A method for estimating friction of a steer-by-wire system

Abstract

The application discloses a method for estimating friction of a steer-by-wire system, and establishes LuGre friction models for an upper steering wheel assembly and a lower steering actuator assembly respectively, so as to estimate the friction of the steer-by-wire system. The estimated friction torque at the upper steering wheel assembly is used to make the motor compensate for part of the friction torque, so as to improve the driving feeling of the driver. The estimated friction at the lower steering actuator assembly is used to make the driver feel the friction from the steering actuator. Finally, a genetic algorithm is used to identify the to-be-determined parameters in the LuGre friction model. The application can estimate the internal friction of the steer-by-wire system by using a fixed model, so that the driver can perceive or filter the internal friction.

Description

TECHNICAL FIELD

[0001] The present application relates to a method for estimating the friction of a steer-by-wire system, belonging to the technical field of steer-by-wire system control of vehicles. BACKGROUND

[0002] Steer-by-wire system (SBW) is a new product of steering system development, which has the trend of replacing electric power steering system (EPS) in the future. It realizes the decoupling between the wheels and the steering wheel, and arranges sensors such as displacement sensors and torque sensors, and shares data in real time through CAN bus, so as to have the characteristics of fast control response and variable transmission ratio, which is consistent with the development trend of high intelligence and integration of automobiles in the future.

[0003] Since the SBW system cancels the intermediate shaft of steering, the driver cannot perceive the road feel information, which is very important for the driver to drive the vehicle safely. Therefore, the road feel motor needs to simulate the road feel. At present, there are many studies on road feel simulation, most of which can better feedback the information from the ground and improve the driving confidence of the driver. However, road feel simulation should not only consider the feedback information from the ground, but also include the inherent properties of the steering system, such as inertia, damping and friction, etc. Among them, friction is a complex and variable property of the system, and most studies only briefly consider the system friction. Friction exists in each part of the steer-by-wire system. The friction of the upper turning module can be directly perceived by the driver, while the friction of the lower turning module cannot be directly fed back to the steering wheel. In order to make the feel of the steer-by-wire system closer to that of the traditional steering system, the friction of the lower turning module needs to be compensated. The simplest way is to load a force sensor in the lower turning (i.e. steering actuator) to measure the friction of the steering actuator in real time, and then feed back to the steering wheel through the road feel motor. However, this increases the manufacturing cost, and due to the many variable factors, the error of real-time measurement is large, so establishing a fixed friction model to estimate the friction force is a feasible way. In addition, the friction at the upper turning (i.e. steering wheel assembly) should support the individual selection of the driver. The driver should be able to choose whether to perceive or filter it, so as to improve the steering feel.

[0004] Therefore, when simulating the road feel of the steer-by-wire system, the friction characteristics of the system should be considered, so as to simulate the road feel closer to the traditional steering system. SUMMARY

[0005] Invention purposes: In view of the deficiencies in the prior art, the application provides a method for estimating the friction of a steer-by-wire system, which can estimate the friction of the upper steering wheel assembly and the lower steering actuator in the steer-by-wire system, so that the driver can better feel the road sense and improve the safety of driving.

[0006] Technical scheme: A method for estimating the friction of a steer-by-wire system, comprising the following steps:

[0007] S1, measuring the actual friction torque and friction on the steer-by-wire system bench;

[0008] S2, establishing a friction estimation model of the upper steering wheel assembly;

[0009] S3, using a genetic algorithm to identify the static parameters and dynamic parameters of the upper friction model,

[0010] S4, establishing a friction estimation model of the lower steering actuator assembly;

[0011] S5, using a genetic algorithm to identify the static parameters and dynamic parameters of the lower friction model,

[0012] S6, superimposing the friction feedback torque of the upper and lower turns to obtain the friction torque of the steer-by-wire system that the road sense motor needs to simulate.

[0013] The friction to be measured in S1 includes:

[0014] The friction torque when the steering wheel starts to move;

[0015] The friction torque when the steering wheel rotates at a constant speed, the friction torque at different speeds is measured, so as to obtain the Stribeck curve;

[0016] The friction of the rack when it starts to move;

[0017] The friction of the rack of the steering actuator when it moves at a constant speed, the friction at different speeds is measured, so as to obtain the Stribeck curve.

