Smart car research is a key topic, and path tracking refers to the core technology of safe, accurate and stable driving of smart cars along the preset highway, and the strength of its performance plays a decisive role in the driving stability of smart cars [1] . To enhance the performance of path tracking, some scholars have adopted a variety of research strategies, such as applying the four-wheel robot path tracking method to smart car target tracking, pure + pursuit method was adopted in reference [2] , the geometric shape method of circular look-ahead for tracking was adopted in reference [3] , because of the constraints of car size, driving condition, turning radius and driving dynamics, the pure geometric path tracking method is not able to solve driving constraint problems of smart cars. At present, the control strategy is often used to study high precision control of smart car path tracking, on the basis of the deviation of car position, course and target path, incremental PID control algorithm was adopted [4] , reflecting better path tracking performance. In the reference [5] , a linear time-varying model predictive control strategy was used to transform three-DOF dynamic model into two programming problems, and the simulation results showed the control strategy has high accuracy, and good driving stability in middle, low speed. Reference [6] established a feedback fuzzy controller by using the feedback prediction error method, and achieved good simulation results when the speed less than 60 km/h.

The performance of path tracking is related to the driving condition and driving parameters of the cars, and all parameters are interconnected. Under the condition of normal driving, how to cooperatively calculate the interconnection between parameters and achieve high-precision control is current key research. Some research achievements have been made in solving multi-objective control by game theory. Xie Nenggang et al. [7] - [9] successfully applied game theory to engineering design, and gave the key solving techniques of game theory in multi-objective decision-making, the formulation of game rules, the division and solving steps of strategy space of each player. In game theory, the design system variables are mapped to the spatial strategy sets that can be selected by each player through fuzzy cluster analysis. Combined with bionic game theory, this paper gives a path tracking method for smart cars. Through calculating interaction factors, it established mapping factor indicators between design variables and objective functions, solving the respective strategy spaces of each game participant, and ensuring that all game participants follow a common constraint protocol. The behavior and survival mechanism of side-blotched lizard were studied; Opportunism, egoism and collectivism were defined according to it’s own color, and the three behaviors were considered as set by the corresponding player. Based on the behavioral characteristics exhibited by each species during the evolutionary process, they establish mapping relationships between their respective adaptation functions and three objective functions to evaluate the ability of three evolved lizard individuals to adapt to nature. In the process of evolution, three types of male lateral spotted lizards evolve games with their respective fitness functions as the objective. After each round of evolutionary games, the optimal genes of the three lizards can be solved, and new chromosomes can be constructed. Convergence criteria are used to determine convergence, and after multiple evolutionary iterations, the optimal chromosome, i.e. multi-objective solution, can be obtained. The bio-mimetic lizard evolutionary game algorithm was used to solve the smoothness index, and the simulation results showed the effectiveness of this algorithm.

Neglecting air resistance during the operation of smart cars, assuming tires and ground contact is point contact, the front wheel deflection angle is small, and there is no relative sliding, combined with the reference [10] [11] , the four-wheel car model used in the paper as shown in Figure 1 , where XOY is road coordinate system, xoy is local coordinate system of smart car, and the positive x-axis is defined as the direction of car travel.

The kinematics equations of the smart cars around x-axis:

$m\ddot{x}=m\stackrel{\dot{}}{y}\stackrel{\dot{}}{\theta }+2F_{xf}+2F_{xr}$ (1)

The lateral kinematics mechanics equation around y-axis:

$m\ddot{y}=m\stackrel{\dot{}}{x}\stackrel{\dot{}}{\theta }+2F_{yf}+2F_{yt}$ (2)

The transverse oscillation equation around z-axis:

$I_z\ddot{\theta }=2a{F}_{yf}-2b{F}_{yr}$ (3)

where m is car mass, a and b are the distances from car’s center to front and rear axles. I_z is car inertia moment, θ is car heading angle, F_xf is X-axis force on the front wheel, F_xr is X-axis force on the rear wheels ; F_yf is X-axis force on the front wheel, F_yr is X-axis force on the rear wheels.

