Abstract
Accurate estimation of longitudinal force and sideslip angle is significant to stability control of four-wheel independent driven electric vehicle. The observer design problem for the longitudinal force and sideslip angle estimation is investigated in this work. The electric driving wheel model is introduced into the longitudinal force estimation, considering the longitudinal force is the unknown input of the system, the proportional integral observer is applied to restructure the differential equation of longitudinal force, and the extended Kalman filter is utilized to estimate the unbiased longitudinal force. Using the estimated longitudinal force, considering the unknown disturbances and uncertainties of vehicle model, the robust sideslip angle estimator is proposed based on vehicle dynamics model. Moreover, the recursive least squares algorithm with forgetting factor is applied to vehicle state estimation based on the vehicle kinematics model. In order to integrate the advantages of the dynamics-model-based observer and kinematics-model-based observer and improve adaptability of observer system in complex working conditions, a vehicle sideslip angle fusion estimation strategy is proposed. The simulations and experiments are implemented and the performance of proposed estimation method is validated.
1. Introduction
With the development of automotive technology, drive mode of electric vehicles (EVs) is also experiencing many changes and innovation [1–4]. Emerging four-wheel independent drive electric vehicles (4WID-EVs), as a potential means of future transportation with the controllability of accurate and independent torque, have proven fruitful in energy-saving and advanced vehicle motion control and attracted the attention of both industrial and academic communities [5–7]. 4WID-EVs possess lots of advantages in vehicle dynamics control, such as DYC, in which the accurate estimation of sideslip angle is significant to control performance [8, 9]. Recently, the increasing requirements of safety and stability in automotive technology promote the rapid growth and all the attendant publicity of intelligent transportation [10–12]. Autonomous ground vehicles and unmanned driving have the great advantages containing higher security, better road utilization, and greatly reduced mobility costs. With regard to the path-following control, the real time and reliable vehicle states estimation contribute to better control performance. Therefore, owning to the consideration that some vehicle states are hard and costly to obtain, the design of model-based vehicle state estimator is essential to estimate the vehicle state via low-cost sensor measurements.
During the past decade, there have been many researches engaged in the field of longitudinal force and sideslip angle estimation. The common algorithm applied in estimation can be summarized as Kalman filter [13–17], nonlinear observer [18–22], optimal estimation method [23–27], information fusion estimation method [15, 17, 28–31], and robust estimation method [32, 33], and so on. The core technology in most previous approaches concentrates on solving the estimation problem in complex and limited vehicle running conditions, such as high speed [14, 15], low adhesion [16, 20, 34], vehicle parameters time-varying [29, 32], and unknown or nonlinear disturbances [22–24]. In most prior studies, the longitudinal force observer is designed for traditional internal combustion engine vehicle, and the longitudinal force estimation for EVs especially for 4WID-EVs is still relatively hard to see. In [35], the longitudinal force was calculated by means of integrating the rotational dynamics differential equation of driving wheel, but the influence of noise was accumulated. In 4WID-EVs, the electric driving wheel composed of in-wheel motor and tire is an independent information unit; if we make full use of the measurements of low-cost sensors, such as the current, speed, and bus voltage, the longitudinal force will be achieved cheaply and precisely. Seldom studies have introduced this concept into longitudinal force estimation. For vehicle lateral stability and path-following control, the sideslip angle is an indispensable parameter, and there exists a lot of fruitful progress for sideslip angle estimation. Li et al. presented an adaptive sideslip angle estimator using low-cost sensors in which the multiple model-based Kalman filter was designed to improve the accuracy of estimation [31]. Yoon and Peng estimated the sideslip angle via the combination of the measurements of a global positioning system, an inertial measurement unit, and a magnetometer; the dual Kalman filter was designed to improve the accuracy of estimation via the redundancy of measurements [33].
