slip angle estimation model by combining velocity kinematics, yaw dynamics, and lateral translation dynamics. We also develop an adaptation scheme to dealing with stiffness coefficient variation. The slip angle estimation scheme is realized by applying the slip angle estimation model to develop a state estimator using measurements of lateral acceleration and yaw rate. We are to evaluate the proposed slip angle estimation scheme through test vehicle experiments using RT3000 (see OTS (2004)) to measure slip angle. The contribution of this paper is in developing a unique slip angle estimation scheme that is less prone to the well known problems of sensor drift (in velocity kinematics model that needs to be integrated) and parameter variation effect (in dynamics models), without using expensive sensors. The paper also presents a unique cornering stiffness coefficient adaptation to dealing with parameter variation. also presents a unique cornering stiffness coefficient adaptation to dealing with parameter variation. Moreover, the scheme based on such a unique slip angle estimation model even requires no complex tuning process such as frequency dependent weights. NOMENCLATURE {XY Z} inertial coordinate frame {xyz} local coordinate frame y lateral position of the origin of {xyz} coordinate to the rotation center ‘O’ along the lateral axis ydes lateral position of the desired (reference) lane to the rotation center ‘O’ along the lateral axis x˙ = Vx = vx longitudinal velocity at c.g. of vehicle y˙ = Vy = vy lateral velocity at c.g. of vehicle ay inertial acceleration of vehicle at c.g. m total mass of vehicle Iz yaw moment of inertia of vehicle lf (lr ) longitudinal distance from c.g. to front (rear) tires lf r = lf + lr wheelbase ψ yaw, heading, angle of vehicle in global axes ψ˙ yaw rate β vehicle slip angle at c.g. ψ˙ yaw rate of vehicle δ (δf , δr ) (front, rear) steering angle αf (αr ) slip angle at front (rear) wheel tires Cα (Cαf , Cαr ) cornering stiffness of (front, rear) tire θV tire velocity angle (angle of tire velocity vector with longitudinal axis) φ load bank angle PARAMETER DEFINITIONS
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2Cαf + 2Cα r , a23 = −a22 Vx , mVx 2Cαf lf − 2Cαr lr a24 = −1 − , mVx2 2Cαf lf2 + 2Cαr lr2 2Cαf lf − 2Cαr lr , a44 = − , =− Iz Iz Vx 2Cαf ′ 2Cαf lf b21 = , b21 = b21 Vx , b41 = . mVx Iz a22 = −
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2. MODEL FOR SLIP ANGLE ESTIMATION In this section, we are to develop a slip angle estimation model under the assumption that acceleration and yaw rate are measurable using inertial sensors (accelerometer and gyro) on the vehicle. 2.1 Lateral Force Model µV f fyg
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Fig. 1. Tire slip angle Simple models for the lateral tire forces (Fig. 1) are given by Fyf = 2Cαf αf = 2Cαf (δ − θVf ), Fyr = 2Cαr αr = 2Cαr (−θVr ) with small tire velocity angles θVf and θVr described by y˙ + lf ψ˙ −Vy + lf ψ˙ lf ψ˙ θVf = = = −β + , Vx Vx Vx −Vy − lr ψ˙ lr ψ˙ y˙ − lr ψ˙ = = −β − θVr = Vx Vx Vx with Vy β ≈ tan β = . Vx 2.2 Slip Angle Model An algebraic way to estimate the vehicle slip angle is to use the kinematical relationship of slip angle velocity, yaw rate, lateral acceleration, longitudinal velocity and road bank angle. For developing velocity kinematics-based model, we use ˙ ay + g sin φ = y¨ + Vx ψ. The effect of longitudinal acceleration has been disregarded due to its slow dynamics compared to lateral dyV namics. Using y˙ = −Vy and β ≈ tan β = Vyx , we have ˙ ay + g sin φ = −V˙y + Vx ψ.
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia
Observe that (1) does not involve vehicle parameters, thus robust against variations in vehicle parameters, tire-road conditions, and driving operations. It should be noted that the kinematic model (1) is robust against variations in the vehicle parameters that change according to tire-road conditions and driving operations. Observe that the slip angle rate is directly computed from kinematical relationship of slip angle velocity, yaw rate, lateral acceleration, longitudinal velocity and road bank angle. An additional integration operation is required for a direct measurement of the slip angle. However, the slip angle estimation by integrating (1) becomes prone to drift due to bias errors in the accelerometers and gyroscopes Farrelly and Wellstead (1996); Hac and Simpson (2000); Tseng et al. (1999b); Van Zanten (2000). The estimated slip angle will drift over time no matter how small the bias is.
