On-Site Implementation of External Wrench Measurement via Non-Linear Optimization in Six-Axis Force–Torque Sensor Calibration and Crosstalk Compensation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Auhtors  study  a novel calibration method for accurate external wrench measurement using a 6-axis FT sensor.

Some visions should be complected as follows before accepted.

1. FT and CoM in abstract  should be explained more clearly.

2. Gnerally,  FT sensor calibration  is used for providing measuring base for Sensor. Auhtor should explined clearly the relation between 

However,  The measuring error can not be reduced by sensor calibration.

Author using existing 6-axis FT sensor measure external wrench of robot by   proposed calibration method .

Gnerally,   the calibration of FT sensor should be conducted in the Regulation condition, and  calibration result can be used to  provide a measuring base (coefficient K of FT ) of Sensor. Thus, based on K and measuring values, reality value of  FT can be obtained.  

Therefore,  authors should explined relation between reality value of  FT,  calibration result and measuring values.

3. Regulation condition must be satisfy most work conditions which is not  given by author. 

However,  The measuring error can not be reduced by sensor Calibration. 

4. The measuring error can  be reduced by sensor with high accuracy sensor or new model sensor.

5. What are new theoretical or new sensor should are explained in this paper. 

Comments on the Quality of English Language

English Language should be improved.

Author Response

In the revised submission, we have highlighted all changes made from the previous version in blue. While this has temporarily affected the readability of the layout, including the placement of tables and figures, we will ensure that these issues are fully addressed and resolved in the final version for better readability and presentation.

• Comments 1: FT and CoM in abstract should be explained more clearly.
• Response 1: Thank you for pointing this out. We have added the full terms for the abbreviations FT and CoM on the abstract.

 

• Comments 2: Gnerally,  FT sensor calibration  is used for providing measuring base for Sensor. Auhtor should explined clearly the relation between 

However,  The measuring error can not be reduced by sensor calibration.

Author using existing 6-axis FT sensor measure external wrench of robot by   proposed calibration method .

Gnerally,   the calibration of FT sensor should be conducted in the Regulation condition, and  calibration result can be used to  provide a measuring base (coefficient K of FT ) of Sensor. Thus, based on K and measuring values, reality value of  FT can be obtained.  

Therefore,  authors should explined relation between reality value of  FT,  calibration result and measuring values.

• Response 2: We agree with you. Indeed, manufacturers of FT sensors typically calibrate the internal parameters of the sensor, enabling users to obtain six-axis force and torque measurements. However, our goal is not to recalibrate these internal parameters using raw data but rather to accurately decompose the forces and torques in real-world scenarios where the sensor is used while using sensor data as-is, such as when an end tool is attached to the sensor or when the base is inclined. To clarify this point, we have added content to the end of the fourth paragraph in the introduction.

 

• Comments 3: Regulation condition must be satisfy most work conditions which is not  given by author. 

However,  The measuring error can not be reduced by sensor Calibration.

• Response 3: Similarly to the previous response, our aim was to accurately determine the forces and torques during actual sensor usage. Therefore, we conducted experiments in scenarios resembling real-world conditions where the sensor would typically be employed.

 

• Comments 4: The measuring error can be reduced by sensor with high accuracy sensor or new model sensor.
• Response 4: We agree with your opinion. To reduce measurement errors, more precise sensors should be used. However, our focus is not on improving sensor precision itself but rather on enhancing accuracy under the specific conditions in which the sensor is used. For instance, even if the sensor is mounted at the same orientation on a manipulator's end effector, it will output different measurements if the base is inclined. From the perspective of users who aim to perform tasks with an FT sensor, they wish to obtain the actual forces acting on the sensor, regardless of the robot's orientation, the base's inclination, or the type of end tool. Therefore, we approached this from a practical perspective, aiming to identify and compensate for factors that significantly affect the sensor's measurements, such as the CoM of the end tool, crosstalk, and the base inclination, to accurately determine the forces and torques acting on the sensor.

