A cooperative robust control method for vehicle motion considering model uncertainties

Abstract

This invention discloses a robust vehicle motion cooperative control method considering model uncertainty, relating to the field of vehicle cooperative control. A two-degree-of-freedom model of the vehicle is established; a nonlinear tire model is used to describe the interaction between the tire and the ground, and the lateral force of the wheel is calculated using the nonlinear tire; based on the driver's steering wheel angle, a control tracking reference quantity is generated through the two-degree-of-freedom model; vehicle state constraints and control input constraints are set, the control objective is defined as the tracking reference quantity and model uncertainty is suppressed, and an error tracking system is constructed; with minimizing a preset objective function as the optimization objective, the nominal trajectory and nominal control sequence are solved, and tightened state constraints and control constraints are introduced; a robust control law composed of feedforward and feedback components is designed, and control commands for the rear wheel steering angle and the additional motor torque of the wheel are generated; robust vehicle motion cooperative control is implemented according to the control commands. This invention aims to improve the handling stability of vehicles under complex operating conditions.

Description

Technical Field

[0001] This invention relates to the field of vehicle cooperative control, and more specifically to a vehicle motion cooperative robust control method that takes into account model uncertainties. Background Technology

[0002] Vehicle motion control technology plays a crucial role in improving driving comfort and enhancing stability under complex road conditions, serving as a fundamental pillar for ensuring safe driving and achieving all-weather autonomous driving. However, on icy roads or during emergency turns, tires can easily enter a saturation zone, resulting in insufficient tire force to guarantee safe vehicle operation. Many studies employ torque vector control (TVC) to guide the vehicle to track the desired yaw rate and stabilize the sideslip angle, thereby ensuring overall vehicle stability. However, existing methods have two significant limitations: first, they do not adequately consider the coordination between the vehicle's drive and steering subsystems; second, they do not account for the errors and uncertainties between the control-oriented model and the actual vehicle.

[0003] Among existing vehicle handling stability control methods, patent "CN109398361B" provides a handling stability control method for four-wheel independent drive vehicles. This method designs a vehicle motion controller including a longitudinal controller and a yaw controller, controls the actual angular velocity of the wheels to track the target angular velocity, and coordinates the torque distribution of each motor. This method can improve the handling stability of four-wheel independent drive vehicles; however, it only utilizes the vehicle's drive subsystem. Patent "CN116279409A" invented a cooperative control method for four-wheel independent drive and steering electric vehicles. It constructs an optimization problem based on the current vehicle motion state and the expected value of the yaw rate, solves the optimization problem, and obtains the rear wheel steering angle and additional yaw moment. Although this invention coordinates the vehicle drive and rear wheel steering subsystems, it does not consider the errors and uncertainties between the established mathematical model and the actual vehicle. Summary of the Invention

[0004] In view of this, the present invention provides a vehicle motion cooperative robust control method that considers model uncertainties, aiming to improve the handling stability of vehicles under complex operating conditions. This method simultaneously considers vehicle state constraints, input constraints, and modeling errors.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A cooperative robust control method for vehicle motion considering model uncertainties includes the following steps:

[0007] A two-degree-of-freedom model of the vehicle is established based on its lateral and yaw motions.

[0008] A nonlinear tire model is used to describe the interaction between the tire and the ground, and the lateral force of the wheel is calculated using a nonlinear tire model.

[0009] Based on the driver's steering wheel angle, a control tracking reference quantity is generated using a two-degree-of-freedom model;

[0010] Set vehicle state constraints and control input constraints, define the control objective as tracking the reference quantity and suppressing model uncertainty, and construct an error tracking system;

[0011] The nominal trajectory and nominal control sequence are solved by minimizing the preset objective function, and tightened state constraints and control constraints are introduced.

[0012] Design a robust control law consisting of feedforward and feedback components, and generate control commands for the rear wheel steering angle and the torque of the wheel-attached motor.

[0013] Robust control of vehicle motion coordination is achieved based on control commands.

