Cascaded Vehicle State Estimation Method of 4WIDEVs Considering System Delay and Noise

Abstract

Considering the negative effects of time delay and noise on vehicle state estimation, a cascaded estimation means for the vehicle sideslip angle is proposed utilizing the ODUKF algorithm. To achieve strong-correlation decoupling between state variables and model interference of the EDWM, an augmented EDWM was constructed by introducing the tire relaxation length dynamic equation, which enables the precise model relationship between the longitudinal and transverse tire force relaxation length to be constructed while also achieving the decoupling of the system state from the unknown input. To achieve a vehicle driving state estimation, a hierarchical estimation architecture was adopted to design a cascading estimation method for the vehicle driving state. By using tire force estimation values as input for the vehicle driving state estimation, the required vehicle body postures can be estimated. At the same time, facing the problems of system delay and noise, an estimator derived from the ODUKF is designed by combining the model and cascade estimation strategy. The simulation comparative analysis and quantitative statistical results under multiple operating conditions provide evidence that the developed means effectively heighten the estimation accurateness and real-time performance while considering system time delay and noise.

1. Introduction

The research on vehicle electrification and intelligence has become recognized as an important development direction of automotive technology and has gradually become a strategic high ground for many countries to achieve breakthrough growth and technological innovation in the automotive industry, with significant market prospects and research value [1,2,3]. With the deepening development of vehicle electrification and intelligent technology architecture, the relevant technology accumulation and industrial chain supporting the supply system are gradually maturing. The richness of vehicle control functions and the expansion of usage scenarios have significantly improved the integration of vehicle control systems in collaborative operation and interactive coupling and have also made the chassis dynamics and electronic control systems of vehicles increasingly precise and complex [4,5]. Under this development trend, the precision and real-time requirements for controlling information input in vehicle active safety and motion control systems are gradually increasing [6,7].

The acquisition accurateness of key vehicle body postures is an important factor to ensure the good operation and reliable performance of the vehicle motion control system. However, many vehicle states are currently difficult to directly measure through existing onboard sensors [8,9,10]. Alternatively, many in-car sensors are too expensive. If too many in-car sensors are configured for state acquisition, it will inevitably increase the cost of vehicle generation and manufacturing, adding resistance to control system development and diversified functional design, which is not conducive to product design and development [11,12,13,14]. Meanwhile, overly inexpensive sensors and simple vehicle state conversion methods cannot meet the increasingly precise control system’s demand for controlling input mass [15,16,17,18]. Therefore, the accurate acquisition of vehicle status has become an agenda of vehicle motion control, which is of great significance for improving control performance.

At present, model-based vehicle state estimation methods are one of the main ways to obtain critical driving states of vehicles [19,20,21,22]. By using prior vehicle model formulas to construct mapping relationships between different vehicle parameters, and then using corresponding observation algorithms to design model-based vehicle state observers, the required difficult-to-measure vehicle state values can be calculated using known vehicle state variables [23,24]. The accurate representation ability for parameter mapping relationships and the dynamic estimation ability of the observer towards system noise and interference have become important breakthroughs and technical difficulties in improving the effectiveness of vehicle state estimation [25,26,27,28]. System time delay affects the representation ability of vehicle models. The model uncertainty and system interference caused by a time delay will significantly affect the vehicle application effect of model-based vehicle state observers [29,30]. For four-wheel independent drive electric vehicles (4WIDEVs), adopting a distributed-drive chassis architecture directly driven by hub motors can provide more design space and also provide a carrier for the integration of vehicle chassis control technology and intelligent driving technology. However, at the same time, the multi-degrees-of-freedom dynamic characteristics and electromechanical coupling driving form of the 4WIDEV chassis system will inevitably exacerbate the time delay problem of the vehicle dynamics system model [31,32,33]. Therefore, for the state estimation problem of 4WIDEV, the computation ability in the presence of time delay interference is one of the crucial issues, which has significant research value and urgency. In addition, the driving environment of vehicles is complex and variable, and system noise interference is also one of the most direct and common factors affecting vehicle state estimation, and its negative impact on the observer cannot be ignored. The Kalman filter algorithm is a practicable method for solving the negative impact of noise on vehicle state estimation [34,35,36]. Its applicability in observation is extremely strong and the effect is significant, which has important research significance and application prospects.

In the increasingly complex and refined development trend in vehicle control systems, the communication delay of system signals and the lag problem of mechanical dynamic transmission systems are important factors affecting the real-time performance of control systems. At this point, considering the vehicle state estimation problem with system delay and noise can further improve the real-time performance and accuracy of vehicle information acquisition, which is of great value for enhancing the performance of vehicle control systems.

The time delay and noise of the system are important factors affecting the performance of 4WIDEV driving state estimations. However, in existing research, it is not common to study the cascaded estimation of 4WIDEV driving states that simultaneously consider system delay and noise. Therefore, when facing the negative effects of system delay and noise simultaneously, this paper proposes a cascaded estimation method for 4WIDEV driving states utilizing the observation delay unscented Kalman filter (ODUKF). The vehicle body model, electric drive wheel model (EDWM), and tire model are established, and considering the nonlinear characteristics of the EDWM with input interferance, the tire slack length is incorporated into the dynamic tire force equation, and an augmented EDWM is constructed. Based on this, longitudinal and transverse tire force observers (TFOs) are designed. Then, a vehicle state cascade estimation method is proposed using tire force observer information as input to the vehicle state observer. Moreover, considering the effects of system delay and noise on vehicle state estimation, a vehicle observation delay Kalman filtering algorithm was designed to further address the unavoidable issues of noise and time delay during vehicle operation.

