A Neural Network Semi-Analytical Method for Solving Vehicle Dynamics Models

Abstract

The present invention belongs to the field of intelligent transportation technology, and relates to a neural network semi-analytical method for solving vehicle dynamics models, which can be applied to various intelligent transportation fields related to vehicle dynamics models, such as vehicle assisted driving, vehicle automatic driving planning, and vehicle trajectory optimization. The present invention can use the constraint relationship in the vehicle dynamics equations to randomly select data points, which can greatly reduce the amount of calculation, improve the calculation speed, and ensure the accuracy of the solution while avoiding the dimensionality disaster, thanks to the reasonable use of the constraint relationship in the vehicle dynamics equations in the algorithm. In addition, the method can even give an accurate symbolic analytical solution to the vehicle dynamics equations, which is unimaginable in classical numerical algorithms.

Description

Technical Field

[0001] The present invention belongs to the field of intelligent transportation technology and relates to a neural network semi-analytical method for solving a vehicle dynamics model. The method can be applied to various intelligent transportation fields related to vehicle dynamics models, such as vehicle assisted driving, vehicle automatic driving planning, and vehicle trajectory optimization. Background Art

[0002] Vehicle dynamics models play a key role in various intelligent transportation fields, such as vehicle assisted driving, vehicle autonomous driving planning, and vehicle trajectory optimization. Measuring road safety plays an important role in reducing the risk of road traffic casualties. Road risk measurement is usually based on historical collision data, and risk metrics calculated using past situation data cannot correctly reflect road safety issues caused by rapidly changing events such as heavy traffic and weather conditions. In addition, due to the lack of pre-collision data, it is difficult to study conflict probabilities, predict conflicts, and issue warnings in advance. In addition, more than 90% of vehicle collisions are caused by human errors such as speeding, fatigue, drunkenness, and distracted driving. However, many traffic data analysis platforms hardly consider the driver's behavior in the accident due to the lack of vehicle information. With the rapid development of communication and vehicle-to-infrastructure technologies, dynamic traffic conflict data and vehicle internal information can be retrieved and analyzed. The performance of wireless technologies, such as dedicated short-range communications, Wi-Fi, and 5G, has been evaluated and proven to be able to carry communications between vehicles and roadside infrastructure. Using roadside infrastructure such as cameras and lidar, vehicle detection and tracking can effectively generate vehicle speed and trajectory profiles. In addition, driver behavior can be collected using cameras placed inside the vehicle and then uploaded to a roadside server for further analysis. There is a significant gap between theoretical advances and real-world practice in analyzing driving safety: 1. Driving performance measurements lack driver behavior data because the information is difficult to capture in real time or requires additional equipment inside the vehicle. 2. Road safety prediction models rarely consider the state of individual drivers, such as their emotions and attention. 3. The feature engineering step of road safety analysis requires examining multiple inputs, including driver behavior, road characteristics, speed curves, and vehicle status. Appropriate models must be established to combine different types of inputs to enhance predictive performance. Data constraints based on physical models will become the mainstream technical direction in the field of intelligent transportation. Therefore, solving the vehicle kinematic model has always been a top priority in the field of intelligent transportation.

[0003] In addition, the vehicle kinematic model can be used in autonomous driving virtual simulation technology. As a necessary tool to promote the implementation of autonomous driving, autonomous driving virtual simulation technology has been widely used in the industry. Similar to reproducing the virtual real world in the game, the autonomous driving virtual simulation system creates a virtual environment with geometric images, physical laws and even operating logic that are infinitely close to the real world, allowing autonomous driving cars to run in it and accumulate massive data required for algorithm training and verification, so as to solve the problems of high testing costs, long time, high risks and large scene limitations in reality.

[0004] Existing solutions to vehicle kinematic models are often based on Euler format, Runge-Kutta format and other research methods, or consider finite difference methods or finite element discretization in spatial direction for given geometric shapes, and first-order and second-order algorithm discretization in time. However, such numerical algorithms require gridded data points, and huge amounts of data require a lot of time. These classic algorithms are fatal flaws for high-speed vehicles that need to quickly calculate predicted data.