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2.3 Output measurement model Considering that modern passenger vehicles have inertia sensors installed to measure longitudinal and lateral accelerations and yaw rate, we assume ay and ψ˙ are measurable. Let us consider the vehicle lateral translational motion described by ˙ = Fyf + Fyr may + mg sin φ = m(¨ y + Vx ψ) with Fyf + Fyr = a22 my˙ + (a24 + 1)Vx mψ˙ + b′21 δ. It is immediate to find the lateral translational dynamics model y¨ = a22 y˙ + a24 Vx ψ˙ + b′21 δ. Using y¨ = ay − Vx ψ˙ and y˙ = −βVx , we have (4) y¨ = ay − Vx ψ˙ = a23 β + a24 Vx ψ˙ + b′ δ. 21
Equation (4) is being used for a model-based estimate of slip angle, which is relatively robust against road bank angle and sensor bias. However, one can find that it is a function of the vehicle parameters in (4). The slip angle estimate thus tends to deviate from the actual values due to mismatch between the vehicle actual parameters and those used by the model. Moreover, unmodeled lateral disturbances causes such model mismatch. 288
(a) Slip angle [deg]
By considering the vehicular yaw dynamics model scribed by Iz φ¨ = lf Fyf − lr Fyr equivalently ψ¨ = −a42 β + a44 ψ˙ + b41 δ, we finally have a slip angle model consists of (1) and as follows β˙ = ψ˙ − ay − g sin φ Vx Vx ¨ ψ = −a42 β + a44 ψ˙ + b41 δ.
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ay 1 β˙ = ψ˙ − − g sin φ. Vx Vx
In the paper, we do not use the lateral translational dynamics model for slip angle estimation. Instead, we are to use the model for a measurement model to correct model output utilizing measured output so as to improve slip angle estimate (This is considered in the estimator design section). Then, the slip angle estimate can become more robust against such aforementioned model mismatch.
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Assuming Vx is constant (or very slow varying such that its time derivative becomes negligible), we have a velocity kinematics-based model
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Fig. 2. Delay and amplitude changes of slip angle and yaw rate: (a) at high µ; (b) at low µ. Considering that lateral acceleration and yaw rate are measurable using an accelerometer and an inertia sensor, we can use ′ a (a + 1)Vx β b y = 23 24 + 21 δ. (5) 0 1 0 ψ˙ Slip Angle Estimation Model It is our observation that there exists a time delay between steering angle and slip angle. Figure 2 demonstrates that the time delay increases as µ decreases. Let us simply describe the time delay between the steering angle and slip angle as follows β 1 = βo s/τβ + 1
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia where τβ denotes the time delay between the steering angle input and the slip angle.
Observing Fig. 3, we find the tire cornering stiffness model (7) is quite reasonable for µFz ≤ µFz = 3kN .
Considering the time delay, we immediately find a model for slip angle estimation in terms of the state vector T x = β βo ψ˙ ay x˙ = Ax + B δ + B δ aφ sin φ (6) y = Cx + Dδ δ. where # " # " 0 −τ τ 0 0 1 , Bδ = 0 , A= 0 b41 0 −a42 a42 # " 0 0 Baφ = −1/Vx −g/V x , 0 0 ′ a 0 (a24 + 1)Vx b C = 23 , Dδ = 21 δ. 0 0 1 0
We are to estimated the tire road friction coefficient η ˙ online, using ψ˙ and δ. Utilizing our observation on ψ/δ and µ in Fig. 3, we develop eη = η ψ˙ − δ. The unknown coefficient η need to be estimated in the way to minimize 1 1 X T JT (k) = eη (k)eη (k) T
2.4 Stiffness Coefficient Adaptation ˙ It is worthwhile to observe that the ratio ψ/δ can be used to monitor variation of tire stiffness coefficient (Fig. ˙ becomes dominant at 2). Such variation in the ratio ψ/δ sufficiently large steering angles. We also observe the ratio ˙ ˙ ψ/δ increases as µ decreases. We find ψ/δ ≈ 1 at high ˙ µ, whilst ψ/δ ≈ 1.33, for a steering angle of 15 degree. ˙ at large Thus utilizing the measurement of the ratio ψ/δ steering angle, we can take into account variation. Other parameter variation due to such as time delay can also be ˙ modeled as a function of the ratio ψ/δ
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where T denotes an accumulation interval. Using the current measure (T = 1) ∂J1 = 2eη (k)ψ˙ ∂η leads to a simple adaptation law ˙ η(k + 1) = sat+ (η(k) − ρeη (k)ψ(k)) (8) where sat(·)+ : (−∞, ∞) → [0, 1] denotes the positive saturation function and ρ > 0 is a relaxation gain. The unknown coefficient µ is now estimated on-line from (8) using the measurements of δ and ψ˙ at sufficiently large δ. Using the estimate of η, the cornering stiffness values can be updated on line. It is easy to show that η(·) in (8) converges to its true value as k → ∞, provided it is constant. In practice, the adaptation scheme works for slow varying µ. 3. SLIP ANGLE ESTIMATOR DESIGN Given a sampling rate Tc , one can obtain the ZOH discrete-time equivalent model matrix set: (Φ, [Γ, Γaφ ]) to (A, [B, Baφ ]). ay (k) x(k + 1) = Φx(k) + Γu(k) + Γ aφ sin φ(k) (9) ˆ (k) = Cx(k) + Du(k). x We are to design a state estimator to estimate the slip a angle β using measurements y = y . φ Constructing a state estimator type slip angle estimator is helpful in reducing the effect of parameter variation, which utilizes the differences between the model output and the measured output signals.