[0014] Optionally, a two-degree-of-freedom model of the vehicle can be established based on its lateral and yaw motions, where the two-degree-of-freedom model is represented as follows:

[0015]

[0016] Where x = [β, γ] T Given the vehicle's state, β is the sideslip angle, γ represents the vehicle's yaw rate, and F... yf and F yr These represent the lateral forces on the front and rear wheels, respectively, where m is the vehicle mass, V is the vehicle velocity, and l is the lateral force on the front and rear wheels. f and l r I represents the distance from the front and rear axles to the vehicle's center of gravity, respectively. z It is the moment of inertia of the vehicle rotating about its center of mass, and the control quantity is u = [F]. yr [M] T , where F yr The rear wheel lateral force is M, and the additional yaw torque is w = [w β ,w γ ] T This represents the unmodeled error, and satisfies |w|≤w max =[w β,max ,w γ,max ] T w β,max ,w γ,max These represent the model error w, respectively. β and w γ The maximum value.

[0017] Optionally, a nonlinear tire model is used to describe the interaction between the tire and the ground. The lateral force of the wheel is calculated using a nonlinear tire model, as shown below:

[0018] A nonlinear brush tire model is used to calculate the lateral force on the wheel, thereby describing the tire-ground relationship of the vehicle, as shown below:

[0019]

[0020] Where α is the tire slip angle, C represents the tire slip stiffness, and F... z This represents the normal vertical load on the tire, and the front wheel slip angle α. f and rear wheel slip angle α r The calculation is as follows:

[0021]

[0022] Where β is the sideslip angle, γ represents the yaw rate of the vehicle, and l f and l r These represent the distances from the front and rear axles to the vehicle's center of gravity, respectively; V is the vehicle speed; and δ... f and δ r These refer to the steering angles of the front and rear wheels of the vehicle, respectively.

[0023] Optionally, based on the driver's steering wheel angle, a control tracking reference value is generated using a two-degree-of-freedom model, as shown below:

[0024] Based on the two-degree-of-freedom model of the vehicle:

[0025]

[0026] The following reference values ​​are generated:

[0027]

[0028]

[0029] l = l f +l r

[0030]

[0031] Where m is the vehicle mass, β is the sideslip angle, V is the vehicle speed, and δ f For the front wheel steering angle, C f For the front wheel lateral stiffness, C r For the rear wheel lateral stiffness, l f and l r Representing the distances from the front and rear axles to the vehicle's center of gravity, respectively, and γ is the yaw rate. r (s) is the reference yaw rate in the s-domain, δF (s) represents the steering wheel angle in the s-domain, τ γ ω is the time constant, s is the sign of the change in the s-domain, and ω is the variable. n Let ζ be the vehicle's natural frequency, ζ be the damping coefficient, l be the wheelbase, and K be the stability factor. γ This is the yaw rate gain.

[0032] Optionally, vehicle state constraints and control input constraints can be set, as follows:

[0033] The yaw rate constraint is |γ|≤γ max The reference yaw rate constraint is Side slip angle constraint is Where γ max It is the maximum yaw rate, β max The maximum value of the centroid sideslip angle, l f and l r Let represent the distances from the front and rear axles to the vehicle's center of gravity, respectively; the upper bound of the state is defined as:

[0034] x max =[β max ,γ max ] T ;

[0035] The upper limit of the control input is:

[0036] u max =[F yr,max M max ] T ;

[0037] Where F yr,max M represents the maximum lateral force of the tire. max This is the maximum value of the additional yaw torque.

[0038] Optionally, the control objective is to track the reference yaw rate while adhering to state and control constraints. The nominal system, neglecting the error w, is:

[0039]

[0040] and These are nominal status and control input, respectively;

[0041] The actual tracking error e and the nominal tracking error Defined as

[0042]

[0043] The error tracking system is as follows:

[0044]

[0045] in

[0046]

[0047] Let x be the error between the real vehicle and the established mathematical model. e ,get

[0048]

[0049] Optionally, the objective function J is as follows:

[0050]

[0051] in F yrs The current lateral force; Q∈R 2×2 It is a symmetric positive semi-definite matrix, R∈R 2×2 It is a symmetric positive definite matrix, R H =H T RH.