[0018] The S2 is specifically:

[0019] The friction estimation model of the upper steering wheel assembly is established, since the movement of the upper turn is rotation, the rotational movement needs to be considered in the friction model, at this time the friction is friction torque, the LuGre friction model is established as:

[0020]

[0021] In the formula, s0 is the bristle stiffness coefficient; s1 is the bristle damping coefficient; T c is the Coulomb friction torque; Ts is the maximum static friction torque; is the Stribeck velocity; s2 is the bristle viscous friction coefficient; represents the Stribeck effect; is the relative rotation speed of the contact surface; T f is the LuGre friction torque at the upper rotation; z is the average deformation of the bristles of the contact surface.

[0022] The S3 specifically is:

[0023] S3.1, the static parameters of the upper rotation friction model are identified by using a genetic algorithm, and the identification value of the static friction parameter obtained after each iteration is:

[0024]

[0025] The corresponding friction torque identification value is obtained by the following formula:

[0026]

[0027] Wherein, i=1, 2, 3, ···, N, N is the total number of points on the Stribeck curve;

[0028] S3.2, the dynamic parameters of the upper rotation friction model are identified by using a genetic algorithm, and the identification value of the dynamic friction parameter obtained after each iteration is:

[0029]

[0030] The corresponding friction force identification value is obtained by the following formula:

[0031]

[0032] S3.3, the reluctance torque output by the road feel motor after the driver's individual selection is used to offset the friction torque:

[0033]

[0034] Wherein, η is the individualization coefficient of the driver.

[0035] The S4 specifically is: a friction estimation model of the lower rotation steering actuator assembly is established, since the lower rotation is a translation, the friction generated at this time is a friction force, and the LuGre friction model is established as:

[0036]

[0037] Wherein, s3 is the bristle stiffness coefficient; s4 is the bristle damping coefficient; F c is the Coulomb friction force; F s is the maximum static friction; vs is the Stribeck velocity; s5 is the bristle viscous friction coefficient; g(v) is the Stribeck effect; v is the relative rotational speed of the contact surface; F f is the LuGre friction force at the upper turning; z is the average deformation of the bristles of the contact surface.

[0038] The S5 is specifically:

[0039] S5.1, the static parameters of the lower turning friction model are identified by using a genetic algorithm, and the identification value of the static friction parameter obtained after each step iteration is:

[0040]

[0041] The corresponding friction force identification value is obtained by the following formula:

[0042]

[0043] Wherein, i=1, 2, 3, ···, N, N is the total number of points on the Stribeck curve.

[0044] S5.2, the dynamic parameters of the lower turning friction model are identified by using a genetic algorithm, and the identification value of the dynamic friction parameter obtained after each step iteration is:

[0045]

[0046] The corresponding friction force identification value is obtained by the following formula:

[0047]

[0048] S5.3, the road feel torque generated by friction can be obtained by transmission ratio conversion, that is:

[0049] T ef2 =F f / G

[0050] In the formula: F f is the estimated friction force, and G is the transmission ratio from the lower turning to the upper turning.

[0051] S6, the friction feedback torque of the upper turning and the lower turning is superimposed, and the friction torque of the steer-by-wire system that the road feel motor needs to simulate is:

[0052] T friction =T ef1 +T ef2

[0053] In the formula: T ef1 is the upper turning friction torque to be simulated by the road feel motor, and T ef2 is the lower turning friction torque to be simulated by the road feel motor.

[0054] Beneficial effects: the method for estimating the friction of the steer-by-wire system, the main feature is to consider the friction simulation in the road feeling simulation of the steer-by-wire system, and the friction is estimated by using the LuGre friction model, so that the cost of the sensor for real-time measurement is saved, the driver obtains the road feeling closer to the traditional steering system, and the driving safety is improved.

[0055] The method for estimating the friction of the steer-by-wire system, the main feature is to establish a friction torque estimation model for the upper steering wheel assembly of the steer-by-wire system based on the LuGre friction model, so as to estimate the friction torque generated when rotating, the parameters in the actual measured friction torque are identified by using the genetic algorithm, and the driver's individual selection is carried out, so as to obtain the final road feeling motor compensation to offset the electromagnetic torque of the friction torque, and the driving feeling of the driver is improved.