2. Model Establishment of 4WIDEVs

To achieve the vehicle driving state observer design, a mathematical expression between the key vehicle motion parameters is constructed. This article constructs a dynamics relationship in the longitudinal, transversal, and yaw directions, ignoring the behavior in the pitch and roll directions. Meanwhile, for 4WIDEVs, the mechanical characteristics of the four drive wheels and hub motors are assumed to be consistent. The four tire numbers are denoted as a–d. Under this setting, the three-degrees-of-freedom (3DOF) vehicle body dynamics model is

$$\dot{v}=\phi \mu +m_l,$$

(1)

$$\dot{u}=-\phi v+m_t,$$

(2)

$$\begin{array}{l}{N}_z\phi =\left(F_{xa}+F_{xb}\right)L_f\sin \delta -(F_{yc}+F_{yd})L_r+\left(F_{ya}+F_{yb}\right)L_f\cos \delta \\ +(F_{ya}-F_{yb})L_b\sin \delta -(F_{xa}-F_{xb})L_b\cos \delta -(F_{xc}-F_{xd})L_b\end{array},$$

(3)

where v is the longitudinal speed, u is the lateral speed, $\phi$ is the yaw rate, M is the vehicle mass, F_xj/F_yj (j = a,b,c,d) are longitudinal/lateral tire forces, δ is the front wheel steering angle, N_z is the inertia moment, L_f/L_r are the distances from the mass center point to the front/rear axle, L_b is the half wheelbase, and m_l/m_t are longitudinal/lateral accelerations. The relationship between longitudinal and lateral acceleration and tire force can be expressed as

$$m_l={\displaystyle \frac{1}{M}}\left(\left(F_{xa}+F_{xb}\right)\cos \delta -\left(F_{ya}+F_{yb}\right)\sin \delta +F_{xc}+F_{xd}\right),$$

(4)

$$m_t={\displaystyle \frac{1}{M}}\left(\left(F_{xa}+F_{xb}\right)\sin \delta +\left(F_{ya}+F_{yb}\right)\cos \delta +F_{yc}+F_{yd}\right),$$

(5)

The vehicle sideslip angle β is

$$\beta =v/u,$$

(6)

Considering the unique structural characteristics and driving forms of distributed-drive electric vehicles, the drive unit formed by the single hub motor and wheels is regarded as an independent electric drive wheel module. A single electric drive wheel can be regarded as an independent information unit, and different electric drive wheels are also related to changes in vehicle dynamics parameters through tire force. For the EDWM, its rotational dynamics equation is

$$I_1{\dot{\eta }}_j=T_{Mj}-F_{xj}r,$$

(7)

where $\eta$ is the wheel speed, I₁ is the wheel inertia moment, r is the wheel radius, and T_M is the motor torque. For in-wheel motors, the torque output equation at the drive shaft can be expressed as

$$I_2{\dot{\eta }}_j+D_e{\eta }_j=k_l{i}_j-T_{Mj},$$

(8)

where I₂ is the rotor inertia moment, D_e is the equivalent damping constant, k_l is the load–torque proportional coefficient, and i_j is the bus current. The in-wheel motor is usually a permanent magnet brushless DC motor, and its equivalent voltage equation is

$$V_j=R_e{i}_j+E_i{\dot{i}}_j+k_e{\eta }_j,$$

(9)

where V is the bus voltage, R_e is the equivalent resistance, E_i is the equivalent inductance, and k_e is the back electromotive force coefficient. Combining Equations (7)–(9), the EDWM is

$${\dot{i}}_j=-{\displaystyle \frac{R_e}{E_i}}{i}_j-{\displaystyle \frac{k_e}{E_i}}{\eta }_j+{\displaystyle \frac{1}{E_i}}{V}_j,$$

(10)

$${\dot{\eta }}_j={\displaystyle \frac{K_l}{I}}{i}_j-{\displaystyle \frac{D_e}{I}}{\eta }_j-{\displaystyle \frac{r}{I}}{F}_{xj},$$

(11)

where I = I₁ + I₂ is the equivalent inertia moment of the EDWM.

The electric driven wheel model reflects the transient feature of tire force driving with electromechanical coupling driving. The magic formula is introduced to characterize and further reflect the nonlinear tire–force particularity. Its expression is denoted as

$$f_{xy}=f_D\sin \{f_C\arctan [f_B\alpha -f_E(f_B\alpha -\arctan (f_B\alpha ))]\},$$

(12)

where f_xy is the tire force, f_B is the stiffness factor, f_C is the curve shape factor, f_D is the peak factor, f_E is the curve factor, and α is the tire sideslip angle. The parameters in the nonlinear tire model are all bound up with vertical tire load, which is written as

$$\begin{array}{l}{F}_{za}=L_r({\displaystyle \frac{Mg}{2L}}+{\displaystyle \frac{M{m}_{t}H}{2L_{b}l}})-{\displaystyle \frac{M{m}_{l}H}{2L}},\\ F_{zb}=L_r({\displaystyle \frac{Mg}{2L}}-{\displaystyle \frac{M{m}_{t}H}{2L_{b}L}})-{\displaystyle \frac{M{m}_{l}H}{2L}}\\ F_{zc}=L_f({\displaystyle \frac{Mg}{2}}+{\displaystyle \frac{M{m}_{t}H}{2L_{b}L}})+{\displaystyle \frac{M{m}_{l}H}{2L}},\\ F_{zd}=L_f({\displaystyle \frac{Mg}{2}}-{\displaystyle \frac{M{m}_{t}H}{2L_{b}L}})+{\displaystyle \frac{M{m}_{l}H}{2L}}\end{array},$$