Fig. 3. Cornering stiffness as a function of µFz (Piyabongkarn et al. (2009))
The state estimator type slip angle estimator is described by ay (k) x(k + 1) = Φx(k) + Γu(k) + Γ aφ sin φ(k) (10) ˆ (k) = x ¯ (k) − L(−y(k) + Cx(k) + Du(k) x ¯ (·) and x ˆ (·) denote the predicted state by model where x and the state corrected by output measurements, respectively, L is the state estimator gain to be determined.
The tire cornering stiffness is thus modeled by µFz = Cα η (7) Cα = Cα µFz where Cα is the value of cornering stiffness at µFz , η = µFz , 0 < η < 1, is here referred to be as the cornering µFz stiffness correction parameter which is a function of µFz .
Given weights for state estimates V1 ≥ 0 and noise covariance (weight) V2 > 0, the state estimator gain L is simply computed from L = −Y C T (CY C T + V2 )−1 (11) where Y > 0 is the maximal solution of the discrete-time algebraic Riccati equation Y = ΦY ΦT − ΦY C T (CY C T + V2 )−1 CY ΦT + V1 .
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2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia It is easy to find the state estimation error e(k) = x(k) − ¯ (k) is described by e(k + 1) = (Φ − ΦLC)e(k), under the x assumption that ay (k) and φ(k) are known to use. (13)
parameter variation (because it utilizes the differences be-
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Although the slip angle estimator (10) is less prone to parameter variation (because it utilizes the differences be- the ), under tween the model output and the measured output signals for state correction), its state estimation performance may thepresence slip angleof estimator is less prone to become Although poor in the parameter(11) variation.
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the model for output and the measured output signals Activelytween compensating the effect of parameter variation for state state estimation performance will make the correction), slip angle it’s estimator much more robustmay against parameter variation. State estimator gains can be determined/switched such that the state estimator shows robust convergence wideangle range of µ values will make thein slip estimator much (8). moreGain robust scheduling state estimator design is beyond scopegains of this against parameter variation. State estimator can be paper and thus not considered determined/switched suchhere. that the state estimator shows scheduling state estimator EVALUATION design is beyond scope of this 4. EXPERIMENTAL
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Figures 4 and 5 compare slip angle estimates with its measurements using RT3000 (OTS (2004)) on the test vehicle at high and low µ, respectively. We roughly estimate high and low µ to be 0.8 and 0.3, respectively. The proposed scheme exhibits quite accurate estimates even with simple design, whilst slip angle computation from the velocity kinematics shows poor performance.
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The proposed slip angle estimation scheme is experimentally evaluated. with side slip angle measurement using RT3000 (OTS (2004)) on a test vehicle (KIA Morning). An accelerometer and an inertia sensor on the test vehicle measure lateral acceleration ay and yaw rate ψ˙ a sampling period of 7msec. The state estimator gain L in (10) is simply determined from (11) such that the resulting state estimator is convergent for 0.3 < µ < 1 so as to apply the adaptation scheme (8) in real time.
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This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (20121480).
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Bevly, D., Ryu, J., and Gerdes, J. (2006). Integrating INS sensors with GPS measurements for continuous estimation of vehicle sideslip, roll, and tire cornering stiffness.
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ACKNOWLEDGMENT
REFERENCES
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A slip angle estimation method has been developed, which uses measurements of lateral acceleration and yaw rate. A unique model for slip angle estimation was developed by combining velocity kinematics, yaw dynamics, and lateral translational dynamics, so as to avoid sensor drift and parameter variation effect. The model was used in designing a state estimator estimate slip angle, compensates combining to velocity kinematics, yawwhich dynamics, and lateral slip angle prediction error using the measurements of lateral acceleration and yaw rate. The estimation scheme was validated through experimental tests using RT3000 on a test vehicle. The proposed slip angle estimation scheme was shown to provide reliable slip angle and can thus be effectively used in vehicle stability control.