[0005] There has been a lot of recent work on solving partial differential equations using physically-informed neural networks (PINNs). This work approximates the solution of a partial differential equation with a neural network that takes x and t as input and outputs the solution u(x, t). Although PINNs have achieved great success in many fields, PINNs can get stuck in trivial solutions that are local minima. Just because the residual of a partial differential equation is equal to 0 does not mean that the partial differential equation solution has been found. There are many trivial solutions that satisfy the condition that the residual of the partial differential equation is 0. Adding boundary conditions and initial conditions can help PINNs escape into trivial solutions. However, if the partial differential equation is really irregular, PINNs can face propagation failures characterized by highly unbalanced partial differential equation residual fields, where very high residuals are observed in very narrow regions of the domain. Adaptive techniques are powerful tools to avoid pinning failures. Identifying high residual points in the domain can be used to prevent the propagation failure of pinning. For example, Selectnet is one of them, but it relies on a neural network that plays the role of weights on each training point to identify the importance of each point. However, SelectNet requires solving the min-max problem for both the weighted network and the solver network, and theoretically, the weight function in the min-max problem only supports the locations where the residual is maximized. Some careful training practices to balance the two networks are crucial for the success of efficient adaptive training. The simplest heuristic weight selection is to follow the residual value. In traditional grid-based adaptive PDE methods, a very classic and successful refinement strategy grid size is based on this choice. On the other hand, min-max adversarial adaptive training in SelectNet shows that the theoretical singular distribution only supports the maximum value. However, in network-based PDE solvers such as PINN, the exact assignment of weights proportional to the residual or only supporting the maximum value is far from achieving satisfactory performance.

[0006] At the same time, the precise analytical algorithm based on conventional symbolic reasoning cannot ensure the solution of arbitrary constraint problems, and there is currently no universal ODE analytical algorithm. Therefore, the precise analytical algorithm based on symbolic reasoning cannot be well applied to various intelligent transportation fields related to vehicle dynamics models such as vehicle assisted driving, vehicle automatic driving planning, and vehicle trajectory optimization. Summary of the invention

[0007] To solve the above problems, this patent provides a neural network semi-analytical method for solving the vehicle dynamics model. By using a neural network to approximate the dynamic parameters in autonomous driving, a semi-analytical solution of the dynamics in autonomous driving is obtained.

[0008] The technical solution of the present invention is as follows:

[0009] A neural network semi-analytical method for solving vehicle dynamics models is as follows:

[0010] (1) Vehicle dynamics model

[0011]

[0012]

[0013] Among them, F xr is the longitudinal force on the rear wheels of the vehicle, F yr is the lateral force on the rear wheels of the vehicle, F xf is the front wheel longitudinal force of the vehicle, F yf is the lateral force on the front wheels of the vehicle, m represents the weight of the vehicle, δ represents the turning angle of the front wheels, and v x is the longitudinal velocity of the vehicle at the center of mass. Similarly, v y is the lateral velocity of the vehicle's center of mass, represents the lateral acceleration of the vehicle's center of mass, r represents the vehicle's yaw rate, is the vehicle yaw rate, L f is the distance from the center of mass of the vehicle to the front wheel, L r is the distance from the vehicle's center of mass to the rear wheel, I z represents the moment of inertia around the z-axis. f is the cornering stiffness of the front wheel, c r is the cornering stiffness of the rear wheel, α f is the slip angle of the front wheel, where α r is the slip angle of the rear wheel.

[0014] (2) Establish a neural network model for solving vehicle dynamics models

[0015] The fully connected neural network model can be used to represent a mathematical mapping, which is the following mathematical expression:

[0016] u=∑wF(...∑wF(∑wF(ξ)+b)+b))+b where F represents the activation function, w is the weight, and b is the threshold. ξ represents the neurons of the previous layer. After the neurons ξ of the previous layer are input into this layer, they are output through the activation function F(.) of this layer and then added according to a certain weight w and a constant threshold b. The result ∑wF(ξ)+b is used as the neurons of this layer and input into the network of the next layer. The neurons ξ of the first hidden layer are obtained by adding the input layer [x, y, z, …, t] according to a certain weight w. The specific expression is as follows:

[0017] ξ=∑w(x+y+z+...+t)

[0018] where x, y, z, …, t represents each element of the input layer.

