Comparative Study of Two Dynamics-Model-Based Estimation Algorithms for Distributed Drive Electric Vehicles

Featured Application

The proposed estimators can be applied in many vehicle active safety systems. They are capable of estimating vehicle states that are normally not measurable or that are too complicated to measure directly using additional expensive sensors.

Abstract

The effect of vehicle active safety systems is subject to the accurate knowledge of vehicle states. Therefore, it is of great importance to develop a precise and robust estimation approach so as to deal with nonlinear vehicle dynamics systems. In this paper, a planar vehicle model with a simplified tire model is established first. Two advanced model-based estimation algorithms, an unscented Kalman filter and a moving horizon estimation, are developed for distributed drive electric vehicles. Using the proposed algorithms, vehicle longitudinal velocity, lateral velocity, yaw rate as well as lateral tire forces are estimated based on information fusion of standard sensors in today’s typical vehicle and feedback signals from electric motors. Computer simulations are implemented in the environment of CarSim combined with Matlab/Simulink. The performance of both estimators regarding convergence, accuracy, and robustness against an incorrect initial estimate of longitudinal velocity is compared in detail. The comparison results demonstrate that both estimation approaches have favourable coincidence with the corresponding reference values, while the moving horizon estimation is more accurate and robust, and owns faster convergence.

1. Introduction

Safety and energy conservation are two eternal themes for automotive design [1]. Over the past few decades, active safety control has become one of the most effective accident avoidance technologies [2]. Anti-lock braking system (ABS), traction control system (TCS) and electronic stability program (ESP), etc., all key components of the active safety control system [3], rely heavily on accurate knowledge of vehicle states such as vehicle velocity, yaw rate, side-slip angle, tire forces and so on. Furthermore, vehicle state information is crucial not only for safety but also for the energy management system of electric vehicles [4]. Generally speaking, vehicles with standard stability control systems are normally equipped with a steering wheel angle sensor and inertial sensors (INS, including longitudinal and lateral acceleration sensor, yaw rate sensor). As the research subject in this article, distributed drive electric vehicle (DDEV) is an electric vehicle with four independently controllable motorized wheels. This kind of power configuration provides us with many remarkable advantages, such as short transmission chain and high transmission efficiency, precise torque generation, and easy implementation of torque vectoring. More importantly, for our study, DDEV is able to provide us with the accurate torque and rotational speed information of each wheel from the feedback signals of four individual motors [5]. However, the other vehicle dynamics states are still difficult to measure or cannot be measured economically [6].

Notable attempts have been made to work out different estimation approaches. Currently, existing state estimation algorithms are classified into three categories: “non-model-based”, “kinematic-model-based” (KMB) and “dynamic-model-based” (DMB) approaches. “Non-model-based” approaches mainly include fuzzy logic, support vector machine and artificial neural network, etc. [7,8,9,10], where a vehicle is regarded as a black box and the nonlinear relationship between inputs and outputs is mapped based on a mass of training data. The accuracy of these methods strongly depends on the quality and quantity of the training data. Meanwhile, without vehicle models, this method is difficult to give out a convincing mathematical explanation of the mapping relationship [11]. As for the KMB approach, it does not contain the physical parameters of the vehicle, and therefore this kinematic approach is not affected by parametric uncertainties. Hac and Simpson [12] developed a preliminary estimation of the yaw rate using kinematic relationships, and subsequently fed this initial estimate into a nonlinear observer to generate the final estimate of yaw rate. Farrelly and Wellstead [13] proposed a vehicle lateral velocity estimator using both physical and kinematic modeling. As for the kinematic part, the estimator has been shown to provide satisfactory performance. However, the drawbacks of the KMB approaches are also apparent. This method only works when the yaw rate is non-zero, because the kinematic model is unobservable when the yaw rate is zero. Additionally, the estimates produced by the kinematic approach are more noisy than those produced by the dynamic-model-based approach [13].

Unlike the aforementioned methods, DMB approaches include specific and detailed vehicle mathematical models as well as physical parameters such as mass, yaw moment inertia, COG (center of gravity) position and so on. The DMB estimation design can be carried out using sliding mode observer [14,15,16], Luenberger observer [17,18], Kalman filter (KF) and its extensions, and moving horizon estimation (MHE), etc. We will provide a more detailed review of the latter two approaches.

The Kalman filter [19] is widely applied for linear system estimation under the assumption of Gaussian-distributed state and measurement noise. In the nonlinear case of automobile field, the extended Kalman filter (EKF) [20,21,22] and unscented Kalman filter (UKF) [23,24,25,26] take up a large percentage. Pengov [20] compared EKF and a higher gain observer, the results of which demonstrated that EKF has more robustness and accuracy. Wenzel et al. [21] proposed the dual EKF, which has two Kalman filters running in parallel to estimate vehicle states and parameters simultaneously. However, the main drawback of the EKF is Jacobian matrices calculation, which requires costly computation. Moreover, EKF only employs the first-order Taylor expansion on a nonlinear system, which may lead to great error or even divergence of the filter if the model is seriously nonlinear. Addressing these issues, the UKF utilizes a deterministic sampling technique known as the unscented transform (UT) to pick a minimal set of sample points (called sigma points) around the mean, which is a derivative-free alternative to EKF and avoids the expensive update of the Jacobian matrix on each iteration. Meanwhile, UKF is able to achieve higher-order Taylor series expansion accuracy [27,28]. Thus, UKF should be more suitable for vehicle state estimation application in consideration of the high nonlinearity of vehicle dynamics, particularly during critical conditions. Hamann and Hedrich [23] developed a robust method to estimate the tire forces using UKF. Simulation results demonstrated a high convergence rate and good stability properties of the UKF estimator. Based on the piecewise linear tire model, Ren et al. [24] achieved vehicle state estimation with UKF. Antonov et al. [25] constructed an UKF observer for the vehicle state using an advanced vertical tire load calculation method. Apart from the Kalman filter methods, MHE is another powerful method able to provide a good solution to nonlinear estimation systems. At each sampling time, MHE estimates the states or parameters by minimizing a cost function over the previous finite time horizon. In addition, constraints can be added to this optimization problem in a very natural way, which makes it possible for MHE to handle constrained processes very well [29]. Zanon [30] estimated the friction coefficient in autonomous driving using MHE. Kraus [31] proposed an MHE-nonlinear model predictive control (NMPC) framework to control field vehicles using an adaptive nonlinear kinematic model.

