Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. Line 74: I cannot find any mention of the PRAXIA robot on the Internet or through AI. Is Rover the manufacturer? The manufacturer and the type of robot should be specified precisely, including its dimensions, mass, and an actual image of the robot.
2. Figure 2 should be translated into English.
3. In line 92, it is written "It is worth noting that no filtering is applied to the derivative action, due to the limited encoder resolution."

Firstly, if no filter is applied to the derivative action, it becomes difficult to make meaningful comparisons, as the PID controller is, in practice, ineffective. Secondly, the encoder resolution is actually quite high, not limited. Having 9600 pulses per revolution is excellent. The omission of the controller filter in the PID controller is inexplicable in practical implementation. In my opinion, the controller noise could become very high without any practical reason.

4. What is tss in equation (6)? It is probably steady-state time, but this should be stated explicitly, with equations if needed.
5. In line 120 it was written: "Experimental observations of the motor show that the system operates in an overdamped regime, under which the eigenvalues are real and negative". In line 147, it is written: "the natural frequency of the actuators lies in the range of 32–33 Hz".

Can you explain what the "natural" frequency of a system with real negative poles is? Perhaps the issue is with nomenclature - what exactly is meant by "natural frequency"? Please be more precise.

6. Expression (18): Is it really necessary to use mathematical nomenclature to define nq[k]? Is it not sufficient to state that nq[k] is the nearest integer to n[k]? It can also be written as nq[k] = round(n[k]), where round denotes the function that rounds to the nearest integer.
7. Table 2: All tables and images should be placed after they are mentioned in the text. Please ensure this is corrected throughout the entire text. Otherwise, it is difficult to follow what the values mean.

What kind of tuning method has been used? This is crucial information. The text, “The tuning is carried out with the objective of obtaining a fast response, with reduced overshoot and disturbance rejection capability, prioritizing a stable performance of the robot actuator.” does not provide much detail. As will be discussed later, uploading the scientific data to a repository is essential.

8. Can you be more specific about what the letter P in (19) represents? If it denotes probability, it must be clearly defined. Unfortunately, P(B=1)=1/2 is unclear to me. Please use a clearer definition.

For example, it can be stated that nq[k] is calculated as the integer value of n[k] plus a stochastic signal with an average value of 1/2 and a standard deviation of ...

9. Line 208: "The overdamped nature of the plant prevents the use of classical empirical methods such as Ziegler–Nichols, whose applicability requires a step response with a well-defined inflection point in the transition region, a condition that the system does not satisfy."

This statement does not appear to be correct. For overdamped or underdamped second-order processes, the inflection point can be easily identified. I am wondering where this information was obtained. There was no citation, and even if there were, it would still not be true.

10. In line 230, it is mentioned "The controller structure includes two compensation filters...". Therefore, the filters are part of the controller transfer function and should be mentioned together with (21), not in two separate subsections. However, if this is a separate method, then the original PID should also include similar filters. Otherwise the comparison is not fair. This is crucial.
11. In equation (22), the error (controller) filter has a steady-state gain of 1/(1+a). This is quite unusual, as it means the controller gains are affected by a. Why not use the classic filter (1-a)*z/(z-a)?
12. Figure 8: Again, as with the initial PID controller, signal filtering is missing. It is difficult to expect that a controller without any filtering, especially on the D-term, could perform better than one with filter. This is important. If the filter is implemented as in the compensated PID, this should be mentioned.
13. Equation (27): Controller parameters were obtained, among other methods, by using the ISE criterion, while other criteria (27) were used to evaluate closed-loop control performance. This is unusual. At least one of the criteria used for evaluation should be used in the search for controller parameters instead of ISE. Did I miss something?
14. Figure 10: Measuring low speeds with a time resolution of 0.01 seconds is not the correct approach. A controller algorithm sampling time of 0.01 seconds is acceptable, however. The sampling time and the time resolution of the speed measurement should be treated as two separate variables at these speeds. Specifically, a microprocessor can measure the time between two pulses with microsecond or sub-microsecond resolution using interrupts (especially on RP2040). This high-resolution timing should be used instead of Ts in equation (16). The actual control sampling time can remain unchanged at Ts = 0.01. Therefore, at each Ts, the most recent and highly accurate speed value from (16) is used in the control algorithm. This is the standard implementation in practice.

Thus, it appears that the paper is addressing a problem that does not exist in practice, as this is not how it is typically done.

15. Figure 10: It is clear that the absence of a reference filter in a classic PID leads to a kick in the control signal at the start. If reference and error filter are applied, the closed-loop response can be made faster without the initial kick (although this may seem counter-intuitive at first, it is achievable). For example, the error filter can be considered part of the process when calculating controller parameters.
16. Table 6: Please provide the actual equation used to calculate jerk, as jerk cannot be measured without appropriate filtering. The equation for RMS should also be explicitly given.
17. Line 351: "In terms of tracking error (IAE), the fuzzy controller achieves the lowest mean value and the ranking is preserved in 87,6 % of the samples,". The sentence ends with a comma. Moreover, the difference between 0.21 and 0.219 is, in practice, very small. According to the actual numbers, all the controllers can be considered similar. The same applies to RMS.
18. Where is the scientific data, such as the Matlab/Simulink scripts used in section 5 or the raw measurements in section 6? This data can easily be published on the Zenodo repository, especially as the research is funded by a public institution. Publishing the data is crucial for other researchers to compare approaches directly without spending additional time.
19. Many references lack the year of publication. Some references also appear incomplete, for example, missing the journal name.

Author Response

Comments 1:Line 74: I cannot find any mention of the PRAXIA robot on the Internet or through AI. Is Rover the manufacturer? The manufacturer and the type of robot should be specified precisely, including its dimensions, mass, and an actual image of the robot.

Response 1:We thank the reviewer for the clarification. The term 'rover' refers to the functional configuration of the platform as a six-wheeled mobile robot with rocker-bogie suspension, rather than a commercial manufacturer. PRAXIA is an in-house development by the GIIRA research group at Universidad Politécnica Salesiana. We have added Section 2.1 to the revised manuscript to provide a comprehensive technical description of the platform. 

Comment 2: Figure 2 should be translated into English.

Response 2: Figure 2 has been fully translated into English. All labels, axis titles, and captions now appear in English throughout the manuscript. 

Comment 3: a) In line 92, it is written "It is worth noting that no filtering is applied to the derivative action, due to the limited encoder resolution." b) Firstly, if no filter is applied to the derivative action, it becomes difficult to make meaningful comparisons, as the PID controller is, in practice, ineffective. Secondly, the encoder resolution is actually quite high, not limited. Having 9600 pulses per revolution is excellent. The omission of the controller filter in the PID controller is inexplicable in practical implementation. In my opinion, the controller noise could become very high without any practical reason.

Response 3: a)The reviewer is correct, and the sentence has been removed. The encoder resolution (R = 9600 PPR, Δωc = 0.0654 rad/s) is high, not limited. More importantly, the sentence was misleading in its premise: the fixed-window frequency estimation method adopted in this work accumulates encoder pulses over a fixed sampling interval, which inherently attenuates quantization jitter and provides a smooth velocity measurement without requiring additional filtering at the controller level. The original sentence incorrectly framed this deliberate methodological choice as a hardware constraint, when in fact it was a design decision aimed at improving measurement reliability. This has been clarified in the revised manuscript. b)The omission of the derivative filter was a deliberate design choice. In the discrete implementation, the derivative contribution is u_d[k] = (Kd/Ts)·Δe[k] = 1.2·Δe[k]. Since the maximum intersample error variation at steady state is bounded by the encoder quantization step Δωc = 0.0654 rad/s, the derivative term contributes at most 0.08 V against a total control effort of 1.5–2 V, representing less than 5% of the control signal. Under these conditions, derivative noise amplification is negligible and a filter produces no observable effect on performance. Additionally, introducing a filter coefficient exclusively in the classical PID would add an asymmetric tuning parameter absent from the compensated PID and fuzzy designs, compromising the fairness of the comparison. A clarifying sentence reflecting this justification has been added to the manuscript

Comment 4: What is tss in equation (6)? It is probably steady-state time, but this should be stated explicitly, with equations if needed.