[0019] When the solution is a system of equations, there are n dependent variables, then n neural networks are needed to express the corresponding dependent variables. Of course, a special case can be used here, that is, 1 neural network with n outputs. Such a network is equivalent to multiple neural networks with the same weight threshold structure in front, but different output layer weights. For the nonlinear vehicle dynamics model proposed in this invention, it is obvious that r and v y There are two dependent variables, so two neural networks can be selected to express the two dependent variables respectively, that is, the following undetermined solutions are obtained:

[0020] r=∑wiFi(...∑wiFi(∑wiFi(ξ)+bi)+bi))+bi (2.a)

[0021] v y =∑wjFj(...∑wjF(∑wjFj(ξ)+bj)+bj))+bj (2.b)

[0022] Where wi, Fi, bi represent the weights, activation functions and thresholds in the first neural network model respectively. wj, Fj, bj represent the weights, activation functions and thresholds in the second neural network model respectively.

[0023] (3) Semi-analytical method based on neural network model

[0024] Substituting the expressions (2.a) and (2.b) corresponding to the above two neural network models into the vehicle dynamics models (1.a) and (1.b), we get a nonlinear equation system with two nonlinear equations:

[0025] Eqs1(wi,wj,bi,bj,Fi,Fj,vars,t)=0 (3.a)

[0026] Eqs2(wi,wj,bi,bj,Fi,Fj,vars,t)=0 (3.b)

[0027] Where t is the independent variable and vars represents the parameter F in the vehicle dynamics model (1.a) and (1.b). xr , F yr , F xf , F yf , m, δ, v x , ,L f , L r , I z , c r , α f .

[0028] To obtain the undetermined solutions (2.a) and (2.b), we only need to obtain the weights wi, wj and thresholds bi, bj in (2.a) and (2.b), where the activation functions Fi(.) and Fj(.) can be selected arbitrarily. When the activation function and specific parameters are given to (3.a) and (3.b), the equation group containing only weights wi, wj, thresholds bi, bj and independent variables t is obtained as follows,

[0029] Eqs1(wi,wj,bi,bj,t)=0 (4.a)

[0030] Eqs2(wi,wj,bi,bj,t)=0 (4.b)

[0031] Only by solving the nonlinear equations (4.a) and (4.b) can we obtain the unknown coefficients wi, wj, bi, bj in the unknown solution. By bringing the unknown coefficients wi, wj, bi, bj back into the expressions (2.a) and (2.b) corresponding to the two neural network models, we can get the required analytical solution.

[0032] In order to obtain the solution of wi, wj, bi, bj in the nonlinear equations (4.a) and (4.b), it is necessary to eliminate the parameter t and obtain a system of equations containing only wi, wj, bi, bj, as follows:

[0033] Eqs1(wi,wj,bi,bj)=0

[0034] Eqs2(wi,wj,bi,bj)=0

[0035] By randomly or uniformly selecting points (considering that the derived nonlinear equations (4.a) and (4.b) for the vehicle dynamics model only contain one independent variable t, points on the time series are randomly or uniformly selected). Assuming that n points are selected, a new equation system containing only wi, wj, bi, bj is obtained:

[0036] eq1(wi,wj,bi,bj)=0

[0037] eq2(wi,wj,bi,bj)=0

[0038] eq3(wi,wj,bi,bj)=0 (5) ...

[0040] eqn(wi,wj,bi,bj)=0

[0041] Case 1: The system of equations has an exact solution

[0042] For the system of equations (5), we first solve them directly. If we can obtain the coefficients of weights and thresholds that satisfy all n equations, then we can bring these weights and thresholds back to the solutions to be determined (2.a) and (2.b) to obtain an exact analytical solution.

[0043] Case 2: The system of equations has no exact solution

[0044] For equation group (5), if there is no such wi, wj, bi, bj that can make equation group (5) fully satisfied, then we can obtain approximate parameter results by seeking the least squares solution. Bring these weights and thresholds back to the undetermined solutions (2.a) and (2.b) to obtain approximate analytical solutions.

[0045] At this point, the exact analytical solution or approximate analytical solution (i.e., semi-analytical solution) of Vy and r constructed by the neural network is obtained.

[0046] Beneficial effects of the present invention: The present invention can utilize the constraint relationship in the vehicle dynamics equations to randomly select data points, which can greatly reduce the amount of calculation, improve the calculation speed, and ensure the accuracy of the solution while avoiding the dimensionality disaster, thanks to the reasonable use of the constraint relationship in the vehicle dynamics equations in the algorithm. In addition, the method can even give an accurate symbolic analytical solution to the vehicle dynamics equations, which is unimaginable in classical numerical algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 The force analysis diagram of bicycle tire is simplified from the force analysis of vehicle tire;

[0048] Figure 2 is a neural network model;

[0049] Figure 3(a) and Figure 3(b) are two neural network images selected for the dependent variables in the vehicle dynamics model;

[0050] Figure 4 This is a flow chart of the neural network semi-analytical method of the present invention.