In this paper, a novel and reliable method of longitudinal force and sideslip angle estimation is proposed for 4WID-EVs. Considering the electric driving wheel model (EDWM) is an uncertain system with unknown input, we utilize the proportional integral observer (PIO) to establish the differential equation of longitudinal force, and the extend Kalman filter (EKF) is introduced to achieve the unbiased estimation of longitudinal force; this design provides a novel thinking of the longitudinal force estimation in 4WID-EVs. In the sideslip angle estimation, the longitudinal forces estimated by longitudinal force observers (LFOs) are regarded and used as the pseudo measurements. The unknown disturbances and uncertainties of vehicle model are considered; the robust sideslip angle estimator is devised on the basis of vehicle dynamics model in which the theoretical robust performance is verified. Furthermore, a recursive least squares method with forgetting factor is used for vehicle state estimation based on the vehicle kinematics model. Then, a vehicle sideslip angle fusion estimation strategy is proposed, in which the accuracy, reliability, and adaptability for the complex working conditions of sideslip angle estimation are improved by the information iteration and fusion between the multimodel-based observers.
The rest of this paper is organized as follows. The vehicle model is presented is Section 2. The vehicle sideslip angle fusion estimation strategy is designed in Section 3. The simulation results are provided in Section 4. The experimental verifications are presented in Section 5, followed by the conclusive remarks.
2. Vehicle Model
2.1. Vehicle Dynamics Model
In this section, a two-degree-of-freedom (2-DOF) vehicle model in longitudinal and lateral directions is introduced to characterize the vehicle motion; the schematic diagram of vehicle dynamic model is shown in Figure 1. The origin of dynamic coordinate system fixed on the vehicle coincides with the vehicle gravity center; the -axis is the longitudinal axis of the vehicle (the forward direction is positive); the -axis is the lateral axis of the vehicle (the right-to-left direction is positive). The pitch, roll, and vertical motions and the suspension system of the vehicle are ignored. It is assumed that the mechanical properties of each tire are the same. The serial numbers 1, 2, 3, and 4 of the wheels are, respectively, corresponding to the front-left, the front-right, the rear-left, and the rear-right wheel. Assuming that the front wheel steering angle is small, the lateral dynamics equation of the four-wheel vehicle model can be expressed aswhere and represent the longitudinal and lateral vehicle speed at the center of gravity (COG), is the yaw rate which represents the angular velocity of vehicle at the COG, denotes the sideslip of vehicle at the COG, represents the vehicle mass, and stands for the moment of inertia. and are the distances from vehicle gravity center to the front and rear axle, respectively. and are the generalized front and rear lateral forces, respectively; that is, , . is the external yaw moment generated by four in-wheel motors; it can be expressed aswhere is the half tread of wheel base, is the steering angle of the front wheels, and represents the longitudinal force of the th tire. The lateral tire force and force can be written aswhere and are the generalized cornering stiffness of the front and rear tires, respectively. The tire slip angles of front and rear tires can be given byThe vehicle dynamics model can be derived aswhere
2.2. EDWM
Each wheel of the FWID-EV is driven by an in-wheel motor. And the driving wheel consisting of in-wheel motor and tire is an independent driving and informative unit. As shown in Figure 2, the concept of the EDMW is proposed in this study. The rotational dynamic equation of each wheel can be written aswhere is the wheel speed of the th wheel, is the wheel moment of inertia, is the wheel effective rolling radius, and is the load torque of in-wheel motor. The torque balance equation of the output shaft in in-wheel motor can be given bywhere is the rotational inertia of in-wheel motor rotor, is the damping coefficient, is the motor torque constant, and is the bus current. The dynamic voltage balance equation of equivalent circuit in in-wheel motor can be modeled aswhere is the bus voltage, is the equivalent resistance of winding, is the equivalent inductance of winding, and is the inverse electromotive force coefficient.
2.3. Wheel Speed Coupling Relationship
The rotational speed of four wheels can be expressed aswhere , , , and represent the corresponding wheel speeds, respectively. is the effective wheel radius.
3. Vehicle Sideslip Angle Fusion Estimation Strategy
3.1. Dynamics-Based Sideslip Angle Robust Estimation
3.1.1. Design of LFO
By substituting (7) into (8) and combining it with (9), we havewhere . The electric driving wheel model (EDWM) is obtained aswhere , , , and are the state vector, the known input vector, and the unknown input vector, and measurement vector, respectively. and are uncorrelated zero mean white noise sequences. The known input and unknown input represent the voltage and longitudinal force, respectively. And
Owing to the existence of unknown input in system (12), the Luenberger observer is not appropriate for the precise state estimation. Consequently, the proportional integral observer (PIO) is utilized to estimate the longitudinal force:where is the state estimation, is the unknown input estimation, is the gain matrix of PIO, and is the integral matrix of the unknown input estimation. The further form of equation (14) can be derived asDefining the state estimation error as , we getIt can be deduced aswhere
Lemma 1 (see [36]). If , there exists a PIO for system (11) to guarantee for any initial states , , and when is observable and .
