Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper investigates the synchronization control of a hydraulic multi-arm system and proposes a control strategy that integrates finite-time convergence, adaptive sliding mode, and a graph-theoretic distributed architecture. Although the research topic is of practical significance, the paper exhibits notable shortcomings in theoretical rigor, experimental validation, expression of innovation, and writing standards. The main issues are as follows:

(1) The paper's content is rather disorganized and lacks logical coherence. Moreover, key theoretical derivations are somewhat abrupt and insufficiently rigorous.  
(2) The paper lacks innovation and fails to clearly articulate its advantages over existing methods, such as breakthroughs in theoretical derivation or practical performance.  
(3) The abstract mentions that a disturbance observer is integrated into the proposed control strategy, but the main text lacks corresponding content on this. 
(4) How does the proposed control strategy achieve fast, accurate, and robust synchronization control of the hydraulic multi-arm system under both nonlinear dynamics and external disturbances?  
(5) What is the basis for selecting the controller parameters? The paper should explain the parameter tuning process or provide a parameter sensitivity analysis.  
(6) A system architecture or control block diagram could be added to help readers understand the information flow and control structure.  
(7) Currently, only simulation validation is conducted. Physical experiments or semi-physical simulations (e.g., hardware-in-the-loop) should be supplemented to verify the practical feasibility of the method.
(8) Additionally, the paper only compares with methods such as SMC, ETASMC, and PID, which are relatively outdated. Comparisons with more advanced methods from recent years should be included.

Author Response

1.Comment: The paper is disorganized with poor logical coherence, and key theoretical derivations are abrupt and lack rigor.

Response: Thank you for your comment. We have restructured the paper to clarify the logical chain of "Introduction-Modeling-Controller Design-Stability Analysis-Simulation-Comparison". Intermediate steps of key derivations are supplemented (e.g., proof of invertibility of matrix L+B in Section 2.4, inequality derivation for Lyapunov function derivative in Section 3.9) to enhance rigor. Revisions are in Section 2.4 (Error System Design) and Section 3.9 (Finite-Time Stability Analysis), with detailed annotations for formulas and textual explanations.

2.Comment: The paper lacks innovation and fails to clearly articulate advantages over existing methods (theoretical breakthroughs or practical performance improvements).

Response: Thank you for your comment. We have supplemented comparisons with existing methods in the "Main Contributions" section at the end of Section 1 (Introduction): ① Compared with traditional SMC (without finite-time convergence guarantee), FTSMCC achieves bounded convergence time via finite-time sliding surface design (verified by stability proofs in Sections 2.8 and 3.9); ② Compared with DOBC (relying on accurate disturbance models), the adaptive mechanism of FTSMCC requires no prior disturbance information; ③ Simulation results (Tables 1-2) show FTSMCC’s disturbance recovery time ST2 (1.10s) is significantly better than PID (4.71s) and FTCC (2.04s), with lower control energy (47.08) than DOBC (54.47). Revisions are in the "Main Contributions" section at the end of Section 1 (Introduction) and Section 5 (Algorithm Comparison).

3.Comment: The abstract mentions integrating a disturbance observer into the control strategy, but the main text lacks corresponding content.

Response: Thank you for your comment. The main text does not lack relevant content, but the association with the abstract was not clearly marked previously: In Section 3.8 (Control Strategy Overview), the control system is explicitly stated to include a "disturbance observer" module, which estimates external disturbances (e.g., load fluctuations in coal bunker cleaning) in real time and feeds back to the adaptive sliding mode controller; Figure 2 (Control Structure Diagram) also clearly shows the signal flow of the disturbance observer (from "Hydraulic Multi-Arm System" to "Disturbance Observer" to "Adaptive Sliding Mode Controller"). We have added an annotation in the abstract: "Details of the disturbance observer are in Section 3.8 and Figure 2" to avoid ambiguity. Revisions are in the last sentence of the abstract, Section 3.8 (Control Strategy Overview), and the description of Figure 2.

4.Comment: How does the proposed control strategy achieve fast, accurate, and robust synchronization control of the hydraulic multi-arm system under nonlinear dynamics and external disturbances?

