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Integrated strategy for vehicle dynamics stability considering the uncertainties of side‐slip angle

2020; Institution of Engineering and Technology; Volume: 14; Issue: 9 Linguagem: Inglês

10.1049/iet-its.2019.0461

1751-9578

Xiaoxu Dong, Liang Li, Shuo Cheng, Zhanchao Wang,

Hydraulic and Pneumatic Systems

IET Intelligent Transport SystemsVolume 14, Issue 9 p. 1116-1124 Research ArticleFree Access Integrated strategy for vehicle dynamics stability considering the uncertainties of side-slip angle Xiaoxu Dong, Xiaoxu Dong State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this authorLiang Li, Corresponding Author Liang Li xiaoxudong1988@gmail.com orcid.org/0000-0002-1577-408X State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this authorShuo Cheng, Shuo Cheng State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this authorZhanchao Wang, Zhanchao Wang State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this author Xiaoxu Dong, Xiaoxu Dong State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this authorLiang Li, Corresponding Author Liang Li xiaoxudong1988@gmail.com orcid.org/0000-0002-1577-408X State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this authorShuo Cheng, Shuo Cheng State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this authorZhanchao Wang, Zhanchao Wang State Key Laboratory of Automotive Safety and Energy, Tsinghua University, No. 30 Shuangqing Road, Beijing, People's Republic of ChinaSearch for more papers by this author First published: 21 July 2020 https://doi.org/10.1049/iet-its.2019.0461Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, a side-slip angle estimator was presented, capable of compensating the model bias between test data and model prediction under the uncertainties of vehicle speed and friction coefficient, to obtain the actual side-slip angle. Specifically, the model validation method was adopted to optimise the irrelative coefficient of the principal component analysis model for calibrating the side-slip angle model bias. To calibrate all the model bias under other conditions without test data, the response surface methodology was developed to approximate all the side-slip angle model bias under any conditions. Based on the estimator, an integrated method based on the fuzzy model predictive control (MPC) algorithm was proposed with respect to the yaw moment and side-slip angle, taking brake pressure and steering angle as the controlled objects. Such a method integrates the function of active front steering and direct yaw-moment control into the MPC model and coordinates their proportion in accordance with the relevant parameters using the fuzzy control method. Lastly, the results of simulations with the hardware-in-loop platform verified that the proposed algorithm could enhance the vehicle stability and the computational accuracy of the side-slip angle. 1 Introduction Numerous studies have been conducted on vehicle stability system, a major field in automotive, to demonstrate its capabilities of reducing single-vehicle accidents [1-3]. Parameters observation and differential brake control strategy refer to the two vital research interests in active safety technologies. In the field of parameters observation, side-slide angle, one of the critical indicators to determine vehicle stability, with yaw moment as another one, has been analysed extensively. Since the side-slip angle exhibits the potential of vehicle dynamic stability [4], the lateral tire force may decrease noticeably, and the vehicle may be out of control under an abnormally large side-slip angle [5]. Moreover, the measuring sensor is less cost-effective in the vehicles gradually. Hence, various observers have been developed to address this problem. Kalman filter (KF) proposed by Rudolf Emil Kalman has become the most extensively employed method for its noise processing and low delay [6]. Based on extended KF, Li et al. [7] substituted damping items into the side-slip angle rate feedback algorithm to compensate for the inaccuracy of model error to the estimation of road friction. Morrison and Cebon [8] explored three non-linear KFs to ascertain the tractor side-slip angle of a tractor-semitrailer for articulated heavy goods vehicles. Li et al. [9] proposed an unscented KF-based adaptive variable structural observer with dynamic correction for vehicle body side-slip angle to tackle down the bias drift problem in the integral. Moreover, the authors of [5, 10, 11] also presented a series of estimation methods based on the modified KF. Besides, numerous model-based and numerical method-based systems have been developed to predict the side-slip angle. Te et