Vehicle Driving State Estimation Using an Improved Adaptive Unscented Kalman Filter

: This paper proposes an improved adaptive unscented Kalman filter (iAUKF)-based vehicle driving state estimation method. A three-degree-of-freedom vehicle dynamics model is first established, then the varying principles of estimation errors for vehicle driving states using constant process and measurement noises in the standard unscented Kalman filter (UKF) are compared and analyzed. Next, a new type of normalized innovation square-based adaptive noise covariance adjustment strategy is designed and incorporated into the UKF to derive our expected vehicle driving state estimation method. Finally, a comparative simulation investigation using CarSim and MATLAB/Simulink is conducted to validate the effectiveness of the proposed method, and the results show that our proposed iAUKF-based estimation method has higher accuracy and stronger robustness against the standard UKF algorithm. the results show that compared to the UKF approach, our proposed iAUKF-18


Introduction
Currently, automotive active safety control systems (ASCSs) including rollover prevention system (RPS) [1], adaptive cruise control (ACC) [2], and lanedeparture avoidance system (LDA) [3], have been extensively investigated and developed in response to the continuously growing demand on vehicle driving safety and dynamics performances [4]. It is well known that the accurate vehicle states like sideslip angle, yaw rate, and longitudinal speed can significantly affect an ASCS's performance. How to accurately obtain these vehicle states has become a core technical premise in the development of automobile ASCSs [5].
However, considering the limits and costs of sensors, only a few of vehicle driving states can be directly measured. Consequently, it is necessary to propose an appropriate estimation method to achieve accurate and effective estimation of vehicle driving states using fewer on-board sensors [6].
At present, there exists two types of vehicle state estimation methods: kinematics and dynamics methods [7]. For the former one, the vehicle states are usually predicted by integrating the related vehicle states measured by sensors [8].
Note that the common kinematics methods include the closed-loop nonlinear observer [9] and neural network methods [10], etc. However, these methods have high dependency on the sensor accuracy, and the cumulative errors caused by continuous integration also limit their applications [11][12]. For the latter one, a large number of driving state estimation methods such as Kalman filter (KF) family algorithms, particle filters, and robust observers have been proposed by lots of researchers and scholars. In those methods, the KF family approaches are widely employed to estimate vehicle states due to its conveniences of providing optimal solutions and suppressing the effect of measurement and sensor noises.
For instances, the extended Kalman filter (EKF) was usually used to estimate vehicle slip angle and lateral tire force, and a variable model covariance method was introduced to design a stable estimator in order to improve local observability of the observer [13]. In the work of [14], an effective state estimation was proposed for a four-wheel-drive vehicle based on the minimum model error criterion and the EKF algorithm. Besides, an augmented EKF algorithm was utilized to simultaneously estimate the common vehicle states and the lateral stiffness of tires, thus the impacts of the time-varying parameters in this vehicle system on the estimation accuracy can be eliminated [15].
Although the EKF algorithm maintains the elegant and efficient recursive update form by computing Jacobian matrices, while the local linearization may unavoidably lead to large cumulative estimation errors and divergence [16][17][18].
Fortunately, the unscented Kalman filter algorithm (UKF) can avoid this problem by approximating the nonlinear probability distribution using a sampling method instead of evaluating the Jacobian [19]. Recently, the UKF has been broadly used to estimate vehicle driving states. In [20], both EKF and UKF algorithms were applied to estimate the sideslip angle and the tire lateral force using real car test data. It was conformed that the performance of the UKF algorithm was far superior to that of the EKF algorithm. Moreover, a new type of online estimation method based on joint UKF was presented in [21] to estimate vehicle sideslip angle, body mass, along with moment of inertia, and the effectiveness of this estimation method were verified using the results from real car measurements.
Similarly, an adaptive state estimator was proposed for all-wheel-drive electric vehicles based on UKF algorithm [22], and this estimator could provide an accurate estimation of the longitudinal and lateral speed, tire slip angle, as well as tire friction coefficients. Different from the above literatures, a multi-sensor optimal data fusion method was proposed in [23] to estimate vehicle states, in which the integrated Inertial Navigation System (INS)/Global Navigation Satellite System (GNSS)/Celestial Navigation System (CNS) were integrated with the common UKF algorithm together. Afterwards, the global optimal state estimation of vehicle states was achieved according to the linear minimum variance principle.
Compared with the EKF, the UKF can greatly improve the estimation accuracy of the state variables for a nonlinear system. However, since the noise covariances of the EKF and UKF are often set as constant values that may be not accorded with the practical situation, the estimation accuracy of vehicle states cannot be guaranteed. Additionally, it is difficult to obtain accurate noise covariances under the changeable working conditions, which will reduce the robustness of common UKF algorithm. Therefore, it is an interesting and challenging issue to derive an improved UKF algorithm in conducting vehicle driving state estimation.
For example, in the work of [24], an adaptive EKF algorithm approach was proposed to achieve state-of-charge estimation for lithium-ion battery packs used in electric vehicles, in which the normalized innovation square (NIS) was used to validate the effectiveness of the designed estimation method. To compensate for the uncertainties caused by model errors, an adaptive sideslip angle observer was developed to fulfill vehicle body estimation [25] by combing the adaptive technology with the UKF along with a sensor-based integration approach. The simulation and real-car experimental results demonstrated that this method exhibited better estimation performances than the common UKF. Furthermore, in [26], a composite estimation procedure of the side slip angle was presented to reduce the interference of noise using an adaptive neuro-fuzzy inference system and the UKF. A novel adaptive UKF-based state estimation method was also presented in [27], wherein the measured signals were categorized into different road levels such that the noise covariance of different roads can be adaptively adjusted. Moreover, in the work of [28], an adaptive square-root UKF approach was proposed for the state estimation/detection of nonlinear systems, in which process noise and measurement noises were unknown. By summarizing the above related research work, this paper proposes an improved adaptive unscented Kalman filter (iAUKF)-based estimation method of vehicle driving states with adjustable noise covariance. The main contributions of this work are summarized as follows: (1) By comparing and analyzing the influences of process and measurement noises on the estimation accuracy of vehicle states using the UKF algorithm, the varying principles of estimation errors for vehicle driving states are obtained.
(2) A NIS-based adaptive noise covariance adjustment strategy is designed and combined with the UKF algorithm to adaptively adjust process and measurement noise covariances, thus the proposed estimation method can improve the accuracy and adaptability of vehicle driving state estimation.
The rest of this paper is organized as follows. In Section 2, a three-degree-offreedom (3-DOF) vehicle dynamics model is constructed, and the problem statement of vehicle driving states is described. In Section 3, an adaptive adjustment strategy of the noise covariance is designed, and based on this, a novel iAUKF-based vehicle driving state estimation method is proposed. In Section 4, simulation investigations based on CarSim and MATLAB/Simulink software are presented to illustrate the effectiveness of the proposed method under different working conditions. Finally, the concluding remarks are summarized in Section 5.