Response 4: We thank the reviewer for this observation. The variable tss has been explicitly defined upon its first appearance as the 2% settling time.  This definition has been added to the manuscript immediately preceding Equation (6).

Comment 5: In line 120 it was written: "Experimental observations of the motor show that the system operates in an overdamped regime, under which the eigenvalues are real and negative". In line 147, it is written: "the natural frequency of the actuators lies in the range of 32–33 Hz". Can you explain what the "natural" frequency of a system with real negative poles is? Perhaps the issue is with nomenclature - what exactly is meant by "natural frequency"? Please be more precise.

Response 5: We thank the reviewer for this observation. For a strictly overdamped system, the eigenvalues are real and negative, and the system exhibits no oscillatory behavior. Consequently, the concept of natural frequency   associated with underdamped second-order systems  is not applicable and its use was misleading. The reference to natural frequency has been removed from the manuscript. The sampling frequency is now determined solely from the dominant time constant τ = 1/|λ2|, selecting fs = 100 Hz to achieve approximately 5 samples per time constant. This criterion provides sufficient temporal resolution to capture the exponential dynamics of the overdamped system without relying on oscillatory parameters or introducing ambiguous terminology. The revised text reads: "The dominant eigenvalue corresponds to a time constant τ = 50.2 ms. The selected sampling frequency fs = 100 Hz provides approximately 5 samples per time constant and 20 samples over the full transient response, which is sufficient to capture the exponential dynamics of the overdamped system.

Comment 6: Expression (18): Is it really necessary to use mathematical nomenclature to define nq[k]? Is it not sufficient to state that nq[k] is the nearest integer to n[k]? It can also be written as nq[k] = round(n[k]), where round denotes the function that rounds to the nearest integer. Response 6: The reviewer's suggestion has been adopted. The expression has been simplified to nq[k] = round(n[k]), where round(·) denotes rounding to the nearest integer. The previous formal notation has been removed in favor of this cleaner formulation, which conveys the same meaning more directly.   Comment 7: Table 2: All tables and images should be placed after they are mentioned in the text. Please ensure this is corrected throughout the entire text. Otherwise, it is difficult to follow what the values mean. Response 7:The placement of all tables and figures has been revised throughout the manuscript to ensure that each element appears after its first mention in the text. The authors thank the reviewer for this observation.    Comment 8: What kind of tuning method has been used? This is crucial information. The text, “The tuning is carried out with the objective of obtaining a fast response, with reduced overshoot and disturbance rejection capability, prioritizing a stable performance of the robot actuator.” does not provide much detail. As will be discussed later, uploading the scientific data to a repository is essential. Response 8:. The previous version of the manuscript only reported the final selected PID gains without detailing the tuning process. The manuscript has been revised to address this. Section 3.1 now describes the complete tuning procedure: a model based approach using MATLAB's PID Tuner aligned with Internal Model Control (IMC) , in which a systematic sweep over two parameters was conducted  the closed-loop response time τc, which sets the target bandwidth, and the transient behavior parameter α ∈ [0, 1], which balances aggressiveness against robustness. Table 2 now documents representative operating points from this sweep, allowing the reader to follow the progression from initial configurations to the final selection. The chosen configuration (α = 0.9, τc = 0.2 s) was selected based on three explicit criteria: zero overshoot, acceptable rise time (0.22 s), and a phase margin of 85°, ensuring a stable and reproducible design.    Comment 9: Can you be more specific about what the letter P in (19) represents? If it denotes probability, it must be clearly defined. Unfortunately, P(B=1)=1/2 is unclear to me. Please use a clearer definition. For example, it can be stated that nq[k] is calculated as the integer value of n[k] plus a stochastic signal with an average value of 1/2 and a standard deviation of ... Response 9: The reviewer is correct. The notation P(B=1) = 1/2 was unnecessarily formal and unclear. It has been removed and replaced with the following plain-language description: "For the boundary case n[k] = ⌊n[k]⌋ + 0.5, a randomized rounding rule is applied with equal probability assigned to both outcomes, so as to avoid systematic bias in the quantization error." This formulation conveys the same meaning without requiring probabilistic notation.  Comment 10: Line 208: "The overdamped nature of the plant prevents the use of classical empirical methods such as Ziegler–Nichols, whose applicability requires a step response with a well-defined inflection point in the transition region, a condition that the system does not satisfy."This statement does not appear to be correct. For overdamped or underdamped second-order processes, the inflection point can be easily identified. I am wondering where this information was obtained. There was no citation, and even if there were, it would still not be true. Response 10:  We thank the reviewer for this observation. The reviewer is correct, and we apologize for this inaccurate statement. For a second-order overdamped system, the step response does exhibit an inflection point, and the Ziegler-Nichols process reaction curve method is applicable in principle. The sentence has been removed from the manuscript.  Comment 11: In line 230, it is mentioned "The controller structure includes two compensation filters...". Therefore, the filters are part of the controller transfer function and should be mentioned together with (21), not in two separate subsections. However, if this is a separate method, then the original PID should also include similar filters. Otherwise the comparison is not fair. This is crucial.   Response 11: We thank the reviewer for this important observation. The proposed controller should not be interpreted as a conventional PID augmented with independently designed compensation filters. The precompensators (P_e(z)) and (P_r(z)) are an intrinsic part of the pole-placement formulation and do not introduce additional tuning parameters. Their coefficients are uniquely determined by the analytical solution of the pole-placement equations and therefore form part of the controller synthesis procedure itself. to avoid this misunderstanding, the manuscript has been revised to explicitly state that the complete controller structure includes the PID gains together with the precompensators as a single pole-placement-based design. The description of the precompensators has also been relocated and integrated into the controller derivation section.

Comment 12:In equation (22), the error (controller) filter has a steady-state gain of 1/(1+a). This is quite unusual, as it means the controller gains are affected by a. Why not use the classic filter (1-a)*z/(z-a)? Response 12: We thank the reviewer for this observation. The proposed precompensator (P_e(z)) was not introduced as a conventional low-pass filter but as an analytical element required by the pole-placement formulation to cancel the plant zero at (z=-a). Consequently, its structure is directly determined by the plant model and not by independent filtering considerations. While the filter ((1-a)z/(z-a)) provides unit steady-state gain and is commonly employed when filtering is the primary objective, such normalization is not required in the present derivation. The overall controller parameters are obtained from the pole-placement equations, and any scaling effect introduced by (P_e(z)) is inherently accounted for in the resulting controller coefficients. Therefore, the steady-state gain (1/(1+a)) does not constitute an additional design constraint but rather a natural consequence of the analytical zero-cancellation procedure.   Comment 13:Figure 8: Again, as with the initial PID controller, signal filtering is missing. It is difficult to expect that a controller without any filtering, especially on the D-term, could perform better than one with filter. This is important. If the filter is implemented as in the compensated PID, this should be mentioned.
Response 13: We thank the reviewer for this observation. The proposed precompensators should not be interpreted as derivative filters or as signal-conditioning stages introduced to improve noise rejection. Their purpose is fundamentally different. Both (P_e(z)) and (P_r(z)) arise directly from the analytical pole-placement formulation and are required to achieve the desired closed-loop dynamics. In particular, (P_e(z)) is introduced to compensate the plant zero, whereas (P_r(z)) shapes the closed-loop numerator to obtain the specified second-order response. Therefore, the proposed method does not rely on the addition of a derivative filter to improve performance relative to the classical PID controller. Any attenuation of high-frequency components is a secondary consequence of the analytical controller structure and not an independent design objective. Furthermore, the identified controller parameters show that the derivative gain of the compensated PID is significantly smaller than that of the classical PID. As a result, the derivative contribution to the control action is already substantially reduced, further limiting the practical impact that an additional derivative filter would have on the reported performance metrics.