[0056] The method for estimating the friction of the steer-by-wire system, the main feature is to establish a friction torque estimation model for the upper steering wheel assembly of the steer-by-wire system based on the LuGre friction model, so as to estimate the friction torque generated when rotating, the parameters in the actual measured friction torque are identified by using the genetic algorithm, and the driver's individual selection is carried out, so as to obtain the final road feeling motor compensation to offset the electromagnetic torque of the friction torque, and the driving feeling of the driver is improved.

[0057] Summarizing, the LuGre friction model can comprehensively reflect the rotation and translation friction characteristics of the steer-by-wire system, the genetic algorithm can identify the parameters in the model, the problem that the driver is sensitive to the upper friction and lacks the lower friction when driving the vehicle equipped with the steer-by-wire system is solved, and the steering feeling and driving safety are improved. BRIEF DESCRIPTION OF DRAWINGS

[0058] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description only belong to the embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor on the basis of the provided drawings.

[0059] Figure 1 The structure diagram of the steer-by-wire system of the embodiment of the present application.

[0060] Figure 2 The method schematic diagram of the embodiment of the present application.

[0061] Figure 3 The bristle movement schematic diagram of the embodiment of the present application.

[0062] Figure 4 Stribeck curve of the embodiment of the present application. DETAILED DESCRIPTION

[0063] The technical solutions in the embodiments of the present application will be clearly and completely described in combination with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of protection of the present application.

[0064] In the description of the present application, it should be understood that the terms "upper", "lower", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer" and the like indicate the orientation or positional relationship shown in the drawings based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply that the device or element referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as a limitation on the present application.

[0065] In the present application, unless otherwise explicitly specified and limited, "on" or "under" of a first feature to a second feature can include that the first and second features are in direct contact, or that the first and second features are not in direct contact but are in contact through another feature between them. Moreover, "on", "above" and "above" of the first feature to the second feature include that the first feature is directly above and obliquely above the second feature, or only indicates that the horizontal height of the first feature is higher than that of the second feature. "Below", "below" and "below" of the first feature to the second feature include that the first feature is directly below and obliquely below the second feature, or only indicates that the horizontal height of the first feature is less than that of the second feature.

[0066] As shown in Figure 1 The steer-by-wire system is divided into an upper steering wheel assembly and a lower steering execution mechanism, and the two are not connected by mechanical structure, so they cannot directly transmit force, and they contain complex components, including steering wheel, torsion bar, road feel motor, steering motor, rack, synchronous belt and nut, etc. Friction exists in each component and is complex and variable.

[0067] A method for estimating the friction of a steer-by-wire system, comprising the following steps:

[0068] S1, measuring the actual friction torque and friction force on the steer-by-wire system bench;

[0069] S2, establishing a friction estimation model of the upper steering wheel assembly;

[0070] S3, using a genetic algorithm to identify the static parameters and dynamic parameters of the up-conversion friction model,

[0071] S4, establishing a friction estimation model of the down-conversion steering actuator assembly;

[0072] S5, using a genetic algorithm to identify the static parameters and dynamic parameters of the down-conversion friction model,

[0073] S6, superimposing the friction feedback torques of up-conversion and down-conversion to obtain the friction torque of the steer-by-wire system that the road feeling motor needs to simulate.

[0074] The friction to be measured in S1 includes:

[0075] The friction torque when the steering wheel is initially moved;

[0076] The friction torque when the steering wheel is rotating at a constant speed, which requires measuring multiple sets of rotational speed friction torques, and the rotational speed interval is selected as

[0077] [-360deg / s, 360deg / s], set an interval of 36deg / s, measure 21 sets, and thus obtain a Stribeck curve;

[0078] The friction of the rack when it is initially moved;

[0079] The friction of the steering actuator rack when it is moving at a constant speed, which requires measuring multiple sets of rotational speed friction torques, and the rotational speed interval [-1m / s, 1m / s], set an interval of 0.1m / s, measure 21 sets, and thus obtain a Stribeck curve.

