It is necessary to acquire the accurate information of vehicle driving states for the implementation of automobile active safety control. To this end, this paper proposes an effective co-estimation method based on an unscented Kalman filter (UKF) algorithm to accurately predict the sideslip angle, yaw rate, and longitudinal speed of a ground vehicle. First, a 3 degrees-of-freedom (DOFs) nonlinear vehicle dynamics model is established as the nominal control plant. Then, based on CarSim software, the simulation results of the front steer angle and longitudinal and lateral acceleration are obtained under a variety of working conditions, which are regarded as the pseudo-measured values. Finally, the joint simulation of vehicle state estimation is realized in the MATLAB/Simulink environment by using the pseudo-measured values and UKF algorithm concurrently. The results show that the proposed UKF-based vehicle driving state estimation method is effective and more accurate in different working scenarios compared with the EKF-based estimation method.
As we all know, a variety of vehicles has become a popular and common device in people's daily lives; meanwhile, an active safety system (ASC) of a ground vehicle plays an important role in avoiding traffic accidents as a means of guaranteeing passenger safety. However, the effective operation of ASC systems is inseparable from the accurate acquisition of vehicle driving states (Zhang et al., 2019). In general, most vehicle state information is obtained by onboard sensors. However, due to the limitation of cost and measurement methods, it is difficult for some vehicle state signals to be measured directly by sensors, which makes the estimation of vehicle states a hot topic in the field of vehicle ASCs.
Among the vehicle state signals, the sideslip angle, yaw rate, and longitudinal speed are important input variables for the vehicle ASC systems, such as the Electronic Stabilization Program (ESP), Adaptive Cruise Control (ACC), and Lane Keeping Assist (LKA), as well as the state signals that often need to be estimated (Li et al., 2020).
Currently, the common vehicle state estimation methods are usually divided into kinematics methods and dynamics methods (Selmanaj et al., 2017). Based on the relationship between the known state and the unknown state, the kinematics model-based methods calculate the unknown states by directly integrating the kinematics equation (Yamamoto et al., 1995). Generally, these methods have good accuracy for vehicle parameters, road adhesion coefficient, and driving operation. Under the condition of accurate sensor signals, these methods have high estimation accuracy for the sideslip angle in both linear and nonlinear regions (Li et al., 2014). However, the accuracy of these estimation methods is heavily dependent on the measurement accuracy of sensors, which will directly affect the estimation result of the kinematics method. What is more, due to the error accumulations in the integral process, the estimation error caused by measurement noise will increase gradually, and this will still result in the deviation of the estimated value of kinematics from its corresponding values (Piyabongkarn et al., 2009; Kim et al., 2020).
Based on the vehicle dynamics model and tire model, the dynamics methods use modern observation technology to estimate the unknown state, which can reduce the dependence on the sensor precision. The Kalman filter (KF) algorithm family are the common dynamics model-based methods, and lots of scholars have conducted extensive research on the Kalman filter family-based state estimation method. Based on an extended Kalman filter (EKF), Zong et al. (2009) established an information fusion algorithm, which gave out fusion results of vehicle state at minimum square error, and an offline simulation with real vehicle site test data in the MATLAB/Simulink environment was carried out. In Naets et al. (2017)'s research, a nonlinear least square tire parameter estimator was designed to estimate the sideslip angle and the side force of the tire based on the EKF approach, and the covariance caused by the model changes and the variation of driving conditions were also considered. Besides, Katriniok and Abel (2016) proposed an EKF-based estimator to estimate the dynamic parameters of electric vehicles, such as the longitudinal and lateral speeds, along with the yaw rate. Meanwhile, the effectiveness of this EKF-based estimator was validated through MATLAB/Simulink. Moreover, a model-based state observer (Reina et al., 2017) was developed to estimate the key motion states and the vehicle mass online; based on this, a type of vehicle parameter estimation approach was proposed by integrating with the EKF algorithm, and a comparative study between the proposed method and the KF approach was also conducted. Considering that the steering torque signal has faster and more direct response characteristics than the steering angle signal, Ma et al. (2018) proposed an EKF estimation method for the sideslip angle based on the steering torque and verified the accuracy of this method by real vehicle tests.
Like the study above, Liu et al. (2016) proposed a state estimation method for four-wheel drive vehicles based on the minimum model error (MME) criterion by combing with the EKF algorithm. It should be noticed that this method can effectively find out the dynamic tire force errors and then update the system model parameters, which improves the estimation accuracy of the vehicle state.
It is worth pointing out that the EKF algorithm is usually used to estimate the weak-nonlinear systems and simple systems. When the system is strongly nonlinear, the EKF estimation results may lead to large errors or even diverge. In addition, when the estimated system is too complex, the computational load of the EKF algorithm will increase dramatically, which may cause no solution of the Jacobian matrix (Zhou et al., 2019; Strano and Terzo, 2018). Fortunately, compared with the EKF algorithm, the unscented Kalman filter (UKF) algorithm approximates linearization by sampling instead of calculating the Jacobian matrix, which can avoid the above-mentioned problems.
In recent years, some scholars have used the UKF algorithm to estimate the driving states of vehicles. Heidfeld et al. (2019) proposed a state estimation method for all-wheel drive electric vehicles based on the UKF algorithm, which realized the comprehensive estimation of longitudinal and lateral speed, tire slip angle, and tire friction coefficient on each wheel. Based on the UKF algorithm, Song et al. (2020) designed a state observer to realize the joint estimation of vehicle states and parameters and carried out the simulation verification on the Simulink/Carsim platform. The results showed that the joint observer could effectively estimate and identify the relevant vehicle states and parameters and had a good convergence effect.
Inspired by these studies, in this paper, an effective estimation method is conducted based on the UKF algorithm in order to accurately estimate the driving states of vehicles.