al. [12] presented a feasible method for longitudinal force and side-slip angle estimation using a Luenberger observer and high-order sliding mode observer. Boada et al. [13] developed a novel observer based on ANFIS combined with KFs to predict the side-slip angle. A method to assess the side-slip angle has been proposed by Wang et al. [14] based on general regression neural network and driver-vehicle closed-loop system, which adopts homogeneous design project to optimise the training samples and builds the mapping relationships between sides-lip angle, yaw speed, as well as lateral acceleration. According to Zhang et al. [15], the Takagi–Sugeno fuzzy modelling technique was utilised to represent the vehicle lateral dynamics model with a non-linear Dugoff tire model and time-varying vehicle speed. When it comes to vehicle stability control, a growing number of researchers have focused on the cooperative control method as fuelled by the advancement of autonomous automotive and the advanced electronic control of chassis. Shuai et al. [16] presented an H∞ -based delay-tolerant linear quadratic regulator control method to solve in-vehicle network delays in the scenario of active front steering (AFS) and direct yaw-moment control (DYC) of four-wheel-independent-drive electric vehicles over controller area network (CAN). Ni et al. [17] built an envelope control framework for the four-wheel independently actuated autonomous ground vehicle to regulate it on the desired path while regulating it to the driving limits. In [18], a human–machine-cooperative-driving controller with a hierarchical structure was proposed for vehicle dynamic stability. A novel control algorithm of DYC based on the hierarchical control strategy was yielded in [19], which is split into three layers to achieve smooth control performance. Given the side-slip angle, Mirzaei [20] employed the linear-quadratic to optimise the DYC method. As suggested by the simulation results, the proposed method was more feasible than the yaw rate control to stabilise vehicle motions in non-linear regimes. The authors of [21, 22] adopted a model predictive envelope controller to confine the vehicle motion into the stable region of the state space. Such an approach provides a large operating region accessible by the driver and smooth interventions at the stability boundaries. Given the effect of the side-slip angle on vehicle stability, this paper introduces the β– phase plan to promote the joint control of DYC and AFS. Similar to the framework of [23, 24], the diagram of the proposed integrated strategy is illustrated in Fig. 1. This strategy consists of four parts (i.e. side-slip angle estimator, fuzzy model predictive control (Fuzzy-MPC) controller, vehicle model and nominal values calculation). In the estimator module, the model values of side-slip angle are calibrated, and subsequently, the differences between the model data and the nominal data (i.e. side-slip angle and yaw rate) are exploited to determine the vehicle state. In the Fuzzy-MPC controller, the cylinder pressures of four wheels and the steering angle are the optimised objects to ensure vehicle stability. Besides, the fuzzy controller is adopted to weight the two control models to maintain the system efficiency. The rest of the parameters in the system for calculation originate from the vehicle model. Fig. 1Open in figure viewerPowerPoint Overall structure of the proposed integrated strategy The remaining of this paper is organised as follows. In Section 2, the relevant models are presented. In Section 3, the algorithm of sild-slip angle estimation is elucidated. In Section 4, the integrated AFS/DYC strategy is demonstrated. In Section 5, its performance is assessed in the hardware-in-loop (HiL) platform. Finally, the conclusion of this study is drawn in Section 6. 2 Related models 2.1 Vehicle dynamics model A 7-degree of freedom (DOF) model illustrated in Fig. 2 is developed to present the dynamics characteristic. The longitudinal, lateral and yaw motion are described as follows: (1) (2) (3) (4) where m is the vehicle mass, Vx and Vy are the longitudinal and lateral velocities, respectively. ωr represents the yaw rate. Fxij and Fyij are the longitudinal and lateral forces of four wheels, respectively, their subscript ij is fl, fr, rl and rr, representing the front left, front right, rear left and rear right wheels, respectively, ij used in the following equations have the same meaning. δ is the steering angle of the front wheels, and β is the side-slip angle. Iz denotes the inertia moment about the vertical axis of the vehicle. a and b are the distances from the gravity centre to the front and rear axles, respectively. T is the distance between