In this study, to further explore the potential application of UKF and MHE, we compare their performance regarding to convergence, accuracy as well as robustness, for the DDEV state estimation. First of all, a 3-DOFs vehicle model is presented serving for the state estimation. The longitudinal tire force is calculated based on the wheel rotational equation, while the lateral tire force is obtained from a semi-empirical tire model. After that, the estimator designing based on UKF and MHE is described in detail. Simulations are carried out using a 27-DOFs high-fidelity CarSim vehicle model equipped with vehicle stability controller (VSC), which is believed to be accurate enough to inspect the precision of the presented approaches. Furthermore, the accidental error of estimated results is taken into account to test the robustness of the two estimation algorithms. Finally, simulation results are analyzed in terms of tracking accuracy and convergence behavior.

The rest of this paper proceeds as follows. Section 2 gives the measurement, control and state vectors of a DDEV followed by a detailed modeling of a 3-DOFs vehicle state estimation model. Estimation algorithms and the implemented procedures are described in Section 3. The computer simulation is conducted under a double lane change (DLC) maneuver, the results of which are compared and analyzed in Section 4. Section 5 concludes the paper.

2. Vehicle Modeling for State Estimation

A planar two-track vehicle model is presented in this section. It is a 3-DOFs vehicle model, which contains the longitudinal velocity u, the lateral velocity v, and the yaw rate r. Figure 1 shows the vehicle model and coordinate systems. The wheel positions are numbered with the subscript ij = fl,rl,fr,rr denoting front left, rear left, front right and rear right, respectively.

The following assumptions are made before the modeling work:

(a)

Vehicle is moving on a flat horizontal plane;

(b)

Vertical, roll and pitch dynamics are omitted;

(c)

longitudinal acceleration, lateral acceleration and the yaw rate is measured with white Gauss noise;

(d)

INS sensors are mounted on the vehicle COG.

2.1. Measurement, Control Input, and State Vectors

As for the DDEV equipped with the standard active safety sensors suit, the signals that can be directly measured consist of the steering wheel angle ${\delta }_{sw}$, the longitudinal acceleration a_x, the lateral acceleration a_y, yaw rate r, torque output T_{m_ij} and rotational speed w_{m_ij} of each motor.

As the reduction ratio i between motor and wheel is known, the wheel speed is simply obtained,

$${\omega }_{ij}={\omega }_{m\_ij}·i.$$

(1)

Therefore, the measurement vector is given as

$$\mathit{y}={\left[\begin{array}{ccccccc}{a}_x& a_y& r& {\omega }_{fl}& {\omega }_{rl}& {\omega }_{fr}& {\omega }_{rr}\end{array}\right]}^T.$$

Additionally, the longitudinal tire force can be calculated by the rotational dynamic equation instead of complicated tire models, which is shown below:

$$F_{t\_ij}=\frac{T_{m\_ij}·i-J_w·{\dot{\omega }}_{ij}}{R},ij=fl,fr,rl,rr,$$

(2)

where J_w is the wheel rotational inertia; R is the tire radius and in this study it is assumed to be a constant.

These four forces and steering wheel angle form the control input vector as follows:

$$\mathit{u}={\left[\begin{array}{ccccc}{\delta }_{sw}& F_{t\_fl}& F_{t\_fr}& F_{t\_rl}& F_{t\_rr}\end{array}\right]}^T.$$

Finally, the vehicle states that need to be estimated include the longitudinal velocity, the lateral velocity, yaw rate and the lateral force of each tire.

$$\mathit{x}={\left[\begin{array}{ccccccc}u& v& r& F_{s_{fl}}& F_{s_{fr}}& F_{s_{rl}}& F_{s_{rr}}\end{array}\right]}^T$$

2.2. Planar Vehicle Model

According to Newton’s second law, the 3-DOFs vehicle body motion equations can be expressed as follows. Longitudinal and lateral motions along the x and y-axis:

$$m·a_x={\displaystyle \sum }{F}_{x\_ij}-\frac{1}{2}{C}_{d}A\rho u^2$$

(3)

$$m·a_y=\sum F_{y\_ij}.$$

(4)

Rotational motions of yaw about z-axis:

$$I_z·\dot{r}=a\left(F_{y\_fl}+F_{y\_fr}\right)-b\left(F_{y\_rl}+F_{y\_rr}\right)-T/2\left(F_{x\_fl}+F_{x\_rl}\right)+T/2\left(F_{x\_fr}+F_{x\_rr}\right),$$

(5)

where a_x is the longitudinal acceleration; a_y is the lateral acceleration, r is the yaw rate and T is the vehicle wheel track; C_d, A, and ρ denote the air resistance coefficient, the frontal projected area, and the air density, respectively. Moreover, the acceleration terms are defined as

$$a_x=\dot{u}-vr$$

(6)

$$a_y=\dot{v}+ur.$$

(7)

