Multistage Estimators for the Distributed Drive Articulated Steering Vehicle

+e distributed drive articulated steering vehicle (DDASV) has a broad application prospect in the field of special operations. It is essential to obtain accurate vehicle states for better effect of active control. DDASV dynamic model is presented. To improve robustness, an adaptive strong tracking algorithm is applied to the singular value decomposition unscented Kalman filter (SVDUKF). Divided by yaw rate sensors and the tire models, two multistage estimators are established for DDASVs. Stable steering condition is simulated to investigate the influence on the estimated accuracy about the sensors and tire models. +e velocities and tire forces are the key parameters to be estimated. +e performance of each estimator regarding the practicability and accuracy is compared. +e results show that all estimators are practicable. However, the accuracy of the estimated velocities based on yaw rate sensors is better and the transient tire model can improve the accuracy of estimated lateral forces more effectively for the estimator established with yaw rate sensors.


Introduction
e distributed drive articulated steering vehicle (DDASV) is a kind of articulated steering vehicle, which can realize active control and energy efficiency. It has a bright prospect on mine haulage, construction, and agriculture machinery [1,2]. Accurate state estimation of the key parameters like velocities and tire forces is more beneficial for vehicle control [3]. DDASVs have a smaller turning radius and better flexibility, and state estimation of DDASVs can improve the manipulation property, especially on the steering condition.
It is mature for passenger vehicles and trailers to estimate state values such as velocities, yaw rate, and tire forces [4,5]. Electronic stability program (ESP) sensors (yaw rate, longitudinal acceleration and lateral acceleration, wheel speeds) are the key to get the measurement parameters. An improved adaptive unscented Kalman filter was developed to estimate the longitudinal and lateral velocities with ESP sensors [6]. In references [7,8], the tire forces were estimated with the measurement vector combined with ESP sensors. In reference [9], the tire forces are estimated with the measurement of yaw rate, longitudinal acceleration, and lateral acceleration only, with a difficulty that the observer gains were necessary but inaccessible. e lateral tire forces were estimated by neural network based on the simulated data from IPG CarMaker in reference [10]. It could realize the estimation without sensors but need more rigorous experimental data. For state estimation of articulated vehicles, the studies are focused on the free articulated vehicle like a tractor with a trailer. e sideslip based on Cheng and Cebon's linear single-track 5-DOF yawroll model was estimated in reference [11]. e lateral velocity and yaw rate for articulated heavy vehicle were estimated in reference [12].
Kalman filter (KF) is taken as a basis. e KF obtains the feedback gain to correct the forecast error. For the nonlinear vehicle model, unscented Kalman filter (UKF) can develop the feasibility. UKF approximates the probability density distribution of nonlinear function by sampling method which realizes Bayesian recursion. ere exist two potential problems for UKF. One is for the error covariance matrix, which may be negative on Cholesky decomposition. Here are some methods for this problem such as SR decomposition [13], singular value decomposition (SVD) [14], and adaptive noise variance [15]. SVD is more robust than SR decomposition and less complex than the adaptive noise variance method, so it is adopted in this study [16]. e other problem is to restrain the divergence of the error covariance matrix and improve the accuracy. Two methods are presented for this problem. One is taking the strong tracking filter, like the freezing K(k) method and the S method [17,18]. e other is taking the adaptive unscented Kalman filter (AUKF) [19], which takes more computing resources. Strong tracking filter methods need less CPU resources, but the optimal tracking coefficient is difficult to be confirmed.
An adaptive strong tracking filter is applied for singular value decomposition unscented Kalman filter (SVDUKF). e SVDUKF overcomes the disadvantages of traditional AUKF and strong tracking UKF.
An 8-DOF DDASV model is presented for state estimation on the steering condition. e adaptive strong tracking SVDUKF is taken as the filter algorithm. Based on the vehicle dynamic model and the filter algorithm, two multistage estimators are established to estimate the velocities and tire forces. Two multistage estimators divided by the yaw rate sensors and the tire models are presented to estimate the velocities and tire forces of the DDASV. e performances regarding the estimated accuracy are compared, and conclusions are presented in the end.