 

• Comments 5: What are new theoretical or new sensor should are explained in this paper. 
• Response 5: These points constitute the primary contributions of our paper:

1. We propose a generalized sensor model consisting of parameters such as the CoM, crosstalk matrix, and inclination, ensuring that sensor compensation is complete once all parameters are identified. By using the force and torque data provided by the sensor, rather than relying on raw sensor data, our model enhances generality and applicability across various sensor types and manufacturers.
2. Instead of directly utilizing the sensor's raw data, we perform calibration using only the force and torque data provided by the sensor. This ensures that our proposed method can be applied universally, regardless of the sensor type or manufacturer.
3. Our method enables "on-site" calibration without requiring additional sensors or equipment, even when precise information about the operating environment of the manipulator-mounted FT sensor (such as the CoM of the end tool or the inclination of the base) is unknown. This is particularly relevant in scenarios where the FT sensor is used with an end tool, as users often lack accurate knowledge of the end tool's physical properties (e.g., weight and CoM) or the degree of ground inclination. Our approach allows these environmental factors to be determined through calibration and compensated for in the sensor data, enabling accurate force and torque measurements. Unlike other studies that require attaching an object with a precisely known CoM for calibration, our method does not rely on such prior information, offering a more flexible and practical solution.

To clarify this point, we have added content to the sixth paragraph in the introduction.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents an innovative 6-axis force-torque (FT) sensor calibration method that combines bias correction, center of mass (CoM) estimation, crosstalk compensation, and nonlinear optimization for inclination calibration. The method effectively addresses the limitations of traditional calibration techniques in mobile and inclined environments, showing significant engineering applicability. However, there are several aspects that could be improved to enhance the overall quality and relevance of the manuscript:

 

1. Methodological Details

The design of the nonlinear optimization objective function (Equation 10) is clear, but more details about parameter initialization and the convergence behavior could improve reproducibility. For example, discussing how the initial parameters are selected and their impact on the final accuracy would help practitioners implement this method.

2. Experimental Design Expansion

The experiments primarily focus on a single type of FT sensor (AFT200) and tool (Robotiq 3F gripper). Adding tests with different sensors and tools (e.g., from other manufacturers or with varying capacities) could demonstrate the method’s generalizability.

Inclination experiments are limited to a 5-degree tilt. Expanding the tests to include a wider range of tilt angles (e.g., up to 10 degrees or more) would provide a more comprehensive validation of the method.

3. In-depth Discussion of Results

Although the results indicate significant reductions in mean error (14% in non-inclined conditions, 42% in inclined conditions), the underlying reasons for these improvements could be discussed further. For instance, the impact of inclination calibration on CoM estimation or the specific role of crosstalk compensation in reducing torque errors.

4. Figures and Language Improvements

Figures 3 and 5 showing robot postures could include clearer annotations, such as marking the CoM position and force/torque directions, to make the experimental design more intuitive.

This paper presents a solid approach with innovative elements, well-structured experiments, and clear results, making it relevant to practical applications. Incorporating the suggested improvements would further strengthen the paper’s scientific soundness and enhance its academic value. It is recommended to address these points to better showcase the study's contributions and potential.

Author Response

In the revised submission, we have highlighted all changes made from the previous version in blue. While this has temporarily affected the readability of the layout, including the placement of tables and figures, we will ensure that these issues are fully addressed and resolved in the final version for better readability and presentation.

• Comments 1: Methodological Details

The design of the nonlinear optimization objective function (Equation 10) is clear, but more details about parameter initialization and the convergence behavior could improve reproducibility. For example, discussing how the initial parameters are selected and their impact on the final accuracy would help practitioners implement this method.

• Response 1: Thank you for pointing this out. We have added an explanation of how and why the initial values and step size were set to the second paragraph of "2.3.2 Non-linear Method."

The parameters and initial values for nonlinear optimization are set as follows: the step size for CoM is 0.00005, the step size for alpha and beta is 0.0001, and the step size for crosstalk is 0.001. For the ground inclination, since it is measured in radians and the base inclination is generally small, it is advantageous to optimize with a small step size. Similarly, for CoM, as its value is in meters and relatively small, a small step size is also appropriate. For initial values, the CoM can converge easily by directly using the values provided in the datasheet, as they are close to the optimal point. For ground inclination, even if the inclination is unknown, setting it to 0 allows convergence due to the small step size. It is acceptable to input approximate angles, such as 10° or 20°, based on estimation. For crosstalk, it is recommended to set the initial value to 0, as sensor manufacturers usually perform internal calibration. As will be shown in the experimental results discussed later, the inclination significantly impacts accurate force-torque decoupling. Therefore, setting the tolerance to 10e-6 to ensure sufficient convergence is advantageous for obtaining precise results. 