In Equations (3)–(5), $F_{x\_ij}$ and $F_{y\_ij}$ represent the resultant force of the longitudinal and lateral tire forces along the x and y axis in the vehicle coordinate system, which could be expressed by the following equations:

$$F_{x\_ij}=F_{t\_ij}·cos{\delta }_{T\_ij}-F_{s\_ij}·sin{\delta }_{T\_ij}$$

(8)

$$F_{y\_ij}=F_{t\_ij}·sin{\delta }_{T\_ij}+F_{s\_ij}·cos{\delta }_{T\_ij}.$$

(9)

As seen in Figure 1, $F_{t\_ij}$ and $F_{s\_ij}$ are the longitudinal and lateral tire force in the wheel coordinate system. Without regard to the roll motion, the steering angle of each wheel is simplified as

$$\{\begin{array}{c}{\delta }_{T\_fl}={\delta }_{T\_fr}={\delta }_{sw}/i_s\\ {\delta }_{T\_rl}={\delta }_{T\_rr}=0\end{array},$$

(10)

where $i_s$ is the transmission ratio from the hand wheel to front wheels.

2.3. Load Transfer

Due to both longitudinal and lateral acceleration, quasi-static load transfer is formulated in the vehicle model. The normal load expression for each wheel is written as

$$F_{z\_fl}=mg\frac{b}{2l}-m{a}_x\frac{h}{2l}-m{a}_y\frac{b}{l}\frac{h}{T}$$

(11)

$$F_{z\_rl}=mg\frac{b}{2l}+m{a}_x\frac{h}{2l}-m{a}_y\frac{a}{l}\frac{h}{T}$$

(12)

$$F_{z\_fr}=mg\frac{b}{2l}-m{a}_x\frac{h}{2l}+m{a}_y\frac{b}{l}\frac{h}{T}$$

(13)

$$F_{z\_rr}=mg\frac{b}{2l}+m{a}_x\frac{h}{2l}+m{a}_y\frac{a}{l}\frac{h}{T}.$$

(14)

2.4. Tire Force Calculation

Thanks to the advantages of the DDEV, the longitudinal tire force can be calculated based on the wheel dynamic in Equation (2).

Therefore, in this study, the well-known semi-empirical “Pacejka 2002” tire model [32] is only employed for lateral tire force calculation, which is helpful to reduce the computational effort. Lateral tire force is formulated by “Pacejka 2002” in two steps. Firstly, for the pure slip condition [32]:

$$F_{s0}=D_{y}sin\left\{C_{y}arctan\left\{B_y\alpha -E_y\left[B_y\alpha -arctan\left(B_y\alpha \right)\right]\right\}\right\}+S_{Vy}.$$

(15)

Subsequently, for the combined slip condition [32]:

$$F_s=G_y{F}_{s0}+S_{Vyk},$$

(16)

where G_y is the weighting function, which always has a value between 0 and 1. The lateral and longitudinal slip ratio of each tire are given as

$${\alpha }_{ij}={\delta }_{T\_ij}-arctan\left(\frac{v+ar}{u\pm \frac{1}{2}Tr}\right),ij=fl,fr$$

(17)

$${\alpha }_{ij}=-arctan\left(\frac{v-br}{u\pm \frac{1}{2}Tr}\right),ij=rl,rr$$

(18)

$${\lambda }_{ij}=-\frac{u_{w\_ij}-R·{\omega }_{ij}}{u_{w\_ij}},ij=fl,fr,rl,rr.$$

(19)

The wheel center speed u_{w_ij} is given by

$$u_{w\_ij}=\left(u\pm \frac{1}{2}Tr\right)cos{\delta }_{T\_ij}+\left(v+ar\right)sin{\delta }_{T\_ij},ij=fl,fr$$

(20)

$$u_{w\_ij}=u\pm \frac{1}{2}Tr,ij=rl,rr.$$

(21)

For the sake of simplicity, the wheel camber is neglected as a low-effect parameter.

2.5. System Discretization

Discretization must be accomplished before the observer design. The forward Euler difference method is applied to discretize the continuous system described in Section 2.2. Then the nonlinear estimation system in the form of discretization is rewritten as

$$\begin{gathered} \mathit{x}\left(k+1\right)=f\left(\mathit{x}\left(k\right),\mathit{u}\left(k\right)\right)+\mathit{w}\left(k\right) \\ \mathit{y}\left(k\right)=h\left(\mathit{x}\left(k\right),\mathit{u}\left(k\right)\right)+\mathit{v}\left(k\right), \end{gathered}$$

(22)

where $\mathit{x}\left(k\right)$, $\mathit{y}\left(k\right)$, and $\mathit{u}\left(k\right)$ are the system state, measurement input and control input, respectively, at the k − 1 step. Besides, w and v are the process noise and measurement noise vectors.

3. Estimation Algorithms Design for Vehicle State Estimation

Two closed-loop estimation algorithms, UKF and MHE, are designed to implement vehicle state estimation.