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Fig. 4. Experimental test at high µ: (a) steering angle and longitudinal speed; (b) yaw rate; (c) slip angles.
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia
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Fig. 5. Experimental test at low µ: (a) steering angle and longitudinal speed; (b) yaw rate; (c) slip angles.
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IEEE Trans. Intelligent Transportation Systems, 7(4), 483–493. Bevly, D., Sheridan, R., and Gerdes, J. (2001). Integrating INS sensors with GPS velocity measurements for continuous estimation of vehicle sideslip and tire cornering stiffness. In Amer. Contr. Conf., volume 1, 25–30. IEEE. Daily, R. and Bevly, D. (2004). The use of GPS for vehicle stability control systems. IEEE Trans. Ind. Electron., 51(2), 270–277. Farrelly, J. and Wellstead, P. (1996). Estimation of vehicle lateral velocity. In Conf. Control Applications, 552–557. IEEE. Grip, H.F., Imsland, L., Johansen, T.A., Kalkkuhl, J.C., and Suissa, A. (2009). Vehicle sideslip estimation. IEEE Contr. Syst. Mag., 29(5), 36–52. Grip, H., Imsland, L., Johansen, T., Fossen, T., Kalkkuhl, J., and Suissa, A. (2008). Nonlinear vehicle side-slip estimation with friction adaptation. Automatica, 44(3), 611–622. Ref : 21(2012.2). Hac, A. and Simpson, M. (2000). Estimation of vehicle side slip angle and yaw rate. SAE Paper. Hiemer, M., Vietinghoff, A., Kiencke, U., and Matsunaga, T. (2005). Determination of the vehicle body side slip angle with non-linear observer strategies. SAE Paper, 01–0400. Nishio, A., Tozu, K., Yamaguchi, H., K., A., and Y., A. (2001). Development of vehicle stability control system based on vehicle sideslip angle estimation. SAE paper, 01–0137. OTS (2004). RT3000 Inertial and Measurement System User Manual. Oxford Technical Solutions. Phanomchoeng, G., Rajamani, R., and Piyabongkarn, D. (2011). Real-time automotive slip angle estimation with nonlinear observer. In Amer. Contr. Conf., 3942–3947. IEEE. Piyabongkarn, D., Rajamani, R., Grogg, J., and Lew, J. (2009). Development and experimental evaluation of a slip angle estimator for vehicle stability control. IEEE Trans. Control Syst. Technol., 17(1), 78–88. Rodic, A.D. and Vukobratovic, M.K. (1999). Contribution to the integrated control synthesis of road vehicles. IEEE Trans. Control Syst. Technol., 7(1), 64–78. Ryu, J. and Gerdes, J. (2004). Integrating inertial sensors with global positioning system (GPS) for vehicle dynamics control. ASME J. Dyn. Syst. Meas. Control, 126, 243. Ryu, J., Rossetter, E., and Gerdes, J. (2002). Vehicle sideslip and roll parameter estimation using GPS. In Symp. Advanced Vehicle Control, 373–380. Hiroshima, Japan. Tseng, H., Ashrafi, B., Madau, D., Allen Brown, T., and Recker, D. (1999a). The development of vehicle stability control at Ford. IEEE/ASME Trans. Mechatron., 4(3), 223–234. Tseng, H., Madau, D., Ashrafi, B., Brown, T., and Recker, D. (1999b). Technical challenges in the development of vehicle stability control system. In Conf. Control Applications, 1660–1666. IEEE. Ungoren, A., Peng, H., and Tseng, H. (2002). A study on lateral speed estimation methods. Int. J. Vehicle Autonomous Systems, 1(1), 126–144. Van Zanten, A. (2000). Bosch EPS systems: 5 years of experience. SAE paper, 01.1633.
