CN106203684A - A kind of parameter identification for tire magic formula and optimization method - Google Patents

Abstract

The invention belongs to the technical field of vehicle dynamics parameter identification, and specifically relates to a parameter identification and optimization method for tire magic formula, comprising the following steps: 1) performing test operation according to automobile tire dynamics test, based on empirical magic formula tire model , collect the data of the corresponding independent variable and dependent variable in the formula according to different working conditions; 2) use the data collected in step 1) to identify the main parameters in the tire magic formula for the first time through the nonlinear least square method; 3) in the step On the basis of 2), the particle swarm algorithm is used to deeply optimize the parameters obtained from the initial identification to obtain parameters with higher accuracy and precision. Through the identification method, the parameters of the magic formula can be identified efficiently and quickly, which is easy to implement and has good universality.

Description

A Parameter Identification and Optimization Method for Tire Magic Formula

technical field

The invention belongs to the technical field of vehicle dynamics parameter identification, and relates to a parameter identification and optimization method for a tire magic formula.

Background technique

With the rapid growth of the automobile industry and the substantial increase in the number of cars in the market, and the rapid development of vehicle technology and computer information technology, whether it is for traditional cars or new energy vehicles, vehicle design is increasingly biased towards low-cost, high-efficiency, and informatization. etc. Information technology promotes the development of modern automobile design technology, and the intelligent, refined and rapid processing and manufacturing have improved the technological content of automobile manufacturing technology and brought about a qualitative change in traditional manufacturing technology. In particular, the application of computer simulation technology in the automobile industry greatly facilitates product design and development, improves product quality, and provides effective assistance to automobile companies and related scientific research institutions. Automotive simulation analysis technology can predict and analyze the performance and manufacturability of automotive products, thereby shortening product design and manufacturing cycles, reducing product development costs, and improving the ability of R&D and design systems to quickly respond to market changes.

Tire is the only key component that maintains direct contact between the vehicle and the road, and is an important part of the car. As a supporting and transmitting unit between the car and the road surface, its mechanical properties are the basis for studying car dynamics. The longitudinal force, lateral force, vertical load and righting moment on the tire play an important role in the dynamic performance, handling stability, braking stability, ride comfort and driving safety of the vehicle. Establishing a reasonable tire dynamics model plays a key role in product development and vehicle performance analysis, so the accuracy of the tire model can directly affect the follow-up research of automotive products.

The magic formula tire model is an empirical model formula based on test data, which can accurately describe the cornering performance of tires, and is widely used in the research of vehicle dynamics. The magic formula is based on the tire test data and fitted with a combination of trigonometric functions. This series of formulas in the same form can completely express the longitudinal force, lateral force, aligning moment, turning moment, and resistance moment on the tire. Respectively and the joint action condition of longitudinal force and lateral force.

The general expression of the magic formula is:

Y(x)=Dsin(Carctan(Bx-E(Bx-arctan(Bx))))+...

Among them, Y(x) can be the lateral force, the righting moment or the longitudinal force, and x can respectively represent the side slip angle or the longitudinal slip rate of the tire under different working conditions, and the coefficients B, C, D, E etc. are determined by the vertical load and camber angle of the tire. It can be seen that the magic formula has many parameters and is highly nonlinear, so the identification of the parameters is difficult but particularly important. At present, the genetic algorithm is often used for research on the general magic formula parameter identification problem. Although the genetic algorithm can also be identified and can approach the optimal solution in the global range, but the local search ability is poor, the convergence speed is slow, and the algorithm is real-time. Poor.

Contents of the invention

The purpose of the present invention is to overcome the deficiencies of the prior art, and propose a parameter identification and optimization method for tire magic formula. This method has strong purpose, very fast convergence speed, and high solution accuracy.

In order to use actual measurement data to fit and optimize the parameters in the magic formula, the present invention adopts the following technical solutions:

A parameter identification and optimization method for tire magic formula, comprising the following steps:

1) Carry out the test operation according to the automobile tire dynamics test, based on the empirical magic formula tire model, collect the data of the corresponding independent variable and dependent variable in the formula for different working conditions;

2) Using the data collected in step 1) to identify the main parameters in the tire magic formula for the first time through the nonlinear least squares method;

3) On the basis of step 2), the particle swarm optimization algorithm is used to deeply optimize the parameters obtained from the initial identification, to realize the adaptive adjustment of the parameters of the magic formula, and to obtain the optimized identification parameter values.

