This paper describes a valuable linear yaw-roll tractor-semitrailer (TST) model with five-degree-of-freedom (DOFs) for control algorithm development when steering and braking. The key parameters, roll stiffness, axle cornering stiffness, and fifth-wheel stiffness, are identified by the genetic algorithm (GA) and multistage genetic algorithm (MGA) based on TruckSim outputs to increase the accuracy of the model. Thus, the key parameters of the simplified model can be modified according to the real-time vehicle states by online lookup table and interpolation. The TruckSim vehicle model is built referring to the real tractor (JAC-HFC4251P1K7E33ZTF6×2) and semitrailer (Luyue LHX9406) used in the field test later. The validation of the linear yaw-roll model of a tractor-semitrailer using field test data is presented in this paper. The field test in the performance testing ground is detailed, and the test data of roll angle, roll rate, and yaw rate are compared with the outputs of the model with maps of the key parameters. The results indicate that the error of the tractor’s roll angle and semitrailer’s roll angle between model data and test data is 1.13% and 1.24%, respectively. The roll rate and yaw rate of the tractor and semitrailer are also in good agreement.
Tractor-semitrailers (TSTs) are commonly used in road freight transportation all over the world. The popularity of such vehicles is due to their flexibility and transport efficiency. The most popular vehicles of this type consist of a tractor unit or prime mover coupled to a long semitrailer [
The linear simple model is widely used in designing control algorithms and strategies. It offers considerably shorter simulation run time and is simpler and faster to change any parameters [
In order to generate more accurately the nonlinear TST model, the nonlinear tyre models (presented by Pacejka) are used in Kharrazi and Oreh’s researches. Controllers are designed based on the nonlinear vehicle models. A steering-based simple controller is presented in Kharrazi’s research [
Firstly, the lateral force obtained by the nonlinear tyre model is not accurate enough. The sideslip angles of each tyre are essential input variables of the nonlinear tyre model. On account of the flexibility of suspension and a different center location of each tyre, the sideslip angles of each tyre are different. The lateral force increase at the outside tyre is usually smaller than the lateral force decrease at the inside tyre because of the nonlinearity of the tyre and suspension [
Secondly, the nonlinear characteristics of the real vehicle listed below are not exhaustively considered in their models. In a real TST, there are uncertainties in the vehicle dynamics such as the inertia parameters, the friction coefficient, and the tyre’s cornering stiffness [
The nonlinear characteristics of all the components have to be determined by the field test with advanced sensors and are very costly. Therefore, identification of these nonlinear characteristics would be a logical choice [
Primarily, in Nie and Zong, Kim et al., and Guang et al.’s researches, the stationary turning condition and step steering condition are considered to identify parameters. The design variables of parameter identification are longitudinal speed, steering wheel angle, and lateral acceleration. According to their results, in steering and braking conditions, the key parameters of the TST model are identified as the same as those under the same longitudinal speed but different deceleration. According to Newton’s first law of motion, every object in a state of uniform motion tends to remain in that state of motion unless an unbalanced force is applied to it. When the TST was under a steering and braking condition, both the lateral acceleration and the longitudinal deceleration were affected by the lateral force. Actually, when the TST is braking at the same speed (larger than zero), the force, the roll angle of the sprung mass, and the sideslip angle of the tyre were all affected by the deceleration. During braking, the instantaneous velocities are changed over time. It is hard to identify the stiffness at every moment. Hence, we identified the key parameters of the vehicle model under steering and braking conditions. The deceleration and steering wheel angle were set as constant under each condition.
In the second place, the roll stiffness of the fifth wheel was set as constant in their researches. Actually, the complicated structure and the placement of the fifth wheel result in the nonlinear characteristics [
Above all, the condition and object variables of identification were confirmed. Then, the identification problem is solved as an optimization problem, and the search technique is a critical factor in determining the performance of the optimization scheme. Many search techniques are used in identifying characteristics of the TST model, for example, the trial-and-error method [
This paper is organized as follows: Section
In this section, a linear TST model, namely, 5-degree-of-freedom (5-DOF) model, is derived. To ensure that the linear model can reflect the model more accurately, the parameters of the linear model are obtained from the JAC-HFC4251P1K7E33ZTF6×2 tractor and Luyue LHX9406 semitrailer, as shown in Table
Nomenclature.
