Vehicle driving safety is the urgent key problem to be solved of automobile independent development while encountering emergency collision avoidance with high speed. And it is also the premise and one of the necessary conditions of vehicle active safety. A new technique of vehicle handling inverse dynamics which can evaluate the emergency collision avoidance performance is proposed. Based on optimal control theory, the steering angle input and the traction/brake force imposed by driver are the control variables; the minimum time required to complete the fitting biker line change is the control object. By using the improved direct multiple shooting method, the optimal control problem is converted into a nonlinear programming problem that is then solved by means of the sequential quadratic programming. The simulation results show that the proposed method can solve the vehicle minimum time maneuver problem, and can compare the maneuverability of two different vehicles that complete fitting biker line change with the minimum time and the correctness of the model is verified through real vehicle test.
With the continuous development of the automobile industry, the number of car accidents grows accordingly, especially in traffic accidents involving pedestrians and cyclists. In some cases, car accidents can be seen as a collision between vehicles and obstacles. And then the emergency avoidance problem is proposed to avoid the accidents. Today, people pay more and more attention to the problem of high-speed emergency avoidance [
The research methods of vehicle handling dynamics usually include open-loop and closed-loop method. The two methods are called “forward problem” method of vehicle handling dynamics research. Open-loop research method does not consider the function of the driver’s feedback and obtains vehicle response under the condition of mathematical model of vehicle and driver input. But closed-loop method obtains vehicle motion which follows the ideal path based on driver vehicle closed-loop control system model [
In emergency avoidance research, the vast majority of research focused on the shortest path in the process of emergency avoidance. Sundar and Shiller (1997) proposed a method producing the shortest path based on the Hamilton-Jacobi-Bellman equation in a cluttered environment. The method attributed the emergency avoidance problem of shortest distance to the optimal control problem of shortest time, generated the shortest path through the function of negative gradient, and achieved good results [
Vehicle handling inverse dynamics can evaluate the driver’s handling input by the specified handling performance and improve the performance of high-speed vehicle emergency avoidance. The handling performance of different vehicle can be compared with the most efficient way by the vehicle handling inverse dynamics [
In the paper, the optimal control theory is used in the field of vehicle handling inverse dynamics. In order to simplify the problem, the ideal driver handling inputs are considered without consideration of driver response lag and the forward-looking role.
Assuming tire cornering properties in the linear range and considering rotational inertia of the steering system, the vehicle steering motion model is simplified as shown in Figure
4 DOF vehicle steering model.
If driving force/braking force is considered to impact the cornering force, it is
The
Control variable
According to
In the equation, the state variables are
Longitudinal velocity
When the vehicle is under a braking force and the front and rear wheels are assumed in lock state, it is
Control variable
According to the connection between engine speed and the velocity of vehicles and the connection between the engine output torque and the driving force, the regulation between the maximum driving force and the velocity of vehicles can be obtained by the engine external characteristic curve.
According to the literature [
All the constraints are shown by the following equation:
In the process of the vehicle tracking the desired trajectory, the ultimate elapsed time is difficult to determine. In order to solve this problem conveniently, the free terminal time can be transformed into the fixed terminal time for optimal control problem with the following ways.
Longitudinal displacement variable
According to
According to
Similarly,
The state variable, control variable, and time of nodes are assumed at the same time in the direct multiple shooting algorithm. It will increase variable numbers of the transformed nonlinear programming problem, thus making it more difficult to get the answer. Therefore, this paper puts forward an improved direct multiple shooting method; in other words, only control variables of nodes are assumed.
The performance index of
Therefore, as long as
Considering the common nonlinear constrained optimal control problem,
The procedures of sequential quadratic programming are given as follows. Give
One gets
In formula
For the optimal control problem of time-varying system in this paper, it can be converted to the finite dimensional nonlinear programming problem by using the improved direct multiple shooting method. The interval A set of vectors
The gating finite dimensional nonlinear programming problem can be solved by using the sequential quadratic programming (using the fmincon function in the optimization toolbox of MATLAB).