3.1. Unscented Kalman Filter

Addressing the main drawbacks of the EKF, Jacobian matrices calculation and the first order Taylor expansion on nonlinear system, the UKF utilizes a deterministic sampling technique known as the unscented transform (UT) to pick a minimal set of sample points (called sigma points) around the mean, which is a derivative-free alternative to EKF and meanwhile avoids the expensive update of the Jacobian matrix on each iteration [33]. Additionally, UKF achieves higher-order Taylor series expansion accuracy [27,28]. Considering a nonlinear time-discrete y = g(x) with mean $\overline{\mathit{x}}$ and covariance ${\mathit{P}}_{\mathit{x}}$, to calculate the statistics of y, 2L + 1 sigma points ${\chi }_i$ with its corresponding weighting factors is formulated via the following equations:

$$\{\begin{array}{c}{\chi }_0=\overline{\mathit{x}}i=0\\ {\chi }_i=\overline{\mathit{x}}+{\sqrt{\left(L+\lambda \right){\mathit{P}}_{\mathit{x}}}}_{i}i=1,\dots \dots ,L\\ {\chi }_i=\overline{\mathit{x}}-{\sqrt{\left(L+\lambda \right){\mathit{P}}_{\mathit{x}}}}_{i-L}i=L+1,\dots \dots ,2L\\ W_0^{\left(m\right)}=\lambda /\left(L+\lambda \right)\\ W_0^{\left(c\right)}=\lambda /\left(L+\lambda \right)+1-{\alpha }^2+\beta \\ W_i^{\left(m\right)}=W_i^{\left(c\right)}=0.5/\left(L+\lambda \right)i=1,2,\dots ,2L\end{array}$$

(23)

where L is the dimension of x; $\lambda ={\alpha }^2\left(L+\kappa \right)-L$ is a scaling parameter. $\alpha$ determines the spread of the sigma points around $\overline{x}$ and is usually set to a small positive value (e.g., ${10}^{-3}$). $\kappa$ is a secondary scaling parameter that is normally set to a non-negative value to ensure that the covariance matrix is positive definite. $\beta$ is used to incorporate prior knowledge of the distribution of x, which affects the weighting of the zeroth sigma point for the calculation of the covariance. For Gaussian distribution, $\beta =2$ is optimal [28]. These sigma vectors are propagated through a nonlinear function, $y_i=g\left({\chi }_i\right)$, i = 0, 1, …, 2L. The mean and covariance of y are estimated using the weighted sample mean and covariance of the posterior sigma points as follows:

$$\overline{\mathit{y}}={\displaystyle \sum }_{i=0}^{2L}{W}_i^{\left(m\right)}{\mathit{y}}_{\mathit{i}}$$

(24)

$${\mathit{P}}_{\mathit{y}}={\sum }_{i=0}^{2L}{W}_i^{\left(c\right)}\left({\mathit{y}}_{\mathit{i}}-\overline{\mathit{y}}\right){\left({\mathit{y}}_{\mathit{i}}-\overline{\mathit{y}}\right)}^T.$$

(25)

On the basis of unscented transform, the main steps of UKF are as in Figure 2 and put forward in the following:

(a)

Initialize vehicle state and covariance matrix at time step k = 0 with

$${\widehat{\mathit{x}}}_0=E\left[{\mathit{x}}_{\mathbf{0}}\right]$$

(26)

$${\mathit{P}}_{\mathit{x}\mathbf{0}}=E\left[\left({\mathit{x}}_{\mathbf{0}}-{\widehat{\mathit{x}}}_{\mathbf{0}}\right)·{\left({\mathit{x}}_{\mathbf{0}}-{\widehat{\mathit{x}}}_0\right)}^T\right].$$

(27)

(b)

For time step k = 1, 2…, calculate sigma points in sigma vector:

$$\mathit{\chi }\left(k-1\right)={\left[\begin{array}{c}\widehat{\mathit{x}}\left(k-1\right)\\ \widehat{\mathit{x}}\left(k-1\right)+\sqrt{\left(L+\lambda \right)\mathit{P}\left(k-1\right)}\\ \widehat{\mathit{x}}\left(k-1\right)-\sqrt{\left(L+\lambda \right)\mathit{P}\left(k-1\right)}\end{array}\right]}^T.$$

(28)

(c)

Time update

• Propagate the sigma points through Equation (22):

  $$\mathit{\chi }\left(k|k-1\right)=f\left(\mathit{\chi }\left(k-1\right),\mathit{u}\left(k-1\right),\mathit{w}\left(k-1\right)\right).$$

  (29)
• The propagated mean calculation:

  $$\widehat{\mathit{x}}\left(k|k-1\right)={\sum }_{i=0}^{2L}{W}_i^{\left(m\right)}·{\mathit{\chi }}_{\mathit{i}}\left(k|k-1\right).$$

  (30)
• The propagated covariance calculation:

  $${\mathit{P}}_{\mathit{x}}\left(k|k-1\right)={\sum }_{i=0}^{2L}{W}_i^{\left(c\right)}·[{\mathit{\chi }}_{\mathit{i}}\left(k|k-1\right)-\widehat{\mathit{x}}\left(k|k-1\right)]·{[{\mathit{\chi }}_{\mathit{i}}(k|k-1)-\widehat{\mathit{x}}(k|k-1)]}^T+{\mathit{Q}}_{\mathit{k}}.$$

  (31)

(d)

Measurement update

• Propagate sigma points through measurement function:

  $$\mathit{y}\left(k|k-1\right)=h\left(\mathit{\chi }\left(k-1\right),\mathit{u}\left(k-1\right),\mathit{v}\left(k-1\right)\right)$$

  (32)
• The propagated mean calculation:

  $$\widehat{\mathit{y}}\left(k|k-1\right)={\sum }_{i=0}^{2L}{W}_i^{\left(m\right)}·{\mathit{y}}_{\mathit{i}}\left(k|k-1\right).$$

  (33)
• The propagated covariance and the Kalman gain calculation:

  $${\mathit{P}}_{\mathit{y}}\left(k|k-1\right)={\displaystyle \sum }_{i=0}^{2n}{W}_i^{\left(c\right)}·[{\mathit{y}}_i\left(k|k-1\right)-\widehat{\mathit{y}}\left(k|k-1\right)]·{[{\mathit{y}}_{\mathit{i}}(k|k-1)-\widehat{\mathit{y}}(k|k-1)]}^T+{\mathit{R}}_{\mathit{k}}$$