As a further improvement of the present invention, step 1) is specifically: carry out the test of the corresponding mechanical properties of the tire under different working conditions, and obtain Y(x) and x in the magic formula of the tire model under each working condition through sensor detection and collection in the test Data, including lateral force, righting moment or longitudinal force, and the side slip angle or longitudinal slip rate of the corresponding tire under different working conditions.

As a further improvement of the present invention, the nonlinear least squares method in step 2) expands the formula into a linear model through Taylor series, which includes a first-order expansion, and high-order expansions are all included in the error term, and then the minimum The quadratic regression uses the obtained estimator as a new expansion point, and then estimates the linear part, and iterates until convergence.

As a further improvement of the present invention, the particle swarm algorithm in step 3) refers to initializing a group of particles within the scope of feasible solutions, each particle represents a solution; the characteristics of the particles are represented by position and speed, and the fitness function is introduced To calculate the fitness value of the particle to represent the quality of the particle; in the solution space, update the individual position by tracking the individual extremum and the group extremum; every time the particle updates its position, recalculate the fitness value, and compare The fitness value of the new particle, the individual extremum, and the fitness value of the group extremum update the positions of the individual extremum and the group extremum;

Among them, the individual extremum refers to the optimal position of the fitness value in the position experienced by the individual, and the group extremum is the optimal position of fitness searched by all particles in the population.

As a further improvement of the present invention, the specific steps of the particle swarm algorithm are as follows:

3.1) Initialize and set the initialization factor in the particle swarm optimization algorithm;

3.2) Particle fitness value calculation;

3.3) In each iteration process, the particle updates the speed and position of the particle by searching for the individual extremum and the group extremum;

3.4) Continuously update the particle velocity and position, and update the individual extremum and group extremum according to the fitness value of the new particle; at the same time, perform a loop according to the iteration threshold and the set loop conditions set in the initialization;

If the number of iterations reaches the threshold, the iteration ends, otherwise, go to step 3.3), and update the velocity and position of each particle in turn, and finally obtain the optimized parameter value.

As a further improvement of the present invention, step 2) specific algorithm is as follows:

Suppose there are (k+1) parameters in the model β=(β ₀ ,β ₁ ,K,β _k );

First select a set of initial values: β ₀ = (β _0,0 ,β _1,0 ,K,β _k,0 ), and expand f(X,β) at β ₀ to get:

f(X,β)=f(X,β ₀ )+g ₍₀₎ ′(β-β ₍₀₎ )+R

Where g ₍₀₎ means that the first-order derivative takes value when β ₀ = (β _0,0 ,β _1,0 ,K,β _k,0 ), R is the high-order part, but only the linear part of β is reserved, the The high-order part is subsumed into the error term, and it can be obtained:

y=f(X,β)+u=f(X,β ₀ )+g ₍₀₎ ′(β-β ₍₀₎ )+R+u

＝g ₍₀₎ ′β+f(X,β ₀ )-g ₍₀₎ ′β+u ₁

Among them, the random disturbance term u1 contains u and the high _- order part in the Taylor series expansion, and a new regression model is obtained:

yf(X,β ₀ )+g ₍₀₎ ′β ₍₀₎ ＝g ₍₀₎ ′β+u

The new objective function is The least squares estimator of the model is:

β ₁ =(g ₍₀₎ g ₍₀₎ ′) ^-1 g ₍₀₎ (yf(X,β ₀ )+g ₍₀₎ ′β ₍₀₎ )

=β ₍₀₎ +(g ₍₀₎ g ₍₀₎ ′)g ₍₀₎ (yf(X,β ₍₀₎ ))

Therefore, the nonlinear least squares iterative estimation formula is:

β _j+1 = β _(j) + (g _(j) g _(j) ′)g _(j) (yf(X,β _(j) )).

As a further improvement of the present invention, step 3.1) initializes the particle population according to the magic formula tire model that needs to be identified:

In a D-dimensional search space, a population X=(X ₁ ,X ₂ ,L,X _n ) composed of n particles; the parameters that need to be set include the number of particles in the particle swarm n, acceleration factors c ₁ , c _2. Inertia weight ω and number of iterations k;

Among them, the i-th particle is expressed as a D-dimensional vector _{Xi = (X i1 ,} X _i2 ,L,X _iD ) ^T , which represents the position of the i-th particle in the D-dimensional search space and also represents a potential untie;

The initial position of the particle is initialized with reference to the parameters obtained by the nonlinear least squares method.