 

• Comments 2: Experimental Design Expansion

The experiments primarily focus on a single type of FT sensor (AFT200) and tool (Robotiq 3F gripper). Adding tests with different sensors and tools (e.g., from other manufacturers or with varying capacities) could demonstrate the method’s generalizability.

Inclination experiments are limited to a 5-degree tilt. Expanding the tests to include a wider range of tilt angles (e.g., up to 10 degrees or more) would provide a more comprehensive validation of the method.

• Response 2: Thank you for pointing this out. We conducted additional experiments by varying the tilt angles and using different types of grippers. Specifically, we utilized the Robotiq 3F gripper and the Barrett Hand. The experiments included scenarios with roll angles of -7° and -15°, and pitch angles of -8° and -17°, distinguishing between cases where a 2 kg object was grasped and where it was not. Most FT sensors either follow the standards set by ATI or incorporate modules based on these standards. Therefore, we believe that using the AFT200 sensor, which adheres to ATI's standards, is sufficient for our experiments.

The results of these additional experiments have been compiled into a table and added to the main text as Table 9.

 

• Comments 3: In-depth Discussion of Results

Although the results indicate significant reductions in mean error (14% in non-inclined conditions, 42% in inclined conditions), the underlying reasons for these improvements could be discussed further. For instance, the impact of inclination calibration on CoM estimation or the specific role of crosstalk compensation in reducing torque errors.

• Response 3: Thank you for pointing this out. We have added an additional discussion to the third paragraph of "3.3 Inclination Calibration."

From Eq. (3) and (4), it can be observed that the predicted force and torque are dependent on the ground inclination and the CoM. In the case of the crosstalk matrix used in Eq. (8) with LSM or in Eq. (9) with non-linear form, it can be interpreted as a constant value multiplied by the error of the predicted force and torque. Therefore, in the absence of inclination, multiplying the torque by the crosstalk matrix can reduce the error. However, when inclination is present, the inclination and CoM are coupled, making it impossible to reduce the error using only the crosstalk matrix. Furthermore, as seen in Eq. (3), when there is no inclination, the sine term becomes 0, eliminating many terms. However, when inclination is present, a sine value appears, increasing the nonlinearity. Hence, the influence of inclination can be considered the most significant factor. This is notable by the experimental results shown in Figure 8, where the error remains large until the inclination is compensated. 

 

• Comments 4: Figures and Language Improvements

Figures 3 and 5 showing robot postures could include clearer annotations, such as marking the CoM position and force/torque directions, to make the experimental design more intuitive.

• Response 4: Thank you for pointing this out. Since displaying all forces and torques was challenging, we revised the illustrations to include the sensor axes and the CoM of the gripper, highlighting how they change with different orientations.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This work proposes and tests a method for the calibration of a robotic arm on a robotic platform on the field, through solving a non-linear optimization problem using data gathered in a series of calibration positions.

I find this study interesting from a practical point of view, but some problems need to be addressed.

1.       Acronyms should be expanded in their first occurrence, although both FT and CoM are pretty common, they should be introduced at their very first use.

2.       The abstract is weak. It should be self-contained. Apart from acronyms, there are some vague sentences “reducing error” (which kind of error?) “non linear optimization” (in which kind of problem?), and overall, the novelty is not well presented.

3.       Going further in the text I find more vague forms: “solving the fundamental dynamics” (of what?) moreover, please contextualize better your work (it’s a specific branch of robotics dealing with robotic arms).

4.       I’d prefer to see the problems affecting robotic arms in a more systematic way (basically what you do in section 2.1, but in a wider way).

5.       Literature review is quite poor, you may start from some good review/survey to introduce a classification of solutions and contextualize yours among them.