(13)

where F_za-d are vertical tire loads, H is the mass center height, and g is the gravitational acceleration. Meanwhile, the tire sideslip angle is

$$\begin{array}{l}{\alpha }_a=\delta -\arctan {\displaystyle \frac{u+L_f\phi }{v+L_b\phi /2}},\\ {\alpha }_b=\delta -\arctan {\displaystyle \frac{u+L_f\phi }{v-L_b\phi /2}}\\ {\alpha }_c=-\arctan {\displaystyle \frac{u-L_r\phi }{v+L_b\phi /2}},\\ {\alpha }_d=-\arctan {\displaystyle \frac{u-L_r\phi }{v-L_b\phi /2}}\end{array},$$

(14)

3. Cascaded Estimation Method for Vehicle State Considering System Delay and Noise

3.1. Overall Estimation Method and Strategy

3.1.1. Estimation of Tire Forces

Distributed-drive electric vehicles are directly driven by hub motors, with each of the four hub motors being independently controllable. Based on this characteristic, the electric drive wheel model constructed by combining the hub motor model and the wheel rotation dynamics model can be regarded as an independent information unit. From the perspective of vehicle body motion, the longitudinal and transverse tire force information involved in the four electric drive wheel models is directly related to the driving status.

The electric drive model constructed based on the electromechanical coupling driving, particularity of 4WIDEVs, helps to innovate the tire force estimation method for vehicles. On the one hand, the electric drive wheel model that covers the electromechanical coupling driving characteristics can provide better feedback for the dynamic characteristics of the tire force transmission process and has better real-time dynamic characteristics compared to a single dynamic model. On the other hand, the current, voltage, and speed information of the hub motor can be obtained through low-cost sensors, which can provide a large amount of redundant information input for the tire force estimation system, helping to improve the estimation effect. Combining the electric drive wheel models in Equations (11) and (12), it can be concluded that the longitudinal tire force is essentially a variable with dual properties of input and interference. Therefore, before designing a tire force observer based on the EDWM, it is important to decouple the input interference and state variables.

By combining state decoupling requirements of unknown inputs in the EDWM and considering the influence factors of tire slack length, the longitudinal and transverse tire forces are extended to the state variables. Therefore, the dynamic tire force equation incorporating tire slack length can be expressed as

$$\{\begin{array}{l}{\dot{F}}_{xj}={\displaystyle \frac{v-{\left(-1\right)}^j\phi L_b}{{\epsilon }_x}}\left({\overline{F}}_{xj}-F_{xj}\right)\\ {\dot{F}}_{yj}={\displaystyle \frac{u-{\left(-1\right)}^j\phi L_b}{{\epsilon }_y}}\left({\overline{F}}_{yj}-F_{yj}\right)\end{array},$$

(15)

where ${\epsilon }_x$ and ${\epsilon }_y$ are the tire longitudinal/lateral slack lengths and ${\overline{F}}_{xj}$ and ${\overline{F}}_{yj}$ are the nominal longitudinal/lateral tire forces. ${\overline{F}}_{xj}$ and ${\overline{F}}_{yj}$ can be obtained through an empirical tire model.

According to Equations (10), (11), and (15), the tire force can be extended to the state variable of the system equation. The augmented electric drive wheel model that integrates the tire force differential equation can be represented as

$$\{\begin{array}{l}{\dot{x}}_{ej}=f_{ej}\left(x_{ej},u_{ej}\right)+w\\ y_{ej}=h_{ej}\left(x_{ej},u_{ej}\right)+v\end{array},$$

(16)

where $x_{ej}={\left[\begin{array}{cccc}{i}_j& {\eta }_j& F_{xj}& F_{yj}\end{array}\right]}^T$ is the system state value, $y_{ej}={\left[\begin{array}{cc}{i}_j& {\eta }_j\end{array}\right]}^T$ is the system measurement, $u_{ej}={\left[\begin{array}{ccc}{V}_j& {\overline{F}}_{xj}& {\overline{F}}_{yj}\end{array}\right]}^T$ is the system input value, and $w$ and $v$ are noise.

By combining the augmented electric drive wheel model, the discrete form of system state is

$$\{\begin{array}{l}{x}_{ej1,k+1}=i_{j,k+1}=i_{j,k}+\left(-{\displaystyle \frac{R_e}{E}}{i}_{j,k}-{\displaystyle \frac{k_e}{E}}{\eta }_{j,k}+{\displaystyle \frac{1}{E}}{V}_{j,k}\right)\cdot T+w_k\\ x_{ej2,k+1}={\eta }_{j,k+1}={\eta }_{j,k}+\left({\displaystyle \frac{k_l}{I}}{i}_{j,k}-{\displaystyle \frac{D_e}{I}}{\eta }_{j,k}-{\displaystyle \frac{r}{I}}{F}_{xj,k}\right)\cdot T+w_k\\ x_{ej3,k+1}=F_{xj,k+1}=F_{xj,k}+{\displaystyle \frac{v_k-{\left(-1\right)}^j{\phi }_{k}b}{{\epsilon }_x}}\left({\overline{F}}_{xj,k}-F_{xj,k}\right)\cdot T+w_k\\ x_{ej4,k+1}=F_{yj,k+1}=F_{yj,k}+{\displaystyle \frac{u_k-{\left(-1\right)}^j{\phi }_{k}b}{{\epsilon }_y}}\left({\overline{F}}_{yj,k}-F_{yj,k}\right)\cdot T+w_k\end{array},$$