As a further improvement of the present invention, the specific steps of step 3.2) are as follows:

According to the objective function, the fitness value corresponding to each particle position X _i is calculated, and the speed of the i-th particle is:

V _i =(V _i1 ,V _i2 ,L,V _iD ) ^T ,

The individual extremum value is P _i =(P _i1 ,P _i2 ,L,P _iD ) ^T , and the population extremum value is P _g =(P _g1 ,P _g2 ,L,P _gD ) ^T ;

For the general magic formula, set the fitness function as follows:

Y(x)=Dsin(Carctan(Bx-E(Bx-arctan(Bx))))+...

Among them, x is the input of the equation, Y(x) is the output of the equation, and B, C, D, E are the parameters to be fitted;

The parameters of the magic formula obtained according to step 2) can be obtained through particle swarm calculation:

Y(x _j )＝D _i sin(C _i arctan(B _i x _j -E _i (B _i x _j -arctan(B _i x _j ))))+……

Among them, D _i , C _i , _Bi , E _i , etc. represent the parameter values obtained in step 2), x _j represents the jth input value of the equation, and Y(x _j ) represents the output value of the equation; the objective function introduced in combination with the test data is The root mean square of the difference between the output value of the function fitted by the parameters of each optimized identification result and the actual measured value:

$\sqrt{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; \&lsqb;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; D.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \dots </span>}<span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

Among them, n is the number of test data, B _i , C _i D _i , E _i L are the parameters corresponding to the i-th search, x _j , Y(x _j ) are the test results of the j-th test data respectively.

As a further improvement of the present invention, in step 3.3),

The velocity update formula is: V _id (k+1)＝ω×V _id (k)+c ₁ r ₁ ×[P _id (k)-X _id (k)]+c ₂ r ₂ ×[P _gd (k )-X _gd (k)];

The location update formula is: X _id (k+1)=X _id (k)+V _id (k+1);

Among them, ω is the inertial weight, d=1,2,L,D; i=1,2,L,n; k is the current iteration number; Indicates the velocity of the particle numbered id whose current number of iterations is k; Indicates the position of the example numbered id whose current number of iterations is k; Indicates the extremum of the dth population when the current number of iterations is k; c ₁ and c ₂ are non-negative constants, called acceleration factors; r ₁ and r ₂ are random numbers distributed in the [0,1] interval.

Compared with the existing parameter identification technology, the method provided by the present invention has the following advantages:

The method of the present invention is based on the tire dynamics model, collects the required data through test and test, and then based on the empirical magic formula tire model, firstly adopts the nonlinear least square method to fit the data, and iterates the magic formula parameters first identification. Then, the particle swarm optimization algorithm is used to determine the fitness function according to the characteristics of the magic formula, and the parameter identification and optimization are carried out through the particle swarm algorithm identification, so that scientific and reasonable formula parameter values can be obtained. This method has strong purpose, very fast convergence speed, and high solution accuracy. It can realize the identification and optimization of the parameters in the commonly used tire magic formula, and the obtained parameter identification values can be used to calculate and analyze the force of the tire under other working conditions and the product design of the vehicle. The specific advantages are:

1) Combining the nonlinear least squares fitting with the particle swarm optimization algorithm, the first choice is to use the nonlinear least squares method to identify the sampling data for the first time, and then use the particle swarm optimization algorithm to optimize the initial identification results and re-identify, which is different from the general identification The genetic algorithm used is different. Although the genetic algorithm can approach the optimal solution in the global range, the local search ability is poor, the convergence speed is slow, and the real-time performance of the algorithm is not good.

2) The particle swarm optimization algorithm adopted in this invention has the following advantages: the algorithm rule is simple, and the purpose is strong; the convergence speed is very fast, and there are many measures to avoid falling into the local optimum, thereby obtaining the global optimum; the solution accuracy is high, easy to accomplish.