Slip Angle Estimation: Development and Experimental Evaluation Seung-Hi Lee ∗ Youngseop Son ∗∗ Chang Mook Kang ∗∗∗ Chung Choo Chung ∗∗∗∗ Div. of Electrical and Computer Engineering, Hanyang University, Korea (e-mail: [email protected]). ∗∗ Global R&D Center, MANDO Corporation, Korea (email: [email protected]). ∗∗∗ Div. of Electrical and Computer Engineering, Hanyang University, Korea (e-mail: [email protected]). ∗∗∗∗ Div. of Electrical and Biomedical Engineering, Hanyang University, Korea (e-mail: [email protected]). ∗
Abstract: This paper develops a slip angle estimation method using measurements of lateral acceleration and yaw rate. A unique model for slip angle estimation is developed by combining velocity kinematics, yaw dynamics, and lateral translational dynamics, so as to avoid sensor drift and parameter variation effect. The model is used in designing a state estimator to estimate slip angle, which compensates slip angle prediction error using the measurements of lateral acceleration and yaw rate. Experimental performance verification of the proposed slip angle estimation scheme is presented. 1. INTRODUCTION Slip angle estimation is an important technical component in the vehicle stability control system development. Thus, the problems of slip angle estimation has received considerable attention over the decades. In this regard, there has been some work on slip angle estimation (see e.g. Bevly et al. (2006, 2001); Hac and Simpson (2000); Hiemer et al. (2005); Nishio et al. (2001); Piyabongkarn et al. (2009); Ryu et al. (2002); Tseng et al. (1999a); Ungoren et al. (2002); Van Zanten (2000)). In an aspect of models to use, slip angle estimation methods can be classified into two categories: velocitykinematics-based and dynamics-model-based approaches. The velocity-kinematics-based approach make use of kinematics only to describe the slip angle rate in terms of lateral acceleration, yaw rate, longitudinal velocity, it is thus independent of parameter variation. Thus it is robust against variations in the vehicle parameters that change according to tire-road conditions and driving operations, because it involves no vehicle parameters (see e.g. Ungoren et al. (2002)). However, an integration is required to obtain the slip angle. Thus it is prone to sensor drift. For example, direct integral method Van Zanten (2000) estimates the slip angle based on integration of the kinematics model. In this method, estimated slip angle contains steady state error, therefore it cant estimate slip angle exactly. Meanwhile, the dynamics-model-based approach make use of the vehicle lateral dynamics to describe the slip angle rate. The slip angle can be represented by either a differential equation form (in terms of slip angle, yaw rate, steering angle, longitudinal velocity) or an algebraic form (in terms of lateral acceleration, yaw rate, longitudinal velocity). In this case, no integration is required to obtain an expression for the slip angle. This dynamics model based method is 978-3-902823-36-6 © 2013 IFAC
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known to be relatively robust against road bank angle and sensor bias (see e.g. Bevly et al. (2001); Nishio et al. (2001); Van Zanten (2000)). However, the dynamics model is a function of the vehicle parameters. The slip angle estimate thus tends to deviate from the actual values as a result of mismatch between the vehicle actual parameters and those used by the model, or disturbances, such as unmodeled lateral forces and moments due to side wind. Moreover, this estimate will be reasonably good only under somewhat restrictive conditions such as nominal longitudinal tire forces. By dynamically combining the two approaches, one can utilize advantages of each approach. In Piyabongkarn et al. (2009), a method to dynamically combine the velocity kinematics model and dynamics model approaches. They used a combination of model-based estimation and kinematics-based estimation. At low frequencies, the signal from the model-based estimation plays an important role in the estimator. Higher frequency behavior is obtained using the kinematics-based estimation. Applying however a state estimator to the dynamics model based approach is known to be helpful in reducing to some extent the effect of parameter variation, which compensates slip angle prediction utilizing the differences between the model output and the measured output signals. Applying approximate linear tire models (with a small slip angle assumption) causes poor slip angle estimation in the case of large slip angle motion. Nonlinear observers (see e.g. Bevly et al. (2006, 2001); Daily and Bevly (2004); Grip et al. (2009, 2008); Phanomchoeng et al. (2011); Rodic and Vukobratovic (1999); Ryu and Gerdes (2004)) aim to design an accurate model based on actual vehicles dynamics and to estimate the slip angle. In Grip et al. (2009, 2008), they presented nonlinear slip angle observers with slip friction adaptation. However, these methods suffer from complexity of the models, and thus are difficult 10.3182/20130626-3-AU-2035.00071