Comments 14:Equation (27): Controller parameters were obtained, among other methods, by using the ISE criterion, while other criteria (27) were used to evaluate closed-loop control performance. This is unusual. At least one of the criteria used for evaluation should be used in the search for controller parameters instead of ISE. Did I miss something?

Response 14: We thank the reviewer for this important observation. The objective of this work is not to design controllers that explicitly minimize jerk, but rather to compare the kinematic behavior that emerges when different control strategies are tuned to satisfy the same conventional control requirements. The fuzzy controller was therefore optimized using the composite objective function in Eq. (25), which penalizes tracking error, rise time, and overshoot. These criteria were selected because they are the standard performance indicators commonly used in the literature for DC motor speed control and are consistent with the design objectives imposed on both PID-based controllers. In particular, the pole-placement PID is derived from specifications on transient response and overshoot, while the classical PID is tuned to achieve comparable closed-loop behavior. The RMS-J metric introduced in Eq. (27) serves a different purpose. It is not a design criterion but an evaluation metric intended to characterize the kinematic smoothness of the resulting closed-loop response. Incorporating RMS-J directly into the optimization of the fuzzy controller would bias the comparison, since only that controller would be explicitly conditioned to reduce jerk, whereas the PID controllers are designed exclusively from conventional control specifications. The methodological contribution of the paper is precisely to evaluate whether controllers designed under equivalent tracking-performance requirements exhibit different levels of kinematic smoothness when assessed through RMS-J, a dimension that is not captured by traditional control metrics.

Comment 15: Figure 10: Measuring low speeds with a time resolution of 0.01 seconds is not the correct approach. A controller algorithm sampling time of 0.01 seconds is acceptable, however. The sampling time and the time resolution of the speed measurement should be treated as two separate variables at these speeds. Specifically, a microprocessor can measure the time between two pulses with microsecond or sub-microsecond resolution using interrupts (especially on RP2040). This high-resolution timing should be used instead of Ts in equation (16). The actual control sampling time can remain unchanged at Ts = 0.01. Therefore, at each Ts, the most recent and highly accurate speed value from (16) is used in the control algorithm. This is the standard implementation in practice.
Response 15: We agree with the reviewer that high-resolution period measurement using interrupt-based timestamping is a standard and effective approach for low-speed encoder measurements, and that the RP2040 platform is fully capable of implementing such a strategy. However, the objective of the present work was not to maximize measurement resolution but to evaluate controller behavior under the sensing conditions typically encountered in low-cost embedded robotic platforms operating with fixed-rate control loops. For this reason, velocity estimation was intentionally performed using a fixed sampling window synchronized with the controller execution period, resulting in the quantization model described in Section 2.5. This quantization is not an implementation limitation but an explicit component of the experimental framework. The resulting resolution (\Delta\omega_c) is subsequently incorporated into the controller design and analysis, including the definition of the fuzzy controller input domains and the evaluation of steady-state kinematic behavior. Replacing the adopted estimator with a high-resolution period-measurement scheme would substantially alter the measurement noise characteristics and quantization effects that constitute part of the problem investigated in this work. To avoid this misunderstanding, the manuscript has been revised to explicitly state that the selected velocity estimation method was adopted intentionally to preserve consistency between the controller implementation, the quantization model, and the experimental evaluation framework.

Comment 16: Figure 10: It is clear that the absence of a reference filter in a classic PID leads to a kick in the control signal at the start. If reference and error filter are applied, the closed-loop response can be made faster without the initial kick (although this may seem counter-intuitive at first, it is achievable). For example, the error filter can be considered part of the process when calculating controller parameters.

Response 16: We agree with the reviewer that reference filtering and related two-degree-of-freedom PID structures can effectively reduce the derivative kick associated with setpoint changes and may improve transient performance. Such approaches are well established in the control literature. However, the objective of this work was not to identify the best possible implementation of a PID controller through the addition of auxiliary mechanisms, but rather to compare three control strategies under their respective standard formulations. The classical PID was therefore implemented in its conventional form, while the compensated PID incorporates the precompensators that arise naturally from the pole-placement synthesis procedure. The initial control kick observed in Figure 10 is consequently not an implementation artifact but an inherent characteristic of the classical PID structure under step-reference excitation. This behavior is explicitly visible in the experimental results and forms part of the controller dynamics being evaluated. We acknowledge that alternative PID architectures incorporating reference weighting or filtering could reduce this effect. Nevertheless, introducing such modifications would alter the controller structure under study and move the comparison away from the methodological scope of the present work.

Comment 17: Table 6: Please provide the actual equation used to calculate jerk, as jerk cannot be measured without appropriate filtering. The equation for RMS should also be explicitly given.

Response 17: We thank the reviewer for this observation. The revised manuscript now explicitly provides both the jerk estimation equation and the RMS formulation used throughout the study. Specifically, Eq. (26) defines the jerk estimation based on a Savitzky–Golay differentiating filter applied to the angular velocity signal, while Eq. (27) defines the RMS-J metric used for quantitative comparison. Additional details regarding the filter order, window length, and coefficient construction have also been included in Section 5.1 to ensure full reproducibility of the proposed methodology.

Comment 18:We thank the reviewer for this observation. We agree that the absolute differences reported in Table 6 are very small and should not be interpreted as practically significant performance differences between controllers. The purpose of the Monte Carlo analysis is not to establish superiority based on the absolute metric values, but rather to evaluate the robustness of the controller ranking under the parameter variability observed across the rover actuators. To make this point explicit, the revised manuscript now states that the metric values are numerically close and primarily reflect actuator variability rather than controller design differences, and that the analysis focuses on ranking stability rather than absolute performance.

Comment 19: Where is the scientific data, such as the Matlab/Simulink scripts used in section 5 or the raw measurements in section 6? This data can easily be published on the Zenodo repository, especially as the research is funded by a public institution. Publishing the data is crucial for other researchers to compare approaches directly without spending additional time.

Response 19: We thank the reviewer for emphasizing the importance of reproducibility and open scientific data. A Zenodo repository associated with this research project has already been established and is publicly available at:

https://zenodo.org/records/20056863

The repository currently contains the experimental results and datasets generated during the initial stage of this research and the first manuscript version. Following the reviewer's recommendation and considering the substantial developments incorporated during the latest revision process, particularly over the last stages of manuscript improvement, the repository will be expanded and updated to include the additional materials generated in the present study.

 

Comment 20:Many references lack the year of publication. Some references also appear incomplete, for example, missing the journal name.

Response 20: We thank the reviewer for identifying these issues. The reference list has been carefully revised and corrected. Missing publication years, journal names, volume and issue information, page ranges, and other incomplete bibliographic fields have been added wherever necessary. In addition, the entire bibliography has been checked for consistency with the journal formatting requirements to ensure complete and accurate citation information throughout the manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents an experimental comparison among classical PID, compensated PID, and fuzzy controllers for rover actuator speed regulation. However, the current manuscript still has several shortcomings in terms of literature review, result interpretation, and presentation quality. Therefore, a major revision is recommended before further consideration.

