Vehicle driving safety is the urgent key problem to be solved of vehicle 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 for vehicle handling inverse dynamics which can evaluate the emergency collision avoidance performance is proposed. Firstly, the steering angle input of 3-DOF vehicle mode is established. The steering angle input imposed by driver is the control variable, and accurately tracking the expected path was the control object. The optimal control problem can be converted into a nonlinear programming problem while using the state variables conversion, which was solved by the sequential quadratic programming (SQP) algorithm. The results show that vehicle can well track the expected path in high speed

Jingbo Zhao

Kai Zhu

Shaoyi Bei

Wei Wang

Weixing Hang

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 321 
Research on vehicle handling inverse dynamics based on 
optimal control while encountering emergency collision 
avoidance 
Wei Wang1, Shaoyi Bei2, Jingbo Zhao3, Kai Zhu4, Weixing Hang5 
School of Automotive and Traffic Engineering, Jiangsu University of Technology,  
Jiangsu, Changzhou, 213001, China 
1Corresponding author 
E-mail: 1nuaawangwei@126.com, 2beishaoyi@126.com, 3zhaojingbojsut@126.com, 
4zhukaijsut@126.com, 5hangweixingjsut@126.com 
(Received 16 October 2016; accepted 17 October 2016) 
Abstract. Vehicle driving safety is the urgent key problem to be solved of vehicle 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 for 
vehicle handling inverse dynamics which can evaluate the emergency collision avoidance 
performance is proposed. Firstly, the steering angle input of 3-DOF vehicle mode is established. 
The steering angle input imposed by driver is the control variable, and accurately tracking the 
expected path was the control object. The optimal control problem can be converted into a 
nonlinear programming problem while using the state variables conversion, which was solved by 
the sequential quadratic programming (SQP) algorithm. The results show that vehicle can well 
track the expected path in high speed. 
Keywords: collision avoidance, steering angle input vehicle model, handling inverse dynamics, 
optimal control. 
1. Introduction 
On the basis of the known model and vehicle response, vehicle handling inverse dynamics can 
anti-obtain the allowed driver control inputs. Then it analyzes what kind of manipulation is the 
safest and fastest, the most easily accepted by most drivers. It is also known as the “inverse 
problem” of vehicle handling dynamics [1], and belongs to mechanics “dynamic load 
identification” [2]. Vehicle handling inverse dynamics method computes the control inputs that 
the driver applied to the vehicle by specifying the handling performance, and compares to the 
maneuverability of the cars in the most effective way. Handling performances include precisely 
to achieve a given state variable [3-4], to accurately track the given path [5-8], and to pass the 
given path in the shortest time in the case of without departing from the path boundary [9-11]. The 
second case is the problem to be studied in this paper. 
Path tracking always makes the vehicle travel along a desired path, while ensures the vehicle 
safety and comfort. The control layer of path tracking controller operation includes path tracking 
control, speed tracking control, logic components control, emergency management, the 
implementing agencies control and many other modules and components. Path tracking control is 
the core, and currently the most studied problems. With science and technology development and 
the depth research of operation controller for path tracking, many successful control methods have 
appeared: fuzzy control method, PID control method, optimal control method, and neural network 
method, etc. [12-15]. As the vehicle is a very complex nonlinear dynamical systems, it is difficult 
to establish a precise mathematical model, which requires control methods cannot be too 
dependent on the accurate mathematical model. In this paper, the car path tracking controller based 
on fuzzy control method is designed. This control method is suitable for the object which is 
difficult to establish accurate mathematical model, and it is used to control the vehicle path 
tracking. Finally, the simulation results of the control methods are shown. 
RESEARCH ON VEHICLE HANDLING INVERSE DYNAMICS BASED ON OPTIMAL CONTROL WHILE ENCOUNTERING EMERGENCY COLLISION 
AVOIDANCE. WEI WANG, SHAOYI BEI, JINGBO ZHAO, KAI ZHU, WEIXING HANG 
322 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
2. 3-DOF vehicle dynamic model 
In this paper, automotive linear three degrees of freedom angle input model including roll 
degree of freedom is used. Vehicle sports state coordinates are shown in Fig. 1. Fig. 1(a) is the top 
view of vehicle motion, and Fig. 1(b) is the rear view of vehicle motion. 
 