Combined with basic principles of horizontal pre-aiming error and path tracking error [11] to organize (1), (2), (3), the nonlinear dynamic model is obtained:

$\{\begin{array}{l}m\ddot{y}=-m\stackrel{\dot{}}{x}\stackrel{\dot{}}{\theta }+2\left[C_{cf}\left({\sigma }_f-\frac{\stackrel{\dot{}}{y}+a\stackrel{\dot{}}{\theta }}{\stackrel{\dot{}}{x}}\right)+C_{cr}\frac{b\stackrel{\dot{}}{\theta }-\stackrel{\dot{}}{y}}{\stackrel{\dot{}}{x}}\right]\\ m\ddot{x}=m\stackrel{\dot{}}{y}\stackrel{\dot{}}{\theta }+2\left[C_{lf}{s}_f\left({\sigma }_f-\frac{\stackrel{\dot{}}{y}+a\stackrel{\dot{}}{\theta }}{\stackrel{\dot{}}{x}}\right){\sigma }_f+C_{lr}{s}_r\right]\\ I_z\ddot{\theta }=2\left[a{C}_{ef}\left({\sigma }_f-\frac{\stackrel{\dot{}}{y}+a\stackrel{\dot{}}{\theta }}{\stackrel{\dot{}}{x}}\right)\right]\\ \stackrel{\dot{}}{Y}=\stackrel{\dot{}}{x}\sin \theta +\stackrel{\dot{}}{y}\cos \theta \\ \stackrel{\dot{}}{X}=\stackrel{\dot{}}{x}\cos \theta -\stackrel{\dot{}}{y}\sin \theta \end{array}$ (4)

where C_ef is lateral deflection stiffness, C_er is longitudinal stiffness; C_lf is longitudinal stiffness, C_lr is lateral deflection stiffness ; σ_f is tire deflection angle; s_f is the slip rate, s_r is the slip rate. Smart car path tracking has strong real-time performance, nonlinear dynamic tracking model is difficult to achieve real-time control, so the formula (4) is linearized. $\stackrel{\dot{}}{\xi }=f\left(\xi ,u\right)$, where ξ is state variable, $\xi =\left(\stackrel{\dot{}}{y},\stackrel{\dot{}}{x},\theta ,\stackrel{\dot{}}{\theta },Y,X\right)$, $u$ is control variable, $u={\sigma }_f$, we can get:

$\{\begin{array}{l}\stackrel{\dot{}}{\xi }=A\left(t\right)\xi \left(t\right)+B\left(t\right)u\left(t\right)\\ y=C\xi \left(t\right)\end{array}$ (5)

$A\left(t\right)=\left[\begin{array}{cccccc}{a}_{11}& \frac{\partial f_{\stackrel{\dot{}}{y}}}{\partial \stackrel{\dot{}}{x}}& 0& a_{14}& 0& 0\\ a_{21}& \frac{\partial f_{\stackrel{\dot{}}{x}}}{\partial \stackrel{\dot{}}{x}}& 0& a_{24}& 0& 0\\ 0& 0& 0& 1& 0& 0\\ a_{41}& \frac{\partial f_{\stackrel{\dot{}}{\theta }}}{\partial \stackrel{\dot{}}{x}}& 0& a_{44}& 0& 0\\ \cos {\theta }_t& \sin {\theta }_t& a_{53}& 0& 0& 0\\ -\sin {\theta }_t& \cos {\theta }_t& a_{63}& 0& 0& 0\end{array}\right]$; $a_{11}=\frac{-2\left(C_{cf}+C_{cr}\right)}{m{\stackrel{\dot{}}{x}}_t}$,

$a_{14}=-{\stackrel{\dot{}}{x}}_t+\frac{2\left(b{C}_{cf}-a{C}_{cr}\right)}{m{\stackrel{\dot{}}{x}}_t}$; $a_{21}=\stackrel{\dot{}}{\theta }-\frac{2C_{cf}{\sigma }_{f,t-1}}{m{\stackrel{\dot{}}{x}}_t}$; $a_{24}={\stackrel{\dot{}}{y}}_t-\frac{2a{C}_{cf}{\sigma }_{f,t-1}}{m{\stackrel{\dot{}}{x}}_t}$;

$a_{41}=\frac{2\left(b{C}_{cr}-a{C}_{cf}\right)}{I_z{\stackrel{\dot{}}{x}}_t}$; $a_{44}=\frac{-2\left(b^2{C}_{cr}+a^2{C}_{cf}\right)}{I_z{\stackrel{\dot{}}{x}}_t}$, $a_{53}={\stackrel{\dot{}}{x}}_t\cos {\theta }_t-{\stackrel{\dot{}}{y}}_t\sin {\theta }_t$,

$a_{63}=-{\stackrel{\dot{}}{y}}_t\cos {\theta }_t-{\stackrel{\dot{}}{x}}_t\sin {\theta }_t$.