e Vehicle Model.
e dynamic model of the DDASV with longitudinal, lateral, yaw, and articulated steering motions is presented, as shown in Figure 1. In this model, the vertical, roll, and pitch motions are omitted. e driving of each wheel is independent. e longitudinal motion is expressed as follows: e lateral motion is expressed as follows: According to the rigid body kinematics, e yaw dynamic motion is expressed as follows: e articulate motion is expressed as 2 Mathematical Problems in Engineering e vertical tire forces can be expressed as l r ′ � l r cos

e Quasistatic and Transient Tire
Models. e quasistatic tire model of Dugoff is adopted in this study. e longitudinal tire force F x and lateral tire force F y can be given as where λ is given as follows: and f(λ) is given as e transient tire model to calculate the lateral tire forces can be expressed as follows: where v denotes the longitudinal velocity of the wheel, τ denotes the relaxation coefficient, and F y denotes the lateral tire force from the quasistatic tire model of Dugoff.

Singular Value Decomposition Unscented Kalman Filter.
Unlike the extended Kalman filter (EKF), the UKF approximates the probability density distribution of nonlinear functions by sampling, which realizes the Bayesian estimation too. is makes it imperative for UKF to sample and weight. With measurement error and system disturbance, the error covariance matrix might lose positive semidefinite, which makes Cholesky decomposition fail during sampling and weighting. With no limit of positive definiteness of the decomposed matrix, the SVD has higher robustness. e process of SVDUKF can be expressed via the following equations.
e system state function and measurement function can be expressed as follows: Initial conditions: Sampling and weighting based on SVD are expressed as equations (29)-(35). e error covariance matrix P can be expressed by the decomposition matrix U, S, and V: where ρ is the sigma coefficient. e weighting values W m 0 , W p 0 , W m i , W p i can be expressed as follows, and λ is a conversion coefficient, λ � α 2 (n + κ): For equation (29), S � diag(s 1 , s 2 , . . . , s n ). Generally, the error covariance matrix is a symmetric matrix, which means U � V, and the matrix eigenvalue is [s 2 1 , s 2 2 , . . . , s 2 n ]. erefore, the eigenvectors of the error covariance matrix can be substituted by UU T . For Gaussian noise distribution, when the state variable is a single variable, κ � 2. When the state variable is multivariable, κ � 3. e time update function can be expressed as follows: e measurement update function can be expressed as follows:

Adaptive Strong Tracking Algorithm.
Because of the nonlinear model and the deviation between the mathematical model and the physical vehicle model, the theoretical mean square error and the Kalman filter gain become smaller over time. When the new data are used to correct the previous step estimation for extrapolation, the added weight decreases; meanwhile, the weight of the previous data increases. is results in the increase of cumulative error, eventually leading to data saturation and divergence.
According to the closed circuit, the strong tracking algorithm takes the fading factor ξ into the propagated covariance P(k | k − 1). e fading factor increases the proportion of measurement in state estimation and suppresses filter divergence. e propagated covariance P(k | k − 1) based on the adaptive strong tracking algorithm is expressed as follows: Innovation v(k) can be expressed as follows: Covariance matrix correctional parameter ξ(k) can be calculated as follows: For equation (47), r is a reserve coefficient and r ≥ 1. e covariance matrix of system noise Q ∼ can be expressed as follows, where q i is the element of system noise matrix Q:

Initial Settings and Simulation Conditions.
To evaluate the performance and compare the estimated accuracy of different estimators, computer simulation is implemented and the estimate algorithms are established. e process is shown in Figure 3. e noise is added at the road. e vehicle parameters are set as in Table 1.
e simulation condition is steering based on a stable longitudinal velocity at 3 (m/s). e target and actual articulated steering angles are shown in Figure 4.

