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In order to meet the demands of small race car dynamics simulation, a new method of parameter identification in the Magic Formula tire model is presented in this work, based on an analysis of the Magic Formula tire model structure. A high-precision tire model used for vehicle dynamics simulation is established via this method. It is difficult for students to build a high-precision tire model because of the complexity of widely used tire models such as Magic Formula and UniTire. At a pure side slip condition, building a lateral force model is an example, which illustrate the utilization of a multilayer feed-forward neural network to build an intelligent tire model conveniently. In order to fully understand the difference between the two models, a two-degrees-of-freedom (2 DOF) vehicle model is established. The advantages, disadvantages, and applicable scope of the two tire models are discussed after comparing the simulation results of the 2 DOF model with the Magic Formula and intelligent tire model.

The Formula Society of Automotive Engineers (FSAE) race car’s higher speed and greater cornering acceleration make tire characteristics significantly influence the vehicle’s performance [

In 1954, Fiala conducted an analysis and performed various experiments pertaining to cornering characteristics and proposed a dimensionless formula of tire characteristics based on a simplified analytical tire model [

In 1986, Shape proposed a multispoke model, and its central feature is that the carcass flexibility is associated entirely with independently acting radial spokes. Each spoke is radially and laterally flexible [

In the early 1960s, Pacejka simplified the carcass deformation as a stretched “string”, and conducted extensive theoretical and experimental research pertaining to the mechanical characteristics of tires [

In 1986, Guo proposed a pure lateral slip tire model. The contact pressure distribution was assumed to be uniform in the lateral direction and of arbitrary form in the vertical direction [

The intelligent tire model is built on the basis of a neural network and has multi-input, multioutput, and self-adaptive abilities [

In this paper, the FSAE race car of the Harbin Institute of Technology Racing Team (HRT) is taken as a research object. Firstly, a multiple-step parameter identification method is proposed, based on the analysis of the characteristics of the Magic Formula tire model structure. According to this method, all coefficients of Pacejka 2002 [

The FSAE Tire Test Consortium (FSAE TTC) was established to provide high-quality tire data to the participating FSAE teams and was used in the design and setup of race cars [

This machine at the Calspan Tire Research Facility (TIRF) is tested by TTC [

The frequency of test data collection is 50Hz, and the data channels available are elapsed time for the test (ET), road speed (

The test data of the Hoosier 18.0x6-10R25B at 14psi is analyzed in this study. The relationship between slip angle and sample number is pictured in Figure

Relationship between slip angle and sample number.

In order to clarify the relationship between the tire characteristic parameters and the six-dimensional tire force, a right-handed system is necessary. The society of Automotive Engineers (SAE) coordinate system is adopted in this paper [

The content of Pacejka 2002 is exceedingly abundant, which is not convenient to display in this paper. Please refer to [

From (

The force and moment formulas at the combined slip of the Magic Formula are based on formulas at pure side slip and longitudinal slip. Parameters of pure side slip and longitudinal slip should be identified before the combined slip. Figure

Fit the pure side slip test data and then obtain the lateral force

Fit the pure side slip test data and then obtain the aligning moment

Fit the pure longitudinal slip test data and then obtain the longitudinal force

On the basis of

On the basis of

On the basis of

General idea of parameter identification.

Overturning moment

In order to obtain a global optimum solution, a weighting function is applied to the fit. The expression of the objective function is

In the process of fitting the pure slip and combined slip, all coefficients are not fitted together, but step-by-step. Now, the specific fitting process will be introduced by taking the process of fitting the lateral force

Program flow chart.

The fitting steps in the program flow chart are as follows:

Input pure side slip raw data and crop and collapse the raw data.

Select the largest or second-largest vertical load as nominal load

Set the hard boundary of those coefficients unrelated to

Take test data at

Fit and obtain the lateral force model

Calculate the least square error

If

Set 90% and 110% of

Set the hard boundary of those coefficients unrelated to

Add test data at other

Fit all test data at nominal load and obtain the lateral force model

Calculate the least square error

If

Set 90% and 110% of

Set the hard boundary of those coefficients unrelated to

Add test data at another vertical load.

Fit all test data and obtain the lateral force model

Calculate the least square error

If

Output all coefficients of

Based on the general idea of parameter identification and step-by-step fit strategy, raw data can be processed by MATLAB or Optimum Tire [

Lateral force at pure side slip.

Aligning moment at pure side slip.

Longitudinal force at pure longitudinal slip.

Friction ellipse.

The diameter of the test tire, 18 inches, appears smaller compared with the size of the test equipment, which causes the test data points’ dispersion. As shown in Figures

In Figure

The friction ellipse is used to test the global fitting effect. Multidimensional vector space formed by the mapping between the input and output is relatively complex, so most of the fitting process will converge to a local minimum point. If the Magic Formula fitting process converges to a local minimum value, there will be a situation where the fitting results of pure side slip and longitudinal slip condition are better. But an ellipse cannot be obtained in the friction ellipse figure at the combined slip condition. As can be seen from Figure

Magic Formula is a series of composite functions composed of trigonometric and inverse trigonometric functions, which makes it very difficult to identify parameters. Even more difficult is setting the initial value, because the commonly used numerical optimization method is sensitive to the initial value. Rich experience in engineering is needed to properly estimate the initial value range of each parameter. Therefore, it is necessary to find a simpler method of tire modeling. Since Palkovics et al. applied neural network theory to tire modeling for the first time [

The multilayer feed-forward network is one of the most widely used neural networks. The back propagation (BP) network is a kind of multilayer feed-forward network, trained by the back propagation algorithm. It can be proved that a three-layer feed-forward neural network can simulate any complicated nonlinear system [

MATLAB is one of the most convenient programmable tools to train a BP neural network. Users can use the M language to write a program to train a neural network or use the graphical user interface to train a BP neural network.