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia in estimating slip angle in real-time. It is worthwhile to note that GPS measurements from a two-antenna system combined with inertia sensors can also be used to estimate vehicle sideslip angle (see e.g. Ryu et al. (2002)), though such sensors are expensive. The aim is to develop a slip angle estimation scheme to overcome the aforementioned problems of sensor drift in the velocity-kinematics-based approach as well as parameter variation effect in the dynamics-model-based approach. Fully utilizing advantages of each approach, we are to first develop a slip angle estimation model by combining velocity kinematics, yaw dynamics, and lateral translation dynamics. We also develop an adaptation scheme to dealing with stiffness coefficient variation. The slip angle estimation scheme is realized by applying the slip angle estimation model to develop a state estimator using measurements of lateral acceleration and yaw rate. We are to evaluate the proposed slip angle estimation scheme through test vehicle experiments using RT3000 (see OTS (2004)) to measure slip angle. The contribution of this paper is in developing a unique slip angle estimation scheme that is less prone to the well known problems of sensor drift (in velocity kinematics model that needs to be integrated) and parameter variation effect (in dynamics models), without using expensive sensors. The paper also presents a unique cornering stiffness coefficient adaptation to dealing with parameter variation. also presents a unique cornering stiffness coefficient adaptation to dealing with parameter variation. Moreover, the scheme based on such a unique slip angle estimation model even requires no complex tuning process such as frequency dependent weights. NOMENCLATURE {XY Z} inertial coordinate frame {xyz} local coordinate frame y lateral position of the origin of {xyz} coordinate to the rotation center ‘O’ along the lateral axis ydes lateral position of the desired (reference) lane to the rotation center ‘O’ along the lateral axis x˙ = Vx = vx longitudinal velocity at c.g. of vehicle y˙ = Vy = vy lateral velocity at c.g. of vehicle ay inertial acceleration of vehicle at c.g. m total mass of vehicle Iz yaw moment of inertia of vehicle lf (lr ) longitudinal distance from c.g. to front (rear) tires lf r = lf + lr wheelbase ψ yaw, heading, angle of vehicle in global axes ψ˙ yaw rate β vehicle slip angle at c.g. ψ˙ yaw rate of vehicle δ (δf , δr ) (front, rear) steering angle αf (αr ) slip angle at front (rear) wheel tires Cα (Cαf , Cαr ) cornering stiffness of (front, rear) tire θV tire velocity angle (angle of tire velocity vector with longitudinal axis) φ load bank angle PARAMETER DEFINITIONS
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2Cαf + 2Cα r , a23 = −a22 Vx , mVx 2Cαf lf − 2Cαr lr a24 = −1 − , mVx2 2Cαf lf2 + 2Cαr lr2 2Cαf lf − 2Cαr lr , a44 = − , =− Iz Iz Vx 2Cαf ′ 2Cαf lf b21 = , b21 = b21 Vx , b41 = . mVx Iz a22 = −
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2. MODEL FOR SLIP ANGLE ESTIMATION In this section, we are to develop a slip angle estimation model under the assumption that acceleration and yaw rate are measurable using inertial sensors (accelerometer and gyro) on the vehicle. 2.1 Lateral Force Model µV f fyg
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Fig. 1. Tire slip angle Simple models for the lateral tire forces (Fig. 1) are given by Fyf = 2Cαf αf = 2Cαf (δ − θVf ), Fyr = 2Cαr αr = 2Cαr (−θVr ) with small tire velocity angles θVf and θVr described by y˙ + lf ψ˙ −Vy + lf ψ˙ lf ψ˙ θVf = = = −β + , Vx Vx Vx −Vy − lr ψ˙ lr ψ˙ y˙ − lr ψ˙ = = −β − θVr = Vx Vx Vx with Vy β ≈ tan β = . Vx 2.2 Slip Angle Model An algebraic way to estimate the vehicle slip angle is to use the kinematical relationship of slip angle velocity, yaw rate, lateral acceleration, longitudinal velocity and road bank angle. For developing velocity kinematics-based model, we use ˙ ay + g sin φ = y¨ + Vx ψ. The effect of longitudinal acceleration has been disregarded due to its slow dynamics compared to lateral dyV namics. Using y˙ = −Vy and β ≈ tan β = Vyx , we have ˙ ay + g sin φ = −V˙y + Vx ψ.
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia
Observe that (1) does not involve vehicle parameters, thus robust against variations in vehicle parameters, tire-road conditions, and driving operations. It should be noted that the kinematic model (1) is robust against variations in the vehicle parameters that change according to tire-road conditions and driving operations. Observe that the slip angle rate is directly computed from kinematical relationship of slip angle velocity, yaw rate, lateral acceleration, longitudinal velocity and road bank angle. An additional integration operation is required for a direct measurement of the slip angle. However, the slip angle estimation by integrating (1) becomes prone to drift due to bias errors in the accelerometers and gyroscopes Farrelly and Wellstead (1996); Hac and Simpson (2000); Tseng et al. (1999b); Van Zanten (2000). The estimated slip angle will drift over time no matter how small the bias is.