[0080] S2 is specifically:

[0081] Establish a friction estimation model of the up-conversion steering wheel assembly, since the up-conversion motion is rotation, the rotational motion needs to be considered in the friction model, at this time the generated friction is friction torque, the LuGre friction model is established as:

[0082]

[0083] In the formula, s0 is the bristle stiffness coefficient; s1 is the bristle damping coefficient; T c is the Coulomb friction torque; T s is the maximum static friction torque; is the Stribeck speed; s2 is the bristle viscous friction coefficient; represents the Stribeck effect; is the relative rotational speed of the contact surface; T f is the LuGre friction torque of up-conversion; z is the average deformation of the bristles of the contact surface.

[0084] The S3 is specifically:

[0085] The parameter values in the friction model are confirmed, and for the six parameters in the model, T c , T s , and s2 are static parameters, which determine the characteristics of the kinetic friction torque, and s0 and s1 are dynamic parameters, which determine the characteristics of the static friction torque.

[0086] Further, the identification of the static parameters is performed, when the relative motion between the objects is in a steady state, that is, in a uniform motion stage, at this time dz / dt = 0, the bristle deformation does not change, at this time the friction characteristics are determined by the static parameters. The expression of the LuGre friction model becomes:

[0087]

[0088] The above formula is obtained by combining:

[0089]

[0090] According to the positive and negative of the speed, the following can be obtained:

[0091]

[0092] S3.1, the genetic algorithm is used to identify the static parameters of the upper friction model, and the parameter identification needs to be based on the actual measured Stribeck curve. The main process is as shown in Figure 2 , and the main process is as follows:

[0093] (1) First, the evolution number is initialized, that is, t = 0, and the population is randomly initialized, that is, the static parameters of random values are generated;

[0094] (2) The individual fitness f1 is evaluated, so as to evaluate the difference between the identified friction torque and the actual friction torque;

[0095] (3) If t reaches the maximum evolution number, the algorithm is terminated, otherwise, go to (4);

[0096] (4) The optimal individual is generated by screening the individual fitness, and a new generation population is formed;

[0097] (5) The crossover operation is performed with a random probability to generate a new population;

[0098] (6) The individual mutation operation is performed with a random probability to generate a new population;

[0099] (7) Set t + 1→t, go to step (2) and iterate.

[0100] The identified value of the static friction parameter after each iteration of the genetic algorithm is:

[0101]

[0102] The corresponding identified value of the friction torque is obtained from the following formula:

[0103]

[0104] where i = 1, 2, 3, ···, N, and N is the total number of points on the Stribeck curve.

[0105] The identification error is:

[0106]

[0107] where T fi is obtained from the actual Stribeck curve.

[0108] The objective function is:

[0109]

[0110] The individual fitness function is selected:

[0111]

[0112] After continuous iteration and selection, the static parameters closest to the actual Stribeck curve can be obtained.

[0113] S3.2, The identification of dynamic parameters is based on the measurement of pre-sliding displacement, at this time the system motion belongs to the micro category. At this time:

[0114]

[0115] Therefore, the friction model becomes:

[0116]

[0117] Taking s2 in the static parameter identification result as a known condition, similarly, the genetic algorithm identifies s0 and s1, and the steps are similar to the static parameter identification. The identified value of the static friction parameter after each iteration is:

[0118]

[0119] The corresponding identified value of the friction torque is obtained from the following formula:

[0120]

[0121] The identification error, the objective function and the individual fitness function are consistent with the static parameter identification process.

[0122] The estimated model of the upturn friction torque is obtained, the real-time estimated friction torque supports the driver's personalized selection, so that the road feeling motor output is used to offset the magnetic resistance torque of the friction torque, and the driver's driving feeling is improved.

[0123] S3.3, the magnetic resistance torque of the road feeling motor output for offsetting the friction torque after the driver's personalized selection is:

[0124] T ef1 =-ηT f

[0125] Wherein, η is the personalized coefficient of the driver, the greater the value indicates that the driver feels the smaller friction torque; The driver's personalized selection mainly shows the strength of the friction torque perception, if you need to perceive strong, you can adjust the personalized coefficient low, at this time the motor will not too offset the friction torque.