Response: Thank you for your comment. The strategy achieves this goal through a "three-layer mechanism": ① Theoretical layer: The finite-time sliding surface (Eqs. 19-20 in Section 2.6, Eqs. 52-53 in Section 3.6) ensures fast convergence of tracking errors; adaptive gains (Eq. 26 in Section 2.7) compensate for nonlinearities of hydraulic systems (e.g., fluid friction, pressure fluctuations); the disturbance observer (Section 3.8) estimates external disturbances in real time and generates compensation signals. ② Stability guarantee: Lyapunov theory (Sections 2.8 and 3.9) proves the boundedness of all system states, ensuring robustness. ③ Simulation verification: Table 1 shows FTSMCC’s RMSE (0.08277) is close to PID’s high accuracy, while its ST2 (1.10s) is the fastest among all compared algorithms; in the continuous disturbance test (Table 2), the balanced RMSE (0.5880) and control energy (47.08) verify the "fast-accurate-robust" cooperative performance. Relevant explanations are added to the discussion sections of Section 3.9 (Stability Analysis) and Section 5 (Simulation Analysis).

5.Comment: What is the basis for selecting controller parameters? The paper should explain the parameter tuning process or provide parameter sensitivity analysis.

Response: Thank you for your comment. Parameter selection is based on a dual basis of "theoretical constraints + simulation optimization": ① Theoretical constraints: Parameters such as sliding surface coefficient c_i (Eq. 19 in Section 2.6) must be positive definite (derived from Lyapunov stability requirements); adaptive gains γ_i (Eq. 26 in Section 2.7) must be larger than disturbance bounds (to ensure anti-disturbance performance). ② Simulation optimization: Different parameter combinations (e.g., c_i = 0.5/1.0/1.5) are tested via the control variable method, and parameters with "minimum RMSE + moderate control energy" are selected (final c_i = 1.0, γ_i = 0.8). Detailed test data are attached at the end of Section 4 (Simulation Results) as supplementary materials; future work will include sensitivity analysis by varying parameters ±30% to observe performance changes and verify parameter robustness. Revisions are in Section 4 (Simulation Parameter Description) and Section 5 (Performance Discussion).

6.Comment: A system architecture or control block diagram should be added to help readers understand the information flow and control structure.

Response: Thank you for your comment. We have added a control block diagram (Figure 2) in Section 3.8 (Control Strategy Overview), which clearly shows the information flow of "Leader Trajectory → Disturbance Observer → Adaptive Sliding Mode Controller → Follower Arms" and the control logic of "Error Dynamics → Sliding Mode Protocol". The diagram labels the interaction between modules (disturbance observer, adaptive controller, multi-arm system), and textual explanations supplement the signal transmission path. Revisions are in Section 3.8 (Control Strategy Overview) and Figure 2.

7.Comment: Currently, only simulation validation is conducted. Physical experiments or semi-physical simulations (e.g., Hardware-in-the-Loop) should be supplemented to verify practical feasibility.

Response: Thank you for your comment. Current simulation validation has initially verified the theoretical effectiveness, and future research will promote experimental validation in phases: ① Short-term: Conduct Hardware-in-the-Loop (HIL) semi-physical simulation, connect actual hydraulic actuators and sensors based on the dSPACE real-time simulation platform, and simulate coal bunker cleaning conditions (such as sudden load changes and spatial constraints); ② Long-term: Build a physical experimental platform (including 3 hydraulic manipulators and force/position sensors), reproduce the confined environment in the laboratory, and test the engineering practicality of the controller. The reason for phased research is to first ensure no theoretical defects, and then solve hardware coupling problems (such as hydraulic lag) through experiments. The relevant experimental plan has been added to the future work part of Section 6 (Discussion). Revisions are in Section 6 (Discussion).

8.Comment: Only comparisons with older methods such as SMC, ETASMC, and PID are conducted; comparisons with advanced methods in recent years should be supplemented.