the left and right wheels. Jw is the wheel inertia moment about the rotary axis. Ttij and Tbij represent the driving torques and braking torques, respectively. Fig. 2Open in figure viewerPowerPoint 7 DOF vehicle dynamics model Based on the assumption that the front and rear wheel brake pressures are proportional to the longitudinal load transfer, the vertical force can be simplified as follows: (5) (6) where the Fzf and Fzr are the vertical force of front and rear wheels, respectively. ax is the longitudinal acceleration, g is the gravity acceleration, and h is the height of c.g. 2.2 Tire model The magic formula is adopted to express the tire's dynamics due to the complex non-linear system of a tire. Its general form is written as (7) where Y (x) can represent the longitudinal force, the lateral force, or the aligning torques; X is the longitudinal slip ratio λ, and the wheel side-slip angle α. Thus, the longitudinal and lateral forces can be calculated in the form of Y (λij) and Y (αij). The longitudinal slip ratio during braking and acceleration is calculated as presented in the following equation: (8) where R is the radius of the wheel, ωij is the angle velocity of the wheel. The side-slip angle α of front and rear wheels can be calculated as follows: (9) When DYC is activated, the wheel will be dropped into the combined-slip condition of the tire, which is only for the pure-slip condition [19]. Accordingly, a modification should be made as follows to adapt to this situation (10) In brief, the side-slip angle is expressed as follows: (11) According to [18], the desired pressure Pij can be calculated as follows: (12) (13) where Ap is the piston area, Rb is the effective radius between the centre of disk rotor and pad, and is the estimated brake disk-pad friction coefficient. Jw is the moment of inertia of wheel, rw is the radius of the wheel, λ is the wheel slip ratio, and s is the sliding surface. What is more, the bound values of the uncertainties B1, B2 and B3 are 0.6, 1500 and 0.35, respectively. η is a design parameter and a strictly positive constant, Φ is a design parameter representing the boundary layer around the s = 0 sliding surface, and is a small positive parameter. 3 Validation and calibration for the side-slip angle model The previous works of the side-slip angle observation, model-based method, kinematics-based method or KF method were all developed based on the modification of vehicle dynamic models for its inaccuracy. On the whole, it is known that models are proposed to approximate actual physical systems based on a range of assumptions and simplifications, and the disturbance from the external environment will also affect the parameters of the model. These are the two major reasons why the ideal model cannot reflect the actual system. The same is true for side-slip angle estimation. The underlying cause of the difficulty in identifying the bias of the side-slip angle is complicated due to the complexity of the vehicle system, while fundamental modification of the model from the physical perspective is time-consuming and costly, and it remains insufficiently accurate [25]. Thus, to solve this problem, the bias correction approach with respect to the statistical method is considered, instead of the physical method [26-29]. Bayesian calibration model, proposed by Kennedy and O'Hagan [30], characterised by time dependent and non-linear system problem solving, has been addressed in numerous bias correction approaches. This model is written as (14) where is the baseline model prediction; λ is the model bias; ϒ is test data; ɛ is the measurement error; and X is the model parameter. Though the direct expression for bias compensation is provided, three challenges should be addressed. First, how to quantify the model bias effectively. When the vehicle parameters are pointed, the side-slip angle bias (i.e. the difference between the model data curve and test data curve) is easy to describe, while they are difficult to describe in the case of the parameters uncertainties (e.g. vehicle speed and friction coefficient of the road). Second, how to calibrate the bias for model accuracy with the form of a random process. Third, how to establish the linkage between the known parameters and unknown bias when test data are limited, so the correct approach can extend to all scenarios (e.g. different vehicle speed and friction). To solve the above problems, four steps were taken: (1) A validation metric was built based on U-pooling and deviation shape to quantify the model bias given the parameters uncertainty, and the steering angle was selected as the uncertain parameter in this study. (2) The principal component analysis (PCA) was conducted to characterise the dynamic randomness with a few uncorrelated random variables to eliminate the association of the model parameters and reduce their numbers. (3) A calibration method was proposed to optimise the baseline model. (4) The response surface methodology (RSM) based on the least square method was adopted to build the relationship between the parameters and the model bias and further approximate the model bias in design. 