a) 
 
b) 
Fig. 1. Vehicle sports state coordinates graph 
Vehicle motion state response is represented by three generalized coordinates: yaw angle ߰, 
sideslip angle ߚ, car body side angle Φ. Let automobile state vector ܺ = ሼ߱௥, ߚ, ݌, Φሽܶ, ߱௥ is the 
yaw rate, ݌  is the roll angular speed. ߜ௦௪  is the steering wheel angle, the front wheel angle  
ߜ = ߜ௦௪ ݅⁄ , ݅ is the steering gear ratio. The state equation of the car is obtained by the force balance 
equation: 
ሶܺ = ܣܺ + ܤߜ௦௪, (1)
where, ܣ = ܯିଵܥ, ܤ = ܯିଵܰ and: 
ܯ = ൦
ܫ௓ 0 ܫ௑௓ 0
0 ݉ݑ −݉௦ℎ 0
ܫ௑௓ −݉௦ℎݑ ܫ௑ 0
0 0 0 1
൪ , ܥ =
ۏ
ێ
ێ
ۍ ఠܰೝ ఉܰ 0 ܰ஍
ఠܻೝ − ݉ݑ ఉܻ 0 ஍ܻ
݉௦ℎݑ 0 ܮ௣ ܮ஍
0 0 1 0 ے
ۑ
ۑ
ې
,
ܰ = [ ఋܰ/݅ ఋܻ/݅ 0 0]்.
The parameters in Eq. (1) are defined in Reference 1. 
3. The vehicle’s optimal control model for three degrees of freedom angle input  
In this paper, the optimal control theory is used to solve in vehicle handling inverse dynamics 
while encountering emergency collision avoidance. In order to simplify the problem, the driver 
control input is regarded as the ideal one without the effect of driver’s forward-looking, the neural 
response lag and steering response lag. The control variables are selected as the steering wheel 
angle ߜ௦௪(ݐ). Because of considering the path tracking, the driver’s control burden, the effects of 
sideslip and the dangerous of overturn synthetically, the ideal path tracking performance index is: 
ܬ(ݖ) = න ൥ቆݕ(ݐ) − ݕௗ(ݐ)ܧ෠ ቇ
ଶ
+ ቆߜ௦௪(ݐ)ߜመ௦௪
ቇ
ଶ
+ ቆܨ௬(ݐ) ܨ௭⁄̂ߤ ቇ
ଶ
+ ቆΦ(ݐ)Φ෡ ቇ
ଶ
൩ ݀ݐ
௧೐
௧బ
     = න ܮ[ܺ(ݐ), ߜ௦௪(ݐ), ܨ௬(ݐ)]
௧೐
௧బ
݀ݐ,
(2)
RESEARCH ON VEHICLE HANDLING INVERSE DYNAMICS BASED ON OPTIMAL CONTROL WHILE ENCOUNTERING EMERGENCY COLLISION 
AVOIDANCE. WEI WANG, SHAOYI BEI, JINGBO ZHAO, KAI ZHU, WEIXING HANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 323 
where, ݐ଴, ݐ௘  is the initial time and the end time; ݕௗ(ݐ) is the ideal path tracking for collision 
avoidance; ܧ෠  is the standard threshold value of path tracking error, it takes 0.3 m in this paper; 
ߜመ௦௪ is the standard threshold value of steering angle, it takes 600° in this paper; ܨ௬ଵ, ܨ௬ଶ is the 
side force of the front and the rear wheel; ܨ௭ଵ, ܨ௭ଶ is the vertical load of the front and the rear 
wheel; ܨ௬(ݐ)/ܨ௭ = ݉ܽݔ(ܨ௬௜/ܨ௭௜), ݅ = 1, 2; ̂ߤ is the standard threshold value of ܨ௬௜/ܨ௭௜, it takes 
0.3 in this paper; Φ෡  is the standard threshold value of vehicle roll angle, it takes 3° in this paper. 
The state variable of vehicle model is: 
ܺ(ݐ) = [ݒ(ݐ) ߱௥(ݐ) ݌(ݐ) ߔ(ݐ) ݔ(ݐ) ݕ(ݐ) ߰(ݐ)]்.
From the Eq. (1), the state equations can be expressed as that: 
ሶܺ = ݂ൣܺ(ݐ), ߜ௦௪(ݐ), ܨ௬(ݐ)൧. (3)
The condition need to be met in the terminal time on vehicle sport state: 
[ݒ(ݐ) ߱௥(ݐ) ݌(ݐ) Φ(ݐ) ߰(ݐ)]் = [0 0 0 0 0]். (4)
In addition, the control variables of steering angle input need to meet the boundary condition 
due to the limit of driver’s physiology: 
ߜ௦௪(ݐ)௟ ≤ ߜ௦௪(ݐ) ≤ ߜ௦௪(ݐ)௛, (5)
where, ߜ௦௪(ݐ)௛, ߜ௦௪(ݐ)௟ is the upper and the lower boundary of the steering angle input, it takes 
the limit of the steering angle input ±600° in this paper. 
4. The transformation of state variables 
For the terminal time of the above optimal control problem is not fixed, in order to facilitate 
the solution, utilize the unitized longitudinal displacement variable ̅ݔ  to transform the state 
variables. Above problem is transformed into an optimal control problem of fixed terminal  
time [16]. 
Unitized longitudinal displacement variable: 
̅ݔ = ݔ − ݔ଴ݔ௘ − ݔ଴ ,     (0 ≤ ̅ݔ ≤ 1), (6)
where, ݔ଴  is the longitudinal displacement of the initial motor sports; ݔ௘  is the longitudinal 
displacement of the final motor sports. 
Derivate of time for Eq. (6) and combine with Eq. (3): 
݀̅ݔ
݀ݐ =
1
ݔ௘ − ݔ଴ ݔሶ =
ݑcos߰ − ݒsin߰
ݔ௘ − ݔ଴ . (7)
Let the state variable തܺ(̅ݔ) = [ݒ(̅ݔ) ߱௥(̅ݔ) ݌(̅ݔ) Φ(̅ݔ) ݕ(̅ݔ) ߰(̅ݔ)]் . According to 
the Eq. (7), the performance index Eq. (2) transforms to: 
ܬ(ܼ) = න ߣܮൣ തܺ(̅ݔ), ߜ௦௪(̅ݔ), ܨ௬(̅ݔ)൧