[0051] Figure 5(a) and Figure 5(b) show the time-varying images of the two dependent variables r and Vy in the vehicle dynamics model. DETAILED DESCRIPTION

[0052] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0053] First, the simplified bicycle tire force analysis diagram ( Figure 1 ), according to Newton's second law and the torque balance equation, we have

[0054]

[0055]

[0056] Among them, F xr is the longitudinal force on the rear wheels of the vehicle, F yr is the lateral force on the rear wheels of the vehicle, F xf is the front wheel longitudinal force of the vehicle, F yf is the lateral force on the front wheels of the vehicle, m represents the weight of the vehicle, δ represents the turning angle of the front wheels, and v x is the longitudinal velocity of the vehicle at the center of mass. Similarly, v y is the lateral velocity of the vehicle's center of mass, represents the lateral acceleration of the vehicle's center of mass, r represents the vehicle's yaw rate, is the vehicle yaw angular velocity, l f is the distance from the center of mass of the vehicle to the front wheel, l r is the distance from the vehicle's center of mass to the rear wheel, I z represents the moment of inertia (around the z-axis). Next, the lateral forces of the front and rear wheels of the vehicle are analyzed as follows:

[0057] F yf =-c f α f

[0058] F yr =-c r α r

[0059] Among them, c f is the cornering stiffness of the front wheel, c r is the cornering stiffness of the rear wheel, α f is the slip angle of the front wheel, where α r is the side slip angle of the rear wheel, which can be expressed as follows:

[0060]

[0061]

[0062] The lateral force F of the front and rear wheels of the vehicle yf , F yr Substituting into equations (6.a) and (6.b) and rearranging them, we can obtain the vehicle dynamics models (2.a) and (2.b):

[0063] As we all know, fully connected neural network models (such as Figure 2 ) can be used to represent a mathematical mapping;

[0064] Next, two neural network models are used to represent Vy and r in the vehicle dynamics model respectively. Two hidden layers are selected, with two neurons in each layer, as shown in Figure 3(a) and Figure 3(b). Figure 3(a) represents the neural network corresponding to r, and Figure 3(b) represents the neural network corresponding to Vy. Let the activation function be tanh(.) to construct the expressions of the neural networks of Vy and r, and we can get:

[0065] r=b5+w 3，u tanh(tanh(tw T，2 +b2)w 2，3 +tanh(tw T，1 +b1)w 1，3 +b3)+w 4，u tanh(tanh(tw T，2 +b2)w 2，4 +tanh(tw T，1 +b1)w 1，4 +b4)

[0066] V y =b9+w 7，u tanh(tanh(tw T，6 +b6)w 6，7 +tanh(tw T，5 +b5)w 5，7 +b7)+w 8，u tanh(tanh(tw T，6 +b6)w 6，8 +tanh(tw T，5 +b5)w 5，8 +b8)

[0067] Substituting the expressions corresponding to the above two neural network models into the vehicle dynamics model and setting all coefficients (except weights and thresholds) to 1, we can obtain the following two nonlinear equations:

[0068] eq1: =1-2w 7，u tanh(tanhtw T，6 +b6)w 6，7 +tanh(tw T，5 +b5)w 5，7 +b7)-2w 8，u tanh(tanhtw T，6 +b6)w 6，8 +tanh(tw T，5 +b5)w 5，8 +b8)-w 5，8 w 8，u W T，5 -w 6，7 w 7，u w<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> -w<h2 style=";text-align:left;direction:ltr"> 6，8 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 8，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> -2b9-w<h2 style=";text-align:left;direction:ltr"> 5，7 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 7， <h2 style=";text-align:left;direction:ltr"> u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> -sin(1)+tanhtw<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +b6)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 6，7 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 7，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +tanhtw<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +b6)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 6，8 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 8，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 5，7 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 7， <h2 style=";text-align:left;direction:ltr"> u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +tanhtw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 5，8 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 8，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +tanh(tanhtw<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +b6)w<h2 style=";text-align:left;direction:ltr"> 6，7 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)w<h2 style=";text-align:left;direction:ltr"> 5，7 <h2 style=";text-align:left;direction:ltr"> +b7)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 5，7 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 7，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +tanh(tanhtw<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +b6)w<h2 style=";text-align:left;direction:ltr"> 6，7 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)w<h2 style=";text-align:left;direction:ltr"> 5，7 <h2 style=";text-align:left;direction:ltr"> +b7)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 6，7 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 7，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +tanh(tanhtw<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +b6)w<h2 style=";text-align:left;direction:ltr"> 6，8 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)w<h2 style=";text-align:left;direction:ltr"> 5，8 <h2 style=";text-align:left;direction:ltr"> +b8)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 5，8 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 8，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +tanh(tanhtw<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +b6)w<h2 style=";text-align:left;direction:ltr"> 6，8 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)w<h2 style=";text-align:left;direction:ltr"> 5，8 <h2 style=";text-align:left;direction:ltr"> +b8)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 6，8 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 8，uw T，6 -tanhtw T，6 +b6) 2 fishy T，6 +b6)w 6，7 +tanh(tw T，5 +b5)w 5，7 +b7) 2 w 6，7 w 7，u w T，6 -tanhtw T，6 +b6) 2 fishy T，6 +b6)w 6，8 +tanhtw T，5 +b5)w 5，8 +b8) 2 w 6，8 w 8，u w T，6 -tanhtw T，5 +b5) 2 fishy T，6 +b6)w 6，7 +tanh(tw T，5 +b5)w 5，7 +b7) 2 w 5，7 w 7，u w T，5 -tanhtw T，5 +b5) 2 fishy T，6 +b6)w 6，8 +tanhtw T，5 +b5)w 5，8 +b8) 2 w 5，8 w 8，u w T，5

[0069] eq2: 1-5b5-5w 3，u fishy T，2 +b2)w 2，3 +tanh(tw T，1 +b1)w 1，3 +b3)-5w 4，u fishy T，2 +b2)w 2，4 +tanh(tw T，1 +b1)w 1，4 +b4)+w 7，u fishy T，6<h2 style=";text-align:left;direction:ltr">+b6)w<h2 style=";text-align:left;direction:ltr"> 6，7 <h2 style=";text-align:left;direction:ltr"> +tanhtw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)w<h2 style=";text-align:left;direction:ltr"> 5，7 <h2 style=";text-align:left;direction:ltr"> +b7)+w<h2 style=";text-align:left;direction:ltr"> 8，u <h2 style=";text-align:left;direction:ltr"> tanh(tanhtw<h2 style=";text-align:left;direction:ltr"> T，6 <h2 style=";text-align:left;direction:ltr"> +b6)w<h2 style=";text-align:left;direction:ltr"> 6，8 <h2 style=";text-align:left;direction:ltr"> +tanhtw<h2 style=";text-align:left;direction:ltr"> T，5 <h2 style=";text-align:left;direction:ltr"> +b5)w<h2 style=";text-align:left;direction:ltr"> 5，8 <h2 style=";text-align:left;direction:ltr"> +b8)-w<h2 style=";text-align:left;direction:ltr"> 1，3 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 3，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> -w<h2 style=";text-align:left;direction:ltr"> 1，4 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 4，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> -w<h2 style=";text-align:left;direction:ltr"> 2，3 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 3，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> -w<h2 style=";text-align:left;direction:ltr"> 2，4 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 4，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +b9+2sin(1)+tanhtw<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +b2)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 2，3 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 3，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +b2)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 2，4 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 4，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> +b1)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 1，3 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 3，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> +b1)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 1，4 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 4，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> +tanh(tanh(tw<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +b2)w<h2 style=";text-align:left;direction:ltr"> 2，3 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> +b1)w<h2 style=";text-align:left;direction:ltr"> 1，3 <h2 style=";text-align:left;direction:ltr"> +b3)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 1，3 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 3，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> +tanh(tanh(tw<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +b2)w<h2 style=";text-align:left;direction:ltr"> 2，3 <h2 style=";text-align:left;direction:ltr"> +tanh(tw<h2 style=";text-align:left;direction:ltr"> T，1 <h2 style=";text-align:left;direction:ltr"> +b1)w<h2 style=";text-align:left;direction:ltr"> 1，3 <h2 style=";text-align:left;direction:ltr"> +b3)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 2，3 <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> 3，u <h2 style=";text-align:left;direction:ltr"> w<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +tanh(tanh(tw<h2 style=";text-align:left;direction:ltr"> T，2 <h2 style=";text-align:left;direction:ltr"> +b2)w<h2 style=";text-align:left;direction:ltr"> 2，4 <h2 style=";text-align:left;direction:ltr"> +tanh(twT，1 +b1)w 1，4 +b4) 2 w 1，4 w 4，u w T，1 +tanh(tanh(tw T，2 +b2)w 2，4 +tanh(tw T，1 +b1)w 1，4 +b4) 2 w 2，4 w 4，u w T，2 -tanh(tw T，2 +b2) 2 fishy T，2 +b2)w 2，3 +tanh(tw T，1 +b1)w 1，3 +b3) 2 w 2， 3w 3，u w T，2 -tanh(tw T，2 +b2) 2 fishy T，2 +b2)w 2，4 +tanh(tw T，1 +b1)w 1，4 +b4) 2 w 2，4 w 4，u w T，2 -tanh(tw T，1 +b1) 2 fishy T，2 +b2)w 2，3 +tanh(tw T，1 +b1)w 1，3 +b3) 2 w 1，3 w 3，u w T，1 -tanh(tw T，1 +b1) 2 fishy T，2 +b2)w 2，4 +tanh(tw T，1 +b1)w 1，4 +b4) 2 w 1，4 w 4，u w T，1