Vehicle dynamics modeling and problem formulation 2.1 Vehicle dynamics modeling
In this section, a 3-DOF "bicycle" or "single-track" model is used to describe the motion characteristics of vehicle in the yaw, lateral, and longitudinal directions, as shown in Fig. 1. This 3-DOF dynamics model can reflect the vehicles dynamics behaviors in real driving conditions, which has been extensively utilized in the previous literature [29][30][31]. The tire is here assumed to exhibit linear elastic behavior and complies to the small-angle approximation rule.
The dynamics equations of this vehicle are governed by 2 ( 1) where a and b are the distances from the vehicle's center of gravity (CG) to the front axle and the rear axle, respectively; Kf and Kr are the equivalent cornering stiffness coefficients for the front axle and the rear axle, respectively; m is vehicle body mass; Iz is the moment of inertia for the yaw motion; vx is the longitudinal velocity; r is the yaw rate; β is the sideslip angle of CG; δf is the steering angle of the front wheel; and ay is the lateral acceleration.

Problem formulation of vehicle state estimation
Generally, Kf and Kr are constant in the vehicle's lateral dynamics modeling.
However, in the practical operating conditions of vehicles, Kf and Kr are continuously varied with the change of its internal parameters. Therefore, Kf and Kr are herein treated as variable parameters and recursively adjusted by estimator.
Considering the relationship between β, r, vx, ay, as well as Kf and Kr, the nonlinear vehicle dynamics model can be constructed by the state transition equation and the measurement equation [32], which are given as follows: In order to employ the first-order Euler approximation with sampling time Δt to discretize Eq. (5) [15], both f (.) and h (.) can be represented as follows: : : The state-space form of Eq. (7) and Eq. (8) can be written as where k A , k B , k C , and k D are provided in the Appendix.