[0052] Optionally, the robust control law is defined as follows:

[0053]

[0054] in It is a positive definite diagonal matrix. and These represent the control sequence and state sequence obtained from solving the optimization problem, respectively.

[0055] Optionally, the formula for calculating the rear wheel steering angle is as follows:

[0056]

[0057] The additional motor torque is as follows:

[0058]

[0059] Where d is the distance between the left and right wheels, and R e For the tire radius, To provide the optimal additional yaw torque.

[0060] As can be seen from the above technical solution, compared with the prior art, this invention provides a vehicle motion cooperative robust control method that considers model uncertainty, coordinates the vehicle's four-wheel independent drive and rear-wheel steering subsystems, explicitly considers modeling uncertainty, and solves the system nonlinearity problem caused by the tire model. It designs a composite control law by integrating model predictive control and feedback linearization. Tightened state and control constraints are formulated to ensure robust constraint satisfaction and robust vehicle control even in the presence of uncertainty. Attached Figure Description

[0061] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0062] Figure 1 This is a schematic diagram of the technical solution provided by the present invention;

[0063] Figure 2 This is a schematic diagram of the two-degree-of-freedom vehicle model of the present invention;

[0064] Figure 3 This is a schematic diagram of the yaw rate tracking and center of mass sideslip angle during the DLC scene HIL test of the present invention;

[0065] Figure 4 This is a schematic diagram of the four-wheel additional motor torque and rear wheel rotation angle in the DLC scene HIL test of the present invention. Detailed Implementation

[0066] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0067] This invention discloses a vehicle motion cooperative robust control method considering model uncertainties, comprising the following steps:

[0068] A two-degree-of-freedom model of the vehicle is established based on its lateral and yaw motions.

[0069] A nonlinear tire model is used to describe the interaction between the tire and the ground, and the lateral force of the wheel is calculated using a nonlinear tire model.

[0070] Based on the driver's steering wheel angle, a control tracking reference quantity is generated using a two-degree-of-freedom model;

[0071] Set vehicle state constraints and control input constraints, define the control objective as tracking the reference quantity and suppressing model uncertainty, and construct an error tracking system;

[0072] The nominal trajectory and nominal control sequence are solved by minimizing the preset objective function, and tightened state constraints and control constraints are introduced.

[0073] Design a robust control law consisting of feedforward and feedback components, and generate control commands for the rear wheel steering angle and the torque of the wheel-attached motor.

[0074] Robust control of vehicle motion coordination is achieved based on control commands.

[0075] Figure 1 A block diagram of the proposed control structure is described. First, a two-degree-of-freedom reference model generates an ideal desired yaw rate based on the steering wheel angle turned by the driver, which serves as the reference input for the controller. This invention explicitly considers the model mismatch problem caused by vehicle model and parameter uncertainties. Figure 1 The composite control law consists of two parts. The first part is the feedforward component obtained by solving the Model Predictive Control (MPC) optimization problem. The second part considers the feedback component of model uncertainty. In addition, tightened state and control constraints were defined to ensure robust constraint satisfaction under uncertainties. The main design process is described below:

[0076] like Figure 2 As shown, Step 1: Model Establishment:

[0077] 1) Establishment of a two-degree-of-freedom model of the vehicle

[0078] The present invention first establishes a two-degree-of-freedom model of the vehicle, which takes into account the lateral motion and yaw motion of the vehicle.

[0079]

[0080] Where x = β, γ] T Given the vehicle's state, β is the sideslip angle, γ represents the vehicle's yaw rate, and F... yf and F yr These represent the lateral forces of the front and rear wheels, respectively. f and l r I represents the distance from the front and rear axles to the vehicle's center of gravity, respectively. z It is the moment of inertia of the vehicle rotating about its center of mass, and the controlling quantities are the lateral force of the rear wheels and the additional yaw torque u = F. yr M T w = [w β ,w γ ] T This represents the unmodeled error, and satisfies |w|≤w max =[w β,max ,w γ,max ] T .

