Particle Swarm Optimization—An Adaptation for the Control of Robotic Swarms

Round 1

Reviewer 1 Report

You see the attached file.

Comments for author File: Comments.pdf

Author Response

We wish to thank both of the reviewers for their comments and suggested changes to our manuscript, we appreciate that there is significant analysis in our work, and we have strived to improve the readability based on the comments. We have also expanded our discussion and explained in more detail the impact of our assumptions. We have revised the manuscript and also address each comment specifically below. Please see the attached DIFF.PDF file for changes in the manuscript.

Reviewer 1

1.The mathematical model of the treated subject being complex and the results presented require many figures, it is necessary many parameters and mathematical equations, however I suggest an optimal use of them, as much as possible.

We agree that the mathematical analysis we propose is quite complex, and we have spent considerable effort to ensure that both the notation and the format are both rigorous and readable. We have placed several proofs in the appendices and have left in the main body of the manuscript only that we think the reader ought to know. We have also made changes that more clearly highlight the important results, as detailed below.

2. In page 3, at the end of the equations (3), (4), (5), (7) and (8), must point. This has been fixed.
3. In page 5, at the end of the equations (15) and (16) must point. This has been fixed.
4. In page 6, the end of the equation (17), must a comma, and must point at the end of the equation above line 145, and to the end of the equation after the one numbered with (17). This has been fixed.
5. At the end of the equation between eq. (20) and eq. (21), must point, and at the end of eq. (21) must point. 5. At the end of the first equation on page 7, must point, and at the end of the equation above line 146, must point. This has been fixed.
6. At the end of the equations (23), (24), (27), (28), (30), (32), (33), (34), (36) and (38), must point. This has been fixed.
7. At the end of the equations (26), (29), (31), (35), (39), (40) and (42), must comma. This has been fixed.
8. At the end of the equation above line 167 and after eq. (37), must point. This has been fixed.
9. At the end of lines 227, 239, 240 and 241, a period is needed. This has been fixed.
10. The important results of the authors must be well highlighted.

We have made the following changes to make the important results clearer.

Line 383: Added “This paper has introduced… by including the timestep size Δt.” to clarify the contributions in the Discussion.

Line 390: Added “To validate the proposed changes… almost always resulted in collisions.” to highlight the results of the simulations in the Discussion.

Line 449: Added “The new algorithm is compared… environment and robot survivability.” to highlight the results of the simulations in the Conclusion.

I am convinced that it is useful for the manuscript to be included in the References section the following paper, which use similar procedures

1. i) Trajectory optimization for mobile robots using model predictive control, Periodicals of Engineering and Natural Sciences, Vol. 7, Nr. 1, p. 242-248, ISSN: 2303-4521, 2019
2. ii) The optimization of intelligent control interfaces using Versatile Intelligent Portable Robot Platform, Procedia Computer Science, DOI: 10.1016/J.PROCS.2015.09.115, 2015

Thank you for the suggested citations on the topic of control with application to examples including bipedal robots. However, it is unclear from the reviewer's comments how these citations are relevant to our manuscript, or where this citation should be included in the manuscript. We do not think that they are appropriate for inclusion without further guidance.

Reviewer 2 Report

Comments to the authors

Manuscript ID: robotics-1156883

Title: Particle Swarm Optimization - An Adaptation for the Control of Robotic Swarms

1) The analysis are presented under the assumption that there is only one particle. Explain why the achieved results are extendable to more general cases.

2) A similar method to the repulsive force (11) has been investigated in 10.1109/TAC.2019.2906467 and 10.1080/00207179.2017.1317832. For the sake of completeness, please discuss these references in the manuscript.

3) Adding the sign function as in (13) can cause chattering. More justifications/details are required.

4) More details on the guideline presented in Section 3.5 are required. Convergence properties? Computational complexity? Real-time implementation?

Author Response

We wish to thank both of the reviewers for their comments and suggested changes to our manuscript, we appreciate that there is significant analysis in our work, and we have strived to improve the readability based on the comments. We have also expanded our discussion and explained in more detail the impact of our assumptions. We have revised the manuscript and also address each comment specifically below. Please see the attached DIFF.PDF file for changes in the manuscript.

Reviewer 2

1) The analysis are presented under the assumption that there is only one particle. Explain why the achieved results are extendable to more general cases.

The single-particle analysis is performed for the following reasons:

• It is a simplification made following previous work when considering stability and convergence, for example [Bonyadi et al, 2016], [Cleghorn et al, 2018]. The effect of the state of other particles on a single particle is only to change the value of the global best point (y_g). This is considered in the analysis by assuming that y_g can change randomly at every timestep by being drawn from a distribution with well-defined mean and variance (line 97). Therefore, the analysis is extendable to swarms of any size assuming collision do not occur.
• An equivalent state-space analysis of the entire swarm (e.g. using the centre of mass of the swarm or another metric) is interesting and would be novel but it is beyond the scope of this manuscript.