(17)

where T is the Kalman filtering period. Similarly, according to the augmented electric drive wheel model, the discrete form of system measurement is

$$\{\begin{array}{l}{y}_{ej1,k}=i_{j,k}+v_k\\ y_{ej2,k}={\eta }_{j,k}+v_k\end{array},$$

(18)

Therefore, according to Equations (17) and (18), the Jacobian matrix and measurement matrix of the discrete augmented electric drive wheel model can be represented as

$$\begin{array}{c}{S}_{ej}={\displaystyle \frac{\partial f_{ej}}{\partial x_{ej,k}}}=\left[\begin{array}{cccc}-{\displaystyle \frac{R_e}{E}}& -{\displaystyle \frac{k_e}{E}}& 0& 0\\ {\displaystyle \frac{k_l}{I}}& -{\displaystyle \frac{D_e}{I}}& -{\displaystyle \frac{r}{I}}& 0\\ 0& 0& {\displaystyle \frac{-v_k+{\left(-1\right)}^j{\phi }_k{L}_b}{{\epsilon }_x}}& 0\\ 0& 0& 0& {\displaystyle \frac{-u_k+{\left(-1\right)}^j{\phi }_k{L}_b}{{\epsilon }_y}}\end{array}\right]\\ H_{ej}={\displaystyle \frac{\partial h_{ej}}{\partial x_{ej,k}}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\end{array},$$

(19)

where $S_{ej}$ is the Jacobian matrix and $H_{ej}$ is the measurement matrix. Based on the augmented electric drive wheel model in Equation (16), a state observer for a single electric drive wheel model is designed using the Kalman filtering algorithm. This allows for the real-time calculation of dynamic tire force information using the current, speed, and voltage information in the EDWM and the nominal tire force information from the tire model.

3.1.2. Estimation of Vehicle Body Postures

After achieving the tire force estimation, according to (1)–(2) and (4)–(5), the vehicle longitudinal and transverse dynamic equations that integrate tire force information is obtained. Then, combined with Equation (3), the dynamic equation used for estimating the vehicle body postures is

$$\{\begin{array}{l}{\dot{x}}_v=f_v\left(x_v,u_v\right)+w\\ y_v=h_v\left(x_v,u_v\right)+v\end{array},$$

(20)

where $x_v={\left[\begin{array}{ccc}v& u& \phi \end{array}\right]}^T$,$y_v={\left[\phi \right]}^T$,$u_v={\left[\begin{array}{ccc}\delta & F_{xa}\cdots F_{xd}& F_{ya}\cdots F_{yd}\end{array}\right]}^T$. The discretized form of Equation (20) is

$$\{\begin{array}{l}{x}_{v1,k+1}=v_{k+1}=v_k+\left(u_k{\phi }_k+{\displaystyle \frac{1}{M}}\left(\left(F_{xa,k}+F_{xb,k}\right)\cos {\delta }_k-\left(F_{ya,k}+F_{yb,k}\right)\sin {\delta }_k+F_{xc,k}+F_{xd,k}\right)\right)\cdot T+w_k\\ x_{v2,k+1}=u_{k+1}=u_k+\left(-v_k{\phi }_k+{\displaystyle \frac{1}{M}}\left(\left(F_{xa,k}+F_{xb,k}\right)\sin {\delta }_k+\left(F_{ya,k}+F_{yb,k}\right)\cos {\delta }_k+F_{yc,k}+F_{yd,k}\right)\right)\cdot T+w_k\\ x_{v3,k+1}={\phi }_{k+1}={\phi }_k+{\displaystyle \frac{1}{N_z}}\left(\begin{array}{l}\left(F_{xa,k}+F_{xb,k}\right)L_f\sin {\delta }_k-(F_{yc,k}+F_{yd,k})L_r+\left(F_{ya,k}+F_{yb,k}\right)L_f\cos {\delta }_k\\ +(F_{ya,k}-F_{yb,k})L_b\sin {\delta }_k-(F_{xa,k}-F_{xb,k})L_b\cos {\delta }_k-(F_{xc,k}-F_{xd,k})L_b\end{array}\right)\cdot T+w_k\end{array},$$

(21)

Meanwhile, its discretized measurement equation can be expressed as $y_{vk}={\phi }_k+v_k$. On this basis, the Jacobian matrix $S_v$ and measurement matrix $H_v$ of (21) is

$$\begin{array}{c}{S}_v={\displaystyle \frac{\partial f_v}{\partial x_{vk}}}==\left[\begin{array}{ccc}0& \phi & u\\ -\phi & 0& -v\\ 0& 0& 1\end{array}\right]\\ H_v={\displaystyle \frac{\partial h_v}{\partial x_{vk}}}=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\end{array},$$

(22)

Then, a vehicle body posture observer is designed utilizing (21). Among them, the estimated value obtained from TFO is the known quantity of vehicle state observers, thereby achieving a cascading estimation of vehicle body postures. The designed cascaded estimation system of vehicle body postures is shown in Figure 1.

3.2. Analysis of Factors Causing System Time Delay

The electric driven wheel model has a faster dynamic response ability. However, there will inevitably be a certain degree of lag between the tire force signal related to dynamic relationships and the electrical signal related to motor drive relationships, which will affect the representation accuracy and real-time behavior. The tire slack length is a key indicator reflecting the transient characteristics of tires, which can effectively reflect the tire force delay phenomenon caused by elastic deformation during the gradual stress process. Therefore, considering tire slack length in modeling and observer design will reduce the delay of tire force.

