A Closed-Form Cooperative Avoidance Control for Multiple m-DOF Manipulators

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Fig. 4 is supposed to o;;ustrate the merit of the proposed approach. The intersection of the trajectories looks like negating the claim. It has to be explained and improved by attaching the time so that it shows that the apparent intersection in Figure 4 actually occurs actualy at a different time, i.e., is only an apparent collision.

Eq. 28 shows a proportional controller in joint space. Explain where vellocities intervine in motion control.

The control approach appears to be a variant of stiffness control in operational space. This would require dynamic linearization in operational spve. Lyapunov stability does not achieve this linearization. 

 

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript studies collision-free cooperative control for multiple Euler–Lagrange manipulators and proposes a closed-form controller that combines trajectory tracking, collision avoidance, disturbance attenuation, and deadlock breaking within one framework. The paper also claims that the avoidance action is activated through relative-velocity information to reduce conservativeness, and provides a Lyapunov-based stability proof together with simulations on a dual 2-DOF manipulator system. Overall, the topic is relevant to cooperative robot control, and the attempt to derive a closed-form avoidance law without online optimization is interesting.However, the current version still has several issues that should be addressed before the work can be considered for publication.

1. The experimental validation is too limited.
   The whole validation is based on simulations of only two 2-DOF planar manipulators, although the method is formulated for multiple (m)-DOF manipulators. This gap between the general theoretical claim and the narrow simulation setting is substantial. The manuscript would be much stronger if it included simulations with more than two manipulators, higher-dimensional manipulators, or at least a more detailed discussion of scalability and computational burden. At present, the evidence is not sufficient to support the claimed generality.
2. Some practical terms in the proposed controller are not actually validated.
   The manuscript introduces both a disturbance compensation term and a deadlock-breaking perturbation term in the control law. However, in the simulation section, the authors explicitly set the disturbance term to zero and state that the deadlock-breaking term is also inactive because the simulated case is designed to avoid local minima. Therefore, two important components of the proposed method are not truly tested in the numerical examples. Additional experiments are needed to show whether these terms work as intended in non-ideal cases.
3. The collision model is simplified to end-effector distance only.
   The collision avoidance function is defined using the Euclidean distance between end-effectors. For multi-link manipulators operating in a shared workspace, collision risk can also occur between links, not only at the end-effectors. This simplification should be discussed more carefully, because it limits the practical scope of the method. The paper should clarify whether the current formulation is intended only for end-effector safety or whether it can be extended to full link-level collision avoidance.
4. The main theorem relies on assumptions that reduce the claimed generality.
   The theorem assumes safe initial conditions and requires the desired configurations to lie outside each other’s detection range. Although a remark later says this assumption can be relaxed in some cases, the formal result is still built on a relatively restrictive condition. The implications of this assumption should be discussed more explicitly, especially for dense cooperative tasks where desired configurations may naturally fall within nearby detection zones.
5. The novelty relative to existing collision-avoidance control methods should be clarified further.
   The paper emphasizes the use of relative velocity information and closed-form design, but the manuscript would benefit from a sharper comparison with existing Lyapunov-based, artificial-potential-field-based, and safety-constrained control methods. At present, the difference in contribution is described mostly in general terms. A more precise explanation of what is mathematically new, beyond combining known ingredients into one controller, would improve the paper.
6. The presentation quality needs improvement.
   There are many language and formatting problems throughout the manuscript. Some equations and figure captions are not cleanly presented, and there are obvious template artifacts such as “Journal Not Specified” and incomplete figure text around the example section. In addition, several sentences are overly long and difficult to follow, especially in the stability proof. The manuscript requires substantial language editing and formatting revision before it is ready for publication.
7. The simulation comparison is insufficient.
   The manuscript mainly shows that the proposed controller can achieve smooth convergence and collision-free motion, but it does not provide convincing comparisons against representative baseline methods. Since one of the claimed advantages is reduced conservativeness and improved efficiency, quantitative comparisons with standard APF-based control, barrier-function-based control, or another existing cooperative avoidance strategy would be very helpful. Without such comparisons, the claimed performance advantage remains only partly supported.

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

1. Novelty of this work is not satisfactory; Author should explain the novelty with respect to earlier reported work
2. Explain the full form of m-DOF in the abstract and main text.
3. The label of Fig. 1 (Page 13) is incomplete. We recommend adding complete label.
4. The grammar and scientific writing should be improved. (1) Fig 4. label contains the repeated ‘Trajectories’ word (2) sequence of equations are inconsistent Equation Eq. 43>28>29>44 (page no. 15)
5. The limitations and future scope of this work should be presented briefly.
6. Out of the 40 cited references 36 are cited in the introduction section. Author should add more recent references to the discussion section.
7. Reference for Each equation is needed to increase the validation quality of this work.

Author Response

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Reviewer 4 Report

Comments and Suggestions for Authors

This manuscript presents a closed-form cooperative avoidance control strategy for multiple multi-degree-of-freedom manipulators. The authors tackle an important topic in robotics: balancing safety with execution efficiency in shared workspaces. The math is sound. The paper reads well, and the simulation results look quite clean. However, I have several major concerns before the manuscript is published.

In equation (11) and the controller design, you present a unified close-form control law. In real case, robotic actuators have strict torque limits. When manipulators get close, the gradient of your avoidance function approaches infinity. In reality, the motors will saturate before that. How does your controller handle actuator saturation? And please discuss how torque bounds affect your Lyapunov stability proofs.

I like the idea of using a 90-degree rotation matrix to inject a perpendicular torque to break local minima. However, you need to explain the physical mechanism. When you inject this, what happens to the tracking error of the primary task? Does it cause sudden joint jerks or high acceleration? Please explain or provide plots.

The dynamic model in equation (2) simplifies the system by assuming gravity is perfectly compensated and external disturbances are bounded. But in real serial manipulators, joint friction (Coulomb and viscous friction) is highly nonlinear and notoriously hard to compensate perfectly. Why did you omit joint friction? I recommend adding a non-zero friction model into your simulation.

Your simulation uses a dual 2-DOF planar setup. It is too simple to support your claim that the method works for general "m-DOF" systems in three-dimensional workspaces. I strongly urge the authors to supplement their work with a simulation of at least two 3-DOF or 6-DOF arms moving in 3D space.

For the figure caption in figure 1, do you mean “Figure 1. Two-link manipulators”. It seems authors forgot to type the number of links.

Author Response

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Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The author explained answers satisfactory way. I recommed to accept this manuscript in current form. 

Comments on the Quality of English Language

Quality of english in satisfactory 

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have solved all the issues.