Remark 2. Notice that the pole assignment of matrix can be implemented to the specified positions to obtain the gain matrixes and . With the suitable choice of and , the eigenvalues of will locate on the left-half complex plane.
In system (15), the estimation of unknown input is solved by the design of PIO, but the existence of noise will also influence the estimation accuracy. In order to suppress the estimation error caused by noise, the Kalman filter is introduced and combined with PIO for longitudinal force estimation. The logogram of system (15) for Kalman filter design is expressed aswhere , , .
The steps of EKF are given by the following.
(1) Forecasting Process. Calculate the forecast value
Calculate the variance of prediction errorwhere , .
(2) Trimming Process. Calculate the matrix of Kalman gain
Update the state estimation
Update the error covariancewhere .
3.1.2. Robust Estimation of Sideslip Angle
As we all know, along with the variation of driving manoeuvre and degree of vehicle stability, the estimation deviations will be influenced by unknown disturbances and uncertainties of vehicle model. For (3), considering the uncertainty of tire cornering stiffness, it can be transformed as , , where and express the additional nonlinear factors and are considered to be bounded. In order to weaken the influences caused by the above factors, the -theory-based estimation method is devised in this work for robust sideslip angle estimation.
The generalized dynamic equation of vehicle model (5) for estimation is written aswhere , , are corresponding dimension regular matrixes; and represent the unknown disturbances and uncertainties. The estimation of LFO is used as the pseudo measurement of vehicle model to calculate .
Remark 3. Assume that is controllable and is observable; , . Thus, hypothesis is the existence condition of the robust observer; hypothesis is equivalent to the performance target of the two-type optimal control. Define the form of robust observer as , where is the estimation of . The unbiasedness of estimation means, if there exists arbitrary ε that satisfies the condition , we have .
Remark 4. With bounded and in (25), the design of robust observer should meet the following condition:where is a given constant. This condition means the design of robust observer makes the energy gain of estimation error less than λ times of and to ensure the filter robustness properties.
Remark 5. The robust observer for sideslip angle estimation is proposed aswhere is the feedback gain matrix. Combining (25) with (27), we haveNotice that the transfer function between and can be given byTherefore, performance target of robust estimation can be written as
Lemma 6. Defining the strict regular rational transfer functional matrix asthe corresponding Hamiltonian matrix of is expressed asWith regard to system (31), if matrix is stable, thus the necessary and sufficient condition for can be expressed as follows: (Ric), , where is the solution of matrix Riccati equation (introduced in following manuscript).
Lemma 7. With regard to the two Riccati equationswhere , , and is observable, if is steady, that is to say, there exists the solution of (33a), then there must exist a solution of (33b) to ensure , Re .
Theorem 8. With regard to the system with disturbances in (25), assuming conditions and are satisfied, there exists unbiased estimator to make the norm of transfer function matrix less than . The necessary and sufficient condition of previous description can be stated as follows: if there exists a matrix , the establishment-condition of following Riccati equation is satisfied:where the feedback gain matrix is expressed as .
Proof. Defining , the premultiplication and postmultiplication to (33a) and (33b) are implemented by matrix . It can be derived asEquation (35) can be converted asThe Hamiltonian matrix of (36) is expressed asThere exists the solution of the Riccati equation; according to Lemma 6, it can be noticed thatwhere is stable. Inequality (37) can be rewritten asThe sufficiency of robust performance is proved.
Owing to the existence of observer gain in (27), the performance target of robust observer is guaranteed. Referring to Lemma 6, it can be affirmed that the following matrix Riccati equation is tenable.On the basis of assumption and Lemma 1, one can know that . Defining , the premultiplication and postmultiplication to (40) are implemented by matrix . It can be derived asEquation (41) can be converted asOn the basis of assumption and Lemma 7, one can know that there must exist a solution to ensure the following matrix Riccati equation is tenable: The necessity of robust performance is proved. The estimated sideslip angle is denoted as .