6.       In formula (1) I'd use “desidered” rather than “ideal”, but it’s just a suggestion

7.       A slight note: gravity acceleration can vary of about 0,7% around the world, I’d rather refer to it as “g” rather than using its conventional value 9,81 m/s²

8.       The assumptions leading to (6) should be explained better

9.       Tables 2 and 3 should report uncertainty

10.  In the caption of figure 4 something is missing (“on an inclined surface”)

11.  The structure of the C matrix should be presented before in the article, then confirmed in section 3

12.  Table 4 pertains to how many tests?

13.  Table 2 caption is not that much clear

14.  In sec 3.2 “without any inclined” should read “without inclination” or “not inclined” or something similar (same elsewhere)

15.  Control is out of scope of your work (and this is perfectly ok), however some brief explanation about its assumptions is needed, i.e. you should specify you just give a series of desired positions, and the control loop(s) provide the required control action, using joints encoders as feedback.

16.  MSE is a more significant measure of error rather than mean and variance

17.  Considerations in the last part of page 11 should be explained better

18.  In weight estimation, during test with the moving the object how do you take into account dynamic (acceleration) effects?

19.  What is the x-axis of figure 9?

20.  Each time you mention an improvement you should mention the baseline: in conclusion “the overall mean error decreased by about 42%” should be “the overall mean error decreased by about 42% with respect to  …”

21.  Overall, the main weakness of this work which I can notice is the apparent lack of novelty, you should provide the necessary literature background and explain in detail how do you detach from existing solutions

22. Last author's ORCID is missing

23. The optimization part is almost forgot, please describe it and deal with the computational effort related to your technique

24. Can you consider inclinations of the base along two angles?

25. Did you consider using (it's just a suggestion, feel free to ignore) a single figure in fig. 2?

Author Response

In the revised submission, we have highlighted all changes made from the previous version in blue. While this has temporarily affected the readability of the layout, including the placement of tables and figures, we will ensure that these issues are fully addressed and resolved in the final version for better readability and presentation.

• Comments 1: Acronyms should be expanded in their first occurrence, although both FT and CoM are pretty common, they should be introduced at their very first use.
• Response 1: Thank you for your feedback. We have ensured that all acronyms , including those in the abstract, are accompanied by their full terms when first mentioned.

 

• Comments 2: The abstract is weak. It should be self-contained. Apart from acronyms, there are some vague sentences “reducing error” (which kind of error?) “non linear optimization” (in which kind of problem?), and overall, the novelty is not well presented.
• Response 2: Thank you for pointing this out. We have revised the vague description in the abstract as your suggestions. Additionally, we clarified our novelty by explicitly stating the following points:

1. 1. We propose a generalized sensor model consisting of parameters such as CoM, crosstalk matrix, and inclination, enabling complete sensor compensation once all parameters are identified.
   2. Instead of directly using the sensor's raw data, we utilize only the force and torque data provided by the sensor for calibration, ensuring applicability regardless of sensor type.
   3. Even without precise knowledge of the operating environment of an FT sensor mounted on a manipulator (e.g., the CoM of the end tool or the inclination of the base), our method allows on-site calibration to identify environmental factors without requiring additional sensors or equipment.

These points have been added to the abstract and the introduction of the revised manuscript.

 

• Comments 3: Going further in the text I find more vague forms: “solving the fundamental dynamics” (of what?) moreover, please contextualize better your work (it’s a specific branch of robotics dealing with robotic arms).
• Response 3: Thank you for your feedback. We have revised the expressions as suggested to clearly explain which dynamics need to be addressed in the first paragraph of the introduction.

 

• Comments 4:I’d prefer to see the problems affecting robotic arms in a more systematic way (basically what you do in section 2.1, but in a wider way).
• Response 4: Thank you for your suggestion. We have added examples outlining the problems that can arise when using FT sensors directly to the "2.1 Background" section for better clarity and context.

If the FT sensor data is directly used by the robot, discrepancies can arise between the actual wrench acting on the robot's end effector and the sensor readings, potentially causing the system to misinterpret these discrepancies as external forces. For instance, the gravity resulting from the end effector's own weight can lead to varying sensor readings depending on the inclination of the base, even when the robot is in the same posture. Additionally, if the CoM of the end tool is inaccurate, external torque may be detected even when no object is grasped. To address these issues, our proposed method estimates the parameters and enables consistent external force estimation.

 

• Comments 5: Literature review is quite poor, you may start from some good review/survey to introduce a classification of solutions and contextualize yours among them.
• Response 5: Thank you for your feedback. While there were no review or survey papers specifically on FT sensor calibration, we enhanced the understanding of FT sensors by adding a review paper on FT sensor applications, particularly focusing on how manufacturers achieve accurate force and torque separation. This has been included as Reference 6.