  (34)

  $${\mathit{P}}_{\mathit{x}\mathit{y}}\left(k|k-1\right)={\displaystyle \sum }_{i=0}^{2n}{W}_i^{\left(c\right)}·[{\mathit{\chi }}_i\left(k|k-1\right)-\widehat{\mathit{x}}\left(k|k-1\right)]·{[{\mathit{y}}_i(k|k-1)-\widehat{\mathit{y}}(k|k-1)]}^T$$

  (35)

  $$\mathit{K}\left(k\right)={\mathit{P}}_{\mathit{x}\mathit{y}}\left(k|k-1\right)·{\mathit{P}}_{\mathit{y}\mathit{y}}{\left(k|k-1\right)}^{-1},$$

  (36)

  where K(k) is the Kalman gain matrix.
• Update the vehicle state estimation and state covariance:

  $$\widehat{\mathit{x}}\left(k|k\right)=\widehat{\mathit{x}}\left(k|k-1\right)+\mathit{K}\left(k\right)·\left[\mathit{y}\left(k\right)-\widehat{\mathit{y}}\left(k|k-1\right)\right]$$

  (37)

  $${\mathit{P}}_{\mathit{x}\mathit{x}}\left(k|k\right)={\mathit{P}}_{\mathit{x}}\left(k|k-1\right)-\mathit{K}\left(k\right)·{\mathit{P}}_{\mathit{y}}·\mathit{K}{\left(k\right)}^T.$$

  (38)

3.2. Moving Horizon Estimation

The MHE algorithm is able to solve the estimation problem of constrained linear or nonlinear systems, using a horizon covering the prior measurement information. In the same manner as UKF, MHE is also based on the least-squares objective function. UKF employs sigma point sampling to estimate the covariance matrices within linear update derived from the objective function via the maximum likelihood estimation [34], while MHE solves the objective function as a mathematical programming problem. Additionally, both MHE Sequential Quadratic Programming (SQP) and UKF algorithm, employ second-order estimates at each iteration; however, the SQP solver continues to iterate until the convergence tolerance is satisfied, which leads to a robust means of guaranteeing local optimality without tuning.

From the Bayesian theory point of view, the constrained state estimation problem can be formulated as the solution of a full information estimation (FIE) problem [35]. Therefore, the FIE objective function for Equation (22) is written as:

$$\underset{{\mathit{x}}_{\mathbf{0}},\left\{{\mathit{w}}_{\mathit{k}}{,}_{k=0}^{T-1}\right\}}{\min }{\Phi }_T\left\{{\mathit{x}}_{\mathbf{0}},\left\{{\mathit{w}}_{\mathit{k}}\right\}\right\}=\underset{{\mathit{x}}_{\mathbf{0}},\left\{{\mathit{w}}_{\mathit{k}}{,}_{k=0}^{T-1}\right\}}{\min }\left\{{\sum }_{k=0}^{T-1}\parallel {\mathit{w}}_{\mathit{k}}{\parallel }_{{\mathit{Q}}^{-1}}^2+{\sum }_{k=0}^T\parallel {\mathit{v}}_{\mathit{k}}{\parallel }_{{\mathit{R}}^{-\mathbf{1}}}^2+\parallel {\mathit{x}}_{\mathbf{0}}-{\tilde{\mathit{x}}}_{\mathbf{0}}{\parallel }_{{{\mathit{P}}_{\mathbf{0}}}^{-\mathbf{1}}}^2\right\},$$

(39)

subject to

$$\begin{gathered} \mathit{x}\left(k+1\right)=f\left(\mathit{x}\left(k\right),\mathit{u}\left(k\right)\right)+\mathit{w}\left(k\right) \\ \mathit{y}\left(k\right)=h\left(\mathit{x}\left(k\right),\mathit{u}\left(k\right)\right)+\mathit{v}\left(k\right), \end{gathered}$$

where $\parallel \ast {\parallel }_{\mathit{A}}^2={\ast }^T\mathit{A}\ast$; ${\left\{{\mathit{w}}_{\mathit{k}}\right\}}_{k=0}^{k=T-1}$ is the state disturbance sequence from time k = 0 to k = T − 1; and ${\tilde{\mathit{x}}}_{\mathbf{0}}$ is the initial estimate of vehicle states with corresponding covariance matrix ${\mathit{P}}_{\mathbf{0}}$.

Additionally, it is worth noting that, in practice, the vehicle speed should be less than $\max \left({\omega }_{ij}\right)·R$ on driving conditions and greater than $\min \left({\omega }_{ij}\right)·R$ on braking conditions. However, measurement noise or some other reasons may occasionally result in accidental error, which affects the accuracy of the subsequent estimation. Thus, the following additional constraints for the vehicle longitudinal velocity are given and applied to reduce this effect:

$$u=\{\begin{array}{c}\le \max \left({\omega }_{ij}\right)·R,\mathrm{driving}\mathrm{conditions}\\ \ge \min \left({\omega }_{ij}\right)·R,\mathrm{braking}\mathrm{conditions}\end{array}.$$

(40)