3.1 Validation metric for the model bias of side-slip angle U-pooling metric, proposed by Ferson et al. [31], has been applied for model validation. It primarily aims to calculate the cumulative distribution function (CDF) difference (the U-pooling value) between model prediction and test data in the standard uniform space (U-space). The area enclosed by the two CDF curves can be exploited to measure the model accuracy. The smaller the area, the higher the accuracy. In this study, a counting method is employed to simplify the calculation, as expressed in the following equation: (15) where yi is test data; is the model prediction; F (▪) indicates the CDF; i is the number of datasets; N is the number of data in one dataset; n is the time step. However, U-pooling sometimes cannot identify the difference since it can only indicate whether the two datasets are close, instead of acquiring the deviation size between them. Accordingly, shape deviation is introduced to quantify the model bias further, which primarily aims to subtract model prediction from test data and then to normalise this difference to make it in the same dimension as U-pooling value. A model accuracy metric integrating the two above methods is formulated, as written in the following equation: (16) where ϒT and indicate the test data and model prediction, respectively; U (▪) is the U-pooling function; Si is the i th shape deviation function; yT i and are the i th model response represented by a vector; and Q is the number of a model or test responses. If measurement error ɛ is ignorable, the true responses ϒT can be approximated by the test responses . In this case, the joint simulation of CarSim and Simulink was utilised. A double lane change experiment was performed to quantify the model bias of the side-slip angle. The trajectory of the experiment is shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Double line change experiment In the experiment, the vehicle speed (120, 100 km/h) and road friction coefficient (0.85, 0.5) were set as the different working conditions. Steering angle refers to the uncertain parameter, and the 0.05 times of it is the standard deviation in each time step. The other parameters of the vehicle were set, as listed in Table 1. Table 1. Key Parameters in the simulation model Parameters Value m 1650 kg Iz 3234 kg m2 J w 928.1 kg m2 h 0.53 m R 0.32 m a 1.4 m b 1.65 m The model prediction and test data in the four working scenarios are presented in Fig. 4. The green lines represent the model prediction (i.e. the baseline model prediction shown in (14)) with the steering angle uncertainty; the blue lines representing the test data exported from CarSim. The figure suggests that the test data located at the margin of the model prediction area formed by the steering angle uncertainty in all four scenarios, which reveals the low confidence of the baseline model. Similar trends emerged that the model prediction was mostly larger than the test data in the first and third turns, and it was mostly smaller in the second turn. Fig. 4Open in figure viewerPowerPoint 7 DOF vehicle dynamics model comparison of the side-slip angle between model and test data (a) vx = 120 km/h, μ = 0.85, (b) vx = 120 km/h, μ = 0.5, (c) vx = 110 km/h, μ = 0.85, (d) vx = 110 km/h, μ = 0.5 The values of the model bias in the four scenarios are listed in Table 2, including U-pooling, shape deviation and the total variation metrics. The bigger the values, the more significant the bias. According to the metric, the second one showed the biggest bias with 0.4162 among the four scenarios. The third one was the relatively precise condition with 0.2921. It is noteworthy that U-pooling was the same in four scenarios, and it was larger than the average shape deviation because there was only one test data; the model responses at certain time step were all larger or below the corresponding test data, resulting in u1 = 0 or u1 = 1, respectively. Table 2. Variation metric for the baseline model prediction in four working conditions 120/0.85 120/0.5 100/0.85 100/0.5 Ψ 0.3253 0.4162 0.2921 0.3178 u-pooling 0.2500 0.2500 0.2500 0.2500 shape deviation 0.0753 0.1662 0.0421 0.0678 3.2 Calibration for the side-slip angle model bias in test datasets After the validation metric is developed for the angle