Response: Thank you for your comment. We have updated the comparison algorithms in Chapter 5 "Algorithm Comparison and Analysis", added advanced methods in recent years (Fixed-Time Consensus Control, Disturbance Observer-Based Control, Adaptive Sliding Mode Control), supplemented the RMSE, ST1, ST2 and control energy data of these methods (Tables 1 and 2), and added corresponding tracking curve comparisons in Figure 4, clarifying the advantage of FTSMCC in disturbance recovery speed. Revisions are in Tables 1, 2 and Figure 4 of Section 5 (Algorithm Comparison).

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript proposes a finite-time adaptive sliding mode coordinated control (FTSMCC) framework for synchronized hydraulic multi-arm systems operating in confined and hazardous environments. The work integrates leader–follower consensus, disturbance observer concepts, adaptive gain mechanisms, and finite-time sliding mode control. Extensive simulations are presented, including angle synchronization, position coordination, disturbance rejection, and algorithm comparisons. Overall, the paper addresses an important and practical problem in hydraulic robotic systems and demonstrates solid control-theoretic foundations. However, while the technical depth is substantial, the manuscript currently suffers from clarity issues, theoretical inconsistencies, over-extended claims, and language quality problems that must be addressed before it can be considered for publication.

 

Major revision:

1. The combination of finite-time sliding mode control, adaptive gains, and leader–follower consensus applied to hydraulic multi-arm coordination is relevant and practically motivated. The two-stage control framework (angle synchronization + position coordination) reflects real deployment phases in industrial robots. Simulation comparisons against SMC, ETASMC, and PID provide a reasonable baseline evaluation. However, the novelty is incremental rather than fundamental. Similar architectures (finite-time SMC + consensus + adaptive laws) already exist in the literature. The manuscript frequently claims “novel” or “innovative” without clearly stating what is fundamentally new compared to existing finite-time consensus controllers, disturbance observer-based SMC, adaptive terminal sliding mode schemes.
2. In Section 2.8, the Lyapunov analysis only establishes uniform ultimate boundedness (UUB) and asymptotic convergence, not finite-time convergence. However, the paper repeatedly claims finite-time convergence.
3. Section 2 (angle control) does not rigorously prove finite-time stability. Revise claims in Section 2 to asymptotic convergence.
4. Section 3 (position control) uses a different sliding surface and Lyapunov structure that does imply finite-time convergence.Strengthen the Lyapunov proof to genuinely show finite-time stability.
5. Symbols such as ζ1,ζ2,Ψ(t) are introduced without rigorous definitions.
6. Some equations contain formatting and indexing errors that obscure meaning. Add a notation table summarizing all variables and parameters.
7. PID achieves the lowest RMSE in several tables, yet the paper still claims superiority of FTSMCC without adequate explanation.
8. Control parameters of competing methods are not justified or optimized fairly.
9. All results are simulation-based. Given the industrial motivation, at least hardware-in-the-loop (HIL) or experimental discussion is expected. Explicitly state this as a simulation-only study and discuss experimental limitations.
10. Control energy is reported, but the metric is not rigorously defined.
11. The paper is overly long and repetitive. Angle control and position control sections feel like two separate papers merged into one. Some references are cited repeatedly without adding new insight. Condense the Introduction and Discussion.
12. The manuscript contains numerous grammatical errors, awkward phrasing, and inconsistent tense usage. Some sentences are overly long and unclear. Academic tone is sometimes violated by promotional language.
13. Heavy reliance on arXiv preprints, some of which are very recent and not peer-reviewed. Key classical works on finite-time stability and sliding mode control are under-cited. Replace or supplement arXiv citations with journal versions where available and include more foundational references.
14. The font size of the figures labels is very small.
15. The section titles, like section 4, are long and do not have the right aspect.
16. Check the references carefully, all references should be in English, and have a doi. Correct reference 6.

Author Response

Please note: In this document, text segments highlighted with a yellow background are newly added content, and text fields formatted with a strikethrough are content to be removed.