Standard unscented Kalman filter algorithm
The unscented transformation (UT) in the standard UKF is used to approximate the nonlinear probability distributions of the covariance and the mean values of vehicle states. For the 3-DOF vehicle model, the estimation procedure of vehicle driving states using the standard UKF is summarized as follows: where 0 x is the initial values of x.
(2) Time updating Step-a: Creating the Sigma points.
Based on the symmetric sampling strategy, the Sigma points k ξ are created where n is the length of x, and it is set as n = 8 in this work; 2 () p n k n and it is used to adjust the distance between the Sigma points and x , and its optimal value is 2 under the Gaussian distribution.
Step-b. Computing the predicted values of k ξ .
The nonlinear transformation of k ξ is performed by using (9), and the predicted values of k ξ are then obtained Step-c. Computing the priori estimation of ˆk − x and , xk where k Q is the process noise covariance.
Step-d. Computing the priori estimation of k z .

Return to
Step-a, and recreate new Sigma points, afterwards, substitute these points into in Eq. (9) to calculate the prior measurements ˆk − z . To facilitate the subsequent derivation process, the calculation formulas are summarized as follows:

Robustness analysis of standard UKF algorithm
In the standard UKF algorithm, Qk and Rk are usually set as constant values to quantify the process and measurement noises. However, the actual process and measurement noises are often varied with the different driving conditions.
Therefore, it is difficult to obtain the accurate Qk and Rk, moreover, using the inaccurate Qk and Rk will result in greater estimation error and may reduce the robustness of standard UKF algorithm.
To assess the influence of process noise and measurement noise on the The estimation errors of β, r, vx, and ay using standard UKF in Case 1 and Case 2 are shown in Fig. 2(a) and Fig. 2(b), respectively.
To further demonstrate the estimation errors of β, r, vx, and ay using the standard UKF, the root mean squared error (RMSE) [33] is used to evaluate the related estimation errors, which is defined by where n is the size of X; Xi is the true value; ˆi X is the estimated value, where i = 1, 2, …, n.
The RMSEs of β, r, vx, and ay for the standard UKF under different process and measurement noises are shown in Table 1. It can be concluded from Fig. 2 and Table 1

The iAUKF-based vehicle driving state estimation
As described in Section 3.2, the variations of Qk and Rk have a significant impact on the accuracy of the estimated vehicle states with the standard UKF.
Therefore, by referring to a related study [24], an adaptive noise covariance adjustment strategy incorporating the UKF algorithm is herein proposed to conduct vehicle driving state estimation. This approach can adaptively adjust Qk and Rk in terms of the errors between the prior and actual measurements, which can reduce the estimation error and the divergence possibility.
First, in order to approximate the process noise, the discrete Qk is set as [34], rather than simply setting it as a diagonal matrix with n dimension. Moreover, Q is the continuous process noise matrix. Since Q is difficult to measure, it is set to be greater than the noise error of the common steering angle sensor and the accelerometer, as well as the standard deviation of the front-wheel and the rear-wheel lateral stiffness coefficients. Thus, we set Q = diag (0.6,0.1,500,500) [15]. In addition, the initial value of Rk is set as Rk,0 = diag (0.01,0.5).
Second, the innovation vk is defined as the error between the prior measurements and actual measurements, i.e., The theoretical innovation covariance is obtained by the UKF algorithm, which is equal to , P zk . Therefore, the theoretical innovation covariance Ck is denoted by Due to the influences of modeling and measurement errors, there usually exists a certain deviation between the actual and theoretical innovation covariance. The actual innovation covariance is obtained by the definition of the error covariance, which is defined as where M is the length of the sliding window.
where d is the adjustment factor of M; max Essentially, d should reflect the changes of vehicle states correctly. When d is greater than a certain threshold, the vehicle states are considered to change rapidly. In contrast, the vehicle states change slowly, and then M can be adjusted adaptively according to the value of d. Therefore, to evaluate the variation rates of vehicle states, the value of d is defined as the NIS that is commonly used in the Inertial Navigation System-Global Positioning System [24], which is expressed When vehicle running in a certain maneuver operation, the results estimated by the UKF algorithm will inevitably have large errors with the rapid change of vehicle driving states, which will eventually lead to the increase of the innovation and its NIS. Therefore, Eq. (29) is used as the judgment criterion of the UKF algorithm, and the adaptive noise covariance adjustment strategy is used to adjust Qk and Rk.
Here, the noise adjustment factor k  is introduced to adjust Qk and Rk by comparing C k and Ĉ k . According to literature [36], k  is defined as where  is the multiplier coefficient of Qk.
In summary, the proposed iAUKF-based vehicle driving state estimation can be established if the above adaptive noise covariance adjustment strategy is combined with UKF algorithm, and the flowchart of the iAUKF-based vehicle driving state estimator is shown in Fig. 3.
Specifically, the inputs and measurements are obtained from the CarSim or the on-board sensors in terms of the steering wheel angle, and the prior estimation of x is then carried out, thus the time updating is completed. Subsequently, the adaptive noise covariance adjustment strategy is designed to realize the adaptive adjustment of Qk and Rk in standard UKF, which will improve the accuracy and adaptability of vehicle driving state estimation. Finally, the posterior states and the corresponding covariance are updated.

