Construction of Ideal Electric Power-Steering Characteristics by Inverse Dynamic Analysis Method

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

  Most control strategies of EPS systems mainly focus on power steering torque control. In this way, the driver's steering torque is analyzed into power steering torque requests or steering system position requests. There is no direct relationship between steering torque and steering motion intensity. To address this issue, this paper proposes a method to represent the driver's steering intention in the form of steering motion intensity based on the analysis of the dynamic characteristics of EPS system and vehicle motion dynamics. The study uses lateral acceleration and steering wheel steering angles as intermediate variables to connect the driver's input information with vehicle dynamics and calculates the steering torque through the inverse dynamics of the steering system and the inverse dynamics of vehicle motion. A new comprehensive model is proposed to analyze the characteristics of EPS steering assist based on "Driving style-steering system dynamics-vehicle dynamics". The simulation tests demonstrate the effectiveness of the proposed method. This paper is well-organized and has significant application value. It provides a foundation for the further advancement of EPS technology.

1. Existing studies primarily rely on experimental tuning to achieve the equivalent conversion between the driver's steering torque and the power steering torque request. The background section should elaborate on this aspect and highlight the limitations of this approach.
2. For Equations (2)–(4), the author should include the derivation process or provide relevant references to facilitate readers in deriving the model.
3. The author should provide a more detailed explanation for the decomposition model of the EPS system to enhance the clarity and coherence of the overall structure of the paper.
4. Some parameters in the simulation test results, such as i=0÷3, may cause confusion for readers. It is recommended that the author provide additional explanations for better clarity. Additionally, each variable in the paper should be properly defined to ensure consistency and readability.
5. Furthermore, the background section should highlight the rapid development of active steering system technology in recent years to emphasize the significance of this study. The related work can refer to “A Distributed Integrated Control Architecture of AFS and DYC Based on MAS for Distributed Drive Electric Vehicles, IEEE Transactions on Vehicular Technology, vol. 70, no. 6, pp. 5565-5577, June 2021”.

Author Response

Reviewer#1’s comments

Reviewer#1, Concern # 1 (please list here): Most control strategies of EPS systems mainly focus on power steering torque control. In this way, the driver's steering torque is analyzed into power steering torque requests or steering system position requests. There is no direct relationship between steering torque and steering motion intensity. To address this issue, this paper proposes a method to represent the driver's steering intention in the form of steering motion intensity based on the analysis of the dynamic characteristics of EPS system and vehicle motion dynamics. The study uses lateral acceleration and steering wheel steering angles as intermediate variables to connect the driver's input information with vehicle dynamics and calculates the steering torque through the inverse dynamics of the steering system and the inverse dynamics of vehicle motion. A new comprehensive model is proposed to analyze the characteristics of EPS steering assist based on "Driving style-steering system dynamics-vehicle dynamics". The simulation tests demonstrate the effectiveness of the proposed method. This paper is well-organized and has significant application value. It provides a foundation for the further advancement of EPS technology.

Existing studies primarily rely on experimental tuning to achieve the equivalent conversion between the driver's steering torque and the power steering torque request. The background section should elaborate on this aspect and highlight the limitations of this approach.

 

Author response: The authors would like to thank the reviewer for the valuable comment.

Author action: No

Reviewer#1, Concern # 2 (please list here): For Equations (2)–(4), the authors should include the derivation process or provide relevant references to facilitate readers in deriving the model.

Author response: The authors would like to thank the reviewer for the valuable comments.

Author action: The authors have been presented in more detail, with the correction highlighted.

Reviewer#1, Concern # 3 (please list here): The author should provide a more detailed explanation for the decomposition model of the EPS system to enhance the clarity and coherence of the overall structure of the paper.

Author response: The authors would like to thank the reviewer for the valuable comments.

Author action: The authors have been presented in more detail, with the correction highlighted in subsection 3.1.

Reviewer#1, Concern # 4 (please list here): Some parameters in the simulation test results, such as i=0÷3, may cause confusion for readers. It is recommended that the author provide additional explanations for better clarity. Additionally, each variable in the paper should be properly defined to ensure consistency and readability.

Author response: The authors would like to thank the reviewer for the comment.

Author action: The authors have clarified the parameters in the simulation test results, to avoid confusing the readers. Besides, each variable in the paper has been properly defined to ensure consistency and readability. The correction is highlighted in the new manuscript.

Reviewer#1, Concern # 5 (please list here): Furthermore, the background section should highlight the rapid development of active steering system technology in recent years to emphasize the significance of this study. The related work can refer to “A Distributed Integrated Control Architecture of AFS and DYC Based on MAS for Distributed Drive Electric Vehicles, IEEE Transactions on Vehicular Technology, vol. 70, no. 6, pp. 5565-5577, June 2021”.

Author response: The authors would like to thank the reviewer for the suggestion.

Research on EPS is an important premise for further in-depth research and application to self-driving car technology, which is a very large area of knowledge. The authors agree and thank the reviewer for his very accurate suggestion. However, within the limits of this research, the authors have not raised the issue of applying EPS to self-driving cars. This will be done in future studies.