Description of drawings

Fig. 1 is the general flowchart of parameter identification and optimization method of tire magic formula provided by the present invention;

Fig. 2 is the change curve of the particle swarm fitness value in the parameter identification process in the specific embodiment provided by the present invention;

Fig. 3 is a comparison curve after optimization using particle swarm optimization algorithm and before optimization in the specific embodiment provided by the present invention.

detailed description

As shown in Figure 1, the present invention is a parameter identification and optimization method for the tire magic formula, based on the tire dynamics model, based on the simplified empirical magic formula tire model, using the least squares method to initially identify the simplified The parameters in the final magic formula, in order to more accurately identify the parameters of the magic formula, further determine the possible value range of each parameter, use the particle swarm optimization algorithm to optimize a series of parameters obtained from the initial identification, and realize the adaptive adjustment of the parameters of the magic formula. Finally, the magic formula parameters with high accuracy and good precision are identified, which lays a solid foundation for the analysis of vehicle performance and follow-up product research.

Its characteristics specifically include the following steps:

Step 1: Data sampling in the tire model magic formula.

The corresponding mechanical properties of the tire under different working conditions are tested, and the Y(x) and x data in the magic formula of the tire model under each working condition are detected and collected through the sensor in the test. Such as lateral force, righting moment or longitudinal force, and the side slip angle or longitudinal slip rate of the tire corresponding to different working conditions.

Step 2: Use the nonlinear least squares method to identify the main parameters in the tire magic formula for the first time. Mainly through nonlinear least squares fitting of the data sampled in step 1, after data fitting, the parameter values in the magic formula are obtained for the first time identification.

The specific method of nonlinear least squares is to expand the formula into a linear model through Taylor series, that is, only the first-order expansion is included, and the higher-order expansions are all included in the error term, and then the least squares regression is performed, and the obtained The estimator is used as a new expansion point, and then the linear part is estimated. Iterates in this way until convergence. The specific algorithm design is as follows:

Suppose there are (k+1) parameters β=(β ₀ ,β ₁ ,K,β _k ) in the model.

First choose a set of initial values:

β ₀ ＝(β _0,0 ,β _1,0 ,K,β _k,0 ), expanding f(X,β) at β ₀ , we can get:

f(X,β)=f(X,β ₀ )+g ₍₀₎ ′(β-β ₍₀₎ )+R

Where g ₍₀₎ means that the first-order derivative takes value when β ₀ = (β _0,0 ,β _1,0 ,K,β _k,0 ), R is the high-order part, but only the linear part of β is reserved, the The high-order part is subsumed into the error term, and it can be obtained:

y=f(X,β)+u=f(X,β ₀ )+g ₍₀₎ ′(β-β ₍₀₎ )+R+u

＝g ₍₀₎ ′β+f(X,β ₀ )-g ₍₀₎ ′β+u ₁

Among them, the random disturbance term u1 contains u and the high _- order part in the Taylor series expansion, and a new regression model is obtained:

yf(X,β ₀ )+g ₍₀₎ ′β ₍₀₎ ＝g ₍₀₎ ′β+u

The new objective function is least squares estimator for the model

for:

Therefore, the nonlinear least squares iterative estimation formula is:

β _j+1 ＝β _(j) +(g _(j) g _(j) ′)g _(j) (yf(X,β _(j) ))

It is easy to know, from the non-linear least squares algorithm flow, it can be seen that if you want to get better results, you need to set better initial values and iteration end rules. Therefore, although the parameters fitted by the nonlinear least squares method are relatively consistent parameters, they are not necessarily the optimal parameters, and there is still room for further optimization. In order to identify the parameters of the magic formula more accurately, we propose to use the particle swarm optimization algorithm to re-optimize the parameter results obtained by the nonlinear least squares method.

Step 3: After obtaining the parameters of the initial identification of the magic formula through step 2, use the basic particle swarm optimization algorithm to re-identify and optimize the parameters of the magic formula to obtain more accurate results.

PSO algorithm first initializes a group of particles within the range of feasible solutions, and each particle represents a solution. Use the position and speed to represent the characteristics of the particle, and introduce the fitness function to calculate the fitness value of the particle to represent the quality of the particle. In the solution space, the individual position is updated by tracking the individual extremum and the population extremum. The individual extremum refers to the optimal position of the fitness value in the position experienced by the individual, and the group extremum is the optimal position of the fitness searched by all particles in the population. Every time a particle updates its position, the fitness value is recalculated, and the positions of the individual extremum and the group extremum are updated by comparing the fitness value of the new particle with the fitness value of the individual extremum and the group extremum.

Step 3.1: Initialize and set the initialization factor in the particle swarm optimization algorithm.

The parameters that need to be set include the particle number n in the particle swarm, acceleration factors c ₁ , c ₂ , inertial weight ω and iteration number k.