ଵ
଴
݀ݐ, (8)
where, ߣ = (ݔ௘ − ݔ଴) (ݑcosߠ − ݒsinߠ)⁄ . 
At this time, the state equation Eq. (3) will transform to that: 
RESEARCH ON VEHICLE HANDLING INVERSE DYNAMICS BASED ON OPTIMAL CONTROL WHILE ENCOUNTERING EMERGENCY COLLISION 
AVOIDANCE. WEI WANG, SHAOYI BEI, JINGBO ZHAO, KAI ZHU, WEIXING HANG 
324 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
݀ തܺ
݀̅ݔ = ߣ݂ൣ തܺ(̅ݔ), ߜ௦௪(̅ݔ), ܨ௬(̅ݔ)൧. (9)
It needs to meet the boundary conditions for the optimal control problem after transforming 
the state variables; 
1. ߜ௦௪(̅ݔ)௟ ≤ ߜ௦௪(̅ݔ) ≤ ߜ௦௪(̅ݔ)௛; 
2. [ݒ(̅ݔ௘) ߱௥(̅ݔ௘) ݌(̅ݔ௘) Φ(̅ݔ௘) ߰(̅ݔ௘)]் = [0 0 0 0 0]். 
Finally, use the discrete element theory to transform to be the finite dimensional nonlinear 
programming problem, and solve it with the sequence quadratic programming (SQP). 
To simulate the vehicle in an emergency avoidance condition, the ideal road input is double 
lane (two yield). Meanwhile, double lane is the typical road model commonly used in evaluating 
vehicle handling stability evaluation. The road trace takes the actual road centerline trajectory. 
5. The path tracking simulation results 
This paper is about a vehicle path tracking simulation test, using Matlab/Simulink to establish 
the computer simulation model. It chooses the road model of double lane and achieves the accurate 
tracking on a given path to control linear three degrees of freedom vehicle system with the optimal 
control algorithm. 
To simulate the vehicle’s collision avoidance with high speed and reflect the steering 
performance truly when the vehicle is avoiding, this paper takes the avoidance path of the double 
lane that model mentioned above. Model one and model two are referred to the Table 1. 
Table 1. Simulation models parameters  
Parameter name Symbol Dimension Model 1 Model 2 
Vehicle Quality ݉ kg 1685 1500 
Suspension quality ݉௦ kg 1365 1180 
Centroid to the front wheelbase ܽ m 1.185 1.17 
Centroid to the rear wheelbase ܾ m 1.283 1.195 
Front suspension roll stiffness ܥఝଵ N·m/rad 60548 60548 
Rear suspension roll stiffness ܥఝଶ N·m/rad 32732 32732 
Moment of inertia around the ܼ axis ܫ௭ kg·m2 2280 1800 
Moment of inertia around the ܺ axis ܫ௫ kg·m2 960 780 
Inertia ܫ௫௭ kg·m2 0 0 
Roll angle damping of the front suspension ܦ௙ N·m·s/rad 3430 3430 
Roll angle damping of the rear suspension ܦ௥ N·m·s/rad 3430 3430 
Front effective cornering stiffness ݇௙ N/rad 40266 40021 
Rear effective cornering stiffness ݇௥ N/rad 75092 74648 
Roll arm ℎ m 0.46 0.46 
Steering gear ratio ݅ none 24.9 24.9 
Two vehicle models shown in Table 1 have simulation of double lane avoidance path tracking. 
Let the speed of vehicle ݑ = 80 km/h. The vehicle path tracking simulation results are as follows. 
Fig. 2(a) is the lateral displacement simulation results of double lane avoidance path tracking 
when two vehicle models are at high speed. As the figure shown that two vehicle models are 
closely matched to the desired avoidance path at high speed during the whole motor process. In 
addition, the absolute error of actual lateral displacement and avoidanc path of tracking is shown 
in Fig. 2(b). The max error of model one in longitudinal displacement of 165 m is 0.1474 m. 
Likewise, the max error of model two in longitudinal displacement of 165 m is 0.1390 m. The 
max error of the model one is slightly larger than the model two. Thus it can be seen that actual 
lateral displacement of two different vehicle models has little difference, and it has small absolute 
error when the vehicle is tracking the avoidance path. Consequently, the optimal control methods 
RESEARCH ON VEHICLE HANDLING INVERSE DYNAMICS BASED ON OPTIMAL CONTROL WHILE ENCOUNTERING EMERGENCY COLLISION 
AVOIDANCE. WEI WANG, SHAOYI BEI, JINGBO ZHAO, KAI ZHU, WEIXING HANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 325 
used in this paper have the better avoidance at high speed. 
 