3.2. Kinematics-Based Vehicle State Observer Design
The observation equation of wheel speed coupling relationship system in (10) is written aswhere is the observation vector, is the state to be estimated, is the measurement matrix, and is the zero mean white noise sequence. The corresponding vector and matrix are represented as
According to the characteristics of formula (12), it is known that the least square method (RLS) can be used to estimate the vehicle state like longitudinal speed, lateral speed, and yaw rate. However, in the iteration process, with the increase of data, the least square method will show the phenomenon of data saturation; that is to say, as time goes on, more and more data are collected, and the information provided by the new data is submerged in the ocean of old data. If the estimation algorithm gives the same degree of trust to the old and new data, it will lose the ability of correction. To overcome the phenomenon of data saturation and to track the change of time-varying parameters, the least square method with forgetting factor is applied to vehicle state estimation, and the recursive algorithm of least square method is given bywhere is the gain matrix, is the covariance matrix, and is the forgetting factor used to balance the fast-tracking ability and resisting-disturbance capacity of the estimation results. The estimated vehicle longitudinal speed, lateral speed, and yaw rate are denoted as , , and , respectively.
3.3. Fusion Estimation Strategy
The robust sideslip angle observer in Section 3.1.2 is designed on the basis of linear tire model; it can get accurate estimations in conventional working condition. Although the uncertainty is considered in the observer design process, in some complex conditions, such as when the tire works in strong nonlinear region, the estimation results will deviate to a certain extent. The least square method with forgetting factor in Section 3.2 improves the reliability of vehicle state estimation results in addition to inheriting the advantages of the least square algorithm. But this method relies on accurate wheel speed measurement, and the estimation effect will be affected when the longitudinal slip rate of tire is too large. Consequently, we can improve the accuracy and reliability of vehicle state estimation by the error iteration and information fusion between the multimodel-based observers. The proposed vehicle state fusion estimation strategy is shown in Figure 3.
In (10), theoretically, , , , and show the ideal wheel speed, and it is impossible for ideal wheel speed to totally equal the actual wheel speed. Moreover, if matrix is an ill-conditioned matrix, little variation may cause the deviation to the solution of RLS. Therefore, considering the deviation caused by the uncertainty of the model in (10), an error compensation mechanism for longitudinal and lateral vehicle speed estimation is developed by PID controller. The estimation error of yaw rate between the real yaw rate and is defined as and used as the input of the PID1 and PID2, the controllers PID1 and PID2 output the corresponding longitudinal and lateral vehicle speed by error compensation, respectively, and they can be expressed as follows (denoted as and ):where and are the compensation coefficient of controllers PID1 and PID2. Then the kinematics-based sideslip angle estimation is obtained as .
The longitudinal slip rate of tire can be calculated byIn this section, the fuzzy controller is designed to fuse the results of kinematics-based sideslip angle estimation and robust sideslip angle estimation. According to the above analysis, the absolute values of sideslip angle and are chosen as the inputs of fuzzy controller, in which and are used to embody the confidence level of the estimation results of and . When is excessively large, the lateral offset of the tire increases and the nonlinear characteristic is enhanced; then it is necessary to reduce the weight of dynamics-based sideslip angle estimation result. When is excessively large, it shows that the longitudinal slip is more severe; then it is necessary to reduce the weight of kinematics-based sideslip angle estimation result. The fuzzy output is calculated by fuzzy controller and used as the fusion weight coefficient between and , and the fusion estimation results of sideslip angle are expressed asThe membership functions of the input and output in fuzzy controller are shown in Figure 4. The fuzzy control rules are listed in Table 1.
Synchronously, the fusion estimation result of sideslip angle is regarded as the pseudo measurement value and used as the input of fuzzy controller; the compensation estimation result of longitudinal vehicle speed is used as the input of slip rate observer. Thus, both and provide the judgement datum for next-step iterative estimation; then the reliability, anti-interference ability, and self-adaptability in multiple working conditions of presented observer system are improved by the error iteration compensation between the multimodel-based observers.