Author action: No

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper suggests a method for representing the driver’s steering intention as steering motion intensity. This is achieved by analyzing the dynamic characteristics of the EPS system and vehicle motion dynamics. The method establishes an optimal relationship between steering torque and motion intensity, grounded in Stevens's law of psychology, thus providing a theoretical basis for optimizing driving feel.

The paper is nice and I enjoyed reading it; however, I have several concerns:

1. There is no related work section and the survey of related work is included in the introduction. I would encourage the author to split out the related work section.
2. in line 215, the authors write “n, n1- are the power exponents of the steering wheel angle and lateral acceleration”. Why do these values have power exponents? An explanation is needed.
3. Figure 1 includes numerous letters that are not explained in the accompanying text.
4. Equation 2 exhibits an inconsistency in the definition of 'i'. While 'i' is introduced as the index for summation, iterating from 1 to 4, it is subsequently defined as 1/4.
5. Equations 13-17 were written one after the other with almost no explanation. It is difficult to understand them that way.
6. In equation 42 and equation 43, could the authors explain their choice of powers from 0 to 3? What factors determined the upper limit of 3?
7. In equation 47, could the authors explain the reasoning behind their choice of 20, 100, and 150 km/h as speed thresholds?
8. The applicability of Stevens's law of psychology is questioned by many. E.g. There are papers like Weiss, D. J., “The impossible dream of Fechner and Stevens. Perception”, Vol. 10(4), pp. 431-434, 1981.‏ The authors should address this issue.
9. This paper investigates a concept in autonomous driving. It is unclear for which levels of autonomy this model is designed. I would encourage the authors to cite the autonomy levels of self-driving vehicles that can be found here: Wiseman Y., "Autonomous Vehicles", Encyclopedia of Information Science and Technology, Fifth Edition, Vol. 1, Chapter 1, pp. 1-11, 2020, available online at: https://u.cs.biu.ac.il/~wisemay/Autonomous-Vehicles-Encyclopedia.pdf . Please clarify the target autonomy levels.
10. A discussion of the potential limitations and future improvements of the proposed model would be beneficial.

 

 

Author Response

Reviewer#2’s comments

Reviewer#2, Concern # 1 (please list here): This paper suggests a method for representing the driver’s steering intention as steering motion intensity. This is achieved by analyzing the dynamic characteristics of the EPS system and vehicle motion dynamics. The method establishes an optimal relationship between steering torque and motion intensity, grounded in Stevens's law of psychology, thus providing a theoretical basis for optimizing driving feel.

The paper is nice, and I enjoyed reading it; however, I have several concerns:

Author response: The authors would like to thank the reviewer for the valuable evaluation.

Author action: No

Reviewer#2, Concern # 2 (please list here): There is no related work section, and the survey of related work is included in the introduction. I would encourage the author to split out the related work section.

Author response:  The authors would like to thank the reviewer for the valuable comment.

 

Author action: The authors supplemented relevant studies [25-29], with the correction highlighted in subsection 1.2 as follows:

1.2. Variable assistance level control strategy:

The EPS system can offer varying levels of assistance based on driving conditions. These control strategies include adjusting motor torque, steering ratio, and response characteristics to align with driver preferences and improve the characteristics of vehicle dynamics [24]. Notable research includes multi-map control, which utilizes different power assist curves according to load and tire-road contact adhesion [16]. Zhang et al. [25] present a mathematical model of the EPS system and analyze its characteristic curves. They discuss typical power curves that define the relationship between assist torque and steering wheel torque, emphasizing the importance of these curves in enhancing vehicle dynamics and driver comfort. The study provides a foundational understanding of EPS behavior. Li et al. [26] focus on designing the assistance characteristics curve for EPS systems. They propose a method where the assist torque is proportional to the steering wheel torque, allowing for a constant road feel intensity. This linear relationship simplifies the control system design and adjustment. However, the approach may not adequately account for varying driving conditions, such as changes in vehicle speed or road surface, potentially leading to suboptimal steering assistance in diverse scenarios. Du and Liming [27] introduce an assist characteristic curve design that considers load variations. Their method adjusts the assist torque based on vehicle load, enhancing steering performance under different loading conditions. Xue-Ping et al. [28] propose a parametric design of the steering characteristic curve, focusing on the relationship between steering torque and vehicle speed. Their approach allows for dynamic adjustment of assist torque, improving steering feel across various speeds. Ciarla et al. [29] explore the genesis of booster curves in EPS systems, analyzing how these curves influence steering feel and performance. They provide insights into the development of assist characteristics that balance driver effort and steering angle respon. A relation between the assistance and the driver's torque is provided, under the hypothesis of a position-oriented control of the movement and the Stevens' power law.

However, these above studies still build the power assist characteristics based on the synthetic relationship between steering torque and power assist torque or steering angle, without separating the inherent attributes of the steering system and the driver's subjective desire for driving experience.

 

Reviewer#2, Concern # 3 (please list here): in line 215, the authors write “n, n1- are the power exponents of the steering wheel angle and lateral acceleration”. Why do these values have power exponents? An explanation is needed.

Author response: The authors would like to thank the reviewer for this attension.

The authors have presented additional general Stevens law. However, the detailed presentation of the method for constructing equation (1) (in the new revised manuscript, equation (2)) has quite a large content, the group of authors has the idea to present it in the next publication.