[0126] The S4 is specifically: the friction estimation model of the down-turn steering actuator assembly is established, since the down-turn motion is translation, the friction generated at this time is friction force, and the LuGre friction model is established as:

[0127]

[0128] Wherein, s3 is the bristle stiffness coefficient; S4 is the bristle damping coefficient; F c is the coulomb friction force; F s is the maximum static friction force; v s is the Stribeck speed; s5 is the bristle viscous friction coefficient; g(v) is the Stribeck effect; v is the relative speed of the contact surface; F f is the LuGre friction force at the upturn; z is the average deformation of the bristles of the contact surface.

[0129] Confirm the parameter values in the friction model, for the six parameters in the model, F c , F s , v s and s5 are defined as static parameters, which determine the characteristics of dynamic friction, and s3 and s4 are defined as dynamic parameters, which determine the characteristics of static friction.

[0130] Further, the static parameter identification is carried out, when the relative motion between objects is in a steady state, that is, in a uniform motion stage, at this time dz / dt=0, the bristle deformation does not change, at this time the friction characteristics are determined by the static parameters. The expression of LuGre friction model becomes:

[0131] The above formula is solved simultaneously to obtain:

[0132]

[0133]

[0134] The positive and negative of the speed are considered separately to obtain:

[0135]

[0136] The S5 is specifically:

[0137] S5.1, the genetic algorithm is used to identify the static parameters of the down-turn friction model, and the Stribeck curve actually measured is used as the basis. The steps and the direction of the steering wheel assembly are consistent with the friction force parameter identification.

[0138] The identification value of the static friction parameter obtained by the genetic algorithm after each iteration is:

[0139]

[0140] The corresponding friction identification value is obtained by the following formula:

[0141]

[0142] Where, i = 1, 2, 3, ···, N, N is the total number of points on the Stribeck curve.

[0143] The identification error is:

[0144]

[0145] Where, F fi The actual Stribeck curve is obtained.

[0146] The objective function is:

[0147]

[0148] The individual fitness function is selected:

[0149]

[0150] After continuous iteration and screening, the static parameters closest to the actual Stribeck curve can be obtained.

[0151] S5.2, the genetic algorithm is used to identify the dynamic parameters of the down-turn friction model, which is based on the measurement of the pre-sliding displacement, at this time the system movement belongs to the micro category. At this time:

[0152]

[0153] Thus the friction model becomes:

[0154]

[0155] With s5 in the static parameter identification result as a known condition, similarly, the genetic algorithm identifies s3 and s4, and the steps are similar to the static parameter identification. The identification value of the static friction parameter obtained by each iteration is:

[0156]

[0157] The corresponding friction identification value is obtained by the following formula:

[0158]

[0159] The identification error, the objective function, and the individual fitness function are all set to be consistent with the static parameter identification process.

[0160] Finally, the identified dynamic and static parameters are substituted into the LuGre friction model to obtain the friction model representing the friction of the lower turning actuator, and the road feel torque generated by the friction can be obtained by the transmission ratio conversion, that is:

[0161] T ef2 = F f / G

[0162] In the formula: F f is the estimated friction, and G is the transmission ratio from the lower turn to the upper turn.

[0163] S6, the friction feedback torques of the upper turn and the lower turn are superimposed to obtain the friction torque of the steer-by-wire system that the road feel motor needs to simulate:

[0164] T friction = T ef1 + T ef2

[0165] In the formula: T ef1 is the upper turn friction torque to be simulated by the road feel motor, and T ef2 is the lower turn friction torque to be simulated by the road feel motor.

[0166] Each embodiment in the specification is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same and similar parts between each embodiment can be referred to each other. For the device disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the related parts can be referred to the method part.