Other comments:

1. Although the authors report several experimental results, the physical significance and engineering implications of the obtained findings are not adequately highlighted. In particular, the contribution of jerk-based indicators to practical actuator durability and vibration reduction should be discussed more systematically.
2. The literature review remains insufficient and lacks coverage of several recent representative studies. More up-to-date works related to intelligent motor control, disturbance rejection, sliding-mode-based approaches, adaptive fuzzy control, and data-driven optimization methods should be included to improve the research depth and contemporaneity of the paper.
3. The manuscript provides extensive experimental figures and tables; however, the analysis of these figures is relatively superficial. Many plots are only descriptively introduced without sufficiently explaining the underlying causes of the observed behaviors. More detailed comparative discussions and quantitative interpretations should be added.
4. The novelty of the manuscript should be clarified more explicitly. The use of PID, compensated PID, and fuzzy controllers for DC motor speed regulation has been widely studied in previous literature. The authors should better emphasize the unique methodological contribution and practical significance of incorporating kinematic indicators such as jerk and acceleration.
5. Some experimental results lack statistical analysis and repeatability validation. The manuscript would benefit from additional discussions regarding robustness, sensitivity to parameter uncertainties, and repeatability under different operating conditions.
6. Several figures contain small fonts and unclear curve distinctions, especially in multi-curve comparison plots. The readability of the figures should be improved by enlarging labels, enhancing line styles, and highlighting key comparison results.
7. Additional validation under more complex trajectories or varying operating conditions would further strengthen the practical relevance of the proposed evaluation framework.
8. The conclusion section is relatively lengthy but still lacks a concise and well-structured summary of the main technical findings, limitations, and future research directions. It is recommended to reorganize the conclusion to better reflect the core contributions of the work.
9. This method also has potential research value for systems such as autonomous vehicles (sine resistance network-based motion planning approach for autonomous electric vehicles in dynamic environments) and suspension systems. It is recommended to further investigate and analyze its applications in these areas in the future work section.
10. The English writing quality should be further improved. Some sentences are overly long and repetitive, which affects readability and clarity. Careful language polishing is recommended.

This reviewer hopes that the above suggestions will be useful for the authors in improving the quality and technical rigor of this manuscript.

Author Response

Comments 1: Although the authors report several experimental results, the physical significance and engineering implications of the obtained findings are not adequately highlighted. In particular, the contribution of jerk-based indicators to practical actuator durability and vibration reduction should be discussed more systematically.

Response 1:We thank the reviewer for this valuable observation. We agree that the original manuscript did not sufficiently emphasize the physical significance and engineering implications of the jerk-based indicators. To address this concern, we expanded the discussion in Sections 7.1 and 7.2 and explicitly linked the reported jerk metrics to their practical mechanical implications. In Section 7.1, we added a paragraph explaining that the 134% higher steady-state RMS jerk produced by the fuzzy controller compared with the classical PID (6.10 vs. 2.61 rad s⁻³, Table 6) indicates more frequent and intense variations in the acceleration rate transmitted through the motor shaft, coupling plate, and bearing assembly. As reported in the literature [30,31], elevated jerk levels are associated with increased dynamic loading, vibration excitation, and fatigue-related degradation mechanisms in rotating mechanical systems. Consequently, the higher RMS-J values observed for the fuzzy controller suggest a greater potential for cumulative mechanical stress during prolonged operation. In Section 7.2, we further expanded the discussion of disturbance rejection results by highlighting that the classical PID generates jerk peaks up to 2.4 times larger than those of the fuzzy controller under external perturbations. These larger jerk excursions indicate a higher potential for transient dynamic loading and vibration excitation when disturbances are present. Furthermore, we clarified that, although direct vibration measurements and fatigue-life assessments were beyond the scope of the present study, jerk is widely recognized in the motion-control literature as a kinematic indicator closely related to vibration generation and the accumulation of dynamic loads in mechanical transmission systems. Therefore, the jerk-based metrics employed in this work provide complementary engineering information beyond conventional tracking-performance indicators and allow a qualitative assessment of the relative mechanical impact of the evaluated controllers. Finally, we explicitly identify direct vibration measurements and fatigue-life characterization as important directions for future work.

Comments 2:The literature review remains insufficient and lacks coverage of several recent representative studies. More up-to-date works related to intelligent motor control, disturbance rejection, sliding-mode-based approaches, adaptive fuzzy control, and data-driven optimization methods should be included to improve the research depth and contemporaneity of the paper.

Response 2:We thank the reviewer for this valuable observation. We agree that the original version could benefit from a broader discussion of recent developments in intelligent motor control and advanced control methodologies. To address this concern, we substantially expanded the literature review in the Introduction and incorporated additional discussion covering several contemporary research directions. The revised manuscript now includes recent studies on intelligent tuning methods for PMDC motor control based on genetic algorithms and particle swarm optimization, hybrid fuzzy–PID control architectures, adaptive and sliding-mode-based control strategies, nonlinear robust control approaches, active disturbance rejection control (ADRC), and data-driven methodologies based on deep reinforcement learning. Furthermore, we incorporated recent work on fractional-order PID controllers optimized through ant colony optimization techniques. These additions were integrated into a dedicated discussion of alternative control strategies and their respective advantages regarding robustness, disturbance rejection, tracking performance, and model-free optimization capabilities.

Comments 3: The manuscript provides extensive experimental figures and tables; however, the analysis of these figures is relatively superficial. Many plots are only descriptively introduced without sufficiently explaining the underlying causes of the observed behaviors. More detailed comparative discussions and quantitative interpretations should be added

Responde 3: We thank the reviewer for this valuable observation. We agree that the original version of the manuscript relied excessively on descriptive presentation of several experimental results and did not always provide sufficient interpretation of the underlying causes of the observed behaviors. To address this concern, we substantially expanded the discussion throughout the Experimental Results and Discussion sections. The revised manuscript now incorporates quantitative and causal interpretations of the principal trends observed in the figures rather than limiting the analysis to a description of the plotted data. For example, the discussion of the RMS jerk results was expanded to explain the influence of the Savitzky–Golay filter window length, encoder quantization effects, and the progressive attenuation of high-frequency components on the estimated jerk values. Additional analysis was also incorporated to explain the persistence of the relative controller ranking under different filtering conditions and the mechanisms responsible for the higher jerk levels observed for the fuzzy controller during unloaded operation. Similarly, under disturbance conditions, the revised manuscript now discusses the changes in controller ranking as a function of the observation window, relating these effects to the different temporal scales of the corrective actions performed by each controller and to the influence of the additional mechanical loading on the system dynamics. Furthermore, we introduced a complementary interpretation based on tracking performance, explaining the limitations of RMS jerk as a standalone indicator under external disturbances and motivating the inclusion of the normalized IAE analysis. Overall, the revised manuscript places greater emphasis on explaining why the observed behaviors occur and on relating the experimental results to the underlying control dynamics and mechanical characteristics of the system, thereby providing a more comprehensive interpretation of the figures and tables.

Comments 4:The novelty of the manuscript should be clarified more explicitly. The use of PID, compensated PID, and fuzzy controllers for DC motor speed regulation has been widely studied in previous literature. The authors should better emphasize the unique methodological contribution and practical significance of incorporating kinematic indicators such as jerk and acceleration.

Response 4: We thank the reviewer for this important observation. We agree that the novelty of the manuscript was not sufficiently emphasized in the original submission. The revised manuscript has therefore been modified to more clearly position the contribution of this work within the existing literature. Specifically, we now explicitly state in the Abstract, Introduction, and Conclusions that the novelty of the study does not lie in the control strategies themselves, since classical PID, compensated PID, and fuzzy controllers were intentionally selected as representative and well-established approaches. Instead, the main contribution of this work is the proposed evaluation framework based on kinematic indicators, particularly jerk-related metrics, and its experimental validation on a real embedded PMDC motor platform operating under both nominal and disturbed conditions. The revised manuscript further highlights that conventional controller comparisons are typically based on transient-response indicators such as overshoot, settling time, and tracking error. In contrast, this work incorporates velocity-, acceleration-, and jerk-based metrics to characterize aspects of controller behavior that are not directly captured by traditional performance measures. Through this analysis, we show that controller rankings obtained from conventional tracking metrics may differ from those obtained through kinematic indicators, particularly under disturbance conditions, revealing information about mechanical smoothness and disturbance-response characteristics that would not be evident from classical transient metrics alone. In addition, the Conclusions section has been reorganized to clearly identify the principal contributions of the work: (i) the development and experimental validation of a jerk-based evaluation methodology for embedded systems affected by encoder quantization, including the analysis and selection of appropriate filtering parameters; (ii) the experimental demonstration that kinematic indicators provide complementary information to conventional control metrics and may lead to different controller assessments under realistic operating conditions; and (iii) the discussion of the practical engineering implications of these findings for controller selection in robotic and mechatronic applications subject to external disturbances. We believe that these revisions more clearly communicate both the methodological novelty and the practical significance of the proposed approach.