Author action: The authors have adjusted the method of presentation to be clearer, with the correction highlighted in subsection 2.2.

Reviewer#2, Concern # 4 (please list here): Figure 1 includes numerous letters that are not explained in the accompanying text.

Author response: The authors would like to thank the reviewer for this attention.

Author action: The authors have adjusted the method of presentation to be clearer, with the correction highlighted in subsection 3.1.

As mentioned above, during the steering process of the car, the driver has a clear perception of the car's driving speed and steering movement intensity through the visual-vestibular system. The steering wheel torque is the driver's demand and expectation for the car's steering movement intensity; on the other hand, from the perspective of vehicle dynamics, when the car's motion state reaches a steady state, the lateral acceleration ay and the pinion angle d are one-to-one corresponding, and the pinion angle d and the total steering torque acting on pinion Tsum are also one-to-one corresponding. For the power steering system, the steering wheel torque Td and the steering assist torque Ta will also be stable at a certain value.

Therefore, this article first uses the lateral acceleration () as the intermediate variable that links the human steering operation input with the vehicle dynamics, uses the steering angle () as the intermediate variable that calculates the total steering torque () through the inverse characteristics of the steering system dynamics, and then calculates the steering assist torque (). The nonlinear function of the steering assist torque in the EPS assist characteristic concerning the vehicle speed and steering wheel torque is decomposed into three functional modules: “Driving style” (), “Steady-state inverse characteristics of vehicle handling dynamics” (), and “Steady-state inverse characteristics of steering system dynamics” (). The decomposition model of the EPS system based on the relationship “Driving style, EPS system dynamics and vehicle dynamics” is shown in Figure 1.

Where: - the power steering characteristic function represents the relationship between ( and (), this is the function that defines the driving style; - the inverse lateral acceleration-driver angle transfer function; - the inverse steering torque - driver steering angle transfer function; The asterisk (*) on symbols represents the desired value of the quantity with the corresponding symbol; the other symbols of the system are shown in appendix.

The “Driving style calibration” module calculates the driver's anticipated steering intensity, represented by lateral acceleration (is also the desired lateral acceleration to be achieved ), based on the torque steering () acting the steering wheel. This reflects the driver’s expected vehicle movement trend.

The “Steady-state inverse characteristic of vehicle handling dynamics" is the inherent relationship between the lateral acceleration and the front wheel angle when the car reaches a steady state. Since the front wheel angle cannot be measured by on-board sensors, for the rack-and-pinion steering gearbox, the steering wheel angle (also the pinion angle) can be estimated by the motor speed, representing the steering Ackerman mechanism angle. The relationship between it and the front wheel angle is the angular steering gearbox ratio (). Therefore, the “Steady-state inverse characteristic of vehicle handling dynamics” module () mentioned in this article is the inherent relationship between the lateral acceleration and steering wheel angle when the car reaches a steady state. This module calculates the steering wheel angle (is also the desired steering wheel angle to be achieved) through the expected lateral acceleration input (is also the desired lateral acceleration to be achieved ) - .

The “Steady-state inverse characteristic of the steering system” module illustrates the equivalent stiffness inherent relationship between desired pinion angle and total torque required () (acting on the pinion) at steady state. This module computes total torque () based on the steering wheel angle input () ().

Thus, the overall solution as shown in Figure 1 can be summarized as follows: Based on the "Driving style calibration" module, determine the desired steering motion intensity to be achieved - represented by the desired lateral acceleration () under the influence of the input steering torque (). Next, the “Steady-state inverse characteristic of vehicle handling dynamics" module calculates the desired steering wheel rotation angle () from input (). Next, the “Steady-state inverse characteristic of the steering system” module calculates the required total torque () from input (). From () and (), calculate the required assist torque () . Based on the required (), the ECU sends the control current signal () to the assist motor, and the assist motor generates torque (), through the worm gear transmission, it generates the actual assist torque (). From there, the actual total torque +=) is transmitted to the pinion and makes the pinion achieve the desired rotation angle  (also equal to ), then through the Ackermann mechanism, rotates the front wheel at an angle (), from which the car achieves the steering motion intensity as the desired steering motion intensity, i.e. . The above modules will be implemented in the EPS controller. The proposed EPS system characteristic separation method has a good effect in separating the driver's intention from the inherent dynamics of the vehicle and the steering system. This simplifies the calibration of the steering feel and facilitates the design of different driving styles on different types and models of vehicles.

Reviewer#2, Concern # 5 (please list here): Equation 2 exhibits an inconsistency in the definition of 'i'. While 'i' is introduced as the index for summation, iterating from 1 to 4, it is subsequently defined as 1/4.

Author response: The authors would like to thank the reviewer for the valuable comment.

The index "i" in the previous equation (2) (Currently in equations (5), (7), (8), (10) and (11)) is the index indicating the order of tangential and normal reactions acting on wheels from 1 to 4 in the four-wheel vehicle dynamics model. As for the equation describing the two-wheel vehicle dynamics (bicycle) model, there are only two elements: the front wheel and the rear wheel, while the rear wheel is not a steering wheel, so the rotation angle is zero, Therefore, we have the equation represented as (12) only the front wheel rotation angle remains detalf. The authors have added a note detal3 = detal4 =0 in the yellow area.