However, real-time implementations of FIE are computationally infeasible due to the infinite growth of the number of sampling points. To make the objective function tractable, the estimation size needs to be bounded. One common strategy to estimate the ${\{{\mathit{x}}_{\mathit{k}},{\mathit{w}}_{\mathit{k}}\}}_{k=T-N}^{T-1}$ instead of the full-state sequence ${\{{\mathit{x}}_{\mathit{k}},{\mathit{w}}_{\mathit{k}}\}}_{k=0}^{T-1}$, where N is the fixed time horizon of Equation (39), just like a moving “sampling window”. Therefore, Equation (39) is rewritten as,

$$\begin{gathered} \underset{{\mathit{x}}_{T-N},\left\{{\mathit{w}}_{\mathit{k}}{,}_{k=T-N}^{T-1}\right\}}{\min }{\Phi }_T\left\{{\mathit{x}}_{\mathit{T}-\mathit{N}},\left\{{\mathit{w}}_{\mathit{k}}\right\}\right\}=\parallel {\mathit{x}}_{\mathbf{0}}-{\widehat{\mathit{x}}}_{\mathbf{0}}{\parallel }_{{\mathit{P}}_{\mathbf{0}}{}^{-1}}^2+{\sum }_{k=0}^{T-N-1}\parallel {\mathit{w}}_{\mathit{k}}{\parallel }_{{\mathit{Q}}^{-1}}^2+{\sum }_{k=0}^{T-N-1}\parallel {\mathit{v}}_{\mathit{k}}{\parallel }_{{\mathit{R}}^{-1}}^2+ \\ {\sum }_{k=T-N}^{T-1}\parallel {\mathit{w}}_{\mathit{k}}{\parallel }_{{\mathit{Q}}^{-1}}^2+{\sum }_{k=T-N}^T\parallel {\mathit{v}}_{\mathit{k}}{\parallel }_{R^{-1}}^2={\Psi }_{T-N}+{\sum }_{k=T-N}^{T-1}\parallel {\mathit{w}}_{\mathit{k}}{\parallel }_{{\mathit{Q}}^{-1}}^2+{\sum }_{k=T-N}^T\parallel {\mathit{v}}_{\mathit{k}}{\parallel }_{{\mathit{R}}^{-1}}^2, \end{gathered}$$

(41)

where ${\Psi }_{T-N}$ is the arrival cost that incorporates estimate information prior to the horizon, namely from time k = 0 to time k = T − N − 1. Therefore, the arrival cost is indeed necessary for the application of MHE. For nonlinear systems like Equation (22), however, general analytical expressions of the arrival cost are unavailable. Methods for formulating the arrival cost include EKF, UKF, or even discarding the arrival cost by taking it as a constant value. In this study, an UKF-based arrival cost calculation approach [36] is applied. Differently from the UKF described in Section 3.1, for the constrained problem, the selection of sigma points takes the parameter constraints into account so that none of the selected sigma points violate the boundaries of the state variables:

$$\begin{gathered} W_i^{\left[m\right]}=\{\begin{array}{c}\frac{\lambda }{L+\lambda },i=0\\ p{r}_i+q,i=1,2,\dots .,2L\end{array} \\ {W}_0^{\left[c\right]}=W_0^{\left[m\right]}+\left(1-{\alpha }^2+\beta \right) \\ {W}_i^{\left[c\right]}={Wi}^{\left[m\right]}i=1,2,\dots ,2L, \end{gathered}$$

(42)

where $p=\frac{2\lambda -1}{2\left(L+\lambda \right)\left(S_r-\left(2L+1\right)\left(\sqrt{L+\lambda }\right)\right)}$, $q=\frac{1}{2\left(L+\lambda \right)}-\frac{2\lambda -1}{2\left(L+\lambda \right)(S_r-\left(2L+1\right)\left(\sqrt{L+\lambda }\right)}$ and $S_r=\sum _{i=1}^{2L}{r}_i$. r_i is the step size in each direction of the set of sigma points with constraints, which is calculated as follows [36]:

$$\begin{gathered} r_{k,i}=\min \left(\sqrt{L+\lambda },\frac{x_{U,i}-{\widehat{x}}_{k|k,i}}{S_{k,i}},\frac{x_{L,i}-{\widehat{x}}_{k|k,i}}{S_{k,i}}\right) \\ {S}_{k,i}=\pm {(\sqrt{P_{k|k}})}_i,i=1,\dots ,L, \end{gathered}$$

(43)

where $x_{U,i}$, $x_{L,i}$ are the upper and lower bounds in the $S_{k,i}$ direction.

As the sigma points are obtained, ${\mathit{K}}_{\mathit{T}-\mathit{N}},{\tilde{\mathit{x}}}_{\mathit{T}-\mathit{N}}$ and ${\mathit{P}}_{\mathit{T}-\mathit{N}}$ can be updated via the UKF equation described in Section 3.1.

The arrival cost can be calculated as follows:

$${\Psi }_{T-N}=\parallel {\mathit{x}}_{\mathit{T}-\mathit{N}}-{\tilde{\mathit{x}}}_{\mathit{T}-\mathit{N}}{\parallel }_{{{\mathit{P}}_{\mathit{T}-\mathit{N}}}^{-1}}^2.$$

Finally, the whole process of MHE-based state estimation algorithm is shown in Figure 3 and explained in the form of pseudocode in Algorithm 1.