Parameter | Symbol | Unit |
---|---|---|
Fifth-wheel stiffness |
|
N·m/rad |
Roll stiffness of the tractor |
|
N·m/rad |
Roll stiffness of the semitrailer |
|
N·m/rad |
Cornering stiffness of the tractor’s front axle |
|
N/rad |
Cornering stiffness of the tractor’s rear axle |
|
N/rad |
Cornering stiffness of the semitrailer’s axle |
|
N/rad |
Tractor’s speed |
|
m/s |
Semitrailer’s speed |
|
m/s |
Tractor’s sideslip angle |
|
Rad |
Semitrailer’s sideslip angle |
|
Rad |
Tractor’s yaw rate |
|
Rad/s |
Semitrailer’s yaw rate |
|
Rad/s |
Articulated angle |
|
Rad |
Tractor’s roll angle |
|
Rad |
Semitrailer’s roll angle |
|
Rad |
Tractor’s roll rate |
|
Rad/s |
Semitrailer’s roll rate |
|
Rad/s |
Tractor’s entire vehicle mass |
|
kg |
Semitrailer’s entire vehicle mass |
|
kg |
Tractor’s sprung mass |
|
kg |
Semitrailer’s sprung mass |
|
kg |
Roll moment of inertia of the tractor’s sprung mass |
|
kg·m2 |
Roll moment of inertia of the semitrailer’s sprung mass |
|
kg·m2 |
Yaw moment of inertia of the tractor’s sprung mass |
|
kg·m2 |
Yaw moment of inertia of the semitrailer’s sprung mass |
|
kg·m2 |
Distance between the tractor CG and the front axle |
|
m |
Distance between the tractor CG and the rear axle |
|
m |
Distance between the tractor CG and the fifth wheel |
|
m |
Distance between the semitrailer CG and the axle |
|
m |
Distance between the semitrailer CG and the fifth wheel |
|
m |
Distance between the tractor CG and the roll axle line |
|
m |
Distance between the semitrailer CG and the roll axle line |
|
m |
Distance between the fifth wheel and the tractor roll axle line |
|
m |
Distance between the fifth wheel and the semitrailer roll axle line |
|
m |
Roll damping of the tractor’s suspension |
|
N·s/m |
Roll damping of the semitrailer’s suspension |
|
N·s/m |
5-DOF TST model: (a) yaw moment; (b) roll moment of the tractor; (c) roll moment of the semitrailer.
To develop a linear TST model, the sideslip stiffness of the tyre, the roll stiffness, and the fifth-wheel stiffness are assumed to be constant. The aerodynamics, road grade, load transfer, and transmission system of the steering system are neglected to simplify the model [
The assumptions of the TST model are listed as follows: The aerodynamics and road grade are neglected The load transfer is neglected The transmission system of the steering system is ignored, and the steering wheel angle is used directly as the system input The longitudinal acceleration is considered the first-order derivative of the longitudinal speed, and the varying longitudinal speed is used directly as the system input The tractor and the semitrailer units have no pitch or bounce The articulation angle of the fifth wheel is small [ The relationship between the tyre force and the sideslip angle is linear The relationship between the roll moment and the roll angle is linear The roll moment transmitted by the fifth wheel is assumed to be proportional to the relative roll angle and relative roll rate between the two units [
The equations of vehicle motions of the tractor unit are built based on Newton’s second law of motion (see nomenclature) [
The equations of semitrailer motions are
The kinematic constraint equation of the fifth wheel is
The tyre side forces acting on the articulated vehicle are generated at the contact patch between the tyre and the road. Note that, in generating the 5-DOF model, the tyre properties are linearized, and the effects of the camber thrust, the roll steer, and the aligning moment are neglected. The tyre side forces are [
The 5-DOF model can be expressed in the state-space form as follows:
The nonzero elements of
To reflect the main characteristic of the actual vehicle state by the simplified vehicle model, the key parameters are identified offline by the GA according to TruckSim data. Steering and braking conditions are adopted to identify the key parameters. The required key parameters include roll stiffness, axle cornering stiffness, and fifth-wheel stiffness.
To reflect the main characteristic of the actual vehicle state by the simplified vehicle model, the key parameters are identified offline by the GA and multistage GA according to TruckSim data. Steering and braking conditions are adopted to identify the key parameters. The required key parameters include roll stiffness, axle cornering stiffness, and fifth-wheel stiffness.
The genetic algorithm program used in this paper is compiled by the genetic algorithm toolbox in the software MATLAB. The flow diagram of the GA program is shown in Figure
Flow diagram of the GA program.
The fitness function is defined as the absolute value of error between the outputs of the simplified 5-DOF tractor-semitrailer model and TruckSim. The optimal values of each parameter are defined as the values which made the fitness function close to zero. The fitness function is defined by [
The nomenclature is shown in Table
Parameters of the fitness function.
