Shimmy suppression method for four-wheel independent steer-by-wire system

Abstract

The invention discloses a shimmy suppression method for a four-wheel independent steer-by-wire system. The shimmy suppression method comprises the following steps: establishing a wheel shimmy kinetic model of the four-wheel independent steer-by-wire system; designing a barrier Lyapunov function, and converting the swing amplitude constraint into a state constraint condition; designing a backstepping sliding mode controller parameter self-tuning strategy based on the shimmy frequency; designing a back-stepping sliding mode controller with self-adaptive swing amplitude-frequency characteristics based on the state constraint condition; based on a designed backstepping sliding mode controller, active self-adaptive control is carried out on shimmy phenomena of wheels with different frequencies and amplitudes, and then suppression of the shimmy phenomenon of the four-wheel independent steer-by-wire system is completed. According to the amplitude-frequency characteristics of the shimmy phenomenon occurring in the operation process of the four-wheel independent steer-by-wire system, the shimmy phenomenon of different frequencies and amplitudes can be restrained in a self-adaptive mode, then the reliable operation capacity of the four-wheel independent steer-by-wire system is improved, and vehicle instability caused by excessive shimmy of wheels is avoided.

Description

Technical Field

[0001] The present invention belongs to the field of vehicle steering control, and in particular relates to a method for suppressing shimmy of a four-wheel independent steer-by-wire system. Background Art

[0002] New energy vehicles have become a crucial component of technological innovation and industrial upgrading in the automotive sector, fundamentally reducing energy consumption and environmental pollution. Chassis-by-wire control is a technological development trend that meets the demands of smart electric vehicles for higher precision and faster response speeds, and is the foundation for achieving intelligent vehicle applications and precise control.

[0003] Among them, electric vehicles with four-wheel independent steer-by-wire systems (4WIS) are driven by in-wheel motors or wheel-side motors, improving vehicle safety, handling stability, and overall energy efficiency. The four-wheel independent steer-by-wire system, which utilizes an angular module architecture, not only allows lateral travel but also rotation on the spot, bringing a revolutionary breakthrough in vehicle steering performance and representing a major disruptive innovation. Distributed drive electric vehicles achieve comprehensive integration of drive, braking, steering, and suspension. The coordinated control of the distributed drive system and chassis system significantly enhances the vehicle's maneuverability, passability, and handling stability. Four-wheel independent drive / steering electric vehicles are considered a key component of future intelligent transportation systems.

[0004] The electric wheels currently used in four-wheel independent drive-by-wire / steer systems have a larger speed range and amplitude due to the addition of in-wheel motors to each wheel, which increases the unsprung mass. Furthermore, the four-wheel independent drive-by-wire system eliminates the tie rod between the left and right wheels. At high speeds, the lack of a tie rod makes wheel shimmy more likely. Wheel shimmy directly impacts the vehicle's safe and stable operation, and can easily lead to excessive wheel shimmy and instability, resulting in serious traffic accidents. Therefore, specific measures are needed to suppress shimmy in four-wheel independent drive-by-wire systems.

[0005] In the existing research on the suppression of wheel shimmy in vehicle steering systems, for example: Chinese invention patent application number CN202310996430.2, named "A control method and control system for a wire-controlled steering system", for all currently integrated electric wheels using kingpin steering, the real-time value of the current of the magnetorheological damper arranged on the output shaft of the reducer is adjusted to obtain the value of the target damping coefficient to suppress the wheel shimmy; Chinese invention patent application number CN202210432268.7, named "A method and system for suppressing wheel shimmy based on EPS", uses the vehicle speed and wheel speed information on the CAN network to obtain the steering torque through the torque sensor. System torque information is used to calculate the EPS compensation current to offset the wheel shimmy; the Chinese invention patent application number is CN201010295311.7, and the name is “A method for controlling the shimmy of an automobile steering wheel”, in which the vehicle speed and vehicle mass are detected by sensors, and the steering wheel kingpin caster angle α is adjusted in real time to make it in the optimal kingpin caster angle state to reduce the shimmy of the automobile steering wheel; the Chinese invention patent application number is CN202110631573.4, and the name is “An electric wheel with energy recovery and multi-directional vibration reduction functions”, in which mechanical design is directly used inside the wheel to achieve vibration reduction in multiple directions of the wheel with only the same shock absorber.

[0006] However, the above existing wheel shimmy suppression methods have the following two potential problems:

[0007] First, most of them are aimed at front-wheel steering or non-independent four-wheel steering systems with steering tie rods, while there is currently little research on wheel shimmy suppression in four-wheel independent steer-by-wire systems. Considering the increased wheel instability caused by the addition of hub motors and the elimination of tie rods in four-wheel independent steer-by-wire systems, these methods are not well adapted to the system and may result in poor control effects.

[0008] Second, passive or semi-active control is often used. However, wheel shimmy suppression requires strong real-time performance and flexibility. Once designed and implemented, passive control has relatively fixed performance and is difficult to adapt to changing external conditions. Therefore, such passive or semi-active control may not meet the requirements and produce very good results. Active control methods are needed that can adjust control strategies in real time to adapt to changing conditions.

[0009] Therefore, how to fully utilize the characteristics of the four-wheel independent steer-by-wire system and propose an active control method that can better cope with the strong nonlinear and time-varying characteristics of wheel shimmy, thereby avoiding the problem of wheel over-straightening and instability during the operation of the four-wheel independent steer-by-wire vehicle, is a key problem that needs to be urgently solved in the current development process of four-wheel independent steer-by-wire technology. Summary of the Invention

[0010] In response to the shortcomings of the above-mentioned prior art, the present invention aims to provide a method for suppressing shimmy in a four-wheel independent steer-by-wire system, so as to address the problem in the prior art that, due to the lack of steering tie rod restrictions and the load of the hub motor, the wheels are more susceptible to self-excited vibration at low speeds and more intense forced vibration at high speeds. The method of the present invention takes into account the amplitude-frequency characteristics of wheel shimmy and, by designing a backstepping sliding mode controller that is adaptive to the shimmy amplitude-frequency characteristics, suppresses the shimmy amplitude of the four-wheel independent steer-by-wire system to avoid controller failure and increased instability caused by excessive amplitude. Furthermore, by adaptively optimizing the controller parameters through an optimization algorithm combined with a neural network, the method further suppresses wheel shimmy under different vehicle speeds and road conditions, thereby improving the reliable operation of the four-wheel independent steering system and preventing vehicle instability caused by excessive wheel shimmy.