Algorithm 1. Pseudocode of moving horizon estimation algorithm
1.	Initialization: set the initial values for ${\tilde{\mathit{x}}}_0,{\mathit{P}}_0$ and the horizon length N.
2.	Prepare for estimation: increment $T⟶T+1$ and obtain the current measurement input $\mathit{y}\left(T\right)$ and control input $\mathit{u}\left(T\right)$.
3.	Estimation: if $T\le N$
	Solve Equation (39) as full information estimation;
	else
	Compute the UKF gain matrix ${\mathit{K}}_{\mathit{T}-\mathit{N}}$ and update ${\tilde{\mathit{x}}}_{\mathit{T}-\mathit{N}},{\mathit{P}}_{\mathit{T}-\mathit{N}}$ via UKF;
	Compute the arrival cost ${\Psi }_{T-N}$;
	Solve Equation (41) with the obtained arrival cost ${\Psi }_{T-N}$.
	end
4.	Results output: compute states ${\widehat{\mathit{x}}}_{\mathit{k}},{\{}_{k=T-N+1}^{T-1}\}$ in the horizon, ${\widehat{\mathit{x}}}_{\mathit{T}}=f\left({\widehat{\mathit{x}}}_{\mathit{T}-\mathbf{1}},{\mathit{u}}_{\mathit{T}-\mathbf{1}}\right)+{\widehat{\mathit{w}}}_{\mathit{T}-\mathbf{1}}$
5.	End of estimation: go to Step 2.

4. Simulation Results and Analysis

In order to evaluate and compare the presented estimation algorithms, computer simulations are implemented in the environment of CarSim combined with Matlab/Simulink, as shown in Figure 4. CarSim is commercial vehicle dynamic software. The vehicle model in CarSim contains a steering system, tires, and driver model, and is extensively validated and correlated to real-world results, as measured and observed by many automotive OEMs around the world. Therefore, in this study, the embedded vehicle model equipped with VSC in CarSim serves as a real vehicle, providing control input, reference vehicle states and measured signals, while the estimation algorithms are built in Matlab/Simulink. The vehicle parameters used in the simulation are listed in

Table 1. Vehicle parameters. COG: center of gravity.

Parameter	Unit	Value
Gross Mass	m (kg)	1280
Height of sprung mass center of gravity	h (m)	0.5
Distance from COG to front wheels	a (m)	1.203
Distance from COG to rear wheels	b (m)	1.217
Wheelbase	l (m)	2.420
Wheel track	T (m)	1.330
Wheel Radius	R (m)	0.298
Vehicle rotational inertia about z-axis	Iz (kg·m2)	2500
Transmission ratio from motor to wheel	i (-)	4.5
Transmission ratio from the hand wheel to front wheels	is (-)	20
Wheel rotational inertia	Jw (kg·m2)	2.5

. 5% differences of these parameters are added to the UKF 3-DOFs vehicle model in the simulation to imitate modeling uncertainties.

4.1. Initial Settings of Simulations

In this section, an emergency double lane change (DLC) test (ISO3888-1:1999) is simulated with a driver preview time of 0.5 s, which is a short but appropriate value for an emergency situation. The desired longitudinal velocity is 100 km/h controlled and the tire/road friction coefficient is set as 0.8. Such a critical driving condition should be persuasive, since the lateral vehicle motion response has a relatively large variation scale. VSC guarantees that the vehicle passes the DLC test successfully.

In the simulation process, the sensor signals are assumed to be obtained in real time. Furthermore, Gaussian noise is added to the simulated measurements to realistically represent the real application. The standard deviation of longitudinal acceleration sensor noise is 5 × 10⁻² m/s²; the standard deviation of lateral acceleration sensor noise is 5 × 10⁻² m/s²; the standard deviation of yaw rate sensor noise is 2.4°/s; the standard deviation of wheel rotational velocity noise is 10⁻³ rad/s [37].

In addition, the parameters used in the UKF and MEH design are very important for the performance of the proposed estimators. Some design principles have been well investigated [28,38,39]. According to the recommended options and our own tests, the parameters are given as follows: $\mathsf{\alpha }={10}^{-3}$, $\beta =2$, $\kappa =0$ and the fixed time horizon $N=10$. Furthermore, the parameters used in the UKF-based arrival cost calculation in MHE are the same as the UKF estimator’s parameters.

4.2. Results Analysis

Figure 5 shows the tracking performance of the targeted driving process. The CarSim vehicle model equipped with VSC is able to accomplish the designed double lane change maneuver with acceptable tracking accuracy.

Using the torque and rotational speed information directly acquired from CarSim, the longitudinal force of each tire can be calculated according to Equation (2). These longitudinal forces combined with the hand wheel steering angle signal comprise the control input, shown in Figure 6. Additionally, the measurements from virtual sensors under DLC test are depicted in Figure 7.

UKF and MHE are implemented as two separate methods in order to directly compare the estimation performance and robustness. Figure 8, Figure 9, Figure 10 and Figure 11 show the estimated results and errors of the longitudinal velocity, lateral velocity, yaw rate, and lateral tire forces, respectively, using these two estimation methods,. Figure 8 shows that both MHE and UKF can estimate the longitudinal speed satisfactorily. However, MHE shows a faster convergence and slightly higher accuracy compared with UKF.

During the lane changing interval around 4 s where the hand wheel input and lateral acceleration vary greatly, it can be seen from Figure 9 that UKF and MHE perform differently; near 4 s the estimated error of UKF is relatively larger and more fluctuating than that of MHE. Regarding the yaw rate estimation, the overall performance of both algorithms is satisfactory, as seen in Figure 10. Similar to the situation of lateral speed estimation, for the lateral tire force estimation in Figure 11, MHE also shows better performance than UKF under the strong nonlinearity caused by the large lateral acceleration.

Table 2. ME and RMS errors comparison of UKF and MHE on the DLC maneuver. ME: maximum estimation; RMS: root mean square.