$B\left(t\right)=\left[\begin{array}{cccccc}c\frac{C_{cf}}{m}& b_{12}& 0& \frac{2a{C}_{cf}}{I_z}& 0& 0\end{array}\right]$; $b_{12}=\frac{2C_{cf}\left(2{\sigma }_{f,t-1}-\frac{{\stackrel{\dot{}}{y}}_t-b{\stackrel{\dot{}}{\theta }}_t}{{\stackrel{\dot{}}{x}}_t}\right)}{m}$;

$C={\left(0,0,0,0,1,0\right)}^{\text{T}}$

Model output equation

For formula (5), set state variable $x\left(k|t=\left[\begin{array}{l}\xi \left(k|t\right)\\ u\left(k-1|t\right)\end{array}\right]\right)$, then the new state space expressed as

$\{\begin{array}{l}x\left(k+1|t\right)=\tilde{A}\left(t\right)\xi \left(k|t\right)+\tilde{B}\left(t\right)\Delta u\left(k|t\right)\\ \eta \left(k|t\right)=\tilde{C}\left(k|t\right)x\left(k|t\right)\end{array}$ (6)

In the formula, $\tilde{A}\left(k\right)=\left[\begin{array}{cc}A\left(k\right)& B\left(k\right)\\ 0_{1\times 6}& 1\end{array}\right]$, $\tilde{B}\left(k\right)=\left[\begin{array}{c}B\left(k\right)\\ 1\end{array}\right]$;

$\tilde{C}\left(k\right)={\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\end{array}\right]}^{\text{T}}$; $\eta \left(k|t\right)$ is the output of tracking prediction horizon. The output can be defined:

$Y\left(k+1|t\right)=\left[\begin{array}{c}\eta \left(k+1|k\right)\\ \eta \left(k+2|k\right)\\ ⋮\\ \eta \left(k+N_p|k\right)\end{array}\right]$ (7)

where N_p is system tracking prediction horizon. The input k is defined:

$\Delta U\left(k+1|t\right)=\left[\begin{array}{c}\Delta u\left(k\right)\\ \Delta u\left(k+1\right)\\ ⋮\\ \Delta u\left(k+m-1\right)\end{array}\right]$ (8)

Therefore, the output equation k expressed:

$Y\left(k+1|k\right)=S_{\xi }\xi \left(k\right)+S_u\Delta \left(k\right)$ (9)

where $S_{\xi }={\left(C\tilde{A}\left(k\right),C\tilde{A}{\left(k\right)}^2,\cdots ,C\tilde{A}{\left(k\right)}^{N_C},\cdots ,C\tilde{A}{\left(k\right)}^{N_p}\right)}^{\text{T}}$

$S_u=\left[\begin{array}{cccc}C\tilde{B}\left(k\right)& 0& \cdots & 0\\ C\tilde{A}\left(k\right)\tilde{B}\left(k\right)& C\tilde{B}\left(k\right)& \cdots & 0\\ ⋮& ⋮& & ⋮\\ C\tilde{A}{\left(k\right)}^{N_c-1}\tilde{B}\left(k\right)& C\tilde{A}{\left(k\right)}^{N_c-2}\tilde{B}\left(k\right)& \cdots & C\tilde{A}\left(k\right)\\ C\tilde{A}{\left(k\right)}^{N_c}\tilde{B}\left(k\right)& C\tilde{A}{\left(k\right)}^{N_c-2}\tilde{B}\left(k\right)& \cdots & C\tilde{A}\left(k\right)\tilde{B}\left(k\right)\\ ⋮& ⋮& & ⋮\\ C\tilde{A}{\left(k\right)}^{N_p-1}\tilde{B}\left(k\right)& C\tilde{A}{\left(k\right)}^{N_p-2}\tilde{B}\left(k\right)& \cdots & C\tilde{A}{\left(k\right)}^{N_p-N_c-1}\tilde{B}\left(k\right)\end{array}\right]$ (10)

where $N_c$ is system control time domain, $N_c\le N_p$.

Construction of model constraints

The manipulation stability directly affects the riding comfort, so in addition to the dynamic constraints, this paper also increases wheel deflection angle, side deflection increment constraint, to ensure the smart car’s driving stability and comfort.

Front wheel deflection angle and its incremental constraint: when tire deflection angle is less than 5˚, there is a linear proportional relationship between lateral deflection force and lateral deflection angle. In this paper, the lateral deflection angle is: $-2^{\circ }\le \delta \le +2^{\circ }$. According to physical position, the range of deflection angle changes at any time, the deflection angle and incremental constraint expressed as:

$\{\begin{array}{l}{u}_{\min }\left(k+t\right)<u\left(k+t\right)<u_{\max }\left(k+t\right)\\ \Delta u_{\min }\left(k+t\right)<\Delta u\left(k+t\right)<\Delta u_{\max }\left(k+t\right)\\ k=0,1,\cdots ,N-1\end{array}$ (11)