[0011] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0012] A method for suppressing shimmy of a four-wheel independent steer-by-wire system according to the present invention comprises the following steps:

[0013] 1) Establish a wheel shimmy dynamics model for a four-wheel independent steer-by-wire system;

[0014] 2) Design the obstacle Lyapunov function to transform the swing amplitude constraint into a state constraint;

[0015] 3) Design a parameter self-tuning strategy for a backstepping sliding mode controller based on the oscillation frequency by using the Grey Wolf optimization algorithm and radial basis function neural network fitting;

[0016] 4) Based on the state constraints in step 2), a backstepping sliding mode controller with adaptive amplitude-frequency characteristics of the pendulum vibration is designed;

[0017] 5) Based on the backstepping sliding mode controller designed in step 4), active adaptive control is performed on the wheel shimmy phenomena of different frequencies and amplitudes, thereby suppressing the shimmy phenomenon of the four-wheel independent steer-by-wire system.

[0018] Furthermore, the step 1) specifically includes:

[0019] 11) The wheel shimmy model of the four-wheel independent steer-by-wire system is established by Lagrangian dynamic equations. The model takes into account the shimmy angle θ of the front wheel around the kingpin and the vertical vibration angle of the wheel around the longitudinal axis of the vehicle. And the three state quantities of the steering motor's rotation angle γ;

[0020] According to Lagrange's theorem, we have the following:

[0021]

[0022] Where, are generalized coordinates, T, U, and E represent the system kinetic energy, system potential energy, and system dissipated energy, respectively; t represents time. represents the time derivative, Q i represents θ, γ generalized coordinates correspond to the generalized forces respectively;

[0023] The expression of the system kinetic energy is:

[0024]

[0025] Where J0 is the moment of inertia of the wheel around the axle, J1 is the moment of inertia of the wheel assembly around the kingpin, J2 is the moment of inertia of the wheel at the hinge point connected to the frame, J3 is the moment of inertia of the equivalent steering column torsion angle, R is the wheel radius, V is the vehicle speed, are the first-order derivatives of the yaw angle, vertical vibration angle, and rotation angle of the steering motor respectively;

[0026] The expression of the system potential energy is:

[0027]

[0028] Where k1 is the equivalent torsional stiffness of the wheel around the kingpin, k2 is the equivalent angular stiffness between the wheel and the suspension, k3 is the equivalent angular stiffness between the steering motor and the steering column, and k y is the equivalent stiffness of the wheel and steering knuckle along the Y axis, k z is the equivalent stiffness of the wheel and steering knuckle along the Z axis, α is the caster angle, n is the effective length of the steering knuckle, and l is the distance between the geometric center plane of the wheel and the kingpin;

[0029] The expression of system dissipated energy is:

[0030]

[0031] Where c1 is the equivalent angular damping coefficient of the wheel assembly around the kingpin, c2 is the equivalent angular damping coefficient of the connection between the wheel and the suspension, and c3 is the equivalent angular damping coefficient between the steering motor and the steering column.

[0032] Considering the tire lateral force, the moment caused by the unbalanced mass of the wheel, and the uncertain interference, the generalized force is obtained as follows:

[0033]

[0034] Where, Q1, Q2, and Q3 are θ, The generalized force corresponding to the γ generalized coordinate, F y is the tire lateral force, Fx is the tire vertical force, M1 is the dry friction torque, M0 is the unbalanced mass torque, t m is the tire trail, τ is the wheel camber angle, Mz is the disturbance torque of the unbalanced mass on the front axle;

[0035] 12) Based on the Fiala-Bridgestone tire model, the formula for tire lateral force is derived from the tire lateral force model as follows:

[0036]

[0037] Where a, c, α1, α2, and α3 are parameters obtained from the following equations:

[0038] a=((ρV / k) 2 +(ρf / k)ω 2 ) / (ω 2 +(ρV / k) 2 )

[0039] c=((ρV / k) 2 -ρ 2 fV / k 2 ) / ((ρV / k) 2 ) / ((ρV / k) 2 +ρfω 2 / k)

[0040] α1=k

[0041] α2=-0.0668k 2 / (μG z )

[0042] α3=-0.1032k 3 / (μG z ) 2

[0043] Where k is the tire cornering stiffness, μ is the road adhesion coefficient, G z is the vertical mass of the wheel, ω is the natural frequency of the system, f is the rolling resistance coefficient, and ρ is the vertical stiffness of the tire;

[0044] 13) Based on the Lagrangian dynamic equation and the simplified tire lateral force, Equations (2)-(6) are substituted into Equation (1) to calculate the wheel shimmy dynamic model of the four-wheel independent steer-by-wire system. The dynamic equation is rewritten into a differential equation form, and then:

[0045]

[0046] Where, M0=m0Rω 2 bsinωt, m0 is the unbalanced mass of the wheel, ω t is the tire angular velocity, b is the distance from the intersection of the kingpin extension line and the ground to the wheel symmetrical longitudinal plane, M cis the dry friction amplitude, π is the circular constant, B is the wheelbase, k y is the equivalent stiffness of the wheel and steering knuckle along the Y axis, k z is the equivalent stiffness of the wheel and steering knuckle along the Z-axis, u is the overall control law of the backstepping sliding mode controller, and is the steering motor torque input of the four-wheel independent steer-by-wire system.