Author action: The authors have adjusted the method of presentation to be clearer, with the correction highlighted in subsection 3.2.1.

Reviewer#2, Concern # 6 (please list here): Equations 13-17 were written one after the other with almost no explanation. It is difficult to understand them that way.

Author response: The authors would like to thank the reviewer for this comment.

Author action: The authors have adjusted the method of presentation to be clearer, with the correction highlighted in subsection 3.2.1.

Reviewer#2, Concern # 7 (please list here): In equation 42 and equation 43, could the authors explain their choice of powers from 0 to 3? What factors determined the upper limit of 3?

Author response: The authors would like to thank the reviewer for the question.

Author action: In principle, it can be represented by an n-degree equation, where n is an unlimited positive integer, i.e. . However, in the optimization process to find , then  is always 0. Therefore, to reduce the complexity of the optimization problem, the authors chose n = 3.

Author action: No

Reviewer#2, Concern # 8 (please list here): In equation 47, could the authors explain the reasoning behind their choice of 20, 100, and 150 km/h as speed thresholds?

Author response: The authors would like to thank the reviewer for the comment.

This is a very interesting and important question. To deeply analyze this issue, another article is needed to explain the theoretical basis and evidence. Within the framework of this article, the authors would like to share the following basic reasons for choosing the above speed values.

- Choose v = 20km/h. According to research and study of the authors, when the car speed is small, about v ≤20km/h, the resistance from the road when turning is large, when the speed v >20km/h, the road resistance will decrease rapidly according to the car speed (this will be analyzed in the upcoming publication)

- Choose v = 100. Car speed range 100-110 km/h - this is the speed at which popular cars on the market that apply linear power steering characteristic maps often force the power steering system to be disconnected (ie there will be no more power steering), in addition to the reason that at this time the road resistance moment is small, so to ensure the driving feeling, the power steering should be disconnected, in addition, in terms of calculation algorithm, at this time the coefficient K(v) has reached the minimum point and continues to increase the car’s speed v, then K(v) will increase. With the calculation method of the article, will continuously decrease (theoretically v can reach very high car’s speed v>350km/h). Thus, with the above calculation method, the power steering is guaranteed to be very small and approaches 0 when the car has a speed of 100-150 km/h.

- Choose v=150km/h: In fact, at this time, the moment of resistance to the road surface when steering is very small, only a small steering moment can create a large front wheel rotation angle, so it is necessary to limit this impact by providing power in the opposite direction to the steering direction, creating a heavy feeling of the steering wheel, at the same time with a fairly large steering moment still giving a small steering angle to ensure safety at very high speeds.

Author action: No

Reviewer#2, Concern # 9 (please list here): The applicability of Stevens's law of psychology is questioned by many. E.g. There are papers like Weiss, D. J., “The impossible dream of Fechner and Stevens. Perception”, Vol. 10(4), pp. 431-434, 1981.‏ The authors should address this issue.

Author response: The authors would like to thank the reviewer for the valuable suggestion.

In "The Impossible Dream of Fechner and Stevens," David J. Weiss critically examines the pursuit of a universal psychophysical function that quantifies the relationship between stimulus intensity and perceived sensation. Weiss argues that the empirical form of such a function is inherently influenced by the arbitrary methods chosen to measure stimuli, leading to variability rather than universality. This critique suggests that Stevens' power law, which proposes a consistent mathematical relationship between stimulus and perception, may not universally apply across different contexts and measurement approaches. Weiss's analysis highlights the complexities of establishing a singular, overarching psychophysical law. Weiss points out that the exponent in Stevens' equation varies depending on the sensory modality and the range of stimuli used. Also, Weiss argues that human perception is not solely a function of stimulus intensity but is also shaped by contextual factors such as background, contrast, and individual differences, complicating the establishment of universal psychophysical law.​

This is also the issue that will be addressed in the upcoming research of the author group. To ensure that the proposed power-law-based assist map remains consistent with Stevens' law (1957) while not contradicting Weiss (1981). The following issues need to be considered:

• Empirical calibration: The exponent and the speed-dependent coefficient  should be determined experimentally for different driving conditions rather than assuming a fixed value.
• Contextual adaptability: The function should be flexible to accommodate different road conditions, driver preferences, and vehicle types, ensuring it does not assume an overly rigid psychophysical function.
• Accounting for variance: Driver perception of steering intensity may vary due to cognitive and biomechanical factors, requiring adaptive tuning mechanisms such as machine learning-based adjustments.

By incorporating these refinements, the power-law-based EPS characteristic curve can be applied effectively while respecting both Stevens' perceptual framework and Weiss' critique of universality.

Author action: We have added a discussion of this issue to the authors' next research in the conclusion, with the correction highlighted in section 6.

 

Reviewer#2, Concern # 10 (please list here): This paper investigates a concept in autonomous driving. It is unclear for which levels of autonomy this model is designed. I would encourage the authors to cite the autonomy levels of self-driving vehicles that can be found here: Wiseman Y., "Autonomous Vehicles", Encyclopedia of Information Science and Technology, Fifth Edition, Vol. 1, Chapter 1, pp. 1-11, 2020, available online at: https://u.cs.biu.ac.il/~wisemay/Autonomous-Vehicles-Encyclopedia.pdf . Please clarify the target autonomy levels.