This paper is organized as follows: 3 degrees-of-freedom (DOFs) vehicle dynamics modeling and problem formulation of vehicle state estimation are provided in Sect. 2, and Sect. 3 synthesizes the proposed UKF-based vehicle driving state estimation method. Besides, in Sect. 4, the validity of this UKF-based vehicle state estimation is verified by the joint simulation of CarSim and MATLAB/Simulink. Finally, Sect. 5 summarizes the conclusions and future works along this research direction.
In this paper, the 3-DOFs nonlinear vehicle dynamics model is selected as the nominal model for vehicle state estimation. Based on the 2-DOFs linear vehicle dynamics model (Li et al., 2017), a dynamic model including longitudinal, lateral, and yaw motion is established, as demonstrated in Fig. 1.
3-DOFs vehicle dynamics model.
The 3-DOFs vehicle dynamics model can be described as follows:
It is noted that we aim to estimate
The system state vector
Based on the 3-DOFs vehicle dynamics model established above in Eq. (8), the UKF algorithm is applied to estimate the key driving states with the measured inputs acquired by the vehicle onboard sensors.
UT is the key of the UKF algorithm, which is a method to approximate Gaussian distribution by using a fixed number of parameter branches (Kim and Park, 2010). The UT approach can realize the linearization process of a nonlinear system by the sampling method, and it can also avoid the complicated calculation of the Jacobian matrix. The flowchart of UT is demonstrated in Fig. 2.
The flowchart of UT.
The detailed procedure of the UT is illustrated as follows.
Constructing the Sigma points According to a certain sampling strategy, a series of sampling points
Nonlinear transformation After the nonlinear transformation of Calculating the weighted sample mean and covariance By calculating the weighted sum of
The estimation flowchart of vehicle states based on the UKF approach.
When the one-step prediction equation of the standard KF algorithm uses UT
to realize the nonlinear transformation of the mean and covariance matrix,
the UKF algorithm is then constructed. The steps of the UKF algorithm are as
follows.
Setting the initial values: Update the timescale states.
Sampling the Sigma points Based on the symmetric sampling strategy, the Sigma points and the weights of Sigma points, together with the corresponding covariance, are calculated by using the estimated state The weights of each Sigma point and the covariance matrix, i.e., When Calculating the sample points of the predicted values The prediction function Calculating the prior state estimate and covariance By using the weight of Sigma points and corresponding covariance obtained from Eqs. (11) and (12), the weighted sum of the predicted sample points
and covariance can be calculated; that is, the prior state and covariance can be given by In fact, the essence of steps (a) to (c) is to perform a UT on
Calculating the prior measurement Return to step (a) and use the measurement function Update the posterior estimation with measured values. By comparing the actual measured value and the estimated measured value in
Eq. (19), the Kalman gain is used to update the prior state and covariance,
and we can obtain the updated value of the posterior state
In order to validate the accuracy and feasibility of the proposed UKF-based
vehicle states estimation approach, a co-simulation and verification in
CarSim and MATLAB/Simulink is performed. First, the response curves of
The related vehicle parameters.
The structure of the simulator based on the UKF algorithm.
In this work, the initial value of the state covariance matrix of the UKF
algorithm is set as
The sine manoeuver test is first carried out in the CarSim environment, and the simulation results of
The results of
The estimated results of
The absolute error curves of
The relative error curves of
In terms of the results in Figs. 6, 7, and 8, the estimated values of UKF on
The similar simulations are conducted under the fishhook manoeuver test I scenario using CarSim software, and the obtained
In terms of Figs. 10, 11, and 12, there is also a high consistency in UKF between the estimated and measured values of
The obtained
The estimated results of
The absolute error curve of
The relative error curve of
The MAPE of UKF on
Similar to the simulation in Case I, the related simulations are carried out
under fishhook manoeuver test II. The obtained results of
The obtained
The estimated results of
The absolute error curve of
The relative error curve of
The average error of the proposed UKF-based estimation method and traditional EKF.
Like the situations in Case I, the proposed UKF-based vehicle estimation
method still maintains a higher accuracy than EKF in Case II, and only the
error of
To further demonstrate the accuracy of the proposed UKF-based vehicle states
estimation method, the mean absolute percentage error (MAPE) (Ma et
al., 2016), a statistical index which can express the estimation error as a
percentage, is given as below:
The MAPE of UKF and EKF on
As shown in Table 2, the MAPEs of UKF on
In this paper, we proposed a type of UKF-based vehicle driving state estimation method with higher accuracy. First, a 3-DOFs vehicle dynamics model is established, and then a vehicle driving state estimation method is
designed based on the UKF algorithm. Finally, by using CarSim and
MATLAB/Simulink software, the co-simulation and validation are carried out
to validate the accuracy of the proposed estimation method under the
sinusoidal and fishhook conditions. Several highlights of this work are
given below.
A complete UKF-based co-estimation method is proposed to predict the values
of Through the co-simulation and validation, we obtain that the average errors
of
In the future works, we would like to focus on the research of the adaptive
UKF-based estimation method and try to use the adaptive algorithm to reduce
the impacts of noise covariance on the accuracy of the desirable estimation
method.
The state matrices of state-space Eq. (8):
All the data used in this paper can be obtained from the corresponding author upon request.
PW was responsible for the conceptualization, methodology and writing of the original draft. HP was responsible for supervision and writing, review and editing. ZX and JJ performed investigation and data curation.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Robotics and advanced manufacturing”. It is not associated with a conference.
This research has been supported by the National Natural Science Foundation of China (grant nos. 51675423 and 51305342) and the Primary Research & Development Plan of Shaanxi Province (grant no. 2017GY-029).
This paper was edited by Haiyang Li and reviewed by two anonymous referees.