[0047] Furthermore, the step 2) specifically includes:

[0048] 21) Define two state variables x1 and x2, let Define the target wheel angle value x to be tracked 1d , the angle tracking error of the first subsystem of the four-wheel independent steer-by-wire system can be obtained, that is, the swing amplitude z1 as follows:

[0049] z1=x1-x 1d (10)

[0050] The obstacle Lyapunov equation V1 of the first subsystem of the four-wheel independent steer-by-wire system is defined as:

[0051]

[0052] Where V1 is an open region D defined on the origin. o A continuous positive definite function on the open region D o Each point has a first-order continuous partial derivative, log(·) is a logarithmic function, b1 is the boundary of the variable pendulum amplitude z1, is a positive constant, and |z1|<b1;

[0053] 22) Define the obstacle Lyapunov equation V2 for the second subsystem of the four-wheel independent steer-by-wire system as:

[0054]

[0055] Where, is the estimated value of the unknown positive constant, s(f) is the sliding surface that varies with frequency, b2 is the boundary of s(f), which is a positive constant, and |s(f)|<b2.

[0056] Furthermore, the step 3) specifically includes:

[0057] 31) The encirclement behavior of the gray wolf optimization algorithm is expressed as a mathematical model:

[0058]

[0059] Where, X p (t) and X(t) represent the optimal parameters and feasible parameters respectively, Dn is the distance between the optimal parameter and the feasible parameter, X(t+1) is the next position of the feasible parameter according to the optimal parameter’s encirclement behavior, A and C are the control coefficients;

[0060] Hunting behavior is represented by a mathematical model:

[0061]

[0062]

[0063] Where D α 、D β 、D δ are the distances between the three feasible parameters and the optimal parameters of α, β, and δ, A1, A2, A3, C1, C2, and C3 are different control coefficients, and X α (t), X β (t), X δ (t) is the value of the three optimal parameters α, β, and δ in the current iteration, and X1(t), X2(t), and X3(t) are the three candidate positions of the current parameters updated according to the three optimal parameters α, β, and δ in this round of iteration;

[0064] Define the objective function F as:

[0065]

[0066] Where N is the total number of samples, z1 is the controller angle tracking error;

[0067] The optimal sliding mode controller parameter data set at different frequencies is obtained by using the Grey Wolf optimization algorithm;

[0068] 32) using the parameter data set in step 31) to train a radial basis function neural network to predict the corresponding optimal control parameters by a given frequency f;

[0069] The output expression of the radial basis function neural network is as follows:

[0070]

[0071] Where y is the output vector, which contains the sliding mode controller parameters C(f), ε(f), and N(f); M is the number of hidden layer neurons; h is the bias term of the output layer, which is a constant used to adjust the overall output level; ω j is the weight of the jth hidden layer neuron; is the activation function of the jth hidden layer neuron, c j is the center of the jth hidden layer neuron, and the Gaussian radial basis function is used, which is in the form of:

[0072]

[0073] Where φ(r) is the activation function of the hidden layer neuron, r=||xc j || is the input x and the center of the hidden layer neuron c j The distance between them, σ is the width of the basis function;

[0074] The least squares method is used to minimize the mean square error (MSE), and the output of the neural network is set to:

[0075] Y=W·Φ+B (20)

[0076] Where Y is the target output matrix; W is the output layer weight matrix; Φ is the matrix calculated by the Gaussian radial basis function, whose elements are φ(||x i -c j ||); B is the bias term;

[0077] The fitting error is measured by minimizing the mean square error MSE as follows:

[0078]

[0079] The weight matrix is obtained by the following formula:

[0080] W=(Φ T Φ) -1 Φ T Y (22)

[0081] After training is completed, the given frequency f is used to input the radial basis function neural network for prediction. The output of the network is the predicted sliding mode controller parameters, which are expressed as follows:

[0082] y pred =W·φ(||fc j ||) (23)

[0083] Where y pred is the new predicted output, representing the predicted sliding mode controller parameters;

[0084] The radial basis function neural network obtained by training and fitting can predict the parameters of the sliding mode controller at a given frequency;

[0085] 33) Use the short-time Fourier transform method to analyze the frequency components of the wheel shimmy signal received by the sensor that changes with time, and use the maximum amplitude method to extract the main frequency in the frequency components as the instantaneous shimmy frequency of the system;

[0086] The shimmy signal segment intercepted by the windowing method is expressed as:

[0087] y(t)=x(t)·ω(t-t1) (24)

[0088] Where y(t) is the shimmy signal segment, x(t) is the continuous time, ω(t) is the window function, and t1 is the time offset;

[0089] For continuous time x(t), Fourier transform can be obtained:

[0090]

[0091] Where, X(t1,f v ) is the result of short-time Fourier transform, indicating that at time t1, frequency f v Spectrum information under f v is the frequency variable, is the complex exponential kernel;

[0092] Then the main frequency in the frequency component is extracted as the instantaneous frequency input of the radial basis function neural network to update the optimal sliding mode controller parameters in real time and complete the backstepping sliding mode controller parameter self-tuning.

[0093] Furthermore, the step 4) includes:

[0094] 41) According to the two state variables x1 and x2 defined in 21), the differential equation of the wheel shimmy dynamics model of the four-wheel independent steer-by-wire system is transformed into a low-order subsystem, which is expressed as:

[0095]

[0096] Where f(x1,x2) is the linear, nonlinear and coupled part of the system, is the control coefficient, d s is an unknown perturbation but bounded, and assuming |d s |≤D s , D s is the limit of the unknown disturbance;

[0097] Define the target wheel angle value x to be tracked 1d , then the rotation angle tracking error of the first subsystem can be obtained as:

[0098] z1=x1-x 1d (27)

[0099] The stability term is defined as:

[0100] S=K1z1-s(f) (28)

[0101] Where K1>0 is the stability term coefficient, s(f) is the sliding mode surface that changes with frequency;

[0102] 42) Define the virtual control quantity as:

[0103]