The biker line performance of two vehicles is researched. The vehicle specific parameter values are shown in Table
Two vehicles parameter list.
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The size of biker line test road is shown in Figure
Biker line test road.
In the actual driving process, driver’s ideal target track should be as shown in Figure
Fitting biker line.
As is shown in Table
After 14 iterations, the minimum time in which model A passes the biker line after optimization is 15.7 s. After 16 iterations, the minimum time in which model B passes the biker line after optimization is 16.2 s. Therefore, the minimum time in which model A passes the biker line after optimization is shorter than that of model B. Figures
Lateral displacement simulation results.
Figure
Figure
The steering wheel torque simulation results.
Figure
Wheel driving force simulation results.
Longitudinal velocity simulation results.
Figure
When the vehicle travels at 25 m/s high speed initially, after 14 iterations, the minimum time in which Model A passes the biker line after optimization is 14.9 s. After 18 iterations, the minimum time in which model A passes the biker line after optimization is 15.2 s. Therefore, when the vehicle run at the fast speed, it drive through the serpentine in short time. But as the vehicle’s speed increases, the driver’s burden increases and the safety reduces. The driver has to reduce vehicle’s speed to a certain level to ensure his safety before the vehicle passes the biker line.
In this paper, two types of off-road vehicles mentioned above are used to test vehicle handling stability. Real vehicle test is very dangerous in high speed. In order to consider the driver’s safety, the method of pavement design point is taken in the test.
The test procedures are as follows. In the test site, stake position marker is designed as in Figure Connect the test instruments; switch instruments power on in order towarm the instruments to normal operating temperature. The vehicle passes the test section with an initial speed of 72 km/h. Running over the marker is not allowed in the running process. At the same time, the time history curve of the measured variables (steering wheel angle and longitudinal velocity) is recorded by the computer. Repeat steps (3) process 12 times (the times of press the marker is not considered). Two vehicle types’ experimental data are obtained by the same test methods above if the vehicle type is changed.
The test site is built as shown in Figure
Vehicle biker line test.
The bollard of biker line
Car running track
The instrument of steering wheel torque
DEWESoft data signal acquisition system
The experimental procedures and protocols are built as shown in Figure
Experimental procedures and protocols.
12 groups of test time were, respectively, 17.8 s, 18.1 s, 17.9 s, 18.5 s, 18.8 s, 18.3 s, 17.9 s, 18.3 s, 18.2 s, 18.9 s, 19.0 s, and 18.8 s. Due to considering a lot of factors, such as driver’s reaction time and road conditions, the experimental test time was generally longer than the time of optimal control. The mean and standard deviation were, respectively, 18.27 s and 0.3743.
Comparison between the simulation value and the experimental value is shown in Figures
The steering wheel torque simulation results.
Wheel driving force simulation results.
In the field of automotive engineering, many researchers are focusing on the development of self-driving technologies. Self-driving vehicles promise to bring a number of benefits to society, including prevention of road accidents, optimal fuel usage, comfort, and convenience. Vehicle handling inverse dynamics is form of the self-driving technologies. The steering wheel torque can be obtained by the vehicle handling inverse dynamics and used to determine the vehicle steering problems in the emergency collision avoidance. So the vehicle handling inverse dynamics can promote the self-driving vehicle development.
In this paper, minimum time approach to emergency collision avoidance is researched by the method of vehicle handling inverse dynamics. Firstly, the optimal control model of the vehicle emergency collision avoidance problem was established. And then the optimal control problem was changed into a nonlinear programming problem using the improved direct multiple shooting method. Finally, the transformed nonlinear programming problem was solved by using sequential quadratic programming method. The correctness of the optimal control model is verified by using real vehicle test. The results show that this method can successfully solve the minimum time problem of vehicle emergency collision avoidance and compare different vehicles in the minimum time through a given path control performance. It can provide guidance for the self-drive research. Intelligent vehicle driving also has certain reference value.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the National Science Foundation of China (Grant no. 51305175) and the National Science Foundation of JiangSu Province (Grant no. BK2012586).