In this observer, both the time delay between the electromechanical coupling system signals and the time delay caused by the tire slack length are considered. At the same time, for the vehicle dynamics observer, the system delay for the observation effect is also objectively present and needs to be taken seriously. On the one hand, the vehicle dynamics model treats the vehicle as a multi-degrees-of-freedom rigid body. Due to the time-varying process of the vehicle’s driving attitude caused by tire force driving, the time delay problem between system signals caused by the redundant transmission chain of the dynamic system will affect the estimation effect to a certain extent. On the other hand, for cascaded observer estimation systems, there is also a certain degree of signal asynchrony in the estimated values between different levels of observers. Due to the inconsistent model mechanisms, there is also a problem of signal delay between tire force observers based on electromechanical coupling models and state observers based on vehicle dynamics models. Therefore, the influence of time delay in the process of designing vehicle body attitude observers is a required technical key point.

Based on the analysis of the time delay vehicle model mentioned above, in order to improve the real-time performance of the state estimation, the Kalman filter algorithm is adopted in this study to design the TFO and vehicle state observer. Discretization is applied to the augmented electric drive wheel model in Equation (16) and the vehicle dynamics model in Equation (20) while considering the effect of observation signal delay. It can be represented as a stochastic nonlinear observation delay discrete system as follows:

$$\{\begin{array}{l}{x}_k=f_{k-1}\left(x_{k-1},u_{k-1}\right)+w_k\\ y_{k-\tau }=h_{k-\tau }\left(x_{k-\tau },u_{k-\tau }\right)+v_{k-\tau }\end{array},$$

(23)

where $f_k$ is the system state transition function, $h_k$ is the system measurement function, $x_k$ is the system state value at time k, $y_k$ is the system measurement value at time k, and τ is the lag time.

The principle of the ODUKF is to compute the vehicle state variable at time k using a discrete measurement data sequence at time $k-\tau$ containing noise and time delay. Therefore, for the observation delay nonlinear system in Equation (23), the conventional KF is used to calculate the estimation results at time $k-\tau$, which cannot achieve real-time tracking of the state value at time k. To achieve more timely results, the observation delay equation in Equation (23) can be transformed into an observation equation without time delay, thereby solving the application problem of the conventional Kalman filtering algorithm in observation delay systems.

3.3. ODUKF Algorithm for Vehicle State Estimation

3.3.1. Observation Delay Vehicle Model Transformation Method Based on State Augmentation

In order to achieve the filtering estimation of the vehicle state while eliminating the negative effects of time delay, it is necessary to first construct the non-time delay state estimation equation of the time delay vehicle system model through the model augmentation method. For the observed time delay system in Equation (23), the system state variable is expanded to $X_k={\left[\begin{array}{cccc}{x}_k^T& x_{k-1}^T& \cdots & x_{k-\tau }^T\end{array}\right]}^T$ so that the original observed time delay system is

$$\{\begin{array}{l}{X}_k=\left[\begin{array}{c}{x}_k\\ x_{k-1}\\ \cdots \\ x_{k-\tau }\end{array}\right]=\left[\begin{array}{c}f\left(x_{k-1}\right)\\ x_{k-1}\\ \cdots \\ x_{k-\tau }\end{array}\right]+\left[\begin{array}{c}{w}_k\\ 0\\ \cdots \\ 0\end{array}\right]\\ y_{k-\tau }=h\left(\left[\begin{array}{cccc}0& \cdots & 0& 1\end{array}\right]\left[\begin{array}{c}{x}_k\\ x_{k-1}\\ \cdots \\ x_{k-\tau }\end{array}\right]\right)+v_{k-\tau }\end{array},$$

(24)

where $X_k$ is the augmented state variable. The key to applying the Kalman filtering algorithm in observing time delay vehicle systems is to transform the observation equation into a form that can be directly applied by conventional filtering algorithms. In Equation (24), the relationship of variable $X_k$ and the time delay $x_{k-\tau }$ can be established through the augmentation of the state variable, but its expression is still according to the relevance of the observed $y_{k-\tau }$ and the time delay augmented variable $X_{k-r}$. Therefore, to achieve the design of the filter, the measurement equation in Equation (24) is constructed in the same way as an augmented measurement equation in the following form:

$$Y_k=\left[\begin{array}{cccc}0& \cdots & 0& I\end{array}\right]{\left[\begin{array}{cccc}{y}_k& y_{k-1}& \cdots & y_{k-\tau }\end{array}\right]}^T,$$

(25)

In augmented systems, the augmented state and time delay observation equations is

$$\{\begin{array}{l}{X}_k=\left[\begin{array}{c}f\left(x_{k-1}\right)\\ \left[\begin{array}{cccc}I& 0& 0& 0\\ 0& \ddots & 0& 0\\ 0& 0& I& 0\end{array}\right]\end{array}\right]+W_k\\ y_{k-\tau }=h\left(\left[\begin{array}{cccc}0& \cdots & 0& 1\end{array}\right]X_k\right)+V_k\end{array}.$$

(26)

where $W_k={\left[\begin{array}{cccc}{w}_k& 0& \cdots & 0\end{array}\right]}^T$,$V_k=v_{k-\tau }$. On this basis, by combining Equations (25) and (26), the observed time delay system in Equation (23) can be further transformed into

$$\begin{array}{l}{X}_k=F_k\left(X_{k-1}\right)+W_k\\ Y_k=H_k\left(X_k\right)+V_k\end{array},$$

(27)

where $X_k$ is the augmented system state, $Y_k$ is the measurement of the augmented system, $F_k$ is the state transition function of the augmented system, and $H_k$ is the measurement matrix of the augmented system. By comparing Equations (23) and (25), it can be seen that the augmented system obtained through the state equation and measurement equation of the augmented system has been transformed into a non-time delay system model, which meets the vehicle application conditions of the Kalman filtering algorithm.