Simulation investigation
In is shown in Fig. 5. Note that the initial value of matrix P is set as P0 = I8×8, and the related parameters in the simulation are listed in Table 2.

Sinusoidal maneuver test
In the first test, the proposed iAUKF is implemented in case of a sinusoidal maneuver. The estimated β, r, vx, and ay obtained by the UKF, SHAUKF and our The curves of δf and ax simulated by CarSim in sinusoidal maneuver are displayed in Fig. 6. Besides, Fig. 7 shows the curves of β, r, vx, and ay using the UKF, SHAUKF and our designed iAUKF algorithm in the same sinusoidal maneuver. These three UKF algorithms have certain local errors, yet the estimation effect of SHAUKF and the proposed iAUKF algorithm are better than those of the traditional UKF, and the estimation results of the iAUKF are closer to the reference values than that of SHAUKF. Although the error of vx generated by the iAUKF is larger than that generated by UKF and SHAUKF in 10-50 s, the entire error of vx generated by the iAUKF exhibits a decreasing tendency, which makes its estimated curve fitting the simulation results of CarSim well.
Additionally, from the enlarged subplots A, B, and C in Fig. 7, it is clear that the corresponding vehicle states estimated by proposed iAUKF algorithm are closer to the reference values obtained by the CarSim, in compared to those obtained by the UKF and SHAUKF algorithm.  Although the RMSE of vx using the iAUKF is larger than that of the SHAUKF, the RMSE reduction effects of proposed iAUKF are more obvious than those of the SHAUKF for other states.

Fishhook maneuver test
Two Fishhook maneuvers are considered, and they are separately discussed in this section.

Fishhook maneuver I
The mathematical expression of δs under fishhook maneuver I is expressed The simulated curves of δf and ax using CarSim are provided in Fig. 9. Fig. 10 shows the curves of β, r, vx, and ay estimated by the UKF, SHAUKF and iAUKF algorithms. Compared to the UKF and SHAUKF, it is obvious that the responses of β, r, vx, and ay generated by the iAUKF are closer to the corresponding ones simulated by CarSim no matter from the subplot A, B, and C, which demonstrates that the proposed iAUKF maintains a relatively higher accuracy during fishhook maneuver I.
Additionally, Fig. 11 shows the absolute error comparisons of β, r, vx, and ay in fishhook maneuver I. It is easily seen that the estimation errors of β, r, vx, and ay using the iAUKF are flatter and smaller than using the other two UKF algorithms. Especially, the peak values of the estimation errors for these four vehicle states using our iAUKF algorithm are much lower than those using the UKF and SHAUKF algorithms. Table 4 summarizes the RMSE values of β, r, vx, and ay using the UKF, SHAUKF and iAUKF algorithms in fishhook maneuver I. Regarding to the UKF, the RMSE reduction rates of β, r, vx, and ay using the iAUKF are 33.1%, 85.1%, 88.7%, and 33.4%, respectively, while the RMSE reduction rates using SHAUKF are only 18.4%, 50.7%, 56.0%, and 18.3%, respectively. It can be observed that the improvements of vehicle states estimation with our proposed iAUKF are entirely greater than those of vehicle states with the SHAUKF, which demonstrates that the proposed iAUKF holds superior estimation performances.