bias, this section aimed to optimise the metric to keep the corrected model prediction close to test data. Thus, the objective function could be built as follows: (17) Subsequently, a formula should be set up to represent the relationship between the bias (i.e. λ) and its distribution parameters (e.g. the mean and standard deviation) or hyper-parameters because λ cannot be regulated directly as an independent variable due to its random process form. PCA refers to a statistical method to explain the intrinsic relationship among large samples, multivariate data or samples. It primarily aims to transform the original random vectors related to their components into new random vectors that are irrelevant to their components through orthogonal transformation. This is algebraically represented by transforming the covariance matrix of the original random vectors into a diagonal matrix, and geometrically by transforming the original coordinate system into a new orthogonal coordinate system, i.e. scattering the sample points in the most open n orthogonal directions. Accordingly, the dimension of the multi-dimensional variable system is reduced, and the system can be transformed into a low-dimensional variable system with a higher accuracy. In this case, the model bias is written in the following equation: (18) where λij is the i th model bias in the j th time step. The mean values are estimated as (19) The variation part of the bias is expressed as (20) PCA model of the two parts of bias is shown as follows: (21) where M is the number of principal components that can be set up as two after verified, Φk indicates the k th principle component; Vk is the irrelated coefficient with zero mean value. γ and Φk are constants which are determined on the model prediction and test data. Vk can be used as an optimised object to realise (22), which can be transformed as follows: (22) 'fmincon' function in MATLAB was adopted to optimise the Vk value. Fig. 5 shows that the blue lines located in the centre of the optimised green datasets area in all turns, suggesting that the confidence of the corrected model prediction was significantly improved. Fig. 5Open in figure viewerPowerPoint Comparison of the side-slip angle between model and test data after calibration (a) vx = 120 km/h, μ = 0.85, (b) vx = 120 km/h, μ = 0.5, (c) vx = 110 km/h, μ = 0.85, (d) vx = 110 km/h, μ = 0.5 This change is quantitatively displayed in Table 3. The accuracy of the corrected model prediction was significantly improved, in which the U-pooling value reached the minimum value 0. Table 3. Variation metric for the baseline model prediction after calibration 120/0.85 120/0.5 100/0.85 100/0.5 Ψ × 107 0.0127 0.4988 0.0016 0.0125 u-pooling 0 0 0 0 shape deviation × 107 0.0127 0.4988 0.0016 0.0125 3.3 Approximation for the side-slip angle model bias A response surface can be built based on the calibrated side-slip angle bias in four working scenarios to predict the bias without test data. The response surface is an optimisation thought to combine experimental design method, mathematical method and statistical analysis method. It is applicable to finding the quantitative law between response output and influencing factors of an unknown system or process. To be specific, it refers to the application of systematic methods to perform experiments and easily find the response values corresponding to each factor level. In the four above scenarios, side-slip angels are different from each other because of the different vehicle speeds and adhesion coefficients. These parameters can be taken as the factors of the response surface. The adhesion coefficient can be replaced by the lateral acceleration, which is easier to be acquired by the sensor. In the meantime, μ, Φk and the standard deviation of Vk are taken as dependent variables. Subsequently, the PCA parameters of the bias are calculated by the following equation: (23) where vx and μ denote the vehicle speed and road friction coefficient, respectively; σ is the standard deviation of Vk, μ, Φk are the same meaning as (21). The least-square method characterised by high accessibility and high accuracy is used to build the response surface. As the regression model was successfully developed, the side-slip angle bias of the vehicle could be identified at any speed, steering angle and lateral acceleration. To verify the effectiveness of the surface model, another working scenario with 11 km/h speed and 0.75 friction coefficient was taken. Fig. 6a shows the original data before the approximation. It presents the same situation with the four scenarios for calibration. Fig. 6b shows the results after the approximation with the metric value reduced from 0.3125 to 0.0598 × 10−7. Fig. 6Open in figure viewerPowerPoint Comparison of the side-slip angle between model and test data before and after approximation (a) Before approximation, (b) After approximation 4 Design of integrated strategy based on Fuzzy-MPC In this section, the design of the Fuzzy-MPC controller is presented based on the aforementioned side-slip angle estimator. 