3.3.2. UKF-Based Observation Delay Filter Algorithm of Vehicle Augmentation System

In the above section, the observation delay vehicle system in (23) has been transformed into the non-time delay-type system model in (27). Thus, the UKF algorithm can be applied for vehicle state estimation according to the vehicle model in (27), even under the influence of time delay factors. Extended Kalman filtering is a widely used and effective nonlinear system-filtering algorithm. To achieve the design of the vehicle observer, the design steps of the vehicle observer based on Kalman filtering is as follows:

(1) State update:

$$\{\begin{array}{c}{\widehat{X}}_{k|k-1}={\mathsf{\Phi }}_{k|k-1}{\widehat{X}}_{k-1}=F\left({\widehat{X}}_{k-1}\right)\\ P_{k|k-1}={\mathsf{\Phi }}_{k|k-1}{P}_{k-1}{\mathsf{\Phi }}_{k|k-1}^T+Q_{k-1}^{·}\end{array},$$

(28)

(2) Measurement update:

$$\{\begin{array}{cc}\hfill K_k=& P_{k|k-1}{H}_k^T{\left(H_k{P}_{k|k-1}{H}_k^T+R_k^{·}\right)}^{-1}\hfill \\ \hfill {\widehat{X}}_k=& {\widehat{X}}_{k|k-1}+K_k\left(Y_k-H\left({\widehat{X}}_{k|k-1}\right)\right)\hfill \\ \hfill =& {\widehat{X}}_{k|k-1}+K_k\left(y_{k-\tau }-H\left({\widehat{X}}_{k|k-1}\right)\right)\hfill \\ \hfill P_k=& \left(I-K_k{H}_k\right)P_{k|k-1}\hfill \end{array}.$$

(29)

where $P_{k-1}$ is the covariance matrix, $P_{k|k-1}$ is the one-step prediction value of the covariance matrix, $K_k$ is the gain matrix, $Q_k^{·}$ is the non-negative definite variance of $W_k$, and $R_k^{·}$ is the positive definite variance of $V_k$. Therefore, for the electric drive wheel model in Equation (16) and the vehicle dynamics model in Equation (20), considering system noise and observation time delay, combined with the observation time delay model augmentation method in Equation (27), the corresponding tire force observer and vehicle state observer can be designed using the Kalman filtering algorithm in Equations (28) and (29). Then, based on the estimation results $X_k$ of the augmented time delay-free system filter, the required system state estimation value can be obtained as

$${\widehat{x}}_k=\left[\begin{array}{cccc}I& 0& \cdots & 0\end{array}\right]{\widehat{X}}_k.$$

(30)

This calculation method of the extended Kalman filter (EKF) observer will, to some extent, affect the estimation accuracy of the filtering algorithm when facing high-order systems, especially for the augmented time delay system model because as the system dimension increases, the conventional EKF algorithm is relatively more susceptible to impact. To address this issue, based on the system state augmentation, the idea of unscented Kalman filtering is introduced, and the unscented transformation method is used to replace the first-order linearization method in the EKF. The UKF design steps are as follows:

(1) System state initialization: the initial value is ${\widehat{X}}_0=E\left(X_0\right)$ and $P_0=E\left(X_0-{\widehat{X}}_0\right){\left(X_0-{\widehat{X}}_0\right)}^T$.

(2) Unscented transformation (UT): select the Sigma sampling strategy in lossless transformation and obtain the corresponding Sigma points through symmetric sampling. The set of Sigma points under symmetric sampling, the mean weight $W_i^m$ corresponding to each point, and the covariance weight $W_i^e$ is

$$\{\begin{array}{l}{X}_{k/k}^0={\widehat{X}}_{k/k},i=0\\ X_{k/k}^i={\widehat{X}}_{k/k}+{\left(\sqrt{\left(n+\lambda \right)P_k}\right)}_i,i=1,2,\cdots ,n\\ X_{k/k}^i={\widehat{X}}_{k/k}-{\left(\sqrt{\left(n+\lambda \right)P_k}\right)}_i,i=n+1,\cdots ,2n\end{array},$$

(31)

$$\{\begin{array}{l}{W}_0^m=\lambda /\left(n+\lambda \right)\\ W_0^e=\lambda /\left(n+\lambda \right)+\left(1+{\alpha }^2+\beta \right)\\ W_i^m=W_i^e=1/\left(2\left(n+\lambda \right)\right),i=1,2,\cdots ,2n\end{array},$$

(32)

where $\lambda ={\alpha }^2(n+\epsilon )-n$; $\alpha$ is scale parameter and its value is usually greater than 10⁻⁴ and less than 1, $\epsilon$ is the third scale factor, and $\beta$ is the state distribution parameter.