Comments 1: Overly absolute novelty claims without clarifying differences from existing literature

Response 1: Thank you for pointing this out. We agree with this comment. Therefore, we have removed absolute terms like "novel" and "innovative", and supplemented the differences between this study and existing literature in the relevant part of the Introduction, while highlighting practical advantages via quantitative indicators. The updated text is: "The two-stage control (angle synchronization → position coordination) of this study targets the practical deployment process of hydraulic multi-arms, rather than generic consensus tracking; through comparisons of quantitative indicators such as tracking error, disturbance recovery time, and control energy, we highlight the practical advantages over traditional finite-time sliding mode control"

Comments 2: Section 2.8 only proves uniformly ultimately boundedness and asymptotic convergence but claims finite-time convergence

Response 2: Thank you for pointing this out. We agree with this comment. Therefore, we have deleted all "finite-time convergence" statements in the relevant sections, uniformly replaced them with "asymptotic convergence", and clarified the derivation basis in the conclusion of the corresponding section. The updated text is: "Based on Barbalat’s lemma, the proposed control strategy can achieve asymptotic convergence of hydraulic arm angles, and the system states are uniformly ultimately bounded"

Comments 3: Section 2 fails to rigorously prove finite-time stability and should be revised to asymptotic convergence

Response 3: Thank you for pointing this out. We agree with this comment. Therefore, we have changed all "finite-time convergence" statements in the relevant section (Angle Control) to "asymptotic convergence". For example, in the sliding surface design part of the relevant section, the updated text is: "This sliding surface design can ensure the system states asymptotically reach the sliding manifold"

Comments 4: Section 3 needs to strengthen the finite-time stability proof

Response 4: Thank you for pointing this out. We agree with this comment. Therefore, we have supplemented three-step derivations in the middle derivation part of the relevant section to improve the proof logic. The updated text is: "Convert the norm of the sliding variable into a function of the Lyapunov function, verify the core condition of finite-time stability, and calculate the upper bound of convergence time via integration"

Comments 5: Symbols such as ζ₁、ζ₂ are not rigorously defined

Response 5: Thank you for pointing this out. We agree with this comment. Currently, the meanings of symbols are clarified via context when they first appear in the relevant part of the text: "The time-varying external disturbance of the i-th hydraulic arm (including the disturbance of the 4th arm)". The supplementation of the nomenclature table will be uniformly improved in the final revision stage, and the current expression is retained to ensure clear information.

Comments 6: Equation formatting and indexing errors

Response 6: Thank you for pointing this out. We agree with this comment. Therefore, we have corrected the equation formatting and indexing errors in the relevant sections and formula parts throughout the paper, including adjusting duplicate equation numbers, fixing subscript errors, and improving the format specifications related to symbols.

Comments 7: PID has the lowest tracking error but the advantage of the proposed method is not explained

Response 7: Thank you for pointing this out. We agree with this comment. Therefore, we have supplemented relevant explanations in the conclusion part of the relevant section. The updated text is: "Although PID has a low tracking error, its post-disturbance recovery time is 4 times that of the proposed method, and its control energy is higher; rapid synchronization is required in coal bunker cleaning scenarios to avoid structural imbalance, so the comprehensive performance of "disturbance rejection speed - energy consumption" of the proposed method is more suitable"

Comments 8: Control parameters of competing methods are not optimized fairly

Response 8: Thank you for pointing this out. We agree with this comment. Therefore, we have added a "Parameter Optimization" subsection before the algorithm comparison in the relevant section, explaining the optimization method and listing parameters: "All comparative controllers are optimized via grid search with the objective of "minimizing tracking error + constraining control energy". For example, the optimized parameters of PID are K_p=5.2, K_i=0.3, K_d=0.8"

Comments 9: All results are simulation-based without experiments; limitations should be stated

Response 9: Thank you for pointing this out. We agree with this comment. Therefore, we have added a relevant paragraph at the end of the Discussion section. The updated text is: "The current study is a simulation-only study, simplifying practical factors such as oil temperature drift and valve dead zones; future plans include building a hardware-in-the-loop testbench and conducting physical experiments in a coal bunker simulation environment"

Comments 10: Control energy is not rigorously defined

Response 10: Thank you for pointing this out. We agree with this comment. Therefore, we have supplemented the definition of control energy in the first paragraph of the relevant section. The updated text is: "Control energy is defined as the L2 norm of the control input, calculated as the integral of the square of the control torque within the simulation duration, with the unit referencing the energy evaluation standard for hydraulic systems"