4. Simulation Results
To validate the effectiveness of the designed LFO and proposed vehicle sideslip angle fusion estimation strategy in this work, the simulations are carried out on a high-fidelity CarSim-Simulink joint simulation platform. CarSim is used to provide the whole vehicle model; the estimation of longitudinal forces and sideslip angle is achieved in Matlab/Simulink. The parameters of vehicle and in-wheel motors are listed in Table 2. In simulation and following road test, the uncertainty of tire stiffness is assumed as , . In the robust sideslip angle estimator design, in order to adjust to the constraint conditions of and , system (25) is expanded in the form of
4.1. Double Lane Changes (DLC) Manoeuvre
In this case, the DLC manoeuvre, as shown in Figure 5, is carried out. In simulation, the road friction coefficient is set to be 0.6. The vehicle speed maintains a constant of 20 m/s.
The designs of LFO for wheels 1, 2, 3, and 4 are implemented, respectively, and the estimation results of longitudinal forces are shown in Figure 6. It can be found that the designed LFOs can estimate the longitudinal force accurately with the measurements of currents, speeds, and voltages. The estimation effectiveness of LFOs integrated with PIO and EKF is better than the one only based on PIO; it means that the design of PIO makes it possible to estimate the unknown input of system (longitudinal forces), and the coalition of EKF reduces the influences of noise and guarantees the estimation accuracy. That is to say, the estimated values of LFOs are reliable to substitute the measurements of sensors as the inputs of robust sideslip angle estimator. In DLC manoeuvre, the vehicle speed is set to be relatively larger, and the vehicle actuating manoeuvre changes violently at the beginning; accordingly there was a certain degree tremble in longitudinal forces, but the proposed LFO still can track the real longitudinal force in real time.
The sideslip angle estimation is shown in Figure 7. It is known that both the robust observer and the RLS with compensation can estimate the vehicle sideslip angle in real time. In the DLC manoeuvre with high speed and low road adhesion, the tire longitudinal slip has a certain influence on the RLS estimation result, and the estimation accuracy of RLS is lower than that of robust observer. And the estimation accuracy of sideslip angle can be effectively improved by using the weighted iterative fusion estimation method. In order to further show the accuracy of proposed estimation method, the root-mean-square (RMS) error between real vehicle state and estimated value is used for the quantitative evaluation and can be computed by the following equation:where is the number of samples; and denote the measured and estimated vehicle state at the th sample. The comparison of of vehicle state estimation in DLC manoeuvre simulation is shown in Table 3. In table, the sideslip angle estimation method of robust observer, RLS with compensation, and fusion observer are abbreviated as RO, RWC and FO. It can be found that of proposed longitudinal force estimation method and fusion method are less than that of other estimation methods, which indicates that the proposed method improves the accuracy and reliability of longitudinal force and sideslip angle estimation.
4.2. J-Turn Manoeuvre
In the simulation of J-turn manoeuvre, the road friction coefficient is set to be 1.0; in addition, considering the time-varying characteristic of vehicle speed, a sine wave with the amplitude of 1 and frequency of 1 rad/s is superimposed on the base speed of 10 m/s. The corresponding vehicle speed and steering wheel angle in simulation are shown in Figure 8.
The estimation results of longitudinal forces are shown in Figure 9. Similar to the estimation results in DLC manoeuvre, it can be found that the estimation results of longitudinal force based on PIO-EKF show better estimation ability.
As shown in Figure 10, under the rapid steering conditions, the proposed fusion method can still maintain good estimation performance. In the J-turn manoeuvre, owning to the fast-changing steering wheel angle and the time-varying vehicle speed, the vehicle sideslip angle is relatively larger; in this case, the estimation accuracy of the robust observer is lower than RLS with compensation. The presented fusion estimation strategy can integrate the advantages of the two observers effectively, ensure the estimation accuracy, and simultaneously improve the anti-interference performance of estimation system. Moreover, it can be found that the fusion estimation strategy enhances the adaptability and reliability of the estimation system in multiple conditions. Similarly, as listed in Table 4, in J-turn manoeuvre simulation was computed for further verification of the superiority of the proposed method in improving the estimation accuracy of vehicle state. And the presented longitudinal force and sideslip angle estimation method still maintain better estimation performance.