Author response: The authors would like to thank the reviewer for the suggestion.

Research on EPS is an important premise for further in-depth research and application to self-driving car technology, which is a very large area of knowledge. The authors agree and thank the reviewer for his very accurate suggestion. However, within the limits of this research, the authors have not raised the issue of applying EPS to self-driving cars. This will be done in future studies.

Author action: No

Reviewer#2, Concern # 11 (please list here): A discussion of the potential limitations and future improvements of the proposed model would be beneficial.

Author response:  The authors would like to thank the reviewer for the valuable comments.

Author action: The authors have supplemented the above discussion in the conclution of section 6, with the correction highlighted.

 

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

It is supposed that Eqn. (1) depends on time but this is not reflected in the  equations themselves.

Section  3: how is it applied the inverse transfer function to this problem and which are the transfer functions referred to? . Note that the whole dynamics is nonlinear.

Which are  explicitly the inverses in the upper blocks of Figure 1?.

Which is "rho" defined in (3) used in the previous equations?.

It is not clear the reason of the assignments of variables  in  (7).

Explain (11)-(12).

Are the torques with "star" suitable references to be tracked or just auxiliary  intermediate variables to better clarify some equations?. Mention them also in the list of symbols.

Eqns. (45): Why are the torques in the various conditions sometimes used  while in other ocassions they appear with absolute value?. Also,  it is not clear apaprently  if the left-hand- sides  options of (45) include all  the  combinations of the possible alternative conditions in the constarints of the right-hand-sides.

Author Response

Reviewer#3’s comments

Reviewer#3, Concern # 1 (please list here): It is supposed that Eqn. (1) depends on time but this is not reflected in the equations themselves.

Author response: The authors would like to thank the reviewer for the attention.

Analysis of time dependency in equation (1) (The correction is equation (2) in the new manuscript).

Equation (1) describes the relationship between steering torque () and lateral acceleration () using Stevens' power law, incorporating vehicle speed () as a modifying factor. However, real-world driving dynamics are inherently time-dependent due to:

Dynamic steering input: Steering torque () varies over time as the driver continuously adjusts steering effort.

Vehicle motion dynamics: Lateral acceleration () evolves based on vehicle response time, road conditions, and suspension characteristics.

Speed variation: Speed () is a function of time, influencing the coefficient   dynamically.

To rigorously prove time dependency, differentiate both sides of equation (1) with respect to time ():

 

This equation confirms that both lateral acceleration () and its rate of change () are time-dependent through (steering input rate) and  (vehicle acceleration).

However, within the framework of this article, the authors have not analyzed the above aspect in depth. It would be interesting to discuss that issue in the next study.

 

Author action: No

Reviewer#3, Concern # 2 (please list here): Section 3: how is it applied the inverse transfer function to this problem and which are the transfer functions referred to? Note that the whole dynamics is nonlinear.

Author response: The authors would like to thank the reviewer for the valuable comment.

The following inverse transfer functions have been applied in the proposed mathematical model (equation order numbers as per revised version).

Determining  according to:                                (32)

Determining  according to :              (38)

Author action: The authors have re-newed section 3 in the new manuscript.

Reviewer#3, Concern # 3 (please list here): Which are explicitly the inverses in the upper blocks of Figure 1?

Author response: The authors would like to thank the reviewer for the valuable comment.

 

Author action: The authors have been presented in more detail, with the correction highlighted in subsection 3.1.

Reviewer#3, Concern # 4 (please list here): Which is "rho" defined in (3) used in the previous equations?

Author response: The authors would like to thank the reviewer for this comment.

The  symbol () is annotated in "Table 1. Symbol of the EPS system" row 27th

Turning radius

 

Author action: No

Reviewer#3, Concern # 5 (please list here): It is not clear the reason of the assignments of variables in  (7)

Author response: The authors would like to thank the reviewer for the valuable comments.

In (7) (in the revised version it is (13)), it is not an assignment variable but a kinetics relationship between .

Author action: The authors have been described more clearly, with the correction highlighted in subsection 3.2.1.

Reviewer#3, Concern # 6 (please list here): Explain (11)-(12)

Author response: The authors would like to thank the reviewer for the valuable comment.

Many studies [41], [10], [6] have shown that there are 3 stages of tire deformation: linear stage, nonlinear stage and finally inelastic deformation and then destruction. The problem model of this study only considers tire deformation in the linear stage, in which the tire sideslip angle and the lateral force acting on the tire follow the linear law.

Author action: The authors have added the citation, with the correction highlighted in subsection 3.2.1.

Reviewer#3, Concern # 7 (please list here): Are the torques with "star" suitable references to be tracked or just auxiliary intermediate variables to better clarify some equations? Mention them also in the list of symbols.

Author response: The authors would like to thank the reviewer for the valuable suggestion.

The torques with "star" are intermediate variables in the equations determining the power steering torque and have been assigned according to the corresponding expression. This assignment is intended to simplify the formulas describing the mathematical model of the power steering characteristic maps and to facilitate analysis. Since they have been determined according to the corresponding assignment expressions and have not been defined by name, the authors do not include them in Table 1.