Assume that in a D-dimensional search space, a population X=(X ₁ ,X ₂ ,L,X _n ) composed of n particles

The i-th particle is expressed as a D-dimensional vector _{Xi = (X i1 ,} X _i2 ,L,X _iD ) ^T , which represents the position of the i-th particle in the D-dimensional search space and also represents a potential solution of the model .

The initial position of the particles in the optimization algorithm is initialized with reference to the parameters obtained by the nonlinear least squares method.

Step 3.2: Particle fitness value calculation.

According to the objective function, the fitness value corresponding to each particle position X _i can be calculated,

The velocity of the i-th particle is V _i =(V _i1 ,V _i2 ,L,V _iD ) ^T ,

Its individual extreme value is P _i =(P _i1 ,P _i2 ,L,P _iD ) ^T ,

The population extremum value of the population is: P _g =(P _g1 ,P _g2 ,L,P _gD ) ^T .

For the general magic formula, set the fitness function as follows, that is, the model to be identified:

Y(x)=Dsin(Carctan(Bx-E(Bx-arctan(Bx))))+...

Where x is the input of the equation, Y(x) is the output of the equation, and B, C, D, E, etc. are the parameters to be fitted

According to the parameters (B, C, D, E, etc.) of the magic formula obtained in step 2, it can be obtained through particle swarm calculation:

Y(x _j )＝D _i sin(C _i arctan(B _i x _j -E _i (B _i x _j -arctan(B _i x _j ))))+……

Among them, D _i , C _i , _Bi , E _i , etc. represent the parameter values obtained in step 2, x _j represents the jth input value of the equation, and Y(x _j ) represents the output value of the equation. Combined with the experimental data, it can be introduced that the objective function is the root mean square of the difference between the output value of the function fitted by the parameters of each optimization identification result and the actual measured value:

$\sqrt{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; \&lsqb;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; D.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \dots </span>}<span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

Where n is the number of test data, B _i , C _i D _i , E _i L are the parameters corresponding to the i-th search, x _j , Y(x _j ) are the test results of the j-th test data respectively.

Step 3.3: During each iteration, the particles update the velocity and position of the particle by searching for the individual extremum and the group extremum, where the particle swarm algorithm formula updates the particle's position and velocity formula:

Velocity update formula: V _id (k+1)＝ω×V _id (k)+c ₁ r ₁ ×[P _id (k)-X _id (k)]+c ₂ r ₂ ×[P _gd (k) -X _gd (k)]

Position update formula:

X _id (k+1)=X _id (k)+V _id (k+1)

Where ω is the inertial weight, d=1,2,L,D; i=1,2,L,n; k is the current number of iterations; Indicates the velocity of the particle numbered id whose current number of iterations is k; Indicates the position of the example numbered id whose current number of iterations is k; Indicates the extremum of the dth population when the current number of iterations is k; c ₁ and c ₂ are non-negative constants, called acceleration factors; r ₁ and r ₂ are random numbers distributed in the [0,1] interval.

Step 3.4: Constantly update the particle velocity and position, and update the individual extremum and group extremum according to the fitness value of the new particle. At the same time, the loop is performed according to the iteration threshold and the loop condition set in the initialization. If the number of iterations reaches the threshold, the iteration ends, otherwise, go to step 3.3, and update the velocity and position of each particle in turn, and finally obtain the optimized parameter value.

A kind of parameter identification and optimization method for tire magic formula proposed by the present invention is described in detail below in conjunction with specific embodiments:

In this embodiment, the tire model is Hoosier18×6-10R25B, the tire test pressure is 82.74kPa, the vertical load is 227N, and there is no camber angle. Taking the longitudinal force of the tire under pure braking conditions as an example, the magic formula It is applied to the relationship model between the longitudinal force on the tire, its longitudinal slip rate and the vertical load of the tire under pure braking conditions.

The relationship model in this case is:

Y(X)＝Dsin(Carctan(BX ₁ -E(BX ₁ -arctan(BX ₁ ))))+S _V

In the formula:

X ₁ ——Longitudinal force combined variable: X ₁ = (κ+S _h );

κ——longitudinal slip ratio (negative value when the vehicle brakes);

C——curve shape factor, take the value of B ₀ when calculating the longitudinal force: C=B ₀ ;

D——crest factor, indicating the maximum value of the curve: D=B ₁ F _Z ² +B ₂ F _Z ;

B——stiffness factor: B=BCD/CD;

BCD—longitudinal stiffness at zero point of longitudinal force,

S _h ——horizontal direction drift of the curve, _Sh ＝B ₉ F _z +B ₁₀ ;

S _v ——the vertical direction drift of the curve, under this working condition, S _v = 0;

E——the curvature factor of the curve, which indicates the shape near the maximum value of the curve, E=B ₆ F _z ² +B ₇ F _z +B ₈

Step 1: Data sampling in the tire model magic formula.