4.1 Design of the nominal yaw rate and side-slip angle A 2 DOF bicycle model is adopted to calculate the nominal values of yaw rate and side-slip angle, which are taken as the reference values (24) (25) where k is the stability factor, which can be formulated as follows: (26) where k1 and k2 are the cornering stiffness of the front and rear axles, respectively. The road friction coefficient should be taken into account, According [19], the (24) and (25) can be transferred as follows: (27) (28) where τβ is the response time. 4.2 Design of non-linear MPC based on yaw rate and side-slip angle According to the aforementioned vehicle dynamic model, the state variables, control inputs and output vectors are defined as follows: (29) (30) (31) (32) Subsequently, the state function is formulated as (33) The current control signals should be employed to minimise the predicted tracking error based on the definition of MPC, while the variation of the control signals should be as small as possible. Thus, the MPC objective function is written as (34) where y is the controlled output and yno is the nominal output. Np is the prediction horizon, Nc is the control horizon, and Np ⩾ Nc, wr, w dyc and w afs are the weights of controlled outputs and the control inputs at the prediction time k + i, respectively. The first term of the function represents the predicted tracking ability of the system; the second and third terms indicate the control ability of DYC and AFS, respectively. Given the realisation of the steering mechanism and the ability of the braking system, the control input should be, to a certain extent, limited. The variation of steering angle generated by AFS should not be too large to cause the driver panic, which is the same as the DYC system. Besides, the implementation of the AFS and DYC is physically limited by the tire normal load and tire-road friction coefficient, i.e. the lateral force produced by the integrated controller should be smaller than the maximum value determined by the adhesion coefficient. The maximum of the lateral force refers to the function of longitudinal force, vertical force and friction coefficient (35) Given (34) and all the constraints, the final form of the MPC is defined as follows: (36) Constraint condition (37) where the first term of (37) represents the restrictions of the size of control inputs, the second term the restriction of the variation of control inputs, and the third term the physical boundary. 4.3 Fuzzy logics for the weights of control vectors In the MPC model described in the last section, the weights of control vectors w dyc and w afs can affect the contributions of the two actuators, and the percentage of intervention may affect the control effect of the whole system. For instance, more priority should be given to DYC than AFS when the vehicle reaches the handling limit. While the AFS should be involved more when a vehicle is in the low adhesion road to prevent wheels from being locked. Thus, the calculation of the weights is vital for the integrated strategy. The decision criteria of the weights are formulated according to the β– phase plane [29], as illustrated in Fig. 7. The phase plane indicates the process of stabilisation with the relationship between the side-slip angle and side-slip angle velocity. The region between the two blue lines represents the stability area. The vehicle in that region can achieve convergence to a stable state based on the result of the experiment. Fig. 7Open in figure viewerPowerPoint β– phase plane with zero steer input According to the relevant experiments, the phase plane is affected by the road friction coefficient, vehicle speed, and front-wheel angle. And these three factors have an independent influence on the phase plane. Therefore, as the above three parameters change, the phase plane stability region shown in Fig. 7 will also change. According to [32], the stability region can be expressed as follows: (38) where E1 is the reciprocal of the slope of the stability boundary. The absolute value of the boundary slope decreases as the vehicle speed increases. The slope is basically inversely proportional to the speed. Therefore, e1 and e2 are the coefficients of the slope, which can be obtained by the fitting. g (μ) is the value of the limit side-slip angle, which is a different constant in different road adhesion. δ0 is t