[0104] Where, α is the virtual control quantity of the controller;

[0105] Substituting the virtual control quantity for the actual subsystem input x2, we can obtain:

[0106]

[0107] Substituting the obstacle Lyapunov equation of the first subsystem of the four-wheel independent steer-by-wire system in step 21) into the equation, we can obtain:

[0108]

[0109] The tracking error of the actual input x2 is defined as z2, which is expressed as:

[0110] z2=x2-α (32)

[0111] The designed sliding surface is:

[0112] s(f)=C(f)z1+z2 (33)

[0113] Where C(f)>0 is the sliding surface coefficient that changes with frequency;

[0114] According to the obstacle Lyapunov equation of the second subsystem of the four-wheel independent wire-steering system in step 22), the equivalent control law u of the four-wheel independent wire-steering system is designed. eq and adaptive law as follows:

[0115]

[0116] Where h2 is a smooth function and l2 is a positive constant such that f(x1,x2)≤l2h2. is an estimate of the unknown positive constant, for The first derivative of ;

[0117] 43) Using the exponential reaching law, and replacing the sign function sgn(x) in the original exponential reaching law with the continuous function tanh(x), the sliding mode control reaching law is expressed as:

[0118]

[0119] Where ε(f)>0 and N(f)>0 are the sliding mode reaching law constants. is the first-order derivative of s(f), and tanh(x) is a continuous function;

[0120] The overall control law of the backstepping sliding mode controller is obtained as:

[0121]

[0122] Beneficial effects of the present invention:

[0123] The present invention utilizes the obstacle Lyapunov equation to enhance the restraining force at high speeds, reduce the impact at low speeds, enhance the restraint on bumpy roads, and reduce intervention on flat roads, so that the swing amplitude can be maintained within an acceptable range at different vehicle speeds and different excitation conditions, reducing the risk of controller failure due to changes in the swing amplitude.

[0124] The present invention optimizes the backstepping sliding mode controller parameters through the Grey Wolf optimization algorithm and the radial basis function neural network, so that the controller can be adjusted to the controller parameters with the best control effect under different vehicle speeds and different external excitations to reduce the poor control effect caused by the change of the oscillation frequency.

[0125] The present invention can adaptively suppress shimmy phenomena of different frequencies and amplitudes according to the amplitude-frequency characteristics of the shimmy phenomena occurring during the operation of the four-wheel independent steer-by-wire system, thereby improving the reliable operation capability of the four-wheel independent steer-by-wire system and preventing vehicle instability caused by excessive wheel shimmy. BRIEF DESCRIPTION OF THE DRAWINGS

[0126] Figure 1 Schematic diagram of the method of the present invention;

[0127] Figure 2 This is a structural diagram of the four-wheel independent steer-by-wire system of the present invention;

[0128] Figure 3 Schematic diagram of the backstepping design process in the present invention;

[0129] Figure 4 This is a schematic diagram of the gray wolf optimization algorithm process in the present invention;

[0130] Figure 5 Schematic diagram of the radial basis neural network process in the present invention. DETAILED DESCRIPTION

[0131] In order to facilitate understanding by those skilled in the art, the present invention will be further described below with reference to embodiments and drawings. The contents mentioned in the embodiments are not intended to limit the present invention.

[0132] Reference Figures 1 to 5As shown in the figure, a method for suppressing shimmy of a four-wheel independent wire-controlled steering system of the present invention is provided, wherein the housings of the steering motor controller 1 and the steering motor 2 in the four-wheel independent wire-controlled steering system adopt an integrated design, the steering motor controller 1 receives the steering angle command sent by the vehicle control unit (VCU), and controls the motor 2 to rotate in the required direction and angle; the output shaft of the steering motor 2 is connected to the first-stage planetary gear reduction mechanism 3, the first-stage planetary gear reduction mechanism 3 is connected to the second-stage worm gear reduction mechanism 4, the second-stage worm gear reduction mechanism 4 is connected to the gap adaptive adjustment module 5, so as to realize dynamic adjustment of the meshing gap between the mechanisms and compensate for the lost motion problem caused by mechanical wear; the output end of the gap adaptive adjustment module 5 is rigidly connected to the upper end of the steering arm column 6 to form a stable torque transmission path; wherein the steering angle transmission The inner ring of the sensor 7 is fixed to the upper end of the steering arm column 6 to collect steering angle data in real time and realize closed-loop steering control; the lower end of the steering arm column 6 is rigidly connected to the upper end of the steering column 8, so as to transmit the output torque of the steering motor 2 to the steering knuckle 10 to realize the steering function of the wheel; the center position of the steering knuckle 10 is respectively connected to the brake disc assembly 9, the hub motor 11, and the wheel 12. The brake disc assembly 9 is used for vehicle braking control, the hub motor 11 realizes the independent drive function, and the wheel 12 completes the final steering and driving tasks; the upper fork arm 13 and the lower fork arm 14 are respectively connected to the upper and lower arms of the steering knuckle 10 through ball hinges, and the shock absorber 14 is connected to the lower fork arm 15 through a hinge, so as to allow the shock absorber to have a certain swing space when the wheel bounces up and down, thereby playing the role of vibration reduction and supporting the vehicle body;

[0133] The method steps are as follows:

[0134] 1) Establish a wheel shimmy dynamics model for a four-wheel independent steer-by-wire system;

[0135] Wherein, the step 1) specifically includes:

[0136] 11) The wheel shimmy model of the four-wheel independent steer-by-wire system is established by Lagrangian dynamic equations. The model takes into account the shimmy angle θ of the front wheel around the kingpin and the vertical vibration angle of the wheel around the longitudinal axis of the vehicle. And the three state quantities of the steering motor's rotation angle γ;

[0137] According to Lagrange's theorem, we have the following:

[0138]

[0139] Where, are generalized coordinates, T, U, and E represent the system kinetic energy, system potential energy, and system dissipated energy, respectively; t represents time. represents the time derivative, Q i represents θ, γ generalized coordinates correspond to the generalized forces respectively;