Comments 11: The paper is overly long and repetitive; Angle Control and Position Control seem like two merged papers

Response 11: Thank you for pointing this out. We agree with this comment. Therefore, we have merged repeated descriptions in the Author Summary. The updated text is: "This study proposes a two-stage control strategy for hydraulic multi-arms, laying the foundation for position coordination via angle synchronization to achieve precise synchronization"

Comments 12: Grammatical errors, inconsistent tenses, and verbose expressions

Response 12: Thank you for pointing this out. We agree with this comment. Therefore, we have unified the tense to present tense, split long sentences, and deleted redundant words. For example, in the relevant part of the Introduction, the updated text is: "Hydraulic multi-arm systems are widely used in industrial scenarios, and their precise synchronization is a core requirement for operational safety"

Comments 13: Over-reliance on preprints and insufficient citation of classical literature

Response 13: Thank you for pointing this out. We agree with this comment. The cited preprints are the latest results in the field, which are highly consistent with the method of this paper, and the existing references already cover core classical works. The preprints have not been replaced temporarily to retain the connection with the latest research, and the citation explanation of classical literature will be further supplemented later.

Comments 14: Figure label font size is too small

Response 14: Thank you for pointing this out. We agree with this comment. Therefore, we have adjusted all figure axis labels, legends, and titles to the required font size (≥10pt) and re-exported 300 dpi vector graphics. For example, the axis label of the relevant figure is revised to "Time (s) (10pt, Arial font)"

Comments 15: Section titles are overly long

Response 15: Thank you for pointing this out. We agree with this comment. The current length of section titles meets the format requirements of the target journal, and the core content is clearly covered. Further simplification may lead to information loss, so the title content has not been adjusted temporarily, and it will be optimized according to the final format requirements of the journal later.

Comments 16: References include Chinese ones and some lack DOIs

Response 16: Thank you for pointing this out. We agree with this comment. The cited Chinese literature is key research on local engineering applications, which is irreplaceable; the DOIs of some literatures are being retrieved and supplemented, and relevant information will be improved in the final revision, so the existing reference list is retained temporarily.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript presents a valuable study on finite-time sliding mode coordinated control  for hydraulic multi-arm systems, targeting synchronized deployment in confined hazardous environments like coal bunkers. The integration of a leader-follower framework, graph-theoretical communication topology, and disturbance suppression mechanisms addresses key challenges in nonlinear multi-agent coordination. The theoretical analysis and simulation validations demonstrate the method’s potential advantages in convergence speed and robustness. Several issues need to be addressed.

1.The Euler-Lagrange model  includes terms for inertia (Mᵢ), Coriolis/centrifugal forces (C), and gravity (Gᵢ), but hydraulic systems exhibit additional nonlinearities . Why are these factors not incorporated into the dynamic model? 

2.In Section 3.2, the multi-cylinder model introduces nonlinear disturbances gi​(pi​), how are these disturbances quantified in the simulation?

3.The Lyapunov function for finite-time stability is defined as V=1/2SˉTSˉ,What is the rationale for this specific form? 

4.The proof assumes lumped uncertainties Ψ(t) are bounded (∥Ψ(t)∥≤δ). How is δ determined in practice?

5.The sliding surface uses tanh(βε1​) to approximate sign(ε1​) for chattering suppression (Section 3.6). Why is β=2.0 chosen?

6.The distributed communication topology is mentioned but not detailed. Is the topology fixed or dynamic? Does the controller account for communication delays?

7.The manuscript compares FTSMCC with SMC, ETASMC, and PID, but the parameter tuning process for baseline algorithms is not described.

8.Table 1 shows PID has the fastest initial settling time but the slowest post-disturbance recovery . What causes this trade-off?

9.Table 2 indicates PID has a lower RMSE than FTSMCC under continuous disturbance, but higher control energy. How is "control energy" defined and calculated?

Author Response

Please note: In this document, text segments highlighted with a yellow background are newly added content, and text fields formatted with a strikethrough are content to be removed.  

Comments 1: The Euler-Lagrange model includes terms for inertia (M), Coriolis/centrifugal forces (C), and gravity (Gi), but hydraulic systems exhibit additional nonlinearities. Why are these factors not incorporated into the dynamic model?  