Comments 5: Some experimental results lack statistical analysis and repeatability validation. The manuscript would benefit from additional discussions regarding robustness, sensitivity to parameter uncertainties, and repeatability under different operating conditions.

Response 5: We thank the reviewer for this valuable observation. We agree that the original manuscript required a more explicit assessment of repeatability, statistical significance, and robustness to parameter uncertainties. To address this concern, the revised manuscript incorporates substantial additions covering all three aspects. Regarding repeatability, the experimental methodology was expanded to include repeated measurements for each controller, reference speed, and disturbance scenario considered in the study. A total of 480 experiments were conducted, comprising 360 trials under nominal conditions and 120 trials under mechanical disturbance. The revised results now report mean and standard deviation values of the RMS-J metric, while the corresponding figures include variability bands derived from repeated measurements. Furthermore, the coefficient of variation was analyzed to quantify experimental repeatability. Under nominal conditions, the classical PID and compensated PID exhibited coefficients of variation below 8%, whereas the fuzzy controller reached approximately 23%, indicating greater trial-to-trial variability. Regarding statistical analysis, the revised manuscript now incorporates non-parametric hypothesis testing for the principal controller comparisons. Kruskal–Wallis tests were performed at all evaluated speeds under nominal conditions, yielding statistically significant differences among controllers ($p < 0.001$ in all cases). In addition, effect sizes were reported through $\hat{\eta}^{2}$, with values ranging from 0.73 to 0.84, indicating strong practical separation among the controller groups. These additions provide quantitative support for the observed differences beyond descriptive comparisons.We believe that these additions substantially strengthen the manuscript by providing explicit evidence of repeatability, statistical significance, and robustness under realistic operating conditions.

Comments 6:Several figures contain small fonts and unclear curve distinctions, especially in multi-curve comparison plots. The readability of the figures should be improved by enlarging labels, enhancing line styles, and highlighting key comparison results.

Response 6: We thank the reviewer for this observation. The readability of the figures has been substantially improved throughout the revised manuscript. First, font sizes of axes labels, legends, and annotations were increased to improve readability in both digital and printed formats. Line styles and marker configurations were also revised to provide clearer visual distinction among controllers, particularly in multi-curve comparison plots. In addition, several complex multi-panel figures were reorganized. Some figures containing a large number of subplots were divided into separate figures to reduce visual clutter and facilitate interpretation of the results. At the same time, a number of redundant plots were removed because the newly incorporated statistical and kinematic analyses provide a more concise and informative representation of the controller behavior. Finally, the revised manuscript places greater emphasis on the key comparative results through the inclusion of summary tables, statistical indicators, and expanded figure discussions, reducing the dependence on visual inspection alone for interpreting the experimental outcomes. We believe that these modifications significantly improve the clarity, readability, and overall presentation of the experimental results.

Comments 7: Additional validation under more complex trajectories or varying operating conditions would further strengthen the practical relevance of the proposed evaluation framework.
Response 7: We thank the reviewer for this valuable suggestion. We agree that validation under more complex trajectories and a broader range of operating conditions would further strengthen the practical relevance and generalizability of the proposed evaluation framework. In the present work, step references and controlled mechanical disturbances were intentionally selected as a baseline experimental scenario because they allow the transient and steady-state behavior of the controllers to be isolated under highly reproducible conditions, facilitating a rigorous statistical comparison. Nevertheless, we acknowledge that real robotic applications often involve ramp trajectories, multi-step commands, and terrain-induced disturbances. To explicitly address this limitation, we have expanded the Future Work section to include the evaluation of the proposed framework under more realistic navigation scenarios involving complex trajectories and varying environmental conditions.

Comments 8:The conclusion section is relatively lengthy but still lacks a concise and well-structured summary of the main technical findings, limitations, and future research directions. It is recommended to reorganize the conclusion to better reflect the core contributions of the work.

Response 8:The Conclusions section has been completely reorganized into five clearly defined subsections: Summary of the Work, Main Technical Findings, Principal Contribution, Engineering Recommendations, and Future Work.

Comments 9:This method also has potential research value for systems such as autonomous vehicles (sine resistance network-based motion planning approach for autonomous electric vehicles in dynamic environments) and suspension systems. It is recommended to further investigate and analyze its applications in these areas in the future work section.
Response 9:We thank the reviewer for this valuable suggestion. We agree that the proposed framework, based on jerk-related kinematic indicators, has potential applications beyond mobile robotics, particularly in autonomous electric vehicles and active suspension systems, where motion smoothness, vibration mitigation, and disturbance rejection are critical design objectives.

Comments 10: The English writing quality should be further improved. Some sentences are overly long and repetitive, which affects readability and clarity. Careful language polishing is recommended.
Response 10:We thank the reviewer for this observation. The manuscript has undergone a comprehensive language revision aimed at improving readability, clarity, and conciseness. Several long and repetitive sentences were rewritten, terminology was standardized throughout the paper, and grammatical and stylistic corrections were applied across all sections. We believe these revisions have substantially improved the overall quality of the English presentation.

 

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript is technically sound in modeling and implementation, and the experimental platform is well described. The idea of explicitly incorporating kinematic indicators (acceleration, jerk, IAJ) into the comparison of PID, compensated PID, and fuzzy controllers is relevant and potentially useful for actuator design in robotics. However, in its current form:
1- The idea of using kinematic indicators (acceleration, jerk, IAJ) as complementary metrics is interesting, but the manuscript does not clearly show what is fundamentally new beyond “we also look at jerk.”
2- You repeatedly motivate IAJ and jerk as proxies for “mechanical wear” and “dynamic effort,” but there is no experimental or analytical evidence connecting these indicators to actual wear, fatigue, or lifetime. 
3- 

The study is based on a single platform (educational rover) and essentially one type of reference trajectory and disturbance scenario. To support your conclusions (e.g., “compensated PID provides the best overall balance”), you should:

• Include more diverse reference profiles (ramps, multi-step, tracking of varying speeds).

• Clarify whether all six actuators were tested with each controller, and if so, report variability across actuators.

4- 

You present single curves and ratios (e.g., “IAJ 2.378 times higher”), but it is unclear whether these are from one run, an average over many runs, or representative values. You should:

• Report mean ± standard deviation over multiple trials for key metrics (rise time, overshoot, IAJ, peak jerk).

• Clarify the number of repetitions and whether results are consistent across actuators.

5- 

The modeling and identification sections are detailed and solid, but the paper sometimes reads more like a modeling/identification paper than a “kinematic indicators as complementary metrics” study. Consider:

• Tightening the general DC motor/control background.

• Expanding the explanation and interpretation of the kinematic metrics and their implications for design choices.

Author Response

Comments 1:The idea of using kinematic indicators (acceleration, jerk, IAJ) as complementary metrics is interesting, but the manuscript does not clearly show what is fundamentally new beyond “we also look at jerk.”

Response 1:

The proposed jerk-based indicators allow these differences to be quantified experimentally, revealing controller-induced mechanical activity that cannot be inferred from tracking metrics alone. Furthermore, jerk is a well-established kinematic indicator in motion-control and mechatronic systems, as it is directly related to vibration excitation, dynamic loading, and motion smoothness. Therefore, in low-speed embedded PMDC actuator systems such as the one studied here, jerk-based metrics provide a physically meaningful assessment of the mechanical consequences of control actions beyond conventional tracking performance measures.