We also emphasized that our proposed method does not rely on raw sensor data but instead utilizes the data provided by the manufacturer. Our approach focuses on refining the accuracy of this data for practical applications, further highlighting the novelty and practicality of our research.

 

• Comments 6: In formula (1) I'd use “desidered” rather than “ideal”, but it’s just a suggestion
• Response 6: Thank you for your suggestion. The reason we used the term "ideal" is that, in the absence of any errors, the sensor readings should perfectly separate the torque generated by the weight and CoM. Our approach does not aim to estimate this "ideal" value but rather compensates for the inclination and crosstalk in the sensor readings to separate the forces and torques accurately. Thus, we employed this terminology when modeling the sensor.

 

• Comments 7: A slight note: gravity acceleration can vary of about 0,7% around the world, I’d rather refer to it as “g” rather than using its conventional value 9,81 m/s²
• Response 7: Thank you for your detailed review. We have updated the 9.81 in Eq (2) to “g” for clarity and consistency.

 

• Comments 8:The assumptions leading to (6) should be explained better
• Response 8: Thank you for your feedback. The reason for setting the crosstalk matrix as shown in Equation (6) has been explained in detail with actual data in the "3.2 Crosstalk Calibration" section. To make this clearer, we have added a reference to this explanation at the end of the final paragraph in the "2.3 Calibration Model" section.

 

• Comments 9: Tables 2 and 3 should report uncertainty
• Response 9: Thank you for your feedback. We have clarified that the kinematically computed inclination angles have an unknown degree of uncertainty. Since the purpose of this data is to demonstrate the reliability of the calibration results rather than measure exact angles, we explicitly indicated these values as approximations in the caption of Table 2. While the CoM specified in the Robotiq gripper's datasheet is stated to be an approximation, its exact range or percentage of accuracy is not disclosed. This has been noted in Table 3 for transparency.

 

• Comments 10: In the caption of figure 4 something is missing (“on an inclined surface”)
• Response 10: Thank you for your feedback. We have revised the caption as per your suggestion.

 

• Comments 11:The structure of the C matrix should be presented before in the article, then confirmed in section 3
• Response 11: The structure of the C matrix in Equation (6) has been explained in the "2.3 Calibration Model" section.

 

• Comments 12: Table 4 pertains to how many tests?
• Response 12: The experimental data in Table 4 corresponds to the data from one experiment. However, we were able to collect a substantial amount of data using the same experimental method, demonstrating that crosstalk does not affect torque. We included a representative dataset in the table.

 

• Comments 13: Table 2 caption is not that much clear
• Response 13: Thank you for your feedback. We have updated the caption for Table 2. Since the angles we measured represent rotation about an arbitrary axis (axis angle), we have adjusted the representation of the estimated roll and pitch angles accordingly.

 

• Comments 14: In sec 3.2 “without any inclined” should read “without inclination” or “not inclined” or something similar (same elsewhere)
• Response 14: Thank you for your feedback. We have revised all relevant expressions in the main text accordingly.

 

• Comments 15: Control is out of scope of your work (and this is perfectly ok), however some brief explanation about its assumptions is needed, i.e. you should specify you just give a series of desired positions, and the control loop(s) provide the required control action, using joints encoders as feedback.
• Response 15: Thank you for your feedback. We have added information about the controller used to operate the robot and how the pick-and-place scenario was configured to the second paragraph of the "3.2 Crosstalk Calibration" section. Specifically, the robot was controlled using an H-infinity controller, and the movements were executed based on pre-stored joint poses.

 

• Comments 16: MSE is a more significant measure of error rather than mean and variance
• Response 16: Thank you for your suggestion. We have updated the analysis of all experimental results by replacing the averages and variances with Mean Squared Error (MSE) for consistency and clarity.

 

• Comments 17: Considerations in the last part of page 11 should be explained better
• Response 17: Thank you for your feedback. We have added a more detailed analysis of how each parameter affects the results and the reasons behind these experimental outcomes in the third paragraph of the "3.2 Inclination Calibration" section.