[0081] 2) Tire model

[0082] A nonlinear brush tire model is used to calculate the lateral force on the wheel, thereby describing the tire-ground relationship of the vehicle, as shown below:

[0083]

[0084] Where C represents the tire lateral stiffness (C for the front wheel). f The rear wheel is C. r ), F z Indicates the normal vertical load of the tire (F for the front axle). zf The rear axle is F zr The tire slip angle is calculated as follows:

[0085]

[0086] 3) Tracking Reference

[0087] To ensure vehicle stability, appropriate reference center of gravity sideslip angle and reference yaw rate need to be generated based on the front wheel steering angle input by the driver, according to the vehicle's two-degree-of-freedom model:

[0088]

[0089] The following reference values ​​can be generated:

[0090]

[0091] Where m is the vehicle mass, β is the sideslip angle, and V x For longitudinal vehicle speed, δ F This refers to the steering angle of the front wheels.

[0092]

[0093] l = l F +l R

[0094]

[0095] 4) Vehicle status and actuator constraints

[0096] The yaw rate constraint is |γ|≤γ max For safety considerations, the reference yaw rate constraint is as follows: Angular constraint is The upper bound of the state is defined as follows:

[0097] x max =[β max ,γ max ] T (7);

[0098] The upper limit of the control input is:

[0099] u max =[F yr,max ,ΔM max ] T (8);

[0100] 5) Definition of control objectives

[0101] The control objective is to track the reference yaw rate while adhering to state and control constraints (7) and (8), even in the presence of uncertainties. The nominal system, neglecting the error w, is:

[0102]

[0103] The actual tracking error and the nominal tracking error are defined as follows:

[0104]

[0105] The error tracking system is as follows:

[0106]

[0107]

[0108] in:

[0109]

[0110] The error between the real vehicle (1) and the established mathematical model (9) is defined as x. e ,get:

[0111]

[0112] Step 2: Controller Design

[0113] The scheme coordinates the four-wheel independent drive and rear-wheel steering subsystems of an electric vehicle to solve the nonlinear tracking problem discussed in step 1. This approach encompasses the definition of the optimization problem, the design of robust control laws, and the tightening of control and state constraints.

[0114] 1) Description of the robust optimization problem

[0115] Question 1:

[0116]

[0117] Constraints:

[0118]

[0119] |x e,t |≤K -1 w max (18);

[0120]

[0121] in

[0122]

[0123] T represents the prediction time domain, and the objective function J is as follows:

[0124]

[0125] in Q∈R 2×2 It is a symmetric positive semi-definite matrix, R∈R 2×2 It is a symmetric positive definite matrix, R H =H T RH.

[0126] 2) Robust control law

[0127] Solving problem 1 yields the nominal trajectory. and nominal control sequence However, since computer systems cannot implement continuous control laws, at time τ∈[t,t+T]... s The control law acting on the system is defined as follows:

[0128]

[0129] Based on formula (24), the required rear wheel lateral force F can be obtained. yfr And the additional yaw moment M. In order to apply lateral force, the rear wheel steering angle is efficiently determined by a linear model, while the yaw moment is evenly distributed on multiple motors to generate motor torque.

[0130] 3) Control Action Generation

[0131] According to the tire model (3), the rear wheel steering angle can be obtained as follows:

[0132]

[0133] The additional motor torque of the wheel is calculated based on the additional yaw moment as follows:

[0134]

[0135] Where d is the distance between the left and right wheels, and R e This is the tire radius.

[0136] To further illustrate the effectiveness of the invention, a driver-in-the-loop simulator was used for verification. The experimental platform integrates a full-view driving simulator with a 180° projection screen (for visual immersion) and a six-degree-of-freedom motion platform (for providing realistic vehicle dynamics feedback). The system architecture comprises three industrial computers (IPCs): IPC1 runs the vehicle dynamics model and provides motion status to the simulator; IPC2 executes the SCANeR program, providing visual cues to the driver; and IPC3, a miniature industrial computer, acts as the controller, executing the proposed control strategy. A laptop computer monitors the entire system and performs controller calibration. Control signals are exchanged between components via a Controller Area Network (CAN) bus, while monitoring signals are forwarded using User Datagram Protocol (UDP).