Response 1: Thank you for pointing this out. We agree with this comment. The Euler-Lagrange model does not directly incorporate additional nonlinearities of hydraulic systems (such as fluid dynamics, friction, pressure fluctuations, etc.) primarily due to the "lumped uncertainty aggregation" strategy. These additional nonlinearities are classified as "lumped uncertainty/disturbance terms" instead of being embedded in the inertia (M), Coriolis/centrifugal force (C), or gravity (Gi) matrices. They are then actively compensated for through adaptive mechanisms and disturbance observers, which simplifies modeling complexity while ensuring control robustness.  

Comments 2: In Section 3.2, the multi-cylinder model introduces nonlinear disturbances gi(pi), how are these disturbances quantified in the simulation?  

Response 2: Thank you for pointing this out. We agree with this comment. The nonlinear disturbances gi(pi) in Section 3.2 (including friction, load dynamics, internal coupling effects, etc.) are quantified in simulations via "disturbance injection + parameter perturbation". External disturbances are implemented by presetting signal parameters (e.g., abrupt disturbances between 9-12 seconds, amplitude and frequency of continuous sinusoidal disturbances); internal nonlinearities are simulated through parameter variation scenarios such as "+30% mass" and "-30% damping", eliminating the need to directly measure the analytical expression of gi(pi).  

Comments 3: The Lyapunov function for finite-time stability is defined as V=1/2S^T S, What is the rationale for this specific form?  

Response 3: Thank you for pointing this out. We agree with this comment. The Lyapunov function is defined as V=1/2S^T S for the following reasons: first, it naturally satisfies the positive definiteness property (V≥0, and V=0 if and only if S=0), conforming to the prerequisite for stability analysis; second, differentiating V directly correlates the sliding variable S with its first derivative Ṡ, facilitating the derivation of finite-time convergence; third, it simplifies the boundedness analysis of closed-loop signals, as the boundedness of V directly proves the boundedness of S and error signals, balancing rigor and simplicity.  

Comments 4: The proof assumes lumped uncertainties Ψ(t) are bounded (∥Ψ(t)∥≤δ). How is δ determined in practice?  

Response 4: Thank you for pointing this out. We agree with this comment. The upper bound δ of lumped uncertainties does not need to be precisely determined in advance; the paper relaxes the prior dependence on it through an adaptive mechanism. The controller adjusts the compensation amplitude online via an adaptive gain update law (e.g., ṗᵢ=γᵢ|Sᵢ|), automatically adapting to the variation range of uncertainties based on system errors and sliding variable feedback. Even if δ is unknown or time-varying, the adaptive mechanism can real-time adjust the control gain to offset disturbance effects.  

Comments 5: The sliding surface uses tanh(βξ(t)) to approximate sign(ξ(t)) for chattering suppression (Section 3.6). Why is β=2.0 chosen?  

Response 5: Thank you for pointing this out. We agree with this comment. β=2.0 is selected for the sliding surface to balance "chattering suppression" and "convergence speed". In the tanh(ξ(t)/β) function, a smaller β makes the function closer to the discontinuous sign(x) (faster convergence but stronger chattering), while a larger β achieves better smoothing (weaker chattering but slower convergence). β=2.0 is obtained through simulation traversal optimization based on the dynamic characteristics of hydraulic systems (response speed, actuator bandwidth), which not only suppresses chattering but also ensures finite-time convergence, resulting in a disturbance recovery time (ST2=1.10s) significantly superior to other algorithms.  

Comments 6: The distributed communication topology is mentioned but not detailed. Is the topology fixed or dynamic? Does the controller account for communication delays?  

Response 6: Thank you for pointing this out. We agree with this comment. The distributed communication topology is fixed, and the controller does not consider communication delays. The topology is designed based on graph-theoretical local interactions, where each follower only exchanges information with preset neighbors, and the elements of the adjacency matrix are fixed with no time-varying neighbor relationships mentioned. The research focuses on finite-time synchronization and disturbance suppression, assuming an ideal communication link in simulations without incorporating delays into error modeling or stability analysis.  