From Eq. (3) and (4), it can be observed that the predicted force and torque are dependent on the ground inclination and the CoM. In the case of the crosstalk matrix used in Eq. (8) with LSM or in Eq. (9) with non-linear form, it can be interpreted as a constant value multiplied by the error of the predicted force and torque. Therefore, in the absence of inclination, multiplying the torque by the crosstalk matrix can reduce the error. However, when inclination is present, the inclination and CoM are coupled, making it impossible to reduce the error using only the crosstalk matrix. Furthermore, as seen in Eq. (3), when there is no inclination, the sine term becomes 0, eliminating many terms. However, when inclination is present, a sine value appears, increasing the nonlinearity. Hence, the influence of inclination can be considered the most significant factor. This is notable by the experimental results shown in Figure 8, where the error remains large until the inclination is compensated.

 

• Comments 18: In weight estimation, during test with the moving the object how do you take into account dynamic (acceleration) effects?
• Response 18: Thank you for your feedback. In our case, we reduced the robot's acceleration to ignore the dynamic effects of the object during the experiments. Initially, we aimed to account for dynamic effects, including acceleration but calculating acceleration through numerical differentiation of joint velocities introduced excessive noise, making it impractical.

 

• Comments 19:What is the x-axis of figure 9?
• Response 19: Thank you for your feedback. We have added the x-axis label indicating time (s) to the figure for clarity.

 

• Comments 20: Each time you mention an improvement you should mention the baseline: in conclusion “the overall mean error decreased by about 42%” should be “the overall mean error decreased by about 42% with respect to  …”
• Response 20: Thank you for your feedback. We have revised the text to clearly specify the reference values against which the performance improvements were measured

 

• Comments 21:Overall, the main weakness of this work which I can notice is the apparent lack of novelty, you should provide the necessary literature background and explain in detail how do you detach from existing solutions
• Response 21: These points constitute the primary contributions of our paper:

1. 1. We propose a generalized sensor model consisting of parameters such as the CoM, crosstalk matrix, and inclination, ensuring that sensor compensation is complete once all parameters are identified. By using the force and torque data provided by the sensor, rather than relying on raw sensor data, our model enhances generality and applicability across various sensor types and manufacturers.
   2. Instead of directly utilizing the sensor's raw data, we perform calibration using only the force and torque data provided by the sensor. This ensures that our proposed method can be applied universally, regardless of the sensor type or manufacturer.
   3. Our method enables "on-site" calibration without requiring additional sensors or equipment, even when precise information about the operating environment of the manipulator-mounted FT sensor (such as the CoM of the end tool or the inclination of the base) is unknown. This is particularly relevant in scenarios where the FT sensor is used with an end tool, as users often lack accurate knowledge of the end tool's physical properties (e.g., weight and CoM) or the degree of ground inclination. Our approach allows these environmental factors to be determined through calibration and compensated for in the sensor data, enabling accurate force and torque measurements. Unlike other studies that require attaching an object with a precisely known CoM for calibration, our method does not rely on such prior information, offering a more flexible and practical solution.

To clarify this point, we have added content to the sixth paragraph in the introduction.

 

• Comments 22: Last author's ORCID is missing
• Response 22: Thank you for your check. We have added the ORCID information as suggested.

 

• Comments 23: The optimization part is almost forgot, please describe it and deal with the computational effort related to your technique
• Response 23: Thank you for pointing this out. We have added an explanation of how and why the initial values and step size were set to the second paragraph of "2.3.2 Non-linear Method."

The parameters and initial values for nonlinear optimization are set as follows: the step size for CoM is 0.00005, the step size for alpha and beta is 0.0001, and the step size for crosstalk is 0.001. For the ground inclination, since it is measured in radians and the base inclination is generally small, it is advantageous to optimize with a small step size. Similarly, for CoM, as its value is in meters and relatively small, a small step size is also appropriate. For initial values, the CoM can converge easily by directly using the values provided in the datasheet, as they are close to the optimal point. For ground inclination, even if the inclination is unknown, setting it to 0 allows convergence due to the small step size. For crosstalk, it is recommended to set the initial value to 0, as sensor manufacturers usually perform internal calibration. As will be shown in the experimental results discussed later, the inclination significantly impacts accurate force-torque decoupling. Therefore, setting the tolerance to 10e-6 to ensure sufficient convergence is advantageous for obtaining precise results.