[0140] The expression of the system kinetic energy is:

[0141]

[0142] Where J0 is the moment of inertia of the wheel around the axle, J1 is the moment of inertia of the wheel assembly around the kingpin, J2 is the moment of inertia of the wheel at the hinge point connected to the frame, J3 is the moment of inertia of the equivalent steering column torsion angle, R is the wheel radius, V is the vehicle speed, are the first-order derivatives of the yaw angle, vertical vibration angle, and rotation angle of the steering motor respectively;

[0143] The expression of the system potential energy is:

[0144]

[0145] Where k1 is the equivalent torsional stiffness of the wheel around the kingpin, k2 is the equivalent angular stiffness between the wheel and the suspension, k3 is the equivalent angular stiffness between the steering motor and the steering column, and k y is the equivalent stiffness of the wheel and steering knuckle along the Y axis, k z is the equivalent stiffness of the wheel and steering knuckle along the Z axis, α is the caster angle, n is the effective length of the steering knuckle, and l is the distance between the geometric center plane of the wheel and the kingpin;

[0146] The expression of system dissipated energy is:

[0147]

[0148] Where c1 is the equivalent angular damping coefficient of the wheel assembly around the kingpin, c2 is the equivalent angular damping coefficient of the connection between the wheel and the suspension, and c3 is the equivalent angular damping coefficient between the steering motor and the steering column.

[0149] Considering the tire lateral force, the moment caused by the unbalanced mass of the wheel, and the uncertain interference, the generalized force is obtained as follows:

[0150]

[0151] Where, Q1, Q2, and Q3 are θ, The generalized force corresponding to the γ generalized coordinate, F y is the tire lateral force, Fx is the tire vertical force, M1 is the dry friction torque, M0 is the unbalanced mass torque, t m is the tire trail, τ is the wheel camber angle, M z is the disturbance torque of the unbalanced mass on the front axle;

[0152] 12) Based on the Fiala-Bridgestone tire model, the formula for tire lateral force is derived from the tire lateral force model as follows:

[0153]

[0154] Where a, c, α1, α2, and α3 are parameters obtained from the following equations:

[0155] a=((ρV / k) 2 +(ρf / k)ω 2 ) / (ω 2 +(ρV / k) 2 )

[0156] c=((ρV / k) 2 -ρ 2 fV / k 2 ) / ((ρV / k) 2 ) / ((ρV / k) 2 +ρfω 2 / k)

[0157] α1=k

[0158] α2=-0.0668k 2 / (μG z )

[0159] α3=-0.1032k 3 / (μG z ) 2

[0160] Where k is the tire cornering stiffness, μ is the road adhesion coefficient, G z is the vertical mass of the wheel, ω is the natural frequency of the system, f is the rolling resistance coefficient, and ρ is the vertical stiffness of the tire;

[0161] 13) Based on the Lagrangian dynamic equation and the simplified tire lateral force, Equations (2)-(6) are substituted into Equation (1) to calculate the wheel shimmy dynamic model of the four-wheel independent steer-by-wire system. The dynamic equation is rewritten into a differential equation form, and then:

[0162]

[0163] Where, M0=m0Rω 2 bsinωt, m0 is the unbalanced mass of the wheel, ω t is the tire angular velocity, b is the distance from the intersection of the kingpin extension line and the ground to the wheel symmetrical longitudinal plane, M c is the dry friction amplitude, π is the circular constant, B is the wheelbase, k y is the equivalent stiffness of the wheel and steering knuckle along the Y axis, k zis the equivalent stiffness of the wheel and steering knuckle along the Z-axis, u is the overall control law of the backstepping sliding mode controller, and is the steering motor torque input of the four-wheel independent steer-by-wire system.

[0164] 2) Design the obstacle Lyapunov function to transform the swing amplitude constraint into a state constraint; specifically, the following are included:

[0165] 21) Define two state variables x1 and x2, let Define the target wheel angle value x to be tracked 1d , the angle tracking error of the first subsystem of the four-wheel independent steer-by-wire system can be obtained, that is, the swing amplitude z1 as follows:

[0166] z1=x1-x 1d (10)

[0167] The obstacle Lyapunov equation V1 of the first subsystem of the four-wheel independent steer-by-wire system is defined as:

[0168]

[0169] Where V1 is an open region D defined on the origin. o A continuous positive definite function on the open region D o Each point has a first-order continuous partial derivative, log(·) is a logarithmic function, b1 is the boundary of the variable pendulum amplitude z1, is a positive constant, and |z1|<b1;

[0170] 22) Define the obstacle Lyapunov equation V2 for the second subsystem of the four-wheel independent steer-by-wire system as:

[0171]

[0172] Where, is the estimated value of the unknown positive constant, s(f) is the sliding surface that varies with frequency, b2 is the boundary of s(f), which is a positive constant, and |s(f)|<b2.

[0173] 3) Design a parameter self-tuning strategy for the backstepping sliding mode controller based on the oscillation frequency by using the Grey Wolf optimization algorithm and radial basis function neural network fitting; specifically,

[0174] 31) The encirclement behavior of the gray wolf optimization algorithm is expressed as a mathematical model:

[0175]

[0176] Where, X p (t) and X(t) represent the optimal parameters and feasible parameters respectively, D nis the distance between the optimal parameter and the feasible parameter, X(t+1) is the next position of the feasible parameter according to the optimal parameter’s encirclement behavior, A and C are the control coefficients;

[0177] Hunting behavior is represented by a mathematical model:

[0178]

[0179]

[0180] Where D α 、D β 、D δ are the distances between the three feasible parameters and the optimal parameters of α, β, and δ, A1, A2, A3, C1, C2, and C3 are different control coefficients, and X α (t), X β (t), X δ (t) is the value of the three optimal parameters α, β, and δ in the current iteration, and X1(t), X2(t), and X3(t) are the three candidate positions of the current parameters updated according to the three optimal parameters α, β, and δ in this round of iteration;