[0137] To verify the effectiveness and advantages of the method proposed in this invention, Model Predictive Control (MPC) and Backstepping (BS) were selected as comparison methods. For fairness, the Nonlinear Robust Model Predictive Control (NRMPC) method proposed in this invention uses the same weight parameters Q, R, and feedback matrix K as the comparison algorithms. All driving experiments were conducted by the same driver who had undergone simulator training before actual driving.

[0138] Choose the weight matrix Q = diag(0.1, 10), R = diag(100, 0.1), and the feedback matrix K = diag(1, 10). Control period T s =0.01s, prediction time domain m=10. w max =10 -4 [1.05,5] T G max =[0.005,0.1] T ,

[0139] The test scenario involved a driver performing a double lane change (DLC) maneuver on a low-adhesion surface. Figure 3 Yaw rate tracking curves under three control strategies are presented. Results show that the proposed NRMPC and MPC can effectively stabilize the vehicle and assist the driver in completing DLC ​​operations. However, when using the BS controller, the driver failed to complete the DLC operation, resulting in a significant difference between the vehicle's yaw rate and the expected value. This indicates that the vehicle is out of control and cannot follow the driver's intentions.

[0140] Compare Figure 3The yaw rate curves of NRMPC and MPC show that NRMPC accurately and smoothly tracks the reference yaw rate with minimal overshoot, ensuring stability and driving comfort. When the driver performs a steering maneuver, a sudden change in yaw rate is expected. While MPC can stabilize the vehicle, it causes significant yaw rate overshoot, affecting driving comfort. NRMPC's peak tracking error for yaw rate is 3.93 deg / s with a maximum overshoot of 14.0%, while MPC's peak tracking error is 9.51 deg / s with a maximum overshoot of 34.7%. NRMPC's peak sideslip angle is only 0.89 degrees, compared to MPC's 0.99 degrees, indicating poorer control performance.

[0141] Figure 4 The additional torque applied to the wheels by the motor and the rear wheel steering angle are shown. The additional torque curves reveal significant differences in motor torque among the three algorithms. Although the additional torque of both NRMPC and MPC satisfies the control constraints, the motor torque of NRMPC is significantly lower than that of MPC, indicating a reduction in energy consumption. This demonstrates that the NRMPC proposed in this invention can better coordinate the four-wheel independent drive and rear-wheel steering systems, achieving superior control performance. It is worth noting that because BS relies on feedback linearization, it tends to reach the maximum additional torque, leading to increased energy consumption and a failure to achieve vehicle stability.