Carry out the tire braking dynamics test under a certain vertical load, and collect the data of different longitudinal force Y(x) and longitudinal slip rate x in the magic formula of the tire model through the sensor detection in the test.

Step 2: According to the formula in Step 1, the parameters to be identified are: {B ₀ , B ₁ , B ₂ , B ₃ , B ₄ , B ₅ , B ₆ , B ₇ , B ₈ , B ₉ , B ₁₀ } , and then the main parameters in the tire magic formula are firstly identified by nonlinear least squares method. Mainly by performing nonlinear least squares fitting on the data sampled in step 1, after data fitting, the parameter values of the magic formula obtained for the first time identification are shown in Table 1:

Table 1

Step 3: After obtaining the parameters of the initial identification of the magic formula through step 2, use the basic particle swarm optimization algorithm to re-identify and optimize the parameters of the magic formula to obtain more accurate results.

The particle population is initialized according to the magic formula tire model that needs to be identified: in a D-dimensional search space, a population X=(X ₁ ,X ₂ ,L,X _n ) composed of n particles, in which the particles in the particle population Number n, acceleration factors c ₁ , c ₂ , inertial weight ω and iteration number k; according to tire model characteristics and working conditions, take n to be 11, in order to avoid falling into local limit extremum, you can take ω=1, c ₁ =2.05, c ₂ =2.05, k=50. After initialization and generation of the initial population, the values of the 11 parameters that need to be identified will move in the solution space, and the particles usually track the individual extremum and group extremum to update the individual position.

Therefore, in this 11-dimensional search space, the population X=(X ₁ ,X ₂ ,L,X _n ) composed of n particles, where the i-th particle is expressed as a D-dimensional vector _Xi =(X _i1 ,X _i2 ,L,X _iD ) ^T , represents the position of the i-th particle in the D-dimensional search space, and also represents a potential solution of the model. According to the objective function, the fitness value corresponding to each particle position X _i can be calculated. The speed of the i-th particle is V _i =(V _i1 ,V _i2 ,L,V _iD ) ^T , and its individual extremum value is P _i =(P _i1 ,P _i2 ,L,P _iD ) ^T , the population extremum value of the population is P _g =(P _g1 ,P _g2 ,L,P _gD ) ^T .

Step 4: Particle fitness value calculation. First set the fitness function as:

Y(X)＝Dsin(Carctan(BX ₁ -E(BX ₁ -arctan(BX ₁ ))))+S _V

In the formula:

X ₁ ——Longitudinal force combined variable: X ₁ = (κ+S _h );

κ——longitudinal slip ratio (negative value when the vehicle brakes);

C——curve shape factor, take the value of B ₀ when calculating the longitudinal force: C=B ₀ ;

D——crest factor, indicating the maximum value of the curve: D=B ₁ F _Z ² +B ₂ F _Z ;

B——stiffness factor: B=BCD/CD;

BCD—longitudinal stiffness at zero point of longitudinal force,

S _h ——horizontal direction drift of the curve, _Sh ＝B ₉ F _z +B ₁₀ ;

S _v ——the vertical direction drift of the curve, under this working condition, S _v = 0;

E——the curvature factor of the curve, which indicates the shape near the maximum value of the curve, E=B ₆ F _z ² +B ₇ F _z +B ₈

Perform particle fitness calculation and search for individual extremum and group extremum:

$\sqrt{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; \&lsqb;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; D.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; V</span>}}<span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

The present invention provides a change curve of the particle swarm fitness value during the parameter identification process as shown in FIG. 2 .

Step 5: Update the speed and position of particles:

Velocity update formula

V _id (k+1)＝ω×V _id (k)+c ₁ r ₁ ×[P _id (k)-X _id (k)]+c ₂ r ₂ ×[P _gd (k)-X _gd ( k)]

Position update formula:

X _id (k+1)=X _id (k)+V _id (k+1)

Step 6: Constantly update the particle velocity and position, and update the individual extremum and group extremum according to the fitness value of the new particle. At the same time, the loop is performed according to the iteration threshold and the loop condition set in the initialization. If the number of iterations reaches the threshold, the iteration ends. Otherwise, go to step five, and update the speed and position of each particle in turn. The updated particle value is the parameter identification result optimized by the particle swarm optimization algorithm, and finally the particle swarm optimization algorithm is obtained. The final identification parameter values are shown in Table 2.