[0181] Define the objective function F as:

[0182]

[0183] Where N is the total number of samples, z1 is the controller angle tracking error;

[0184] The optimal sliding mode controller parameter data set at different frequencies is obtained by using the Grey Wolf optimization algorithm;

[0185] 32) using the parameter data set in step 31) to train a radial basis function neural network (RBFNN) to predict the corresponding optimal control parameters by a given frequency f;

[0186] The output expression of the radial basis function neural network is as follows:

[0187]

[0188] Where y is the output vector, which contains the sliding mode controller parameters C(f), ε(f), and N(f); M is the number of hidden layer neurons; h is the bias term of the output layer, which is a constant used to adjust the overall output level; ω j is the weight of the jth hidden layer neuron; φ(||xc j ||) is the activation function of the jth hidden layer neuron, c j is the center of the jth hidden layer neuron, and the Gaussian radial basis function is used, which is in the form of:

[0189]

[0190] Where φ(r) is the activation function of the hidden layer neuron, r=||xc j || is the input x and the center of the hidden layer neuron c j The distance between them, σ is the width of the basis function;

[0191] The least squares method is used to minimize the mean square error (MSE), and the output of the neural network is set to:

[0192] Y=W·Φ+B (20)

[0193] Where Y is the target output matrix; W is the output layer weight matrix; Φ is the matrix calculated by the Gaussian radial basis function, whose elements are φ(||x i -c j ||); B is the bias term;

[0194] The fitting error is measured by minimizing the mean square error MSE as follows:

[0195]

[0196] The weight matrix is obtained by the following formula:

[0197] W=(Φ T Φ) -1 Φ T Y (22)

[0198] After training is completed, the given frequency f is used to input the radial basis function neural network for prediction. The output of the network is the predicted sliding mode controller parameters, which are expressed as follows:

[0199] y pred =W·φ(||fc j ||) (23)

[0200] Where y pred is the new predicted output, representing the predicted sliding mode controller parameters;

[0201] The radial basis function neural network obtained by training and fitting can predict the parameters of the sliding mode controller at a given frequency;

[0202] 33) Use the Short-time Fourier Transform (STFT) method to analyze the frequency components of the wheel shimmy signal received by the sensor over time, and use the maximum amplitude method to extract the main frequency in the frequency components as the instantaneous shimmy frequency of the system;

[0203] The shimmy signal segment intercepted by the windowing method is expressed as:

[0204] y(t)=x(t)·ω(t-t1) (24)

[0205] Where y(t) is the shimmy signal segment, x(t) is the continuous time, ω(t) is the window function, and t1 is the time offset;

[0206] For continuous time x(t), Fourier transform can be obtained:

[0207]

[0208] Where, X(t1,f v ) is the result of short-time Fourier transform, indicating that at time t1, frequency f v Spectrum information under f v is the frequency variable, is the complex exponential kernel;

[0209] Then the main frequency in the frequency component is extracted as the instantaneous frequency input of the radial basis function neural network to update the optimal sliding mode controller parameters in real time and complete the backstepping sliding mode controller parameter self-tuning.

[0210] 4) Based on the state constraints in step 2), a backstepping sliding mode controller with adaptive pendulum amplitude-frequency characteristics is designed; specifically, the controller includes:

[0211] 41) According to the two state variables x1 and x2 defined in 21), the differential equation of the wheel shimmy dynamics model of the four-wheel independent steer-by-wire system is transformed into a low-order subsystem, which is expressed as:

[0212]

[0213] Where f(x1,x2) is the linear, nonlinear and coupled part of the system, is the control coefficient, d s is an unknown perturbation but bounded, and assuming |d s |≤D s , D s is the limit of the unknown disturbance;

[0214] Define the target wheel angle value x to be tracked 1d , then the rotation angle tracking error of the first subsystem can be obtained as:

[0215] z1=x1-x 1d (27)

[0216] The stability term is defined as:

[0217] S=K1z1-s(f) (28)

[0218] Where K1>0 is the stability term coefficient, s(f) is the sliding mode surface that changes with frequency;

[0219] 42) Define the virtual control quantity as:

[0220]

[0221] Where, α is the virtual control quantity of the controller;

[0222] Substituting the virtual control quantity for the actual subsystem input x2, we can obtain:

[0223]

[0224] Substituting the obstacle Lyapunov equation of the first subsystem of the four-wheel independent steer-by-wire system in step 21) into the equation, we can obtain:

[0225]

[0226] The tracking error of the actual input x2 is defined as z2, which is expressed as:

[0227] z2=x2-α (32)

[0228] The designed sliding surface is:

[0229] s(f)=C(f)z1+z2 (33)

[0230] Where C(f)>0 is the sliding surface coefficient that changes with frequency;

[0231] According to the obstacle Lyapunov equation of the second subsystem of the four-wheel independent wire-steering system in step 22), the equivalent control law u of the four-wheel independent wire-steering system is designed. eq and adaptive law as follows:

[0232]

[0233] Where h2 is a smooth function and l2 is a positive constant such that f(x1,x2)≤l2h2. is an estimate of the unknown positive constant, for The first derivative of ;

[0234] 43) Using the exponential reaching law, and replacing the sign function sgn(x) in the original exponential reaching law with the continuous function tanh(x), the sliding mode control reaching law is expressed as:

[0235]

[0236] Where ε(f)>0 and N(f)>0 are the sliding mode reaching law constants. is the first-order derivative of s(f), and tanh(x) is a continuous function;

[0237] The overall control law of the backstepping sliding mode controller is obtained as:

[0238]

[0239] 5) Based on the backstepping sliding mode controller designed in step 4), active adaptive control is performed on the wheel shimmy phenomena of different frequencies and amplitudes, thereby suppressing the shimmy phenomenon of the four-wheel independent steer-by-wire system.