Formula (11) can be converted into $U_{\min }\le A\Delta U+U_f\le U_{\max }$, where

$A=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 1& 1& \cdots & 0\\ ⋮& ⋮& & ⋮\\ 1& 1& \cdots & 1\end{array}\right]$, $\Delta U={\left[\begin{array}{c}\Delta u_{\min }\left(t+1\right)\\ \Delta u_{\min }\left(t+2\right)\\ ⋮\\ \Delta u_{\min }\left(t+N\right)\end{array}\right]}_{N\times 1}$; $U_t={\left[\begin{array}{cccc}u\left(t\right)& u\left(t\right)& \cdots & u\left(t\right)\end{array}\right]}_{N\times 1}^{\text{T}}$.

In the range of linear lateral deflection angle change, the product of tire side deflection angle and tire stiffness is tire’s lateral force, and expressed as:

$\{\begin{array}{l}{F}_{yf}=2C_f{\delta }_f\\ F_{yr}=2C_r{\delta }_r\end{array}$ (12)

where

$\{\begin{array}{l}{\delta }_f={\sigma }_f-\frac{\stackrel{\dot{}}{y}+a\stackrel{\dot{}}{\theta }}{\stackrel{\dot{}}{x}}\\ {\delta }_r={\sigma }_r-\frac{\stackrel{\dot{}}{y}-b\stackrel{\dot{}}{\theta }}{\stackrel{\dot{}}{x}}\end{array}$ (13)

${\delta }_r$ is rear wheel deflection angle, ${\sigma }_f$, ${\sigma }_r$ are front and rear rotation angle. Taking $u$, $\xi$ as control quantity and state quantity respectively, the formula (12) is linearized, it can be obtained:

$\delta =E\xi \left(k,t\right)+Fu\left(k,t\right)$ (14)

where $\delta ={\left[\begin{array}{cc}{\delta }_f& {\delta }_r\end{array}\right]}^{\text{T}}$ is tire side deflection matrix; $F={\left[\begin{array}{cc}-1& 0\end{array}\right]}^T$ is transfer equations matrix; E is the output matrix.

Spatial transformation of formulas (13): $\delta =\tilde{E}\xi \left(k,t\right)$, $\tilde{E}=\left[\begin{array}{cc}E\left(t\right)& F\left(t\right)\\ 0& I_m\end{array}\right]$.

Construction of Model Predictive controller

Considering the complexity of driving process, when designing objective function, the relaxation factor is introduced, and the tracking objective function expressed:

$\begin{array}{l}J\left(\xi \left(t\right),u\left(t-1\right),\Delta U\left(t\right)\right)\\ ={\displaystyle {\sum }_{i=1}^{N_p}{\Vert \Delta \eta \left(t+i|t\right)\Vert }_Q^2}+{{\displaystyle {\sum }_{i=1}^{N_c-1}\Vert \Delta u\left(t+i|t\right)\Vert }}_R^2+\rho {\epsilon }^2\end{array}$ (15)

where $\Delta \eta \left(t+i|t\right)=\eta \left(t+i|t\right)-{\eta }_r\left(t+i|t\right)$ is the actual driving and reference path difference, $\rho$ is weight coefficient, $\epsilon$ is relaxation factor, Q and R are weighting matrix.

Based on comprehensive constraints and objective function, the smart car path control based on the dynamic can be transformed into an optimization problem, that is, each cycle needs to solve:

$\begin{array}{l}\min \Delta U,\epsilon {\displaystyle \underset{i=1}{\overset{P}{\sum }}{\Vert \eta \left(k+i|k\right)-\eta \left(k+i|k\right)\Vert }_Q^2}+{\displaystyle \underset{i=1}{\overset{N}{\sum }}{\Vert \Delta u\left(k+i|k\right)\Vert }_R^2+\rho {\epsilon }^2}\\ \{\begin{array}{l}\Delta U_{\min }\le \Delta U\le \Delta U_{\max }\\ \Delta U_{\min }\le A\Delta U+U_f\le \Delta U_{\max }\\ {\delta }_{\min }\le \delta \le {\delta }_{\max }\\ \epsilon \ge 0\end{array}\end{array}$ (16)