Author action: No

Reviewer#3, Concern # 8 (please list here): Eqns. (45): Why are the torques in the various conditions sometimes used while in other ocassions they appear with absolute value? Also, it is not clear apaprently if the left-hand- sides options of (45) include all the combinations of the possible alternative conditions in the constarints of the right-hand-sides.

Author response: The authors would like to thank the reviewer for the valuable comment.

Steering torque and power assist torque are understood to have (+) and (-) values ​​respectively when turning to the left and right. Meanwhile, the torques  are all selected as (+) values, so to ensure that the values of  and  are within the set range, it is necessary to take the absolute value sign. However, there are still some editing errors in the article. The authors have edited them and marked them in yellow.

Author action: The authors have been described more clearly, with the correction highlighted.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The description of the background is still weak, and the contribution section does not sufficiently support the work of the paper. The authors should strengthen the description of the contributions. Furthermore, the recommended references focus on applying steer-by-wire technology to vehicles to assist drivers' manipulation, rather than autonomous driving. The authors should discuss this research area in the background section.

Author Response

Reviewer#1’s comments

Reviewer#1, Concern # 1 (please list here): The description of the background is still weak, and the contribution section does not sufficiently support the work of the paper. The authors should strengthen the description of the contributions. Furthermore, the recommended references focus on applying steer-by-wire technology to vehicles to assist drivers' manipulation, rather than autonomous driving. The authors should discuss this research area in the background section.

Author action: The authors would like to thank the reviewer for the valuable comments.

As the authors have answered in the previous response. The research of this article is an important premise in the development of active driving technology and autonomous driving. However, this study aims to focus on the theoretical basis and method of constructing ideal power assist steering characteristics for cars equipped with EPS steering systems. Therefore, the analysis of related studies in the problem introduction mainly follows this objective. The authors thank the reviewer for the valuable comments. It is probably more appropriate to include the reviewers' comments in the conclusion, and it is a very practical research direction for the future.

Reviewer’s response: The authors have added the next research direction in section 6, "Conclusion", with the correction highlighted.

We have also included two additional references to further highlight the contributions and potential research directions of this study as:

46. Liang, Jinhao; Lu, Yanbo; Yin, Guodong; Fang, Zhenwu; Zhuang, Weichao; Ren, Yanjun; Xu, Liwei; Li, Yanjun. A Distributed Integrated Control Architecture of AFS and DYC Based on MAS for Distributed Drive Electric Vehicles, IEEE Transactions on Vehicular Technology 2021, Volume 70( 6), pp. 5565-5577, http://dx.doi.org/10.1109/TVT.2021.3076105.
47. Wiseman, Y.. Autonomous Vehicles, In Encyclopedia of Information Science and Technology, Fifth Edition 2020, 1-11, IGI Global. https://doi.org/10.4018/978-1-7998-3479-3.ch001.

 

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Thanks to the authors for improving the paper.

There are still issues that should be addressed:

 

Reviewer#2, Concern # 7 (please list here): In equation 42 and equation 43, could the authors explain their choice of powers from 0 to 3? What factors determined the upper limit of 3?

Author response: The authors would like to thank the reviewer for the question.

Author action: In principle, it can be represented by an n-degree equation, where n is an unlimited positive integer, i.e. . However, in the optimization process to find , then  is always 0. Therefore, to reduce the complexity of the optimization problem, the authors chose n = 3.

Author action: No

Reviewer’s response: Please insert this explanation into the text of the paper.

 

 

Reviewer#2, Concern # 8 (please list here): In equation 47, could the authors explain the reasoning behind their choice of 20, 100, and 150 km/h as speed thresholds?

Author response: The authors would like to thank the reviewer for the comment.

This is a very interesting and important question. To deeply analyze this issue, another article is needed to explain the theoretical basis and evidence. Within the framework of this article, the authors would like to share the following basic reasons for choosing the above speed values.

- Choose v = 20km/h. According to research and study of the authors, when the car speed is small, about v ≤20km/h, the resistance from the road when turning is large, when the speed v >20km/h, the road resistance will decrease rapidly according to the car speed (this will be analyzed in the upcoming publication)

- Choose v = 100. Car speed range 100-110 km/h - this is the speed at which popular cars on the market that apply linear power steering characteristic maps often force the power steering system to be disconnected (ie there will be no more power steering), in addition to the reason that at this time the road resistance moment is small, so to ensure the driving feeling, the power steering should be disconnected, in addition, in terms of calculation algorithm, at this time the coefficient K(v) has reached the minimum point and continues to increase the car’s speed v, then K(v) will increase. With the calculation method of the article, will continuously decrease (theoretically v can reach very high car’s speed v>350km/h). Thus, with the above calculation method, the power steering is guaranteed to be very small and approaches 0 when the car has a speed of 100-150 km/h.

- Choose v=150km/h: In fact, at this time, the moment of resistance to the road surface when steering is very small, only a small steering moment can create a large front wheel rotation angle, so it is necessary to limit this impact by providing power in the opposite direction to the steering direction, creating a heavy feeling of the steering wheel, at the same time with a fairly large steering moment still giving a small steering angle to ensure safety at very high speeds.

Author action: No

Reviewer’s response: Please insert this explanation into the text of the paper.