More importantly, the experimental results demonstrate that incorporating these kinematic indicators leads to conclusions that differ from those obtained using classical control metrics alone. In particular, the study reveals a complete inversion of the controller performance hierarchy between nominal and disturbed operating conditions, a phenomenon that is not evident from traditional transient-response metrics. This finding constitutes the principal methodological contribution of the work, showing that kinematic indicators are not merely complementary measurements but can fundamentally alter the interpretation of controller performance in electromechanical systems.

Comments 2:You repeatedly motivate IAJ and jerk as proxies for “mechanical wear” and “dynamic effort,” but there is no experimental or analytical evidence connecting these indicators to actual wear, fatigue, or lifetime. 

Response 2: We thank the reviewer for this important observation. We agree that not all jerk-based metrics possess the same degree of physical interpretation regarding mechanical degradation. In particular, the Integral of Absolute Jerk (IAJ) has not been directly linked in the literature to actuator wear, fatigue accumulation, or component lifetime. For this reason, the revised manuscript no longer bases its engineering interpretation on IAJ. Instead, the discussion has been refocused on RMS jerk, which is directly related to jerk energy (JE). Previous studies have associated jerk energy and jerk-related measures with vibration excitation, dynamic loading, actuator wear, fatigue crack initiation, and drivetrain damage. Nevertheless, we acknowledge that the present work does not directly measure wear, fatigue, or lifetime. Consequently, RMS jerk should be interpreted as a relative indicator of controller-induced mechanical activity and vibration-excitation potential, enabling meaningful comparisons among controllers operating under identical experimental conditions.

Comments 3:

The study is based on a single platform (educational rover) and essentially one type of reference trajectory and disturbance scenario. To support your conclusions (e.g., “compensated PID provides the best overall balance”), you should:

• Include more diverse reference profiles (ramps, multi-step, tracking of varying speeds).

• Clarify whether all six actuators were tested with each controller, and if so, report variability across actuators.

Response 3:We thank the reviewer for this valuable observation. We agree that evaluating additional reference profiles and disturbance conditions would further strengthen the generality of the conclusions. The present study intentionally employed step references and a controlled braking disturbance as a baseline experimental scenario because these conditions allow transient and steady-state controller behavior to be isolated under highly reproducible conditions, facilitating rigorous statistical comparison. The objective was not to provide an exhaustive benchmark across all possible operating scenarios, but rather to investigate whether kinematic indicators reveal controller behaviors that remain hidden under conventional evaluation metrics. Nevertheless, we acknowledge that ramp references, multi-step commands, variable-speed trajectories, and more complex terrain-induced disturbances represent important extensions. These directions have therefore been explicitly incorporated into the Future Work section

We agree that assessing the consistency of the proposed framework across different actuators is important for supporting the generality of the conclusions. In the revised manuscript, the experimental campaign was expanded to include a second actuator of the rover platform in addition to the original test actuator. Furthermore, the repeatability analysis was extended through additional experimental trials and statistical evaluation. The results obtained from both actuators exhibited consistent trends regarding the relative performance of the evaluated controllers and the associated kinematic indicators, reinforcing the robustness of the observed findings. Regarding the operating scenarios, the present study intentionally focuses on step references and controlled braking disturbances because these conditions provide a reproducible baseline for isolating controller-induced kinematic effects and enabling rigorous statistical comparisons. Nevertheless, we acknowledge that additional reference profiles, such as ramps, multi-step commands, and variable-speed trajectories, would further broaden the applicability of the framework. This limitation has been explicitly acknowledged in the manuscript and incorporated into the Future Work section.

Comments 4:

You present single curves and ratios (e.g., “IAJ 2.378 times higher”), but it is unclear whether these are from one run, an average over many runs, or representative values. You should:

• Report mean ± standard deviation over multiple trials for key metrics (rise time, overshoot, IAJ, peak jerk).

• Clarify the number of repetitions and whether results are consistent across actuators.

Response 4:

We thank the reviewer for this important observation. We agree that the statistical basis of the reported metrics should be made explicit.

In the revised manuscript, all key performance indicators are now reported using repeated experimental trials and presented as mean ± standard deviation values. The number of repetitions performed for each controller, speed condition, and disturbance scenario has been explicitly stated in the Experimental Methodology section. Furthermore, statistical analyses including Kruskal–Wallis tests, Dunn post-hoc comparisons, effect size estimation, and repeatability assessments have been incorporated to quantify the significance and consistency of the observed differences.

To avoid ambiguity, statements previously expressed using single representative values or ratios have been revised to clearly indicate whether they correspond to averaged metrics derived from the complete experimental dataset.

Comments 5:

The modeling and identification sections are detailed and solid, but the paper sometimes reads more like a modeling/identification paper than a “kinematic indicators as complementary metrics” study. Consider:

• Tightening the general DC motor/control background.

• Expanding the explanation and interpretation of the kinematic metrics and their implications for design choices.