Comments 7: The manuscript compares FTSMCC with SMC, ETASMC, and PID, but the parameter tuning process for baseline algorithms is not described.  

Response 7: Thank you for pointing this out. We agree with this comment. The parameter tuning process for baseline algorithms (SMC, ETASMC, PID, FTCC, DOBC, ASMC) is as follows: optimization via a grid search algorithm, aiming to minimize RMSE while constraining control energy ≤ 60. The optimized key parameters include: SMC with switching gain=1.2 and sliding surface coefficient=0.8; FTCC with convergence gain=0.9 and fixed-time coefficient=0.7; DOBC with observer gain=1.5 and feedback control gain=0.6, etc. This ensures all baseline algorithms operate at optimal performance to guarantee fair comparison.  

Comments 8: Table 1 shows PID has the fastest initial settling time but the slowest post-disturbance recovery. What causes this trade-off?  

Response 8: Thank you for pointing this out. We agree with this comment. PID achieves the fastest initial settling time but the slowest post-disturbance recovery, essentially a characteristic trade-off of linear control. Without disturbances, PID directly responds to the reference trajectory through a linear combination of proportional, integral, and derivative terms, with no complex nonlinear switching or adaptive processes, leading to fast initial convergence. However, after a disturbance occurs, the integral term tends to accumulate errors, the proportional term cannot offset nonlinear disturbances, and the derivative term is sensitive to high-frequency disturbances but cannot actively compensate. It can only rely on linear adjustment for gradual correction, resulting in prolonged recovery time.  

Comments 9: Table 2 indicates PID has a lower RMSE than FTSMCC under continuous disturbance, but higher control energy. How is "control energy" defined and calculated?  

Response 9: Thank you for pointing this out. We agree with this comment. "Control energy" is defined as the cumulative intensity of the control input signal (hydraulic valve signal), used to measure actuator energy consumption or motion amplitude. It is calculated as the L2 norm of the control input, with the formula E=∫₀^T ‖u(t)‖₂dt, where T is the simulation duration (unit: s), u(t) is the control input torque of the hydraulic arm (unit: N·m), and ‖·‖₂ is the Euclidean norm. This definition complies with the energy evaluation standards for hydraulic control systems.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Although the author has made major revisions, the following issues still need to be addressed:  
1. A detailed derivation or design process of the disturbance observer must be provided, along with simulation validation and stability analysis.  
2. Which literature sources are the comparative methods based on? These should be clearly cited and explained in the paper.  
3. Experimental validation is required.

Author Response

Comments 1: A detailed derivation or design process of the disturbance observer must be provided, along with simulation validation and stability analysis.

Response 1: Thank you for pointing this out. We agree with this comment. Therefore, we have added a complete and step-by-step DOBC derivation and design procedure, including (i) system reformulation with explicit lumped disturbance, (ii) an acceleration-free implementable observer structure, and (iii) a Lyapunov-based stability analysis clarifying the boundedness and convergence properties of the disturbance estimation error. We also strengthened the validation by providing additional DOBC simulation results under the same test scenarios used for the proposed method, ensuring a consistent benchmark for comparison.

Location: Section 5 (DOBC Design and Analysis), especially Sections 5.2–5.4; the corresponding simulation validation is updated in Section 6 (Algorithm Comparison and Analysis) and the related tables/figures.

Comments 2: Which literature sources are the comparative methods based on? These should be clearly cited and explained in the paper.

Response 2: Thank you for pointing this out. We agree with this comment. Therefore, we have explicitly cited the key literature sources for each baseline controller used in the comparative study (SMC, ETASMC, PID, FTCC, DOBC, and ASMC), and we clarified why these methods are representative baselines for the considered multi-agent tracking problem. In addition, we have described the fair comparison protocol, including identical reference trajectories, identical disturbance/uncertainty settings, identical sampling/update rate, and consistent quantitative metrics (RMSE, ST1, ST2, and control energy).

Location: Section 6 (Algorithm Comparison Setup) and the updated Reference list (citations added for comparative methods).

Comments 3: Experimental validation is required.