[0070] For this system of equations, let every point in the time series be 0, and by randomly or evenly selecting points to satisfy the equations, you can get the specific results of the parameters. At this time, you get the exact analytical solution. If the above two equations cannot be completely satisfied, you can solve their least squares solution. The whole process is as follows Figure 4 As shown. The approximate parameter results are as follows:

[0071] [3.92499825651698*10^(-18), [b[1]=0.696845183288049, b[2]=0.711573845691163, b[3]=2 .77183848458153, b[4]=9.64544443728096, b[5]=-0.311810495408534, b[6]=1.72799142287 096, b[7]=6.41713967193500, b[8]=2.23978301919359, b[9]=0.664057126942100, w[1,3]=-0 .145819390504738, w[1, 4]=0.477731527163533, w[2, 3]=1.36565019268036, w[2, 4]=0.145222 069614961, w[3, u]=-9.13673471091685*10^(-6), w[4, u]=0.864260924438631, w[5, 7]=0.091 0351074924899, w[5,8]=-1.25945300216702, w[6,7]=0.700192705924716, w[6,8]=9.2819156 2581335, w[7,u]=-0.00683236393292784, w[8,u]=-0.577960264501961, w[T,1]=0.210255832 606108, w[T, 2] = 1.77597579786027, w [T, 5] = 1.36565629654551, w [T, 6] = 0.514898742794797]]

[0072] The first parameter represents the mean square error of the least squares solution, and the remaining parameters are the weights and thresholds corresponding to the two neural network functions. Substituting them into the neural network function, we can get Vy and r to get an approximate analytical solution:

[0073] r=-0.311810495408534-0.00000913673471091685tanh(1.3656501\9268036tanh(1.7759757978602 7t+0.711573845691163)-0.145819390504738tanh(0.210255832606108t+0.696845183288049)+2.77 183848458153)+0.864260924438631tanh(0.145222069614961tanh(1.77597579786027t+0.71157384 5691163)+0.477731527163533tanh(0.210255832606108t+0.696845183288049)+9.64544443728096)

[0074] Vy=0.664057126942100-0.00683236393292784tanh(0.70019270592\4716tanh(0.51489874279479 7t+1.72799142287096)+0.0910351074924899tanh(1.36565629654551t-0.311810495408534)+6.41 713967193500)-0.577960264501961tanh(9.28191562581335tanh(0.514898742794797t+1.727991 42287096)-1.25945300216702tanh(1.36565629654551t-0.311810495408534)+2.23978301919359)

[0075] It can be seen that the analytical solution is composed of nested tanh functions, which is directly related to the tanh activation function we selected for the neural network. Figure 5(a) shows the analytical solution r, that is, the image of the change of the vehicle's yaw rate over time. Figure 5(b) shows the analytical solution Vy, that is, the image of the change of the vehicle's lateral velocity over time. As time changes, the vehicle's yaw rate and the vehicle's lateral velocity tend to 0.