 

 

Reviewer#2, Concern # 10 (please list here): This paper investigates a concept in autonomous driving. It is unclear for which levels of autonomy this model is designed. I would encourage the authors to cite the autonomy levels of self-driving vehicles that can be found here: Wiseman Y., "Autonomous Vehicles", Encyclopedia of Information Science and Technology, Fifth Edition, Vol. 1, Chapter 1, pp. 1-11, 2020, available online at: https://u.cs.biu.ac.il/~wisemay/Autonomous-Vehicles-Encyclopedia.pdf . Please clarify the target autonomy levels.

Author response: The authors would like to thank the reviewer for the suggestion.

Research on EPS is an important premise for further in-depth research and application to self-driving car technology, which is a very large area of knowledge. The authors agree and thank the reviewer for his very accurate suggestion. However, within the limits of this research, the authors have not raised the issue of applying EPS to self-driving cars. This will be done in future studies.

Author action: No

Reviewer’s response: OK. So, please add it as future work in the conclusions section. Your model might fit level 3 or even higher levels. Am I wrong?

 

 

Reviewer#2, Concern # 9 (please list here): The applicability of Stevens's law of psychology is questioned by many. E.g. There are papers like Weiss, D. J., “The impossible dream of Fechner and Stevens. Perception”, Vol. 10(4), pp. 431-434, 1981.‏ The authors should address this issue.

Author response: The authors would like to thank the reviewer for the valuable suggestion.

In "The Impossible Dream of Fechner and Stevens," David J. Weiss critically examines the pursuit of a universal psychophysical function that quantifies the relationship between stimulus intensity and perceived sensation. Weiss argues that the empirical form of such a function is inherently influenced by the arbitrary methods chosen to measure stimuli, leading to variability rather than universality. This critique suggests that Stevens' power law, which proposes a consistent mathematical relationship between stimulus and perception, may not universally apply across different contexts and measurement approaches. Weiss's analysis highlights the complexities of establishing a singular, overarching psychophysical law. Weiss points out that the exponent in Stevens' equation varies depending on the sensory modality and the range of stimuli used. Also, Weiss argues that human perception is not solely a function of stimulus intensity but is also shaped by contextual factors such as background, contrast, and individual differences, complicating the establishment of universal psychophysical law.​

This is also the issue that will be addressed in the upcoming research of the author group. To ensure that the proposed power-law-based assist map remains consistent with Stevens' law (1957) while not contradicting Weiss (1981). The following issues need to be considered:

• Empirical calibration: The exponent and the speed-dependent coefficient  should be determined experimentally for different driving conditions rather than assuming a fixed value.
• Contextual adaptability: The function should be flexible to accommodate different road conditions, driver preferences, and vehicle types, ensuring it does not assume an overly rigid psychophysical function.
• Accounting for variance: Driver perception of steering intensity may vary due to cognitive and biomechanical factors, requiring adaptive tuning mechanisms such as machine learning-based adjustments.

By incorporating these refinements, the power-law-based EPS characteristic curve can be applied effectively while respecting both Stevens' perceptual framework and Weiss' critique of universality.

Author action: We have added a discussion of this issue to the authors' next research in the conclusion, with the correction highlighted in section 6.

Reviewer response: This discussion does not belong to the conclusions but rather to 2.2. The ideal relationship model between steering torque and steering motion intensity based on Stevens' law

 

Author Response

Reviewer#2’s comments

“Thanks to the authors for improving the paper.”

Author action: The authors woud like to thank the reviewer.

“There are still issues that should be addressed:

 Reviewer#2, Concern # 7 (please list here): In equation 42 and equation 43, could the authors explain their choice of powers from 0 to 3? What factors determined the upper limit of 3?

Author response: The authors would like to thank the reviewer for the question.

Author action: In principle, it can be represented by an n-degree equation, where n is an unlimited positive integer, i.e. . However, in the optimization process to find , then  is always 0. Therefore, to reduce the complexity of the optimization problem, the authors chose n = 3.

Author action: No

Reviewer’s response: Please insert this explanation into the text of the paper.”

Author action: The authors would like to thank the reviewer for the valuable comment.

Reviewer’s response: The authors have added this content to the article, with the correction highlighted.

“Reviewer#2, Concern # 8 (please list here): In equation 47, could the authors explain the reasoning behind their choice of 20, 100, and 150 km/h as speed thresholds?

Author response: The authors would like to thank the reviewer for the comment.

This is a very interesting and important question. To deeply analyze this issue, another article is needed to explain the theoretical basis and evidence. Within the framework of this article, the authors would like to share the following basic reasons for choosing the above speed values.

- Choose v = 20km/h. According to research and study of the authors, when the car speed is small, about v ≤20km/h, the resistance from the road when turning is large, when the speed v >20km/h, the road resistance will decrease rapidly according to the car speed (this will be analyzed in the upcoming publication)

- Choose v = 100. Car speed range 100-110 km/h - this is the speed at which popular cars on the market that apply linear power steering characteristic maps often force the power steering system to be disconnected (ie there will be no more power steering), in addition to the reason that at this time the road resistance moment is small, so to ensure the driving feeling, the power steering should be disconnected, in addition, in terms of calculation algorithm, at this time the coefficient K(v) has reached the minimum point and continues to increase the car’s speed v, then K(v) will increase. With the calculation method of the article, will continuously decrease (theoretically v can reach very high car’s speed v>350km/h). Thus, with the above calculation method, the power steering is guaranteed to be very small and approaches 0 when the car has a speed of 100-150 km/h.