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2.3 Output measurement model Considering that modern passenger vehicles have inertia sensors installed to measure longitudinal and lateral accelerations and yaw rate, we assume ay and ψ˙ are measurable. Let us consider the vehicle lateral translational motion described by ˙ = Fyf + Fyr may + mg sin φ = m(¨ y + Vx ψ) with Fyf + Fyr = a22 my˙ + (a24 + 1)Vx mψ˙ + b′21 δ. It is immediate to find the lateral translational dynamics model y¨ = a22 y˙ + a24 Vx ψ˙ + b′21 δ. Using y¨ = ay − Vx ψ˙ and y˙ = −βVx , we have (4) y¨ = ay − Vx ψ˙ = a23 β + a24 Vx ψ˙ + b′ δ. 21
Equation (4) is being used for a model-based estimate of slip angle, which is relatively robust against road bank angle and sensor bias. However, one can find that it is a function of the vehicle parameters in (4). The slip angle estimate thus tends to deviate from the actual values due to mismatch between the vehicle actual parameters and those used by the model. Moreover, unmodeled lateral disturbances causes such model mismatch. 288
(a) Slip angle [deg]
By considering the vehicular yaw dynamics model scribed by Iz φ¨ = lf Fyf − lr Fyr equivalently ψ¨ = −a42 β + a44 ψ˙ + b41 δ, we finally have a slip angle model consists of (1) and as follows β˙ = ψ˙ − ay − g sin φ Vx Vx ¨ ψ = −a42 β + a44 ψ˙ + b41 δ.
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ay 1 β˙ = ψ˙ − − g sin φ. Vx Vx
In the paper, we do not use the lateral translational dynamics model for slip angle estimation. Instead, we are to use the model for a measurement model to correct model output utilizing measured output so as to improve slip angle estimate (This is considered in the estimator design section). Then, the slip angle estimate can become more robust against such aforementioned model mismatch.
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Assuming Vx is constant (or very slow varying such that its time derivative becomes negligible), we have a velocity kinematics-based model
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Fig. 2. Delay and amplitude changes of slip angle and yaw rate: (a) at high µ; (b) at low µ. Considering that lateral acceleration and yaw rate are measurable using an accelerometer and an inertia sensor, we can use ′ a (a + 1)Vx β b y = 23 24 + 21 δ. (5) 0 1 0 ψ˙ Slip Angle Estimation Model It is our observation that there exists a time delay between steering angle and slip angle. Figure 2 demonstrates that the time delay increases as µ decreases. Let us simply describe the time delay between the steering angle and slip angle as follows β 1 = βo s/τβ + 1
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia where τβ denotes the time delay between the steering angle input and the slip angle.
Observing Fig. 3, we find the tire cornering stiffness model (7) is quite reasonable for µFz ≤ µFz = 3kN .
Considering the time delay, we immediately find a model for slip angle estimation in terms of the state vector T x = β βo ψ˙ ay x˙ = Ax + B δ + B δ aφ sin φ (6) y = Cx + Dδ δ. where # " # " 0 −τ τ 0 0 1 , Bδ = 0 , A= 0 b41 0 −a42 a42 # " 0 0 Baφ = −1/Vx −g/V x , 0 0 ′ a 0 (a24 + 1)Vx b C = 23 , Dδ = 21 δ. 0 0 1 0
We are to estimated the tire road friction coefficient η ˙ online, using ψ˙ and δ. Utilizing our observation on ψ/δ and µ in Fig. 3, we develop eη = η ψ˙ − δ. The unknown coefficient η need to be estimated in the way to minimize 1 1 X T JT (k) = eη (k)eη (k) T
2.4 Stiffness Coefficient Adaptation ˙ It is worthwhile to observe that the ratio ψ/δ can be used to monitor variation of tire stiffness coefficient (Fig. ˙ becomes dominant at 2). Such variation in the ratio ψ/δ sufficiently large steering angles. We also observe the ratio ˙ ˙ ψ/δ increases as µ decreases. We find ψ/δ ≈ 1 at high ˙ µ, whilst ψ/δ ≈ 1.33, for a steering angle of 15 degree. ˙ at large Thus utilizing the measurement of the ratio ψ/δ steering angle, we can take into account variation. Other parameter variation due to such as time delay can also be ˙ modeled as a function of the ratio ψ/δ
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where T denotes an accumulation interval. Using the current measure (T = 1) ∂J1 = 2eη (k)ψ˙ ∂η leads to a simple adaptation law ˙ η(k + 1) = sat+ (η(k) − ρeη (k)ψ(k)) (8) where sat(·)+ : (−∞, ∞) → [0, 1] denotes the positive saturation function and ρ > 0 is a relaxation gain. The unknown coefficient µ is now estimated on-line from (8) using the measurements of δ and ψ˙ at sufficiently large δ. Using the estimate of η, the cornering stiffness values can be updated on line. It is easy to show that η(·) in (8) converges to its true value as k → ∞, provided it is constant. In practice, the adaptation scheme works for slow varying µ. 3. SLIP ANGLE ESTIMATOR DESIGN Given a sampling rate Tc , one can obtain the ZOH discrete-time equivalent model matrix set: (Φ, [Γ, Γaφ ]) to (A, [B, Baφ ]). ay (k) x(k + 1) = Φx(k) + Γu(k) + Γ aφ sin φ(k) (9) ˆ (k) = Cx(k) + Du(k). x We are to design a state estimator to estimate the slip a angle β using measurements y = y . φ Constructing a state estimator type slip angle estimator is helpful in reducing the effect of parameter variation, which utilizes the differences between the model output and the measured output signals.