Response 5: We thank the reviewer for this constructive suggestion. We agree that the primary contribution of the manuscript lies in the use of kinematic indicators as complementary performance metrics rather than in the development of a new modeling or control methodology. To better reflect this focus, the revised manuscript has been edited to streamline portions of the general DC motor and control background that were not directly relevant to the proposed evaluation framework. In parallel, the Discussion and Conclusions sections have been substantially expanded to provide a deeper interpretation of the kinematic metrics, including their relationship to vibration excitation, dynamic mechanical loading, controller selection, and the observed performance hierarchy inversion between nominal and disturbed operating conditions. At the same time, the modeling and identification sections have been retained because they provide the foundation for the controller design and ensure that the experimental comparisons are performed on a well-characterized system. In particular, the identified model is required for the compensated PID implementation and establishes a consistent basis for interpreting the observed differences in both conventional and kinematic performance metrics. We believe these revisions improve the balance of the manuscript and more clearly emphasize the methodological contribution of the proposed kinematic evaluation framework.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Comment 3: Authors’ response: “Since the maximum intersample error variation at steady state is bounded by the encoder quantization step Δωc = 0.0654 rad/s, the derivative term contributes at most 0.08 V against a total control effort of 1.5–2 V, representing less than 5% of the control signal. Under these conditions, derivative noise amplification is negligible and a filter produces no observable effect on performance. Additionally, introducing a filter coefficient exclusively in the classical PID would add an asymmetric tuning parameter absent from the compensated PID and fuzzy designs, compromising the fairness of the comparison. A clarifying sentence reflecting this justification has been added to the manuscript”
Comment: 5 % of the control span is not a small value, but it could be manageable in practice. In any case, the derivative noise is not negligible (especially since we are considering a process without noise, only with measurement jitter). Moreover, why would anyone introduce a filter only in the compensated PID? A similar filter can be introduced in all controllers (at least by filtering the control error). Such a filter can be considered part of the process during the tuning stage, so I cannot see any problems. However, see also comment 11, as they are closely related.
Comment 5: Authors’ response: “The revised text reads: "The dominant eigenvalue corresponds to a time constant τ = 50.2 ms. The selected sampling frequency fs = 100 Hz provides approximately 5 samples per time constant and 20 samples over the full transient response, which is sufficient to capture the exponential dynamics of the overdamped system.”
Comment: For closed-loop system, 5 samples per time constant (of the process?) is not enough (in my original comment 15 I did not check the process time constant, though). Usually, some additional closed-loop overshoot can be expected when using 5 samples per closed-loop time constant. I would rather advise authors to use 10 samples or more per closed-loop time constant. If it is not possible in this stage, an additional text should be added with explanation why sampling times are not shorter (for example that the measurement jitter would be even higher).
Comment 11: Authors’s response: “The precompensators (P_e(z)) and (P_r(z)) are an intrinsic part of the pole-placement formulation and do not introduce additional tuning parameters. Their coefficients are uniquely determined by the analytical solution of the pole-placement equations and therefore form part of the controller synthesis procedure itself.”. 
Comment: However, as already stated in Comment 3, similar type of filters should be used in the classical PID. Otherwise, the study is misleading. PID requires a filter. For example, we say that we did not add a filter to the classical PID (which we must), but we add filters in the compensated PID. And, unsurprisingly, the compensated PID works better. I hope this comment is clearer now. You should add a filter to the classical PID and determine the remaining PID parameters as usual. If needed, the filter can be “moved” to the process during the tuning stage. By the way, P_e is a classical first-order PID filter (with one caveat), regardless of what we call it (pre-compensator or otherwise). How we obtain it is not important. For example, the same type of filter (with unity steady-state gain) should be used in the classical PID.
Comment 13: Authors’ response: “The proposed precompensators should not be interpreted as derivative filters or as signal-conditioning stages introduced to improve noise rejection.”. 
Comment: The filter P_e should be considered as the PID filter (frequently, all 3 PID terms are filtered with the same filter, not only the D term!), since it is a filter. Why we get it (for compensation or anything else), does not matter much. If we have it in the compensated PID, we should have a similar filter in the classical PID. We do not have to use the same time constant, but controller output noise should be comparable.
Comment 15: Authors’ response: “However, the objective of the present work was not to maximize measurement resolution but to evaluate controller behavior under the sensing conditions typically encountered in low-cost embedded robotic platforms operating with fixed-rate control loops.”. 
Comment: I suppose that the authors did not understand my original comment. In such kind of implementations, the low speed measurement is usually completely separate from the main sampling time. Speed is measured each time interrupt occurs and then is stored in some variable. This variable is then used by the control algorithm whenever needed. I would like to stress this again: such implementation of speed measurement as in the paper is usually not used in practice because of high jitter at low speeds. Having a method which then “solves a problem of jitter” by using filters (where F_e with some appropriate time constant should be used also in other controllers!) is seeking a solution for a problem which does not exist in practice, since it is solved in a more elegant way.
I suggest that the authors mention that the produced jitter can be easily eliminated using alternative frequency measurement approaches on the PRAXIA robot, such as precisely measuring the intervals between pulses in sub-microsecond ranges with RP2040 controller interrupts. It should also be noted that the study still used an inferior method to examine cases where such improved measurements would be more difficult to implement, for example, when using much less capable microcontrollers.
However, in any case, comparing unfiltered PID and fuzzy controller with filtered one is not acceptable and I expect the new, fairer, experiments. In those experiments, the classical PID filter time constant (additional F_e) should be chosen in such a way that the noise (jitter) amplification of both PID controllers should be comparable in the steady-state. This is important. Since the compensated PID includes an additional reference filter and is tuned using a more targeted approach, it can be expected to deliver better closed-loop results. This is what readers would be interested in. 
Comment 16: Authors’ response: “However, the objective of this work was not to identify the best possible implementation of a PID controller through the addition of auxiliary mechanisms, but rather to compare three control strategies under their respective standard formulations.”.
Comment: I would like to repeat: This is not how the classical controller anywhere is practice is implemented. F_e filter must be applied in practice. 
Overall comments: The revised paper has addressed some of the concerns raised in the initial review. However, there are still a few issues that must be resolved before it is suitable for publication. The most significant issue is that it is not possible to compare the results if the PID without a filter (F_e) is included in the comparison. This is not meaningful in any practical application. If one of the aims of the paper is to demonstrate that a controller with a filter (e.g. similar to F_e) is superior to one without a filter, then, unfortunately, I cannot recommend the paper for publication. It should be noted that adding a filter, which is an essential part of any practical PID controller, is straightforward. Tuning PID controller with the filter is also not problematic, as the filter can be considered part of the process during the tuning procedure. I expect the authors to provide a fairer comparison, which will also substantially improve the quality of the paper. 

Author Response

Comments 1: 5 % of the control span is not a small value, but it could be manageable in practice. In any case, the derivative noise is not negligible (especially since we are considering a process without noise, only with measurement jitter). Moreover, why would anyone introduce a filter only in the compensated PID? A similar filter can be introduced in all controllers (at least by filtering the control error). Such a filter can be considered part of the process during the tuning stage, so I cannot see any problems. However, see also comment 11, as they are closely related.

Response 1:We thank the reviewer for this observation. We agree that the previous discussion, which focused primarily on the magnitude of the derivative noise contribution, did not fully address the more important methodological issue regarding the fairness of the comparison. To address this concern, the manuscript has been substantially revised through the addition of a new subsection entitled “Standardized Signal Conditioning for Comparative Evaluation.” In this section, the frequency response of the error precompensator (P_e(z)) is explicitly analyzed, and its attenuation of high-frequency components is formally acknowledged. To prevent this inherent filtering action from providing an unfair signal-to-noise advantage to the compensated PID controller, a first-order low-pass filter (F_e(z)) has been incorporated into the error path of the alternative controllers. The filter cutoff frequency was selected to match the crossover frequency identified in the frequency-response analysis of (P_e(z)), thereby ensuring comparable attenuation of measurement jitter across all evaluated control strategies. In addition, the classical PID controller was re-tuned with the filter included in the design procedure. Consequently, the revised comparison no longer involves an unfiltered classical PID versus a filtered compensated PID. Instead, all controllers are now evaluated under equivalent signal-conditioning conditions, allowing the observed differences in performance to be attributed to the control strategies themselves rather than to unequal filtering capabilities.

 

 

Comments 2:  For closed-loop system, 5 samples per time constant (of the process?) is not enough (in my original comment 15 I did not check the process time constant, though). Usually, some additional closed-loop overshoot can be expected when using 5 samples per closed-loop time constant. I would rather advise authors to use 10 samples or more per closed-loop time constant. If it is not possible in this stage, an additional text should be added with explanation why sampling times are not shorter (for example that the measurement jitter would be even higher).

Response 2:We thank the reviewer for this valuable observation. We acknowledge that higher sampling frequencies are generally recommended in digital control applications, often corresponding to approximately ten or more samples per dominant time constant. To address this concern, additional explanatory text has been incorporated into Section 2.3 of the revised manuscript. The new text explicitly discusses the trade-off between temporal resolution and measurement resolution in the adopted implementation. Although increasing the sampling frequency would increase the number of samples per time constant, the velocity estimator employed in this work is based on a fixed-window encoder pulse counting method, for which the quantization step is inversely proportional to the sampling period. For this reason, the sampling frequency of 100 Hz was retained as a compromise between dynamic resolution and measurement quality. This clarification has been added to the manuscript to provide a more explicit justification for the selected sampling period and its practical implications.

Comments 3:However, as already stated in Comment 3, similar type of filters should be used in the classical PID. Otherwise, the study is misleading. PID requires a filter. For example, we say that we did not add a filter to the classical PID (which we must), but we add filters in the compensated PID. And, unsurprisingly, the compensated PID works better. I hope this comment is clearer now. You should add a filter to the classical PID and determine the remaining PID parameters as usual. If needed, the filter can be “moved” to the process during the tuning stage. By the way, P_e is a classical first-order PID filter (with one caveat), regardless of what we call it (pre-compensator or otherwise). How we obtain it is not important. For example, the same type of filter (with unity steady-state gain) should be used in the classical PID.