 

• Comments 24: Can you consider inclinations of the base along two angles?
• Response 24: : Thank you for your suggestion. Creating two angles made it difficult to measure the actual angles accurately and caused instability in the robot's base. Instead, we conducted additional experiments using a wider range of angles, and the results have been included in Table 9. 

 

• Comments 25:Did you consider using (it's just a suggestion, feel free to ignore) a single figure in fig. 2?
• Response 25: Thank you for your suggestion. Figure 2 was added to illustrate how the data from the four selected postures eliminate the effects of crosstalk and isolate the bias. To ensure clarity, we explicitly depicted the four postures along with the corresponding forces for each posture.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Author have complected revisions.

However, a relative paper should be mentioned in Introduction.

Development and kinematics/statics analysis of rigid-flexible-soft hybrid
finger mechanism with standard force sensor.  Robotics and Computer Integrated Manufacturing 67 (2021) 101978

Author Response

In the revised submission, we have highlighted all changes made from the previous version in blue. While this has temporarily affected the readability of the layout, including the placement of tables and figures, we will ensure that these issues are fully addressed and resolved in the final version for better readability and presentation.

• Comments 1: Author have complected revisions. However, a relative paper should be mentioned in Introduction.

Development and kinematics/statics analysis of rigid-flexible-soft hybrid
finger mechanism with standard force sensor.  Robotics and Computer Integrated Manufacturing 67 (2021) 101978

• Response 1: Thank you for recommending the great paper. However, the reason we proposed a method for FT sensor calibration in our study is to enable more precise operations when using FT sensor data during tasks performed with an end tool mounted on a manipulator's end effector. The paper you suggested appears to focus primarily on the design and mechanical analysis of a finger mechanism, utilizing force sensors rather than FT sensors for finger control. As this seems to differ from the direction of our research topic, we respectfully decline the recommendation.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Thank you for your replies and effort, I'll comment about my remaining doubts (so you can consider anything I won't comment as ok to me).

1) Base inclination along a single axis is a strong hypothesis, you should make it clear in abstract

2) Non-linear optimization part is still missing, you're not even citing the specific algorithm you've tested. It's plenty of non linear optimization techniques and you should tell which one are you using and the computation effort need. Moreover, details about a specific experiment (i.e. step size) should be given in sec. 3 rather than in sec. 2

3) I highligthed the importance of MSE, but (althoug slightly redundant from a mathematical point of view) the previous measures you made are significant

Comments on the Quality of English Language

Overall it's ok, but it surely needs proofreading

Author Response

In the revised submission, we have highlighted all changes made from the previous version in blue. While this has temporarily affected the readability of the layout, including the placement of tables and figures, we will ensure that these issues are fully addressed and resolved in the final version for better readability and presentation.

• Comments 1: Base inclination along a single axis is a strong hypothesis, you should make it clear in abstract
• Response 1: Thank you for your feedback. We have specified the design of the base inclination experiment in the abstract. The experiment was conducted by introducing steps under the wheels of the mobile base to create tilt. This was achieved by adding height differences to either the front or rear two wheels to generate pitch, or to the two side wheels to generate roll, allowing us to create a single angle for either roll or pitch.

 

• Comments 2: Non-linear optimization part is still missing, you're not even citing the specific algorithm you've tested. It's plenty of non linear optimization techniques and you should tell which one are you using and the computation effort need. Moreover, details about a specific experiment (i.e. step size) should be given in sec. 3 rather than in sec. 2
• Response 2: Thank you for your feedback. For the non-linear optimization, we utilized the gradient descent method, and this information has been added to the abstract and Section 2.3.2 ("Non-linear Method").

Additionally, an analysis of the required computational effort has been included in Section 2.3.2. Specific settings such as step size and tolerance, as you suggested, have been moved to Section 3.

 

• Comments 3: I highligthed the importance of MSE, but (althoug slightly redundant from a mathematical point of view) the previous measures you made are significant
• Response 3: Thank you for your feedback. To clarify that the torque data in Table 4 is unaffected by crosstalk, we have added the mean and variance. For the other data, we believe that the MSE alone is sufficient to provide an adequate explanation.

Author Response File: Author Response.pdf