Claims

1. A neural network semi-analytical method for solving vehicle dynamics models, characterized in that: The details are as follows: (1) Vehicle dynamics model Among them, F xr is the rear wheel longitudinal force of the vehicle, F xf is the longitudinal force of the front wheel of the vehicle, m represents the weight of the vehicle, δ represents the turning angle of the front wheel, v x is the longitudinal velocity of the vehicle at the center of mass. Similarly, v y is the lateral velocity of the vehicle's center of mass, r represents the vehicle's yaw rate, L f is the distance from the center of mass of the vehicle to the front wheel, L r is the distance from the vehicle's center of mass to the rear wheel, I z represents the moment of inertia around the z axis; c f is the cornering stiffness of the front wheel, c r is the cornering stiffness of the rear wheel; (2) Establish a neural network model for solving vehicle dynamics models The fully connected neural network model is used to represent a mathematical mapping, which is the following mathematical expression: u=∑wF(…∑wF(∑wF(ξ)+b)+b))+b Where F represents the activation function, w is the weight, and b is the threshold; ξ represents the neuron of the previous layer. After the neuron ξ of the previous layer is input into this layer, it is output through the activation function F(.) of this layer and then added according to a certain weight w and a constant threshold b. The result ∑wF(ξ)+b is used as the neuron of this layer and input into the network of the next layer. The neuron ξ of the first layer of the hidden layer is obtained by adding the input layer [x, y, z, …, t] according to a certain weight w. The specific expression is as follows: ξ=∑w(x+y+z+...+t) Where x, y, z, …, t represents each element of the input layer; For the vehicle dynamics model, we have r and v y There are two dependent variables, so two neural networks are selected to express the two dependent variables respectively, that is, the following undetermined solutions are obtained: r=∑wiFi(…∑wiFi(∑wiFi(ξ)+bi)+bi))+bi(2.a) v y =∑wjFj(…∑wjF(∑wjFj(ξ)+bj)+bj))+bj(2.b) Where wi, Fi, bi represent the weight, activation function and threshold in the first neural network model respectively; wj, Fj, bj represent the weight, activation function and threshold in the second neural network model respectively; (3) Semi-analytical method based on neural network model Substituting the expressions (2.a) and (2.b) corresponding to the above two neural network models into the vehicle dynamics models (1.a) and (1.b), we get a nonlinear equation system with two nonlinear equations: Eqs1(wi,wj,bi,bj,Fi,Fj,vars,t)=0(3.a) Eqs2(wi,wj,bi,bj,Fi,Fj,vars,t)=0(3.b) Where t is the independent variable and vars represents the parameter F in the vehicle dynamics model (1.a) and (1.b). xr , F xf ,m,δ,v x ,,L f, L r ,I z ,c r ; To obtain the undetermined solutions (2.a) and (2.b), we only need to obtain the weights wi, wj and thresholds bi, bj in (2.a) and (2.b), where the activation functions Fi(.) and Fj(.) can be selected arbitrarily. When the activation function and specific parameters are given to (3.a) and (3.b), the equation group containing only weights wi, wj, thresholds bi, bj and independent variable t is obtained as follows: Eqs1(wi,wj,bi,bj,t)=0(4.a) Eqs2(wi,wj,bi,bj,t)=0(4.b) Only by solving the nonlinear equations (4.a) and (4.b) can we obtain the unknown coefficients wi, wj, bi, bj in the unknown solution; by bringing the unknown coefficients wi, wj, bi, bj back into the expressions (2.a) and (2.b) corresponding to the two neural network models, we can get the required analytical solution. In order to obtain the solution of wi, wj, bi, bj in the nonlinear equations (4.a) and (4.b), it is necessary to eliminate the parameter t and obtain a system of equations containing only wi, wj, bi, bj, as follows: Eqs1(wi,wj,bi,bj)=0 Eqs2(wi,wj,bi,bj)=0 By randomly or uniformly selecting points; assuming that n points are selected, a new set of equations containing only wi, wj, bi, bj is obtained: eq1(wi,wj,bi,bj)=0 eq2(wi,wj,bi,bj)=0 eq3(wi,wj,bi,bj)=0 (5) ... eqn(wi,wj,bi,bj)=0 Case 1: The system of equations has an exact solution For the equation group (5), first directly solve it. If the coefficients of weights and thresholds that satisfy all n equations can be obtained, then these weights and thresholds are brought back to the solutions to be determined (2.a) and (2.b) to obtain an exact analytical solution. Case 2: The system of equations has no exact solution For equation group (5), if there is no wi, wj, bi, bj that can satisfy all equation group (5), then by seeking the least squares solution, an approximate parameter result is obtained; these weights and thresholds are brought back to the pending solutions (2.a) and (2.b) to obtain an approximate analytical solution.

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