Parameter | Symbol | Unit |
---|---|---|
Tractor sideslip angle of the model |
|
Rad |
Semitrailer sideslip angle of the model |
|
Rad |
Tractor yaw rate of the model |
|
Rad/s |
Semitrailer yaw rate of the model |
|
Rad/s |
Tractor roll rate of the model |
|
Rad/s |
Semitrailer roll rate of the model |
|
Rad/s |
Tractor roll of TruckSim |
|
Rad |
Semitrailer roll of TruckSim |
|
Rad |
Tractor sideslip angle of TruckSim |
|
Rad |
Semitrailer sideslip angle of TruckSim |
|
Rad |
Tractor yaw rate of TruckSim |
|
Rad/s |
Semitrailer yaw rate of TruckSim |
|
Rad/s |
Tractor roll rate of TruckSim |
|
Rad/s |
Semitrailer roll rate of TruckSim |
|
Rad/s |
Tractor roll of TruckSim |
|
Rad |
Semitrailer roll of TruckSim |
|
Rad |
Sampling point |
|
— |
Sampling number |
|
— |
The initial population is randomly picked between the stated bound of each parameter. The bound of the fifth-wheel stiffness, roll stiffness, and cornering stiffness is obtained by the output of the vehicle model simulated in the software TruckSim. The TST model in TruckSim is designed based on the parameters of the JAC-HFC4251P1K7E33ZTF6×2 tractor and Luyue LHX9406 semitrailer, which are used in the real vehicle test. The parameters of the TST are shown in Tables
Parameters of the fully loaded JAC-HFC4251P1K7E33ZTF tractor.
Parameter | Unit | Value |
---|---|---|
Entire mass | kg | 9420 |
Sprung mass | kg | 6300 |
Distance between the front axle and the middle axle | m | 3.37 |
Distance between the middle axle and the rear axle | m | 1.38 |
Wheelbase of the front axle | m | 2.04 |
Wheelbase of the middle and rear axles | m | 1.88 |
Height of the CG | m | 1.37 |
Distance between the CG and the middle axle | m | 2.11 |
Distance between the CG and the fifth wheel | m | 2.61 |
Distance between the CG and the roll axle line | m | 1.03 |
Roll moment of inertia of sprung mass | kg·m2 | 17260 |
Yaw moment of inertia of sprung mass | kg·m2 | 18256 |
Minimum turning radius | m | 16.8 |
Steering ratio | — | 25 |
Parameters of the fully loaded Luyue LHX9406 semitrailer.
Parameter | Unit | Value |
---|---|---|
Entire mass | kg | 39140 |
Sprung mass | kg | 36720 |
Distance between axles | m | 1.31 |
Distance between the fifth wheel and the front axle | m | 6.20 |
Wheelbase | m | 1.84 |
Height of the CG | m | 1.80 |
Distance between the CG and the front axle | m | 4.65 |
Distance between the CG and the fifth wheel | m | 3.79 |
Distance between the CG and the roll axle line | m | 1.45 |
Roll moment of inertia of sprung mass | kg·m2 | 21690 |
Yaw moment of inertia of sprung mass | kg·m2 | 346782 |
The estimated fifth-wheel stiffness (
The identification ranges of each stiffness value are set from 50% to 200% of the estimated stiffness in equation (
According to the fitness function, the identified parameters are closer to the optimal value when the value of fitness is smaller. The individuals with a higher fitness value should be more likely to be chosen. The probability of being selected for each individual is shown as
In the next generation, there are still
The crossover and mutation operators are used to turn the chosen individuals to new individuals in the next generation. The operators are designed by referring to the study in [
The multistage genetic algorithm is proposed to improve the precision of identification and avoid the local optimal solution problems. The fitness function, selection, crossover, and mutation in the MGA are the same as those in the GA, as shown in Section
Flow diagram of the MGA program.
The MGA can significantly improve the efficiency and precision of identification. The precision comparison between the GA and the MGA is shown in Figure
Output data of the field test and model based on the GA and MGA: (a) tractor’s roll rate; (b) tractor’s yaw rate.