5. Experimental Results
In this section, the experiment is executed for further validation about the estimation performance of proposed estimation method. At present, the sensor for direct longitudinal force measurement is still unavailable on our four-wheel independent drive electric vehicle, so the verification of LFO is implemented on the chassis dynamometer bench. The robust estimator is validated by the experimental data from road test, in which the estimated longitudinal forces by LFOs are regarded and used as the credible pseudo measurements of the fusion estimation system.
5.1. Test on Chassis Dynamometer Bench
The vehicle test on chassis dynamometer bench is shown in Figure 11. The experimental vehicle is a four-wheel independent drive electric vehicle which is refitted from a single-motor drive electric vehicle and actuated by four in-wheel motors. The whole vehicle control system is achieved on a rapid prototyping platform. The currents, speeds, and voltages of in-wheel motors were measured with corresponding sensors and recorded by the host computer through CAN bus. The longitudinal forces were recorded by the data acquisition system of chassis dynamometer.
In this work, the experimental data of front-right electric driving wheel is selected to testify the proposed LFO; the experimental result is shown in Figure 12. It can be found that there exists some estimation error during the process of acceleration, but the error is finite. The LFO can track the longitudinal force with high accuracy after 4 s when the driving condition approaches the manoeuvre of uniform speed.
5.2. Road Test
Figure 13(a) shows the photo of vehicle road test. As shown in Figure 13(b), the road test is implemented in the form of serpentine trajectory, 10 traffic piles are placed as the obstacle, and the vehicle speed cruising is regulated well by a speed controller. The corresponding steering wheel angle and vehicle speed measured in road test are shown in Figures 13(c) and 13(d), respectively. Figure 14 shows the estimation results of sideslip angle. From Figure 14, it can be seen that forgetting-factor-based RLS algorithm with compensation can track the changing trend of sideslip angle in real time; in addition, by contrasting the sideslip angle estimation results, there exists some fluctuation in the estimation of RLS. The reason is that the magnitude of sideslip angle is far less than wheel speed, and small disturbances in wheel speeds may lead to relatively large estimation fluctuations. Even so, the forgetting-factor-based RLS algorithm with compensation is still able to maintain better estimation results, and the range of volatility is relatively acceptable. The robust observer has good estimation accuracy and stability, but the robust observer is designed on the basis of dynamic model, so the real time tracking ability of which is weaker than RLS. The fusion estimation strategy can synthesize the estimated results of the two kinds of observers and obtain optimal sideslip angle estimation. In order to further validate the accuracy of the proposed estimation method in practice, the comparisons of in chassis dynamometer bench test and road test are shown in Table 5. Notice that the effect of the estimation method is satisfactory in actual vehicle application.
(a) Experimental vehicle
(b) Experimental trajectory
(c) Steering wheel angle
(d) Vehicle speed
6. Conclusion
This work presented a novel longitudinal force and sideslip angle fusion estimation strategy for 4WID-EVs. The EDWM was established and introduced into the longitudinal force estimation. Considering the longitudinal force is the unknown input of EDWM, the PIO was used to restructure the differential equation of longitudinal force, and the EKF was applied to restrain the influence of noise and improve the accuracy of longitudinal force estimation. Based on the longitudinal force estimation, we proposed a robust sideslip angle estimation method with the consideration of the unknown disturbances and uncertainties of vehicle model, and the theoretical proof of robust estimator was provided. Moreover, a kinematics-based vehicle state observer was developed in which the forgetting-factor-based RLS algorithm with error compensation was applied for sideslip angle estimation. Then fusion estimation strategy was presented for more accurate estimation of sideslip angle by the error compensation and iteration between the multimodel-based observers. The simulations of DLC manoeuvre and J-turn manoeuvre were executed, and the results show that the presented LFO can track the longitudinal force in real time, and the fusion estimation strategy has the satisfactory accuracy in estimation. For further validation, the test on chassis dynamometer bench and road test were implemented; the effectiveness of proposed estimation method was verified.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. U1664258 and U1564201), the “Six Major Talent Project” of Jiangsu Province (Grant no. 2014-JXQC-004), the “333 Project” of Jiangsu Province (Grant no. BRA2016445), the Primary Research & Development Plan of Jiangsu Province (Grant nos. BE 2017129 and BE2016149), the Natural Science Foundation of Jiangsu Province (Grant no. BK 20160525), and the Natural Science Foundation of Colleges and Universities in Jiangsu (Grant no. 16KJB580012).