Because road speed

Neural network mapping is an expression of the relationship between slip angle, camber angle, vertical load, and lateral force. It is

The number of neurons in the hidden layer of the network is determined to be 12 by empirical formula [

Structure of adopted neural network.

The number of test data samples is 27442. And 70% of them are randomly selected for training the neural network, 15% for validation, and 15% for testing.

After training, if errors distribute mostly near the zero line in the error histogram, the training process should be stopped. Otherwise, the sample proportion between training, validation, and testing, or the number of hidden neurons, should be changed for further training until the error distribution is satisfied. Then, the training results such as the network structure, network input, target output, network output, and error distribution are loaded into the MATLAB workspace, and the Simulink diagram is generated for Simulink simulation. Figure

Comparisons between test data and output of the neural network.

Vehicle model is the basis of vehicle dynamics research. It should not only be able to accurately and objectively describe the vehicle actual physical system but also be simplified as much as possible in order to calculate easily. The right model for vehicle dynamics analysis should be appropriately selected to include the dynamic motions of the vehicle that need to be studied. The focus of this study is the tire model. In order to be able to make the model accurate and easy to analyze, a 2-DOF vehicle model is adopted. The steering systems, suspension, and pitch and roll motion of vehicle are ignored in this model. In addition, the absolute speed of the vehicle remains unchanged. Assumptions are as follows:

Ignore the influence of the driving force on the tire cornering property.

Ignore the air force.

Ignore wheel load transfer.

Ignore the influence of aligning moment.

In this way, the vehicle model is simplified as a motorcycle model. Figure

Vehicle motion and force analysis diagram.

According to the laws of dynamics for rigid bodies and the SAE coordinate system [

Lateral force is closely related to tire slip angle, and the slip angle can be expressed as a function of the motion parameters of vehicle. In Figure

Simulation diagram of the nonlinear vehicle model with 2 DOF.

According to the tire coordinate system and vehicle coordinate system, the slip angle of each tire can be expressed as follows:

The term

According to previous research on tire characteristics, lateral force

In order to ensure the accuracy, convenience, and easy understanding of the vehicle model, it is divided into six parts: input module, slip angle calculation module, vertical load calculation module, tire module, lateral force conversion module, and governing equations of vehicle motion module. On the basis of the two tire models built in previous sections, the simulation diagram of a nonlinear vehicle model with 2 DOF is established, as shown in Figure

In order to complete the simulation of vehicle dynamic model with different tire models with the stepwise steering input, the specifications of the FSAE race car are listed in Table

Specifications of the FSAE race car.

Vehicle parameters | Symbol | Value |
---|---|---|

Mass | m | 230kg |

Sprung mass | | 199.1kg |

Wheelbase | L | 1.530m |

Front axle distance from C.G. | a | 0.795m |

Rear axle distance from C.G. | b | 0.735m |

Track | T | 1.140m |

Yaw moment of inertia | | 70.046kg⋅m^{2} |

Speed of the HRT race car is designed to be 0 to 100km/h. Range of the hand-wheel steering angle is

Figures

The comparison of simulation results.

Yaw rate response when speed is 5m/s.

Slip angle when tire steering angle is

Yaw rate response when speed is 15m/s.

Slip angle when tire steering angle is

Yaw rate response when speed is 25m/s.

Slip angle when tire steering angle is

As can be seen from Figure

In Figure

In Figure

From Figures

Before applying the two tire models, the maximum range of the side slip angle of the tire should be estimated according to vehicle speed, steering angle, and other parameters. If the maximum tire side slip angle exceeds the tire side slip angle test range, Magic Formula tire model should be selected. Otherwise, neural network intelligent tire model can be used.

According to the structural characteristics of the Magic Formula, an effective method is proposed to identify parameters, namely, the step-by-step fitting method. All coefficients of the Magic Formula are classified into different parts, and all parts are fitted step-by-step. This fitting process is aided by Optimum Tire software. The effectiveness of this method can be verified from the fitting results.

The neural network intelligent tire model for the tire Hoosier 18x6R25B is built on the basis of existing research and test data. It can be inferred that this tire modeling method is more efficient and has higher precision after analyzing the comparison results with the Magic Formula tire model.

After simulating the vehicle model with the Magic Formula tire model and neural network intelligent tire model, and comparing the yaw rate of vehicle, the applicable scope of the two tire models is determined. When the speed is high and the steering angle of the front tire is large, the Magic Formula tire model is more appropriate. When the speed is lower or the steering angle is small, the neural network tire model can be used in the vehicle dynamics simulation.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors are grateful to the Calspan Tire Research Facility which supply the tire test as sponsors of HIT racing team. The authors are also grateful to the HRT Formula Student Team. This study was supported in part by the National Natural Science Foundation of China (51705097), the project funded by China Postdoctoral Science Foundation (2017M621258), the project supported by the Scientific Research Foundation of Harbin Institute of Technology at Weihai (HIT(WH)201601), the Foundation of Chinese State Key Laboratory of Robotics and Systems (Grant no. SKLRS201602B), and the “111 Project” (Grant no. B07018).