a) The lateral displacement simulation results 
 
b) The absolute error of lateral displacement 
 
c) Steering wheel angle simulation results 
 
 
d) The rate of steering wheel  
angle simulation results 
Fig. 2. Double lane path tracking simulation results (ݑ = 108 km/h) 
Fig. 2(c) is the steering wheel angle simulation results, the steering wheel angle gets up to the 
peak when the longitudinal displacement is in 60 m, 105 m, 150 m and 195 m. The steering wheel 
angle of the model one is larger than the model two; Fig. 2(d) is the rate of steering wheel angle 
simulation results, the amplitude is lager when the longitudinal displacement is in 75 m, 120 m, 
165 m and 210 m. The driver has high level of busyness, and the model one’s rate of steering 
wheel angle is faster than the model two’s. As the rate and the angle of steering wheel shown that, 
the amplitude of model one is larger than the model two totally, which the driver in model one has 
heavy control burden. 
6. Conclusions 
This paper presents a research method on vehicle handling inverse dynamics based on optimal 
control while encountering emergency collision avoidance. The method uses steering angle input 
vehicle dynamic model for three degrees of freedom and establishes optimal control model for 
handling inverse dynamics. Transform the optimal control problem to a nonlinear programming 
problem and solve the transformed nonlinear programming problem with the method of sequence 
of quadratic programming. Therefore, optimal control can be an effective control method in the 
process of vehicle path tracking control, and has a wider application value and prospect in applying 
in vehicle path tracking control. 
RESEARCH ON VEHICLE HANDLING INVERSE DYNAMICS BASED ON OPTIMAL CONTROL WHILE ENCOUNTERING EMERGENCY COLLISION 
AVOIDANCE. WEI WANG, SHAOYI BEI, JINGBO ZHAO, KAI ZHU, WEIXING HANG 
326 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
Acknowledgements 
This work is partially supported by the National Science Foundation of Jiangsu University of 
Technology (Grant No. KYY14041) and Policy Guidance Program (Collaboration) – Prospective 
Joint Research Project (BY2015028-02). 
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Research on vehicle handling inverse dynamics based on optimal control while encountering emergency collision avoidance

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Research on vehicle handling inverse dynamics based on optimal control while encountering emergency collision avoidance

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