- Choose v=150km/h: In fact, at this time, the moment of resistance to the road surface when steering is very small, only a small steering moment can create a large front wheel rotation angle, so it is necessary to limit this impact by providing power in the opposite direction to the steering direction, creating a heavy feeling of the steering wheel, at the same time with a fairly large steering moment still giving a small steering angle to ensure safety at very high speeds.

Author action: No

Reviewer’s response: Please insert this explanation into the text of the paper.”

Author action: The authors would like to thank the reviewer for the valuable comment.

Reviewer’s response: The authors have added this content to the article, with the correction highlighted.

 

“Reviewer#2, Concern # 10 (please list here): This paper investigates a concept in autonomous driving. It is unclear for which levels of autonomy this model is designed. I would encourage the authors to cite the autonomy levels of self-driving vehicles that can be found here: Wiseman Y., "Autonomous Vehicles", Encyclopedia of Information Science and Technology, Fifth Edition, Vol. 1, Chapter 1, pp. 1-11, 2020, available online at: https://u.cs.biu.ac.il/~wisemay/Autonomous-Vehicles-Encyclopedia.pdf

 

. Please clarify the target autonomy levels.

Author response: The authors would like to thank the reviewer for the suggestion.

Research on EPS is an important premise for further in-depth research and application to self-driving car technology, which is a very large area of knowledge. The authors agree and thank the reviewer for his very accurate suggestion. However, within the limits of this research, the authors have not raised the issue of applying EPS to self-driving cars. This will be done in future studies.

Author action: No

Reviewer’s response: OK. So, please add it as future work in the conclusions section. Your model might fit level 3 or even higher levels. Am I wrong?”

Author action: The authors would like to thank the reviewer for the valuable comments.

Reviewer’s response: The authors have added this content to the article, with the correction highlighted.

  

 

 “Reviewer#2, Concern # 9 (please list here): The applicability of Stevens's law of psychology is questioned by many. E.g. There are papers like Weiss, D. J., “The impossible dream of Fechner and Stevens. Perception”, Vol. 10(4), pp. 431-434, 1981.‏ The authors should address this issue.

Author response: The authors would like to thank the reviewer for the valuable suggestion.

In "The Impossible Dream of Fechner and Stevens," David J. Weiss critically examines the pursuit of a universal psychophysical function that quantifies the relationship between stimulus intensity and perceived sensation. Weiss argues that the empirical form of such a function is inherently influenced by the arbitrary methods chosen to measure stimuli, leading to variability rather than universality. This critique suggests that Stevens' power law, which proposes a consistent mathematical relationship between stimulus and perception, may not universally apply across different contexts and measurement approaches. Weiss's analysis highlights the complexities of establishing a singular, overarching psychophysical law. Weiss points out that the exponent in Stevens' equation varies depending on the sensory modality and the range of stimuli used. Also, Weiss argues that human perception is not solely a function of stimulus intensity but is also shaped by contextual factors such as background, contrast, and individual differences, complicating the establishment of universal psychophysical law.​

This is also the issue that will be addressed in the upcoming research of the author group. To ensure that the proposed power-law-based assist map remains consistent with Stevens' law (1957) while not contradicting Weiss (1981). The following issues need to be considered:

• Empirical calibration: The exponent and the speed-dependent coefficient  should be determined experimentally for different driving conditions rather than assuming a fixed value.
• Contextual adaptability: The function should be flexible to accommodate different road conditions, driver preferences, and vehicle types, ensuring it does not assume an overly rigid psychophysical function.
• Accounting for variance: Driver perception of steering intensity may vary due to cognitive and biomechanical factors, requiring adaptive tuning mechanisms such as machine learning-based adjustments.

By incorporating these refinements, the power-law-based EPS characteristic curve can be applied effectively while respecting both Stevens' perceptual framework and Weiss' critique of universality.

Author action: We have added a discussion of this issue to the authors' next research in the conclusion, with the correction highlighted in section 6.

Reviewer response: This discussion does not belong to the conclusions but rather to 2.2. The ideal relationship model between steering torque and steering motion intensity based on Stevens' law”

Author action: The authors would like to thank the reviewer for the valuable comments.

Reviewer’s response: The authors have added this content to the article, with the correction highlighted.

  

 

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

The author has addressed my concerns well. There are some grammatical errors and issues with the reference format. Please check carefully. Furthermore, the development of the active steering technology for semi-autonomous driving can refer to the work “ETS-Based Human-Machine Robust Shared Control Design Considering the Network Delays, IEEE Transactions on Automation Science and Engineering, doi: 10.1109/TASE.2024.3383094”.

Author Response

The authors would like to thank the reviewer.
We have reviewed the references and have corrected the yellow highlighted areas in the new manuscript.
As for the reference you suggested, it is very interesting and we thank you. We will refer to this interesting article in future publications.
Thank you very much!

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed all my concerns. The revised manuscript is ready for publication.

 

Author Response

The authors would like to thank the reviewer.