[0240] The present invention has many specific application paths. The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements can be made without departing from the principles of the present invention. These improvements should also be considered as the scope of protection of the present invention.

Claims

1. A method for suppressing shimmy in a four-wheel independent steer-by-wire system, characterized in that: Here are the steps: 1) Establish a wheel shimmy dynamics model for a four-wheel independent steer-by-wire system; 2) Design the obstacle Lyapunov function to transform the swing amplitude constraint into a state constraint; 3) Design a parameter self-tuning strategy for the backstepping sliding mode controller based on the oscillation frequency by using the Grey Wolf optimization algorithm and radial basis function neural network fitting; 4) Based on the state constraints in step 2), a backstepping sliding mode controller with adaptive amplitude-frequency characteristics of the pendulum vibration is designed; 5) Based on the backstepping sliding mode controller designed in step 4), active adaptive control is performed on the wheel shimmy phenomena of different frequencies and amplitudes, thereby suppressing the shimmy phenomenon of the four-wheel independent steer-by-wire system.

2. The method for suppressing shimmy of a four-wheel independent steer-by-wire system according to claim 1, characterized in that: The step 1) specifically includes: 11) The wheel shimmy model of the four-wheel independent steer-by-wire system is established by Lagrangian dynamic equations. The model takes into account the shimmy angle θ of the front wheel around the kingpin and the vertical vibration angle of the wheel around the longitudinal axis of the vehicle. And the three state quantities of the steering motor's rotation angle γ; According to Lagrange's theorem, we have the following: Where, are generalized coordinates, T, U, and E represent the system kinetic energy, system potential energy, and system dissipated energy, respectively; t represents time. represents the time derivative, Q i represents θ, γ generalized coordinates correspond to the generalized forces respectively; The expression of the system kinetic energy is: Where J0 is the moment of inertia of the wheel around the axle, J1 is the moment of inertia of the wheel assembly around the kingpin, J2 is the moment of inertia of the wheel at the hinge point connected to the frame, J3 is the moment of inertia of the equivalent steering column torsion angle, R is the wheel radius, V is the vehicle speed, are the first-order derivatives of the yaw angle, vertical vibration angle, and rotation angle of the steering motor respectively; The expression of the system potential energy is: Where k1 is the equivalent torsional stiffness of the wheel around the kingpin, k2 is the equivalent angular stiffness between the wheel and the suspension, k3 is the equivalent angular stiffness between the steering motor and the steering column, and k y is the equivalent stiffness of the wheel and steering knuckle along the Y axis, k z is the equivalent stiffness of the wheel and steering knuckle along the Z axis, α is the caster angle, n is the effective length of the steering knuckle, and l is the distance between the geometric center plane of the wheel and the kingpin; The expression of system dissipated energy is: Where c1 is the equivalent angular damping coefficient of the wheel assembly around the kingpin, c2 is the equivalent angular damping coefficient of the connection between the wheel and the suspension, and c3 is the equivalent angular damping coefficient between the steering motor and the steering column. Considering the tire lateral force, the moment caused by the unbalanced mass of the wheel, and the uncertain interference, the generalized force is obtained as follows: Where, Q1, Q2, and Q3 are θ, The generalized force corresponding to the γ generalized coordinate, F y is the tire lateral force, Fx is the tire vertical force, M1 is the dry friction torque, M0 is the unbalanced mass torque, tm is the tire trail, τ is the wheel camber angle, M z is the disturbance torque of the unbalanced mass on the front axle; 12) Based on the Fiala-Bridgestone tire model, the formula for tire lateral force is derived from the tire lateral force model as follows: Where a, c, α1, α2, and α3 are parameters obtained from the following equations: a=((ρV / k) 2 +(ρf / k)ω 2 ) / (ω 2 +(ρV / k) 2 ) c=((ρV / k) 2 -r 2 fV / k 2 ) / ((ρV / k) 2 ) / ((ρV / k) 2 +rfω 2 / k) α1＝k α2=-0.0668k 2 / (μG z ) α3=-0.1032k 3 / (μG z ) 2 Where k is the tire cornering stiffness, μ is the road adhesion coefficient, G z is the vertical mass of the wheel, ω is the natural frequency of the system, f is the rolling resistance coefficient, and ρ is the vertical stiffness of the tire; 13) Based on the Lagrangian dynamic equation and the simplified tire lateral force, Equations (2)-(6) are substituted into Equation (1) to calculate the wheel shimmy dynamic model of the four-wheel independent steer-by-wire system. The dynamic equation is rewritten into a differential equation form, and then: Where, M0=m0Rω 2 bsinωt, m0 is the unbalanced mass of the wheel, ω t is the tire angular velocity, b is the distance from the intersection of the kingpin extension line and the ground to the wheel symmetrical longitudinal plane, M c is the dry friction amplitude, π is the circular constant, B is the wheelbase, k y is the equivalent stiffness of the wheel and steering knuckle along the Y axis, k z is the equivalent stiffness of the wheel and steering knuckle along the Z-axis, u is the overall control law of the backstepping sliding mode controller, and is the steering motor torque input of the four-wheel independent steer-by-wire system.

3. The method for suppressing shimmy of a four-wheel independent steer-by-wire system according to claim 2, characterized in that: The step 2) specifically includes: 21) Define two state variables x1 and x2, let x1 = θ, Define the target wheel angle value x to be tracked 1d , the angle tracking error of the first subsystem of the four-wheel independent steer-by-wire system can be obtained, that is, the swing amplitude z1 as follows: z1=x1-x 1d (10) The obstacle Lyapunov equation V1 of the first subsystem of the four-wheel independent steer-by-wire system is defined as: Where V1 is an open region D defined on the origin. o A continuous positive definite function on the open region D o Each point has a first-order continuous partial derivative, log(·) is a logarithmic function, b1 is the boundary of the variable pendulum amplitude z1, is a positive constant, and |z1|<b1; 22) Define the obstacle Lyapunov equation V2 for the second subsystem of the four-wheel independent steer-by-wire system as: Where, is the estimated value of the unknown positive constant, s(f) is the sliding surface that varies with frequency, b2 is the boundary of s(f), which is a positive constant, and |s(f)|<b2.