Line 99: Added “To follow these previous analyses further…”, to make it more understandable that previous analyses are followed.

2) A similar method to the repulsive force (11) has been investigated in 10.1109/TAC.2019.2906467 and 10.1080/00207179.2017.1317832. For the sake of completeness, please discuss these references in the manuscript.

Thank you for this suggestion, having considered the suggested citations we agree that they are relevant and have made the following changes:

Line 104 – below (10): Added “Virtual forces are a well-studied feature in swarm robotics… but more complex functions should be equally applicable.”. Citations 10.1109/TAC.2019.2906467 and 10.1080/00207179.2017.1317832 have been referenced here.

3) Adding the sign function as in (13) can cause chattering. More justifications/details are required.

• We recognise that the use of the sign function could lead to cases with chattering. To highlight this, we have added the citation [Utkin, 2011].
• Nevertheless, the output of the sign function depends on the position of the robot but only appears in the velocity update equation (i.e. it has an accelerating effect). Therefore, these problematic cases can occur only at specific states (e.g. when the robot is close to its personal best point (y), the personal best point is close to the global best point (y_g≈y), and the velocity is close to 0) which are rare and typically happen after convergence has been achieved.
• Order-1 and order-2 stability means that the particle does not diverge from y or y_g. Even if chattering happens, it will not cause the robot to diverge from these points. Therefore, the stability analysis is still valid, even if there exist some cases that can lead to chattering. See [Cleghorn et al, 2018] for more information.
• A smooth function can be used instead of the sign function (e.g. tanh or some type of logistic function) to avoid chattering. Taking y=tanh(x) as an example: 1) For |x|>>1 it approximates y=sign(x) (our PSO). 2) For |x|<<1 it approximates y=x (original PSO). The stability of the original PSO has already been proven. Therefore, proof of the stability of the sign function implies the stability of the tanh function. All of the results of this paper can be applied to these smooth functions too. In this paper, we only consider the sign function as a first step for simplicity.

Line 135: Added “Note that the sgn function… aforementioned smooth functions as well.” to discuss this final point.

4) More details on the guideline presented in Section 3.5 are required. Convergence properties? Computational complexity? Real-time implementation?

Computational Complexity: PSO was designed with simplicity in mind. Generalised Adapted PSO can become more computationally intensive, the more terms are added to the velocity update equation. Nevertheless, due to the simplicity of the algorithm, it should be computationally very simple compared to other tasks that the robot needs to perform (e.g. distance measurement using lidar, communication, vision).

Note that in contrast to parameter optimisation tasks, computation is distributed in swarm robotics (each robot uses the PSO velocity update equation to update its own velocity). Therefore the computational complexity does not scale on each robot, no matter the number of robots used in the swarm.

Real-Time Implementation: Due to the simplicity of PSO, it is not expected that it will interfere with any other tasks. On the other hand, other complicated tasks may interfere with the operation of PSO itself. Here, the timestep size Δt is very important. Δt should be large enough to accommodate delays caused by communications or other complex, time-consuming tasks. In this way, the operation of PSO is not affected by other tasks. This is discussed in Line 129 “Delays cause by inter-robot…”, and it is outlined in the first point of the guidelines. Additional explanation was also added at the end of the Guidelines section.

Convergence Properties: Similar to traditional PSO, it is possible to control the effect of each term of the Adapted PSO velocity update equation, by properly adjusting the values of c₁, c₂, …, c_n. For example, if c₁ > c­₂, the robot will prioritise convergence to the personal best point rather than the global best point. Therefore, traditional PSO rules apply here, but the limitations introduced in this paper should be also satisfied.

According to the analysis presented in this paper, the values of ω, c₁, c₂, …, c_n are directly connected to the maximum velocity and acceleration of a robot. Therefore, these values can only be increased to control the convergence properties of the swarm, as long as it is permitted by the maximum velocity and acceleration of the robots. Increasing them beyond this point will not result in faster convergence. If faster convergence is required, then robots with larger maximum velocity and acceleration need to be used.

Line 194: Added “Traditional guidelines for PSO… should be followed.” to clarify how convergence properties can be controlled using traditional PSO rules.

Line 208: Added “Generalised Adapted PSO can become… ensure optimal control of the swarm.” to discuss computational complexity and real-time implementation.

Round 2

Reviewer 2 Report

No more comments.