Fig. 3. Cornering stiffness as a function of µFz (Piyabongkarn et al. (2009))
The state estimator type slip angle estimator is described by ay (k) x(k + 1) = Φx(k) + Γu(k) + Γ aφ sin φ(k) (10) ˆ (k) = x ¯ (k) − L(−y(k) + Cx(k) + Du(k) x ¯ (·) and x ˆ (·) denote the predicted state by model where x and the state corrected by output measurements, respectively, L is the state estimator gain to be determined.
The tire cornering stiffness is thus modeled by µFz = Cα η (7) Cα = Cα µFz where Cα is the value of cornering stiffness at µFz , η = µFz , 0 < η < 1, is here referred to be as the cornering µFz stiffness correction parameter which is a function of µFz .
Given weights for state estimates V1 ≥ 0 and noise covariance (weight) V2 > 0, the state estimator gain L is simply computed from L = −Y C T (CY C T + V2 )−1 (11) where Y > 0 is the maximal solution of the discrete-time algebraic Riccati equation Y = ΦY ΦT − ΦY C T (CY C T + V2 )−1 CY ΦT + V1 .
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2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia It is easy to find the state estimation error e(k) = x(k) − ¯ (k) is described by e(k + 1) = (Φ − ΦLC)e(k), under the x assumption that ay (k) and φ(k) are known to use. (13)
parameter variation (because it utilizes the differences be-
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Although the slip angle estimator (10) is less prone to parameter variation (because it utilizes the differences be- the ), under tween the model output and the measured output signals for state correction), its state estimation performance may thepresence slip angleof estimator is less prone to become Although poor in the parameter(11) variation.
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the model for output and the measured output signals Activelytween compensating the effect of parameter variation for state state estimation performance will make the correction), slip angle it’s estimator much more robustmay against parameter variation. State estimator gains can be determined/switched such that the state estimator shows robust convergence wideangle range of µ values will make thein slip estimator much (8). moreGain robust scheduling state estimator design is beyond scopegains of this against parameter variation. State estimator can be paper and thus not considered determined/switched suchhere. that the state estimator shows scheduling state estimator EVALUATION design is beyond scope of this 4. EXPERIMENTAL
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Figures 4 and 5 compare slip angle estimates with its measurements using RT3000 (OTS (2004)) on the test vehicle at high and low µ, respectively. We roughly estimate high and low µ to be 0.8 and 0.3, respectively. The proposed scheme exhibits quite accurate estimates even with simple design, whilst slip angle computation from the velocity kinematics shows poor performance.
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The proposed slip angle estimation scheme is experimentally evaluated. with side slip angle measurement using RT3000 (OTS (2004)) on a test vehicle (KIA Morning). An accelerometer and an inertia sensor on the test vehicle measure lateral acceleration ay and yaw rate ψ˙ a sampling period of 7msec. The state estimator gain L in (10) is simply determined from (11) such that the resulting state estimator is convergent for 0.3 < µ < 1 so as to apply the adaptation scheme (8) in real time.
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This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (20121480).
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ACKNOWLEDGMENT
REFERENCES
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A slip angle estimation method has been developed, which uses measurements of lateral acceleration and yaw rate. A unique model for slip angle estimation was developed by combining velocity kinematics, yaw dynamics, and lateral translational dynamics, so as to avoid sensor drift and parameter variation effect. The model was used in designing a state estimator estimate slip angle, compensates combining to velocity kinematics, yawwhich dynamics, and lateral slip angle prediction error using the measurements of lateral acceleration and yaw rate. The estimation scheme was validated through experimental tests using RT3000 on a test vehicle. The proposed slip angle estimation scheme was shown to provide reliable slip angle and can thus be effectively used in vehicle stability control.
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Fig. 4. Experimental test at high µ: (a) steering angle and longitudinal speed; (b) yaw rate; (c) slip angles.
2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia
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Fig. 5. Experimental test at low µ: (a) steering angle and longitudinal speed; (b) yaw rate; (c) slip angles.
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