Response 3:We thank the reviewer for this observation. Taking this consideration into account, a standardized filter was designed to be applied to the classical PID and fuzzy controller without compromising the fairness of the comparison. Specifically, the transfer function of the error precompensator Pe(z) of the compensated PID was analyzed to identify its effective high-frequency attenuation behavior: since Pe(z) incorporates a pole at z = a (with a = −0.052, as established in the text), its unity-gain crossover frequency was determined analytically as fc = 24.58 Hz. A first-order low-pass filter Fe(z) with the same cutoff frequency and unity steady-state gain was then derived, offering a simple and equivalent attenuation dynamic that can be straightforwardly incorporated into the error channel of the remaining controllers. For the classical PID, this filter was included in the plant model during the MATLAB PID Tuner stage, so that the resulting gains already account for its presence and the tuning remains consistent with standard model-based methodology. For the fuzzy controller, Fe(z) was applied directly to the error signal prior to inference. This approach ensures that all three controllers operate under equivalent signal-conditioning conditions, and that any observed performance differences are attributable solely to the mathematical structure of each control law.

Comments 4:The filter P_e should be considered as the PID filter (frequently, all 3 PID terms are filtered with the same filter, not only the D term!), since it is a filter. Why we get it (for compensation or anything else), does not matter much. If we have it in the compensated PID, we should have a similar filter in the classical PID. We do not have to use the same time constant, but controller output noise should be comparable.

Response 4: We thank the reviewer for these observations. To avoid confusion, we have retained the original nomenclature in the manuscript while explicitly clarifying that Pe(z) acts, in effect, as a filter on the error signal.

Acknowledging the methodological weakness identified and guided by the reviewer's suggestions, we have integrated a new term Fe(z), explicitly defined as an intentional filter for the remaining controllers to ensure a fair comparison. Its design was grounded in frequency-domain analysis, replicating the high-frequency attenuation dynamics of Pe(z) so that all controllers operate under equivalent output noise conditions. Full details are provided in Section 3.2 of the revised manuscript.

Comments 5: I suppose that the authors did not understand my original comment. In such kind of implementations, the low speed measurement is usually completely separate from the main sampling time. Speed is measured each time interrupt occurs and then is stored in some variable. This variable is then used by the control algorithm whenever needed. I would like to stress this again: such implementation of speed measurement as in the paper is usually not used in practice because of high jitter at low speeds. Having a method which then “solves a problem of jitter” by using filters (where F_e with some appropriate time constant should be used also in other controllers!) is seeking a solution for a problem which does not exist in practice, since it is solved in a more elegant way.
I suggest that the authors mention that the produced jitter can be easily eliminated using alternative frequency measurement approaches on the PRAXIA robot, such as precisely measuring the intervals between pulses in sub-microsecond ranges with RP2040 controller interrupts. It should also be noted that the study still used an inferior method to examine cases where such improved measurements would be more difficult to implement, for example, when using much less capable microcontrollers.
However, in any case, comparing unfiltered PID and fuzzy controller with filtered one is not acceptable and I expect the new, fairer, experiments. In those experiments, the classical PID filter time constant (additional F_e) should be chosen in such a way that the noise (jitter) amplification of both PID controllers should be comparable in the steady-state. This is important. Since the compensated PID includes an additional reference filter and is tuned using a more targeted approach, it can be expected to deliver better closed-loop results. This is what readers would be interested in. 

Response 5:We thank the reviewer for this detailed clarification and address both points raised.

Regarding speed measurement: we fully agree that interrupt based period measurement using the RP2040 hardware timers represents the standard and more elegant solution that effectively eliminates low speed jitter. Following the reviewer's suggestion, we have strengthened Section 2 (lines 134 to 139) to explicitly state that (i) jitter can be largely eliminated on the PRAXIA platform using this approach, and (ii) the fixed window method was deliberately retained to study controller behavior under the sensing constraints of less capable microcontrollers, where interrupt based measurement is not always feasible. This clarification was also added to the abstract.

Regarding filter comparability: as described in our responses to Comments 3 and 4, new experiments were conducted under equalized filter conditions and all reported results correspond to this fairer configuration. The impact of this correction is directly observable in the results: under nominal conditions, incorporating $F_e(z)$ reduced the jerk excess of the fuzzy controller relative to the classical PID from approximately 134% to 109%, confirming that the previous gap was partially attributable to unequal noise conditioning rather than solely to control law differences. We also agree with the reviewer that the compensated PID carries additional structural advantages, namely the reference prefilter and model based tuning, and have revised the discussion to highlight this distinction explicitly.

Comments 6: I would like to repeat: This is not how the classical controller anywhere is practice is implemented. F_e filter must be applied in practice.

Response 6:We fully agree with the reviewer. The application of a filter to the classical PID is standard practice, and the original manuscript failed to reflect this. As described in our responses to Comments 3, 4, and 5, Fe(z) is now explicitly applied to the classical PID in the revised manuscript, with the controller gains re-tuned accordingly. All reported experimental results correspond to this corrected configuration.

Overall comments: The revised paper has addressed some of the concerns raised in the initial review. However, there are still a few issues that must be resolved before it is suitable for publication. The most significant issue is that it is not possible to compare the results if the PID without a filter (F_e) is included in the comparison. This is not meaningful in any practical application. If one of the aims of the paper is to demonstrate that a controller with a filter (e.g. similar to F_e) is superior to one without a filter, then, unfortunately, I cannot recommend the paper for publication. It should be noted that adding a filter, which is an essential part of any practical PID controller, is straightforward. Tuning PID controller with the filter is also not problematic, as the filter can be considered part of the process during the tuning procedure. I expect the authors to provide a fairer comparison, which will also substantially improve the quality of the paper. 

 

Response:We sincerely thank the reviewer for the time and effort dedicated to improving the quality of this work. As detailed in our responses to Comments 3, 4, and 5, we fully acknowledge this concern and confirm that Fe(z) has been incorporated into the classical PID and fuzzy controller, with the classical PID gains re-tuned with the filter included in the loop model. All experimental results reported in the revised manuscript correspond to this corrected, fairer configuration. We are confident that this revision addresses the reviewer's primary concern and substantially improves the comparability of the three evaluated strategies.

Reviewer 2 Report

Comments and Suggestions for Authors

Accept in present form

Author Response

Comments 1:Accept in present form
Response 1:We thank Reviewer for evaluating the manuscript. We note that the recommendation stated in the comments section is acceptance in present form. As no specific concerns or suggestions were provided, we have no particular points to address. We remain available to respond to any additional clarification the reviewer or editor may require.

Reviewer 3 Report

Comments and Suggestions for Authors

The revised manuscript shows a clear and constructive response to the main concerns raised in the previous review. In particular:

• The novelty is now better articulated, especially the key claim that jerk-based indicators can invert controller ranking under different operating conditions. This is an important point and is now more convincingly supported by the experimental results and discussion.
• The overinterpretation of IAJ has been appropriately corrected, and the shift toward RMS jerk as a relative indicator (rather than a direct proxy for wear) significantly improves the scientific rigor.
• The experimental methodology has been strengthened.
• The manuscript focus is improved, with better balance between modeling and the central contribution (kinematic indicators).

Overall, the revision is substantially improved and technically sound.

 

 

Author Response

Comments 1:

The revised manuscript shows a clear and constructive response to the main concerns raised in the previous review. In particular:

• The novelty is now better articulated, especially the key claim that jerk-based indicators can invert controller ranking under different operating conditions. This is an important point and is now more convincingly supported by the experimental results and discussion.
• The overinterpretation of IAJ has been appropriately corrected, and the shift toward RMS jerk as a relative indicator (rather than a direct proxy for wear) significantly improves the scientific rigor.
• The experimental methodology has been strengthened.
• The manuscript focus is improved, with better balance between modeling and the central contribution (kinematic indicators).

Overall, the revision is substantially improved and technically sound.

Response 1:We sincerely thank Reviewer 3 for the thorough and constructive evaluation. We are pleased that the revisions were well received, particularly regarding the articulation of the novelty, the correction of the jerk interpretation, and the strengthened experimental methodology. We have taken note of the suggestion that the research design can still be improved, and the revisions addressing Reviewer 1's comments, including the standardized filter conditions and the clarification of the speed measurement methodology, directly contribute to this aspect.