(3) Time update:

$$\{\begin{array}{l}{X}_{k+1/k}^i=F\left(X_{k+1/k}^i\right)\\ {\widehat{X}}_{k+1/k}={\displaystyle \sum _{i=0}^{2n}{W}_i^m{X}_{k+1/k}^i}\\ P_{k+1/k}={\displaystyle \sum _{i=0}^{2n}{W}_i^m\left({\widehat{X}}_{k+1/k}-X_{k+1/k}^i\right){\left({\widehat{X}}_{k+1/k}-X_{k+1/k}^i\right)}^T+Q_{k-1}}\\ Y_{k+1/k}^i=H\left(X_{k+1/k}^i\right)\end{array},$$

(33)

(4) Measurement update:

$$\{\begin{array}{l}{P}_{YY,k}={\displaystyle \sum _{i=0}^{2n}{W}_i^m}\left(Y_{k+1/k}^i-{\widehat{Y}}_{k+1/k}\right){\left(Y_{k+1/k}^i-{\widehat{Y}}_{k+1/k}\right)}^T+R_k\\ P_{XY,k}={\displaystyle \sum _{i=0}^{2n}{W}_i^m}\left(X_{k+1/k}^i-{\widehat{Y}}_{k+1/k}\right){\left(X_{k+1/k}^i-{\widehat{Y}}_{k+1/k}\right)}^T\\ K_{k+1/k}=P_{XY,k}{P}_{YY,k}^{-1}\\ {\widehat{x}}_{k+1/k+1}={\widehat{X}}_{k+1/k}+K_{k+1/k}\left(Y_{k+1}-{\widehat{Y}}_{k+1/k}\right)\\ P_{k+1/k+1}=P_{k+1/k}-K_{k+1}{P}_{YY,k}{K}_{k+1}^T\end{array},$$

(34)

where ${\widehat{X}}_{k+1}$ are the updated state values and $P_{k+1}$ is the updated covariance. Then, combining the unscented Kalman filtering algorithm, the vehicle state can be estimated without time delay according to Equation (30).

4. Verification and Analysis

To demonstrate the driving state cascade estimation method, simulation testing and verification of the vehicle state estimation effect under multiple operating conditions were conducted. In order to build an accurate and reliable vehicle dynamics simulation environment and to facilitate the setting of simulation conditions and vehicle parameters, a CarSim-Simulink co-simulation platform was established. The Simulink (Matlab2014a) model includes the in-wheel motor drive model and relevant vehicle state estimation modules with Kalman filters. The CarSim software V8.02 can provide relevant vehicle state information and be considered as reference values.

4.1. Case 1: Sinusoidal Steering Maneuver

Firstly, simulation verification was conducted under a fixed vehicle speed and sinusoidal steering command input. During the simulation process, a speed tracking controller is set to ensure that the vehicle speed is fixed at 20 m/s. The corresponding steering wheel angle command under this steering condition is shown in Figure 2. In the verification results, to reflect the role of the ODUKF in dealing with system time delay and noise, as well as improving estimation performance, the EKF was selected as the comparative verification. The estimated longitudinal force of the tire under the constant speed sine steering condition is shown in Figure 3. Based on the comparisons, the overall change in tire force is visibly influenced by the trend in the vehicle’s turning motion when speed is basically fixed. When the amplitude of the steering wheel angle increases, the longitudinal tire force also increases accordingly, providing sufficient steering resistance for the vehicle to turn. The designed ODUKF algorithm and EKF algorithm can track the trend in longitudinal tire force changes throughout the entire sine steering process and maintain high accuracy in estimation. However, in contrast, the results under the EKF algorithm show a certain lag phenomenon, while the ODUKF algorithm has better real-time estimation ability, which effectively proves that the proposed estimation algorithm can better handle the negative impact of system delay on estimation performance.

The estimated lateral tire force under the constant speed sine steering condition is shown in Figure 4. The trend in lateral tire force variation is mainly influenced by the vehicle turning motion. The lateral tire force is positive with the steering wheel angle being positive. When the steering wheel angle decreases and becomes negative, the lateral tire force also rapidly decreases and stabilizes at a negative value. The comparison results appear as an advolution trend in Figure 3; both the EKF and ODUKF algorithms can also track the trend in lateral tire force changes well at this time. However, through the comparison of locally enlarged images, the results of the EKF have a more significant lag phenomenon, while the ODUKF algorithm constricts the time lag of lateral tire force.

The estimation result of the vehicle’s body postures in the constant speed sine steering condition are shown in Figure 5. According to the speed estimation results graph, the adopted speed control module can basically meet the speed tracking control requirements of 20 m/s. The actual speed during the entire sine steering simulation process is basically maintained between 19.95 and 20 m/s. In vehicle speed estimation, precision performance of vehicle speed under the EKF algorithm and ODUKF algorithm is maintained at a relatively high level, and the overall estimation error is relatively minimal for the amplitude of vehicle speed. Although longitudinal speed under fixed speed conditions generally does not change much, the estimation performance of the EKF algorithm in the case of system delay is still significantly weaker than that of the ODUKF algorithm. The superiority of the ODUKF algorithm in handling system delay characteristics is demonstrated. By comparing the overall curve trend and the local enlarged graph, the ODUKF is also better than the EKF in estimation performance. When facing system noise and time delay situations, the accuracy and real-time performance of vehicle state estimation are effectively improved.

4.2. Case 2: J-Turn Steering Maneuver

For further confirmation of the effectiveness in achieving vehicle state estimation, simulation verification was conducted under variable speed J-turn steering conditions. The changes in state variables of the variable speed J-turn steering condition are shown in Figure 6. Compared to the constant speed sine steering condition, it can be seen that the variable speed J-turn steering condition is relatively more complex, and the vehicle’s driving state changes more dramatically under this condition. Conducting simulation tests under this operating condition can further reflect the effectiveness of the ODUKF.