[0142] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0143] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A vehicle motion cooperative robust control method considering model uncertainty, characterized in that, The method comprises the following steps: a two-degree-of-freedom model of the vehicle is established based on lateral motion and yaw motion of the vehicle; a nonlinear tire model is used to describe the interaction between the tire and the ground, and the wheel lateral force is calculated by the nonlinear tire, and is expressed as follows: a nonlinear brush tire model is used to calculate the wheel lateral force, thereby describing the tire-ground relationship of the vehicle, and is expressed as follows: where a is the tire side slip angle, C represents the tire side slip stiffness, F z represents the tire normal vertical load, the front wheel side slip angle a f and the rear wheel side slip angle a r is calculated as follows: where β is the centroid side slip angle, γ represents the yaw rate of the vehicle, l f and l r represent the distance from the front and rear axles to the vehicle centroid, V is the vehicle speed, δ f and δ r are the front and rear wheel steering angles of the vehicle, respectively. a control tracking reference quantity is generated by the two-degree-of-freedom model based on the steering wheel angle of the driver; vehicle state constraints and control input constraints are set, a control target is defined as the tracking reference quantity and the model uncertainty is suppressed, and an error tracking system is constructed; a nominal trajectory and a nominal control sequence are solved by taking minimization of a preset target function as an optimization target, and tightened state constraints and control constraints are introduced; a robust control law composed of a feedforward component and a feedback component is designed, and control instructions of the rear wheel steering angle and the wheel additional motor torque are generated; vehicle motion cooperative robust control is realized according to the control instructions. 2.The vehicle motion cooperative robust control method considering model uncertainty according to claim 1, wherein, a two-degree-of-freedom model of the vehicle is established based on lateral motion and yaw motion of the vehicle, and the two-degree-of-freedom model is expressed as follows: where x = [β, γ] T is the vehicle state, β is the center of mass side slip angle, γ represents the vehicle yaw rate, F yf and F yr represent the front and rear wheel side forces, respectively, m is the vehicle mass, V is the vehicle speed, l f and l r represent the distance from the front and rear axles to the vehicle center of mass, I z is the moment of inertia of the vehicle about the center of mass, and the control is u = [F yr , M] T , where F yr is the rear wheel side force and M is the additional yaw moment, w = [w β , w γ ] T represents the unmodeled error and satisfies |w|≤w max = [w β,max , w γ,max ] T , w β,max , w γ,max represent the maximum values of the model errors w β and w γ , respectively. 3.The vehicle motion cooperative robust control method considering model uncertainty according to claim 1, wherein, a control tracking reference quantity is generated by the two-degree-of-freedom model based on the steering wheel angle of the driver, and is expressed as follows: according to the two-degree-of-freedom model of the vehicle: the following reference quantities are generated: l＝l f +l r where m is the vehicle mass, β is the mass center side slip angle, V is the vehicle speed, δ f is the front wheel steering angle, I z is the moment of inertia of the vehicle about the mass center, C f is the front wheel cornering stiffness, C r is the rear wheel cornering stiffness, l f and l r are the distances from the front and rear axles to the vehicle mass center, γ is the yaw rate, γ r (s) is the reference yaw rate in the s-domain, δ F (s) is the steering wheel angle in the s-domain, τ γ is a time constant, s is the s-domain variable, ω n is the natural frequency of the vehicle, ζ is the damping coefficient, l is the wheel base, K is the stability factor, K γ is the yaw rate gain. 4.The vehicle motion cooperative robust control method considering model uncertainty according to claim 1, wherein, vehicle state constraints and control input constraints are set, and are expressed as follows: The yaw rate constraint is |γ|≤γmax max , and the reference yaw rate constraint is The side slip angle constraint is where γmax max is the maximum yaw rate, βmax max is the maximum side slip angle, l f and l r denote the distances from the front and rear axles to the vehicle center of mass, respectively; and the state upper bound is defined as: x max = [β max , γ max ] T ; the upper limit of the control input is: u max = [F yr,max , M max ] T ; where F yr,max is the maximum value of the tire lateral force, M max is the maximum value of the additional yaw moment.

5. The vehicle motion cooperative robust control method considering model uncertainty according to claim 1, characterized in that, the control target is to track the reference yaw angular velocity while complying with the state and control constraints, and the nominal system without considering the error w is: and are the nominal states and control inputs, respectively; The actual tracking error e and the nominal tracking error are defined as e = [e β , e γ ] = [β, γ - γ r ] T , the error tracking system is: wherein Define the error of the real vehicle from the established mathematical model as x e , resulting in where F yf represents the lateral force of the front wheels, m is the mass of the vehicle, I z is the moment of inertia of the vehicle about the center of mass, γ r represents the reference yaw rate.

6. The vehicle motion cooperative robust control method considering model uncertainty according to claim 1, characterized in that, the target function J is as follows: wherein F yrs is the current lateral force; Q ∈ R 2×2 is a symmetric positive semi-definite matrix, R ∈ R 2×2 is a symmetric positive definite matrix, R H = H T RH; γ r denotes the reference yaw rate.

7. The vehicle motion cooperative robust control method considering model uncertainty according to claim 1, characterized in that, the definition of the robust control law is as follows: wherein is a positive definite diagonal matrix, and denote the control sequence and state sequence obtained by solving the optimization problem, respectively. 8.The vehicle motion cooperative robust control method considering model uncertainty according to claim 1, wherein, the calculation formula of the rear wheel steering angle is as follows: the additional motor torque is as follows: where d is the distance between the left and right wheels, R e is the tire radius, is the optimal additional yaw moment, F yr denotes the lateral force of the rear wheel, C r is the rear wheel cornering stiffness.

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