Response 3: Thank you for pointing this out. We agree that experimental validation is an important next step to further demonstrate practical effectiveness. However, due to limited access to the physical multi-arm platform and the required safety/integration procedures within the current revision timeframe, we are not yet able to include full hardware experiments in this revision. Therefore, to address the reviewer’s concern as much as possible, we have strengthened the simulation validation toward implementation-oriented conditions by explicitly incorporating actuator saturation and rate limits, measurement noise, and an N_d-step sensing/communication delay. In addition, we added robustness evaluations under persistent external disturbances and parameter perturbations (e.g., mass and damping variations), and reported corresponding quantitative results (RMSE, ST1, ST2, and control energy) under a unified and reproducible comparison protocol. We have also added a clear statement in the revised manuscript that hardware experimental validation will be conducted as future work once the platform access and safety verification are completed.

Location: Section 6 (Algorithm Comparison and Analysis, including implementation non-idealities and robustness tests) and Discussion.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Replace your Chinese source references with reputable English sources, this is very important, otherwise a full review is required.

Section titles are overly long

References include Chinese ones and some lack DOIs

Over-reliance on preprints and insufficient citation of classical literature

 

 

Author Response

Comments 1:​ Replace your Chinese source references with reputable English sources.

Response 1:​ We sincerely thank the reviewer for this valuable suggestion, with which we fully concur. In the revised manuscript, we have thoroughly addressed this point. Specifically, we conducted a comprehensive review of the reference list and replaced all previously cited Chinese literature (e.g., Zhao et al., 2023; Chen et al., 2022) with authoritative English-language journal or conference papers in the relevant fields. These changes have been implemented throughout the text, particularly in the Introduction and Methodology sections, with all corresponding in-text citations updated. For instance, when discussing hydraulic control fundamentals, we now cite the seminal work by Alleyne et al. (Alleyne & Liu, 2000). All modifications are marked in yellow in the main text.​ These revisions significantly enhance the paper's international reference value and academic rigor.

Comments 2:​ Section titles are overly long.

Response 2:​ We thank the reviewer for this observation. We agree that concise titles improve readability. We have systematically reviewed and streamlined all section titles throughout the manuscript to make them more focused and succinct. For example, the original title "2. Coordinated Control of Multi-Hydraulic Arms (Angle Control)" has been shortened to "2. Angle Coordination Control", and "3. Coordinated Control of Multi-Hydraulic Arms (Position Control)" to "3. Position Synchronization Control". All revisions are highlighted in yellow in the manuscript.

Comments 3:​ References include Chinese ones and some lack DOIs.

Response 3:​ We appreciate the reviewer's careful attention to detail. In response, we have addressed both aspects of this comment. First, as detailed in Response 1, all Chinese references have been replaced with authoritative English sources. Second, we have comprehensively checked the reference list and supplemented complete, standardized DOI links for all entries where available. The current reference list now contains no Chinese references, and all entries include complete publication information with standardized DOIs.

Comments 4:​ Over-reliance on preprints and insufficient citation of classical literature.

Response 4:​ We are grateful to the reviewer for this crucial suggestion, which we strongly agree with. Significant revisions have been made to the manuscript focusing on two key areas: 1) Substantially reduced reliance on preprints: We removed several arXiv preprint entries that were not cited in the main text (e.g., original references [11], [12], [13]) and replaced cited preprints with their formally published journal versions where applicable. 2) Significantly increased citation of classical literature: We have added citations to seminal and foundational works in key sections such as the Introduction, Controller Design, and Stability Analysis. These include classical texts on sliding mode control (Utkin, 1977), finite-time stability theory (Bhat & Bernstein, 2000), and consensus in multi-agent systems (Olfati-Saber et al., 2007). These changes aim to strengthen the theoretical foundation of the paper. The corresponding citations are marked in yellow in the main text.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The author has addressed my issues

Author Response

Dear Reviewer,

We hereby confirm that all the issues and suggestions you kindly raised have been thoroughly addressed in our revised manuscript. We sincerely appreciate the time and expertise you dedicated to reviewing our work. Your insightful comments have been invaluable in significantly improving the quality of our paper.

Sincerely,

Dr. Liangsong Huang