The key parameters of the linear 5-DOF model are identified offline during steering and braking at specified initial vehicle speeds, vehicle deceleration, and steering angle. The key parameters are identified by the MGA. The input data are obtained by the nonlinear simulation software TruckSim. The tractor-semitrailer model in TruckSim is designed based on the real vehicle used in the field test. The parameters of the real tractor-semitrailer are shown in Tables
The typical condition for identification covers low speed to high speed, and the area of conditions covers linearity to nonlinearity. The range of the vehicle deceleration is from 2 to 4 m/s2, and the range of steering wheel angle is from −300 to 300°. The initial vehicle speed is 20 and 60 km/h [
Key parameter maps at the initial vehicle speed of 20 km/h: (a) cornering stiffness of the tractor’s front axle; (b) cornering stiffness of the tractor’s rear axle; (c) cornering stiffness of the semitrailer’s axle; (d) roll stiffness of the tractor’s sprung mass; (e) roll stiffness of the semitrailer’s sprung mass; (f) roll stiffness of the fifth wheel.
Key parameter maps at the initial vehicle speed of 60 km/h: (a) cornering stiffness of the tractor’s front axle; (b) cornering stiffness of the tractor’s rear axle; (c) cornering stiffness of the semitrailer’s axle; (d) roll stiffness of the tractor’s sprung mass; (e) roll stiffness of the semitrailer’s sprung mass; (f) roll stiffness of the fifth wheel.
To verify the reliability of the 5-DOF model with identified parameter maps, an experimental TST was used in accordance with the reference vehicle in building the TruckSim model, as shown in Figure
Experimental tractor and semitrailer.
Data acquisition equipment: (a) data acquisition unit; (b, c) AHRSs equipped on the tractor and semitrailer; (d) steering wheel sensor; (e) brake pedal sensor; (f) road friction coefficient sensor.
The test was carried out in the performance testing ground of Dingyuan Proving Ground for Vehicle, General Administration of Quality Supervision, Inspection and Quarantine of the P.R.C. The steering and braking experiment is designed based on the Chinese national standards GB/T 13594-2003 and GB 12676-2014. The testing ground is a flat cement area. The road friction coefficient is 0.62, measured by the Yisai-JN-1 road friction coefficient sensor in Figure
Steering and braking field test: (a) longitudinal speed; (b) steering wheel angle; (c) brake pedal force.
Attitude parameters of the TST.
Attitude parameters | Unit |
---|---|
Tractor speed | km/h |
Steering wheel angle | Rad |
Brake pedal force | N |
Tractor yaw rate | Rad/s |
Semitrailer yaw rate | Rad/s |
Tractor roll angle | Rad |
Semitrailer roll angle | Rad |
Tractor roll rate | Rad/s |
Semitrailer roll rate | Rad/s |
The original outputs of the field test were filtered by the Hamming window filter based on the Filter Design and Analysis Tool in MATLAB. The yaw rate, roll rate, and roll angle of the TST of the 5-DOF model with parameter maps were contrasted with the field test data.
The typical outputs of the TST state variables of the 5-DOF model and test vehicle are shown in Figure
5-DOF model data and field test data: (a) tractor’s roll angle; (b) semitrailer’s roll angle; (c) tractor’s roll rate; (d) semitrailer’s roll rate; (e) tractor’s yaw rate; (f) semitrailer’s yaw rate.
This paper focused on the accurate linear TST model. The following conclusions can be drawn from this study: The 5-DOF linear TST model has been presented. The roll stiffness, axle cornering stiffness, and fifth-wheel stiffness were assumed to be constant at any specified initial longitudinal speeds, longitudinal deceleration, and steering angle. The longitudinal deceleration and steering wheel angle were set as the input variable to figure out their relationships with identification parameters. The GA and MGA were applied to identify the key parameters of the 5-DOF model. The MGA was proved to be more precise than the GA. The objective function (fitness function) is the absolute value of error between the outputs of the simplified 5-DOF tractor-semitrailer model and TruckSim. The key parameter maps were formed at the initial speed of 20 km/h and 60 km/h. The identification results show that the fifth-wheel stiffness is not absolutely constant when the longitudinal deceleration and steering wheel angle change. The fifth-wheel stiffness is stable when the steering wheel angle is small. But when the steering wheel angle is close to the extreme area, the fifth-wheel stiffness is increased dramatically. Running with online interpolation of the key parameters based on the offline maps, the 5-DOF model output is compared with the field test data. The results showed that model outputs of the simplified model and real vehicle agree well.
The nonzero elements of
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The authors would like to thank the School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei, Anhui, P.R.C., as well as the National Natural Science Foundation of China (Grant No. 71431003), the Research Project of University Natural Science of Anhui Province (Grant No. KJ2018A0782), and the General Research Project of Anhui University Management Big Data Research Center (Grant No. 2016015).