Fishhook maneuver II
The δs under fishhook maneuver II is expressed by The output curves of δf and ax from CarSim are provided in Fig. 12. Fig. 13 and Fig. 14 show the estimated curves of β, r, vx, and ay and the corresponding estimation errors, respectively, using the UKF, SHAUKF and iAUKF algorithms. As shown in Fig. 13, it is obvious that the proposed iAUKF could generate a more accurate estimation of vehicle driving states in comparison with the UKF and SHAUKF algorithm when facing a large change of δs.
Moreover, from the subplot A, B, and C, it is seen that the responses of β, r, vx, and ay generated by the iAUKF are closer to the reference values of CarSim compared to the other two UKF algorithms.
Additionally, as shown in Fig. 14, the absolute errors of β, r, vx, and ay generated by the iAUKF are far lower than those by the UKF and SHAUKF algorithm. Meanwhile, the maximum errors of β, r, vx, and ay generated by the iAUKF are smaller than those of the other two UKF algorithms. Moreover, the estimation errors of the vehicle driving states by the iAUKF exhibit a gradually decreasing tendency, i.e., the error curves of the four vehicle states get flatter gradually, which also illustrates the superior robustness of the proposed iAUKF.

Robustness analysis of the iAUKF algorithm
To further demonstrate the robustness of the designed iAUKF algorithm when encountering the variations of process and measurement noises, the same simulations are performed using the UKF and the proposed iAUKF for three different process and measurement noises. The absolute estimation errors of each driving state by the UKF and the iAUKF are provided in Fig. 15.
Under the three different process and measurement noises, the estimation errors of β, r, vx, and ay by the iAUKF are much lower than those by the UKF algorithm. Moreover, with the gradual increase of Qz and Rz, the increase magnitudes of the estimation errors of vehicle states by the proposed iAUKF are far less than those by the UKF algorithm. Besides, the estimated error of vx exhibits a downward trend with the increase of Qz and Rz, which further illustrates the higher robustness of the proposed iAUKF compared to the UKF.
To highlight the robustness of the proposed iAUKF algorithm more visually, the RMSEs of each vehicle driving state estimated by the UKF and iAUKF under different process and measurement noises are compared in Table 6, and its graphical presentation is provided in Fig. 16.
It is clear from Table 6 that the RMSEs of vehicle driving states obtained by the iAUKF are all less than those obtained by the UKF under different process and measurement noises, and the proposed iAUKF could retain a higher global accuracy, even when the process and measurement noises changed significantly.
Furthermore, as shown in Fig. 16, except for the RMSE of r at Qz = 0, the RMSEs of each driving state obtained by the iAUKF are all smaller than those by the UKF. In addition, based on the variation tendency of the RMSEs for the four vehicle states, as the process and measurement noises increased, the increase magnitudes of these four states estimated by the proposed iAUKF are very small, and even present a decreasing appearance in some cases, whereas the RMSEs for the UKF algorithm are all increasing. This further shows that the proposed iAUKF could lower the negative effects of the process and measurement noise changes on the estimation of vehicle driving state. In future work, this designed iAUKF approach will be employed to estimate the driving states of distributed-motor-driven electric vehicles, and those estimated states will be taken as the inputs of direct yaw-moment control.

Conclusions and future work
Meanwhile, the side slip angle is determined to be the control goal to fulfill the stability control of the electric vehicle under a turning maneuver.

Appendix
The state matrices of k A , k B , k C , and k D in Eq. (9) are provided here:

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig.1 A 3-DOF vehicle dynamics model   Fig. 6 Simulated curves of δ f and a x using CarSim in sinusoidal maneuver Fig. 7 The comparison curves of β, r, v x , and a y in sinusoidal maneuver Fig. 8 The absolute errors of β, r, v x , and a y in sinusoidal maneuver Fig. 9 Simulated curves of δ f and a x using CarSim in fishhook maneuver I Fig. 10 The comparison curves of β, r, v x , and a y in fishhook maneuver I Fig. 11 The absolute errors of β, r, v x , and a y in fishhook maneuver I Fig. 12 Simulated curves of δ f and a x using CarSim in fishhook maneuver II Fig. 13 The comparison curves of β, r, v x , and a y in fishhook maneuver II