State	ME Error			RMS Error
	MHE	UKF	Improvement	MHE	UKF	Improvement
$u\left(\mathrm{m}/\mathrm{s}\right)$	0.441	0.428	−3.037%	0.127	0.335	62.090%
$v\left(\mathrm{m}/\mathrm{s}\right)$	0.067	0.130	48.462%	0.024	0.029	17.241%
$r\left(\mathrm{rad}/\mathrm{s}\right)$	0.017	0.022	22.727%	0.006	0.007	14.286%
$F_{s\_fl}\left(\mathrm{N}\right)$	98.429	156.903	37.268%	38.278	43.850	12.707%
$F_{s\_fr}\left(\mathrm{N}\right)$	145.710	195.451	25.449%	44.730	49.941	10.434%
$F_{s\_rl}\left(\mathrm{N}\right)$	93.551	181.288	48.396%	37.152	43.084	13.768%
$F_{s\_rr}\left(\mathrm{N}\right)$	135.059	241.798	44.144%	44.949	55.750	19.374%

summarizes the quantitative analysis for both UKF and MHE algorithms, including ME (maximum estimation) error, RMS (root mean square) error, and improvement percentage. Except that the longitudinal speed ME error of MHE is a bit larger than that of UKF, MHE outperforms UKF in all other cases. The ME errors of estimated states drop significantly when MHE is applied. Meanwhile, compared with UKF, the RMS errors of these seven vehicle states are improved at least 10% by MHE. For the longitudinal speed, the improvement even reaches 62%.

It is noteworthy that in real applications, measurement noise or other reasons may occasionally result in a wrong initial state value, which affects the accuracy of the subsequent estimation. In order to further evaluate the robustness of these two estimation algorithms, simulations with incorrect initial longitudinal speed estimates of 120 km/h and 80 km/h are conducted, the results of which are illustrated in Figure 12 and Figure 13. Compared to UKF, MHE has the advantage of solving this kind of constrained state estimation problem. In this study, the constraint is set as Equation (40) in Section 3.2.

In Figure 12a,c,e are the estimation with the initial value of 120 km/h and Figure 12b,d,f with the initial value of 80 km/h. Due to Equation (40), the estimated longitudinal speed by MHE approach converges to the reference value rapidly when the initial estimate equals 120 km/h. Even under an initial estimate of 80 km/h, MHE still converges faster than UKF. The other vehicle states, such as lateral speed, yaw rate, and lateral tire forces in Figure 13, are not affected too much by the wrong initial longitudinal velocity estimate.

In

Table 3. ME and RMS errors comparison of UKF and MHE on the DLC maneuver with an incorrect initial estimate of longitudinal velocity.

Initial Velocity	$\mathit{u}=120\mathbf{km}/\mathbf{h}$				$\mathit{u}=80\mathbf{km}/\mathbf{h}$
State	ME Error		RMS Error		ME Error		RMS Error
	MHE	UKF	MHE	UKF	MHE	UKF	MHE	UKF
$u\left(\mathrm{m}/\mathrm{s}\right)$	5.555	5.547	0.264	1.408	5.556	5.546	0.752	1.703
$v\left(\mathrm{m}/\mathrm{s}\right)$	0.205	0.279	0.050	0.101	0.084	0.287	0.032	0.090
$r\left(\mathrm{rad}/\mathrm{s}\right)$	0.031	0.046	0.012	0.013	0.025	0.029	0.011	0.013
$F_{s\_fl}\left(\mathrm{N}\right)$	219.867	240.493	49.126	88.098	147.926	292.768	47.576	106.835
$F_{s\_fr}\left(\mathrm{N}\right)$	235.481	299.550	54.423	105.462	201.060	231.921	58.604	83.239
$F_{s\_rl}\left(\mathrm{N}\right)$	263.167	275.681	63.004	89.418	214.612	356.536	50.953	111.919
$F_{s\_rr}\left(\mathrm{N}\right)$	297.239	366.145	59.385	111.808	182.882	275.438	56.724	87.101

the ME error and RMS error of each state variable are listed. Combining Table 2 with the calculated results, we find that, for both the MHE and UKF algorithms, the estimation with wrong initial longitudinal velocity performs slightly worse than that with a correct initial estimate. Even so, MHE still shows more robustness and a faster convergence ability than UKF.

5. Conclusions and Future Work

This study presented and compared two algorithms, UKF and MHE, for the vehicle longitudinal velocity, lateral velocity, yaw rate, and tire lateral force estimating of DDEVs. Even though today some of these vehicle states can be directly measured by optical sensors or GPS sensors, some practical issues such as cost, accuracy, and reliability inhibit production vehicles from using these sensors at present or in the near future. In contrast, our proposed estimation algorithms are designed based on the information fusion combining feedback signals of electric motors and the existing standard sensor unit equipped in today’s typical vehicles. In some vehicle stability controller studies such as [40,41,42,43], it can replace expensive optical or GPS sensors as a cost-efficient way of providing accurate vehicle state information including velocities, yaw rate, sideslip angle, and tire lateral forces.

In this article, the mathematical vehicle model was established based on 3-DOFs planar vehicle model with a simplified “Pacejka 2002” tire model. Computer simulations are conducted under the emergency DLC test in the co-environment of CarSim and Simulink to evaluate the performance of designed estimators. It is supposed to be persuasive, since during this maneuver vehicle states show large variation. Based on the analysis of the simulation results, our findings are summarized as follows.

(1)

The estimation results for vehicle states including the longitudinal velocity, lateral velocity, yaw rate, and lateral tire forces, by both UKF and MHE, are satisfactory.

(2)

Compared with UKF, MHE has better estimation accuracy, especially under serious nonlinear situations.

(3)

MHE has faster convergence ability and is more robust against an incorrect initial estimate of the longitudinal velocity.

As a future research topic, estimation on rolling movement will be taken into consideration, which is of great significance for the rollover prevention control. Furthermore, our prototype vehicle is currently in the design and manufacturing stage. Field tests need to be carried out in the future to verify the proposed estimation algorithms.