Results Analysis.
e estimated results of the DDASV steering on the simulation condition can be shown as follows. Maximum absolute error (MAE) and root mean square error (RMSE) are adopted to compare the Measuring without yaw rate sensors Measuring with yaw rate sensors As shown in Figure 5, the estimated errors of the longitudinal tire forces converge at about 0.15 s. e estimated   e EMSPEs of the estimated longitudinal tire forces are 0.0336%, 0.0419%, 0.0435%, and 0.0418%, which, respectively, denote the results of the front left, front right, rear left, and rear right tires. Figure 6 shows the estimated yaw rates based on the estimator B1. e EMSPEs of the estimated yaw rates are 0.0063%, 0.0543%, and 0.0507%, which, respectively, denote the results of the yaw rate of the DDASV, articulated yaw rate of the front DDASV, and articulated yaw rate of the rear DDASV. Figure 7 shows the estimated lateral tire forces based on the estimator B1. Figure 8 shows the estimated lateral tire forces based on the estimator B2.
e EMSPEs of the          estimated lateral tire forces based on the estimator B1 are 0.0446%, 0.0369%, 0.0223%, and 0.0226%, which, respectively, denote the result of the front left, front right, rear left, and rear right tires. Same as above, the EMSPEs of the estimated lateral tire forces based on the estimator B2 are 0.0446%, 0.0379%, 0.0258%, and 0.0257%. Figure 9 shows the estimated velocities based on the estimator C1. Figure 10 shows the estimated velocities based on the estimator C2. e EMSPEs of the estimated velocities based on the estimator C1 are 0.0008% and 0.0629%, which, respectively, denote the results of the longitudinal and lateral velocities. Same as above, the EMSPEs of the estimated velocities based on the estimator C2 are 0.0003% and 0.0066%. Table 2 summarizes the MAE and RMSE for velocities estimated by estimators C1 and C2. e results demonstrate that the estimated accuracy of the estimator C2 is better than that of the estimator C1. e average improvements about MAE and RMSE from the estimators C1 to C2 are 83.84% and 80.39%. Figure 11 shows the estimated lateral tire forces based on the estimator D1. Figure 12 shows the estimated lateral tire forces based on the estimator D2. e EMSPEs of the estimated lateral tire forces based on the estimator D1 are 0.0103%, 0.0044%, 0.0293%, and 0.0210%, which, respectively, denote the results of the front left, front right, rear left, and rear right tires. Same as above, e EMSPEs of the estimated lateral tire forces based on the estimator D2 are 0.0446%, 0.0042%, 0.0059%, and 0.0060%.

Conclusions
A comparative study about how to estimate the velocities and tire forces of DDAVS is presented. Two factors are compared by multistage estimators, respectively. One is yaw rate sensors, and the other is the tire model.
Without the ESP sensors, the measurement sensor schemes need to be explored. In this paper, two kinds of multistage estimators were established: one was equipped with acceleration sensors only and the other with the additional three yaw rate sensors. For each multistage estimator, this paper compared the estimator based on the quasistatic tire model and transient tire model.
Regarding the EMSPEs, the estimate results for velocities and tire forces of the DDASV based on different multistage estimators differentiated by the yaw sensors and the tire models are all favourable to a certain extent, while the estimated accuracy of different estimators is discrepant.
(1) For the estimated velocities, the yaw rate sensors can improve the estimated accuracy. (2) For the estimated lateral tire forces based on the quasistatic tire model, the performance of the estimator with the yaw rate sensors is not better than that of the estimator without the yaw rate sensors regarding the estimated accuracy. But the improvement is significant if it is based on the transient tire model. (3) For the estimated lateral tire forces, the estimated accuracy based on the transient tire model is better than that based on the quasistatic tire model, whether the estimator has yaw rate sensors or not. (4) For ameliorating the estimated accuracy of the lateral tire forces, the improvement by using the transient tire model is more effective than equipping yaw rate sensors. e simulation in this paper is based on the conventional pavement. Furthermore, the simulation elements can be extended to off-road, which can enhance the availability of the estimator. Meanwhile, more degrees of freedom of DDASV need to be taken into consideration, like rolling movement and pitch movement.

Nomenclature m:
Gross mass R xj : Longitudinal tire resistance, j � 1, 2, 3, 4, represents the position as front left, front right, rear left, and rear right, respectively F xj : Longitudinal tire force, j � 1, 2, 3, 4 represents the position as front left, front right, rear left, and rear right, respectively F yj : Lateral tire force, j � 1, 2, 3, 4 represents the position as front left, front right, rear left, and rear right, respectively δ: Swing angle a x : Longitudinal acceleration m f : Mass of the front part vehicle v ij : Front and rear vehicle velocity, ij � xf represents the longitudinal velocity of front vehicle, ij � yf represents the lateral velocity of front vehicle, ij � xr represents the longitudinal velocity of rear vehicle, and ij � yr represents the lateral velocity of rear vehicle φ: Yaw rate φ f : Yaw rate of the front part vehicle φ r : Yaw rate of the rear part vehicle m r : Mass of the rear part vehicle a y : Lateral acceleration v x : Longitudinal velocity v y : Lateral velocity l mf : Distance from articulated point to the center of front vehicle gravity l mr : Distance from articulated point to rear axle I z : Vehicle rotational inertia about z-axis