Table 2

From Figure 3 and the actual measured data in the known experiment, it can be clearly compared that the error between the optimized identification parameter value and the actual value is smaller than that before optimization, indicating that the method is available and the identification accuracy is better improved.

The above description is only a stress embodiment of the tire under a vehicle dynamics working condition of the present invention, and it does not limit the force analysis and parameter identification under the rest of the working conditions of the present invention. The parameter identification and optimization method based on the tire magic formula is scientific and effective. At the same time, the above-mentioned embodiments are not intended to limit the present invention, and the method can also be applied to parameter identification of other nonlinear models with many parameters, and has certain universality and promotion value.

Claims (9)

1. A parameter identification and optimization method for tire magic formula, is characterized in that: comprise the following steps: 1) Carry out the test operation according to the automobile tire dynamics test, based on the empirical magic formula tire model, collect the data of the corresponding independent variable and dependent variable in the formula for different working conditions; 2) Using the data collected in step 1) to identify the main parameters in the tire magic formula for the first time through the nonlinear least squares method; 3) On the basis of step 2), the particle swarm optimization algorithm is used to deeply optimize the parameters obtained from the initial identification, to realize the adaptive adjustment of the parameters of the magic formula, and to obtain the optimized identification parameter values. 2. The parameter identification and optimization method for the tire magic formula according to claim 1, characterized in that: step 1) is specifically: carry out the test of the corresponding mechanical properties of the tire under different working conditions, and pass the sensor detection in the test and The Y(x) and x data in the magic formula of the tire model are collected under each working condition, including lateral force, righting moment or longitudinal force, and the side slip angle or longitudinal slip rate of the corresponding tire under different working conditions . 3. The parameter identification and optimization method for tire magic formula according to claim 1, characterized in that: the nonlinear least squares method in step 2) is to expand the formula into a linear model by Taylor series, which includes a The first-order expansion and the higher-order expansion are all included in the error term, and then the least squares regression is performed, and the obtained estimator is used as a new expansion point, and then the linear part is estimated, and so forth until convergence. 4. The parameter identification and optimization method for tire magic formula according to claim 1, characterized in that: the particle swarm algorithm in step 3) refers to initializing a group of particles within the scope of feasible solutions, and each particle represents A solution; use the position and velocity to represent the characteristics of the particle, introduce the fitness function to calculate the fitness value of the particle to represent the quality of the particle; in the solution space, update the individual by tracking the individual extreme value and the group extreme value Position: Every time a particle updates its position, it recalculates the fitness value, and updates the positions of the individual extremum and the group extremum by comparing the fitness value of the new particle with the fitness value of the individual extremum and the group extremum; Among them, the individual extremum refers to the optimal position of the fitness value in the position experienced by the individual, and the group extremum is the optimal position of fitness searched by all particles in the population. 5. The parameter identification and optimization method for tire magic formula according to claim 3, characterized in that: the specific steps of particle swarm algorithm are as follows: 3.1) Initialize and set the initialization factor in the particle swarm optimization algorithm; 3.2) Particle fitness value calculation; 3.3) In each iteration process, the particle updates the speed and position of the particle by searching for the individual extremum and the group extremum; 3.4) Continuously update the particle velocity and position, and update the individual extremum and group extremum according to the fitness value of the new particle; at the same time, perform a loop according to the iteration threshold and the set loop conditions set in the initialization; If the number of iterations reaches the threshold, the iteration ends, otherwise, go to step 3.3), and update the velocity and position of each particle in turn, and finally obtain the optimized parameter value. 