Barry Sinervo and Tim Stephens’ research [12] [13] found that the male side spotted lizard’s throat appears in three colors: blue, yellow, and orange, as shown in Figure 2 ; During the evolution of biological populations, different throat colors exhibit different behaviors. The male spotted lizard with yellow throat is weaker in physique than those with blue and orange throats, and its movements are secretive. When competing for sexual power with females, it often plays the role of an adulterer, commonly known as the “mistress”. The male spotted lizard with orange throat has a robust physique, hot tempered and aggressive personality, large habitat, and numerous female partners. The male spotted lizard with blue throat has a physical fitness between the yellow and the orange. When faced with foreign enemy, it can form a defend group with its companions to defend its habitat. During the evolution of biological populations, each male spotted lizard with blue throat only has one female partner and follows the principle of monogamy. In the entire evolutionary biological populations, the male spotted lizard with orange throat often engages in wars with the male spotted lizard with blue throat in search of more female mating rights. the male spotted lizard with blue throat lizard selflessly fights against the invasion for partner and territory. The male spotted lizard with orange throat has wide territory and numerous mates, it can’t fully consider territory and numerous mates. This provides an opportunity for the male spotted lizard with yellow throat lizard to launch a surprise attack. The male spotted lizard with yellow throat lizard will run to places beyond its reach and secretly mate with mates.

During the biological evolution, Barry Sinervo and other scholars found that the populations of three lizards are mutually constrained. When the male spotted lizard with blue throat dominates in reproduction, the male spotted lizard with orange throat adds dramatically; When the male spotted lizard with orange throat has an advantage in reproduction, the male spotted lizard with yellow throat lizard’s population adds dramatically, thereby suppressing the reproductive speed of the male spotted lizard with orange throat. When the male spotted lizard with yellow throat has an advantage in reproduction, the male spotted lizard with blue throat lizard’s population adds dramatically, thereby suppressing the reproductive speed of the male spotted lizard with yellow throat. The three lizard species repeat in a cycle in this way, mutually constraining each other, in order to prevent the population of single throat lizards from reproducing too quickly.

Research results show that these three male side spotted lizards inherit their own behavior patterns and pass on this evolved behavior pattern to the next generation through gene combinations. Although the behavior of the three lizards with different throats during reproduction varies greatly, they ultimately achieved “fittest survival” in Darwin’s theory of evolution, and achieved co-evolution.

A bio-mimetic lizard evolutionary algorithm is proposed based on the phenomenon of population reproduction and male side spotted lizard evolution. The algorithm’s main idea is: ① Through calculating interaction factors, it established mapping factor indicators between design variables and objective functions, solving for the respective strategy spaces of each game participant, and ensuring that all game participants follow a common constraint protocol. ② The behavior and survival mechanism of side-blotched lizard were studied; Opportunism, egoism and collectivism were defined according to it’s own color, and the three behaviors were considered as set by the corresponding player. ③ Based on the behavioral characteristics exhibited by each species during the evolutionary process, establish mapping relationships between their respective adaptation functions and three objective functions to evaluate the ability of three evolved lizard individuals to adapt to nature. ④ In the process of evolution, three male side spotted lizard evolve games with their respective fitness functions as the objective. ⑤ After each round of evolutionary games, the optimal genes of the three lizards can be solved, and new chromosomes can be constructed. Convergence criteria are used to determine convergence, and after multiple evolutionary iterations, the optimal chromosome, multi-objective solution, can be obtained.

In the construction of the evolutionary game algorithm for the three objective bio-mimetic male side spotted lizard, three core skills must be considered: ① Establishing a segmentation method for the chromosome genes of the male side spotted lizard population, as well as the mutual mapping relationship between design variables and subsets of design variables. ② Establish adaptive functions for three lizards. ③ Structure of evolutionary game algorithm for bio-mimetic lizards.

To verify the application of multi-objective bionic game in smart car path tracking, the author combined Fortran and MATLAB software to simulate, and the smart car parameters were shown in Table 1 .

To improve car’s driver-less performance, avoid the side-slip of cars in target path tracking, using the four-wheel car model after linearization as the tracking control model, a method of smart car target path tracking based on multi-objective bionic game theory is proposed. Combining the constraint relationship between front wheel deflection angle, side deflection increment constraint and tire side deflection dynamics, taking the path tracking state model in time domain as the objective function, the competitive cooperative game method is used to simulate, and the results show the validity of the algorithm. Combined with the experimental modification car, the experimental results show that the method can track the smart cars quickly and stably, the distance deviation and azimuth deviation are in a reasonable range with a good real-time performance, which provides a new research for the target path tracking of smart cars.

Project was supported by Anhui Provincial Department of Science and Technology Project (202423a11020012), Major Natural Science Research Project of Anhui Provincial Universities (Grant No.2023AH040300), Anhui Province University Science and Engineering Teachers’ Internship Program in Enterprises (Grant No.2024jsqygz90), the “Discipline Construction Quality Improvement Project” of Chaohu University (kj22zkfp01) and the Research Start-up Fund Project of Chaohu University (KYQD-2023065).