[0167] The foregoing description of the disclosed embodiments enables a person skilled in the art to make or use the application. Modifications of these embodiments will occur to persons of skill in the art, and that the appended claims are intended to cover all such modifications that do not depart from the true spirit and scope of the application. Therefore, the application is not limited to the embodiments shown but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for estimating friction of a steer-by-wire system, characterized by, The method comprises the following steps: S1, measuring the actual friction torque and friction force on the online control steering system bench; S2, establishing an upper turning steering wheel assembly friction estimation model; S3, identifying the static parameters and dynamic parameters of the upper turning friction model by using a genetic algorithm, The output of the road feel motor after the driver's individual selection is used to offset the friction torque, and the output of the road feel motor is: ； wherein, is the individualization coefficient for the driver; S4, establishing a lower turning steering actuator assembly friction estimation model; S5, identifying the static parameters and dynamic parameters of the lower turning friction model by using a genetic algorithm, S6, superimposing the friction feedback torques of the upper turning and lower turning to obtain the friction torque of the steer-by-wire system that needs to be simulated by the road feel motor.

2. The method for estimating friction of a steer-by-wire system according to claim 1, wherein, The friction to be measured in S1 includes: The friction torque when the steering wheel starts to move; The friction torque when the steering wheel rotates at a constant speed, and the friction torques at different speeds are measured to obtain a Stribeck curve; The friction of the rack when it starts to move; The friction of the rack of the steering actuator when it moves at a constant speed, and the friction at different speeds is measured to obtain a Stribeck curve.

3. The method for estimating friction of a steer-by-wire system of claim 1, wherein, S2 specifically includes: Establishing an upper turning steering wheel assembly friction estimation model, since the upper turning motion is rotation, the rotational motion needs to be considered in the friction model, and the friction torque generated at this time is LuGre friction model, which is established as: ； wherein is the bristle stiffness coefficient; is the bristle damping coefficient; is the Coulomb friction torque; is the maximum static friction torque; is the Stribeck velocity; is the bristle viscous friction coefficient; represents the Stribeck effect; is the relative rotational speed of the contact surfaces; is the LuGre friction torque at the upper transition; is the average deformation of the bristles of the contact surfaces.

4. The method for estimating friction of a steer-by-wire system according to claim 3, wherein, S3 specifically includes: S3.1, identifying the static parameters of the upper turning friction model by using a genetic algorithm, and the identification value of the static friction parameter obtained after each iteration is: ； The corresponding friction torque identification value is obtained by the following formula: ； Wherein, i=1, 2, 3, ···, N, N is the total number of points on the Stribeck curve; S3.2, identifying the dynamic parameters of the upper turning friction model by using a genetic algorithm, and the identification value of the dynamic friction parameter obtained after each iteration is: ； The corresponding friction torque identification value is obtained by the following formula: ；。 5. The method for estimating friction of a steer-by-wire system according to claim 4, wherein, S4 specifically includes: establishing a lower turning steering actuator assembly friction estimation model, since the lower turning motion is translation, the friction generated at this time is friction force, and the LuGre friction model is established as: ； wherein, is the bristle stiffness coefficient; is the bristle damping coefficient; is the Coulomb friction force; is the maximum static friction force; is the Stribeck velocity; is the bristle viscous friction coefficient; is the Stribeck effect; is the relative rotational speed of the contact surfaces; is the LuGre friction force at the upper transition; is the average deformation of the bristles of the contact surfaces.

6. The method for estimating friction of a steer-by-wire system according to claim 5, wherein, S5 specifically includes: S5.1, identifying the static parameters of the lower turning friction model by using a genetic algorithm, and the identification value of the static friction parameter obtained after each iteration is: ； The corresponding friction identification value is obtained by the following formula: ； Wherein, i=1, 2, 3, ···, N, N is the total number of points on the Stribeck curve; S5.2, identifying the dynamic parameters of the lower turning friction model by using a genetic algorithm, and the identification value of the dynamic friction parameter obtained after each iteration is: ； The corresponding friction identification value is obtained by the following formula: ； S5.3, the road feel torque generated by friction can be obtained by conversion through the transmission ratio, that is: ； In the formula: F f G is the transmission ratio from downshift to upshift for the estimated friction force.

7. The method for estimating friction of a steer-by-wire system according to claim 6, wherein, The friction feedback torques of the upper turning and lower turning are superimposed to obtain the friction torque of the steer-by-wire system that needs to be simulated by the road feel motor: ； where: T ef1 is the upper friction torque to be simulated by the road feel motor, T ef2 is the lower friction torque to be simulated by the road feel motor.

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