6. The parameter identification and optimization method for tire magic formula according to claim 5, characterized in that: step 2) specific algorithm is as follows: Suppose there are (k+1) parameters in the model β=(β ₀ ,β ₁ ,K,β _k ); First select a set of initial values: β ₀ = (β _0,0 ,β _1,0 ,K,β _k,0 ), and expand f(X,β) at β ₀ to get: f(X,β)=f(X,β ₀ )+g ₍₀₎ ′(β-β ₍₀₎ )+R Where g ₍₀₎ means that the first-order derivative takes value when β ₀ = (β _0,0 ,β _1,0 ,K,β _k,0 ), R is the high-order part, but only the linear part of β is reserved, the The high-order part is subsumed into the error term, and it can be obtained: y=f(X,β)+u=f(X,β ₀ )+g ₍₀₎ ′(β-β ₍₀₎ )+R+u ＝g ₍₀₎ ′β+f(X,β ₀ )-g ₍₀₎ ′β+u ₁ Among them, the random disturbance term u1 contains u and the high _- order part in the Taylor series expansion, and a new regression model is obtained: yf(X,β ₀ )+g ₍₀₎ ′β ₍₀₎ ＝g ₍₀₎ ′β+u The new objective function is The least squares estimator for the model is: β ₁ =(g ₍₀₎ g ₍₀₎ ′) ⁻¹ g ₍₀₎ (yf(X, β ₀ )+g ₍₀₎ ′β ₍₀₎ ) =β ₍₀₎ +(g ₍₀₎ g ₍₀₎ ')g ₍₀₎ (yf(X,β ₍₀₎ )) Therefore, the nonlinear least squares iterative estimation formula is: β _j+1 = β _(j) + (g _(j) g _(j) ′) g _(j) (yf(X, β _(j) )). 7. The parameter identification and optimization method for tire magic formula according to claim 6, characterized in that: step 3.1) initializes the particle population according to the magic formula tire model that needs to be identified: In a D-dimensional search space, a population X=(X ₁ , X ₂ , L, X _n ) composed of n particles; the parameters that need to be set include the number of particles in the particle swarm n, acceleration factors c ₁ , c _2. Inertia weight ω and number of iterations k; Among them, the i-th particle is expressed as a D-dimensional vector Xi = (X _i1 , _Xi2 , L, X _iD ) ^T , which represents the position of the _i -th particle in the D-dimensional search space and also represents a potential untie; The initial position of the particle is initialized with reference to the parameters obtained by the nonlinear least squares method. 8. The parameter identification and optimization method for tire magic formula according to claim 7, characterized in that: step 3.2) The specific steps are as follows: According to the objective function, the fitness value corresponding to each particle position X _i is calculated, and the speed of the i-th particle is: V _i =(V _i1 , V _i2 , L, V _iD ) ^T , Its individual extremum value is P _i = (P _i1 , P _i2 , L, P _iD ) ^T , and the group extremum value of the population is P _g = (P _g1 , P _g2 , L, P _gD ) ^T ; For the general magic formula, set the fitness function as follows: Y(x)＝D sin(C arctan(Bx-E(Bx-arctan(Bx))))+... Among them, x is the input of the equation, Y(x) is the output of the equation, and B, C, D, E are the parameters to be fitted; The parameters of the magic formula obtained according to step 2) can be obtained through particle swarm calculation: Y(x _j )＝D _i sin(C _i arctan(B _i x _j -E _i (B _i x _j -arctan(B _i x _j ))))+…… Among them, D _i , C _i , _Bi , E _i , etc. represent the parameter values obtained in step 2), x _j represents the jth input value of the equation, and Y(x _j ) represents the output value of the equation; the objective function introduced in combination with the test data is The root mean square of the difference between the output value of the function fitted by the parameters of each optimized identification result and the actual measured value: $\sqrt{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; \&lsqb;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; D.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \dots </span>}<span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$ Among them, n is the number of test data, B _i , C _i D _i , E _i L are the parameters corresponding to the i-th search, x _j , Y(x _j ) are the test results of the j-th test data respectively. 9. The parameter identification and optimization method for tire magic formula according to claim 8, characterized in that: in step 3.3), The velocity update formula is: V _id (k+1)＝ω×V _id (k)+c ₁ r ₁ ×[P _id (k)-X _id (k)]+c ₂ r ₂ ×[P _gd (k )-X _gd (k)]; The location update formula is: X _id (k+1)=X _id (k)+V _id (k+1); Among them, ω is the inertial weight, d=1,2,L,D; i=1,2,L,n; k is the current iteration number; Indicates the velocity of the particle numbered id whose current number of iterations is k; Indicates the position of the example numbered id whose current number of iterations is k; Indicates the extremum of the dth population when the current number of iterations is k; c ₁ and c ₂ are non-negative constants, called acceleration factors; r ₁ and r ₂ are random numbers distributed in the [0,1] interval.