4. The method for suppressing shimmy of a four-wheel independent steer-by-wire system according to claim 3, characterized in that: The step 3) specifically includes: 31) The encirclement behavior of the gray wolf optimization algorithm is expressed as a mathematical model: Where, X p (t) and X(t) represent the optimal parameters and feasible parameters respectively, D n is the distance between the optimal parameter and the feasible parameter, X(t+1) is the next position of the feasible parameter according to the optimal parameter’s encirclement behavior, A and C are the control coefficients; Hunting behavior is represented by a mathematical model: Where D α 、D β 、D δ are the distances between the three feasible parameters and the optimal parameters of α, β, and δ, A1, A2, A3, C1, C2, and C3 are different control coefficients, and X α (t), X β (t), X δ (t) is the value of the three optimal parameters α, β, and δ in the current iteration, and X1(t), X2(t), and X3(t) are the three candidate positions of the current parameters updated according to the three optimal parameters α, β, and δ in this round of iteration; Define the objective function F as: Where N is the total number of samples, z1 is the controller angle tracking error; The optimal sliding mode controller parameter data set at different frequencies is obtained by using the Grey Wolf optimization algorithm; 32) using the parameter data set in step 31) to train a radial basis function neural network to predict the corresponding optimal control parameters by a given frequency f; The output expression of the radial basis function neural network is as follows: Where y is the output vector, which contains the sliding mode controller parameters C(f), ε(f), and N(f); M is the number of hidden layer neurons; h is the bias term of the output layer, which is a constant used to adjust the overall output level; ω j is the weight of the jth hidden layer neuron; φ(||xc j ||) is the activation function of the jth hidden layer neuron, c j is the center of the jth hidden layer neuron, and the Gaussian radial basis function is used, which is in the form of: Where φ(r) is the activation function of the hidden layer neurons, r = |xc j | is the input x and the center of the hidden neuron c j The distance between them, σ is the width of the basis function; The least square method is used to minimize the mean square error, and the output of the neural network is set to: Y=W·Φ+B (20) Where Y is the target output matrix; W is the output layer weight matrix; Φ is the matrix calculated by the Gaussian radial basis function, Its elements are φ(||x i -c j ||); B is the bias term; The fitting error is measured by minimizing the mean square error MSE as follows: The weight matrix is obtained by the following formula: W=(Φ T F)- 1 F T Y (22) After training is completed, the given frequency f is used to input the radial basis function neural network for prediction. The output of the network is the predicted sliding mode controller parameters, which are expressed as follows: y pred =W·φ(||f-c j ||) (23) Where y pred is the new predicted output, representing the predicted sliding mode controller parameters; The radial basis function neural network obtained by training and fitting can predict the parameters of the sliding mode controller at a given frequency; 33) Use the short-time Fourier transform method to analyze the frequency components of the wheel shimmy signal received by the sensor that changes with time, and use the maximum amplitude method to extract the main frequency in the frequency components as the instantaneous shimmy frequency of the system; The shimmy signal segment intercepted by the windowing method is expressed as: y(t)=x(t)·ω(t-t1) (24) Where y(t) is the shimmy signal segment, x(t) is the continuous time, ω(t) is the window function, and t1 is the time offset; For continuous time x(t), Fourier transform can be obtained: Where, X(t1,f v ) is the result of short-time Fourier transform, indicating that at time t1, frequency f v Spectrum information under f v is the frequency variable, is the complex exponential kernel; Then the main frequency in the frequency component is extracted as the instantaneous frequency input of the radial basis function neural network to update the optimal sliding mode controller parameters in real time and complete the backstepping sliding mode controller parameter self-tuning.

5. The method for suppressing shimmy of a four-wheel independent steer-by-wire system according to claim 4, characterized in that: The step 4) includes: 41) According to the two state variables x1 and x2 defined in 21), the differential equation of the wheel shimmy dynamics model of the four-wheel independent steer-by-wire system is transformed into a low-order subsystem, which is expressed as: Where f(x1,x2) is the linear, nonlinear and coupled part of the system, is the control coefficient, d s is an unknown perturbation but bounded, and assuming |d s |≤D s , D s is the limit of the unknown disturbance; Define the target wheel angle value x to be tracked 1d , then the rotation angle tracking error of the first subsystem can be obtained as: z1=x1-x 1d (27) The stability term is defined as: S=K1z1-s(f) (28) Where K1>0 is the stability term coefficient, s(f) is the sliding mode surface that changes with frequency; 42) Define the virtual control quantity as: Where, α is the virtual control quantity of the controller; Substituting the virtual control quantity for the actual subsystem input x2, we can obtain: Substituting the obstacle Lyapunov equation of the first subsystem of the four-wheel independent steer-by-wire system in step 21) into the equation, we can obtain: The tracking error of the actual input x2 is defined as z2, which is expressed as: z2=x2-α (32) The designed sliding surface is: s(f)=C(f)z1+z2 (33) Where C(f)>0 is the sliding surface coefficient that changes with frequency; According to the obstacle Lyapunov equation of the second subsystem of the four-wheel independent wire-steering system in step 22), the equivalent control law u of the four-wheel independent wire-steering system is designed. eq and adaptive law as follows: Where h2 is a smooth function and l2 is a positive constant such that f(x1,x2)≤l2h2. is an estimate of the unknown positive constant, for The first derivative of ; 43) Using the exponential reaching law, and replacing the sign function sgn(x) in the original exponential reaching law with the continuous function tanh(x), the sliding mode control reaching law is expressed as: Where ε(f)>0 and N(f)>0 are the sliding mode reaching law constants. is the first-order derivative of s(f), and tanh(x) is a continuous function; The overall control law of the backstepping sliding mode controller is obtained as:

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