The estimated longitudinal tire force under the variable speed J-turn steering condition is shown in Figure 7. As shown in the figure, the trend in longitudinal tire force variation is influenced by both vehicle turning and speed changes, with vehicle speed variation having a more direct impact on longitudinal tire force variation. During the uniform acceleration process, the F_x is relatively large. When speed is constant, the tire force suddenly decreases sharply. With the steering wheel angle gradually reducing, the longitudinal tire force also decreases accordingly. During this process, combined with local zooming in, it can be found that the ODUFK algorithm still has more accurate estimation performance and better real-time estimation results compared to the EKF algorithm, while the EKF algorithm has a certain delay problem. Even when sudden changes in the turning angle and the transition from uniform acceleration to uniform speed, the observer performance is not significantly affected. The anti-interference and dynamic adjustment capabilities of the state estimator is suitable in complex driving application conditions.

The estimated lateral tire force under the variable speed J-turn steering condition is shown in Figure 8. As shown in the figure, the trend in lateral tire force variation in vehicles is mainly influenced by changes in the steering wheel angle. In the initial stage, due to the sharp increase in the steering wheel angle, the lateral tire force also increases rapidly. At this point, the estimation performance of the EKF algorithm and the ODUKF algorithm overall meets the requirements, but the real-time performance of the ODUKF algorithm is better. In the subsequent J-turn steering conditions, the real-time exactitude of the ODUKF algorithm is significantly optimal to the EKF in maintaining stable lateral tire forces and fluctuating vibrations.

The estimated driving state of the vehicle under the variable speed J-turn steering condition is shown in Figure 9. By comparing the estimation results of vehicle body postures, the ODUKF algorithm can still suppress the negative impact of system delay and noise on estimation performance under the more severe variable speed J-turn steering condition and can maintain good estimation performance overall.

Then, the error average and root mean square are used for comparative verification and can be obtained as

$$\begin{array}{c}{e}_{AVE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left({\widehat{x}}_i-x_i\right)}\\ e_{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left({\widehat{x}}_i-x_i\right)}^2}}\end{array}$$

(35)

where $e_{AVE}$ is the error average value, $e_{RMSE}$ is the error root mean square of estimation, N is the sampling amount, $x_i$ is the referenced value, and ${\widehat{x}}_i$ is the estimated value. The results of e_AVE and e_RMSE in the sinusoidal steering maneuver and J-turn steering maneuver are listed in

Table 1. e_AVE and e_RMSE in sinusoidal steering maneuver and J-turn steering maneuver.

Vehicle State	Error	Sinusoidal Steering		J-Turn Steering
		EKF	ODUKF	EKF	ODUKF
Fx	eAVE	1.9121	1.6452	3.6716	2.2654
	eRMSE	0.3654	0.2886	0.3758	0.2655
Fy	eAVE	3.8652	3.0119	3.9672	3.1301
	eRMSE	0.4355	0.3804	0.4389	0.3753
vx	eAVE	0.1459	0.0761	0.1632	0.0777
	eRMSE	0.1798	0.1392	0.1117	0.0962
vy	eAVE	0.1176	0.0694	0.1038	0.0661
	eRMSE	0.1511	0.1007	0.1362	0.0878
β	eAVE	0.1139	0.0628	0.1033	0.0601
	eRMSE	0.1399	0.0886	0.1222	0.0836

. In the comparison results, the error analysis values of tire forces and vehicle body postures obtained by the EKF and ODUKF algorithms under different steering conditions were listed. According to the comparison results, it can be seen that regardless of whether it is a sinusoidal steering condition or a J-turn steering condition, the e_AVE and e_RMSE of different vehicle states under the ODUKF algorithm are smaller than those of the EKF algorithm. This indicates that the cascade estimation method is optimal in terms of the magnitude and fluctuation of estimation errors. Thus, it can further reflect that the ODUKF algorithm has better vehicle application effects in improving estimation accuracy and suppressing error fluctuations. By calculating the estimation accuracy of all vehicle states under the two operating conditions and taking the average, it can be concluded that the proposed ODUKF algorithm has an average estimation accuracy of 93.67%, while the EKF algorithm has an average estimation accuracy of 85.11% under the same conditions. The proposed algorithm improved the overall estimation accuracy by 8.56 percentage points when considering the impact of time delay.

5. Conclusions

In response to the driving state estimation problem of 4WIDEVs, considering the effects of time delay and noise, a cascaded estimation method for the vehicle driving state is presented using model construction and the ODUKF method. In response to the problem of unknown inputs in an EDWM, an augmented EDWM was constructed by combining the electric drive wheel model and the tire slack length dynamic equation, achieving the decoupling of unknown inputs and state variables in the electric drive wheel model. A vehicle state estimation method based on the cascaded estimation strategy is proposed by using a hierarchical estimation approach, where estimated tire force is used as the pseudo-sensor input quantity for vehicle driving state estimation. Then, the time delay and noise issues in the state estimation of 4WIDEVs is analyzed, and an ODUKF algorithm based on the augmented electric drive wheel model and vehicle dynamics model is designed to achieve the effective estimation of vehicle states. The comparative simulation tests and data statistical analysis were conducted under constant speed sine steering and variable speed J-turn steering conditions, which verifies that the proposed ODUKF algorithm has a better estimation effect under the time delay conditions and improved the overall estimation accuracy by 8.56 percentage points.

In future research, the influence of nonlinear disturbances in vehicle dynamics systems on vehicle state estimation can be further considered, and the accuracy of state estimation can be improved through the precise modeling of nonlinear systems. In addition, the integration optimization problem of vehicle state estimation systems and vehicle dynamics control systems can be studied, effectively combining vehicle state estimation with motion control to improve the overall control effect of the vehicle.