Kinematic Modelling of a 3RRR Planar Parallel Robot Using Genetic Algorithms and Neural Networks

Round 1

Reviewer 1 Report

Dear Authors,

the paper is interesting, however some suggestions could improve this paper.

1. Figure 1 and Table 1 should be in the next chapter 2

2. Lines 187-192 - What is the risk of singularities and what should be done to avoid them?

3. Lines 315-317 - Why was this activation function and network training algorithm chosen?

4. Please provide exuation to Mean Square Error.

Author Response

Thank you very much for your revision, which has help us to improve our paper. Your suggestions have been implemented and now appear highlighted in the anuscript. Explanations of hoy it has been done can be find bellow:

Comment 1: Figure 1 and Table 1 should be in the next chapter 2

Response: They have been placed in chapter 2, after line 100.

Comment 2: Lines 187-192 - What is the risk of singularities and what should be done to avoid them?

Response: The paragraph explaining the risk of singularities has been extended, adding, which  a more detailed explanation of why the Numerical Method may exhibit convergence errors near direct singularities. In addition, two possible solutions to this problem have been referenced.

Comment 3: Lines 315-317 - Why was this activation function and network training algorithm chosen?

Response: A paragraph justifying the elections has been included.

Comment 4: Please provide exuation to Mean Square Error.

Response: The reasons of why MSE has been chosen as performance criterion have been presented in the new version of the manuscript.

Reviewer 2 Report

The aim of the work done by the authors is to solve the the direct kinematic problem of parallel manipulators from AI tools. The paper presents two main contributions. On the one hand, a GA has been developed by employing the IK Model as the fitness function. As the authors show, this method is indifferent to the presence of singular points. On a second stage, a feedforward Neural Netwrok has been trained using data generated from the IK model.

The work is well presented and organized, with a clear formulation. However, there are some small errors that the authors should correct.

-     Typos in line 45: “…the have become research tools, awith…”. Typo in line 137, coma instead of dot point. In line 187, it should be “offers”.

-  When explaining Fig. 1, the passive angles of the joints are mentioned, hovewer, in that figure only the first passive angle psi_1 is depicted. Although is quite obvious which are the remaining passive angles psi_2 and psi_3, I recommend to depict them also in the figure so that all variables and parameters appear.

-     Please note that according to Eq. (5), alpha_1 is the sum of phi and gamma_1, but this does not fit with the angle depicted in Figure 2.

-       In line 359, the second referred Eq. should be (23).

The developed methods presented by the authors are quite interesting, and it is worth mentioning that they have make an exhaustive comparison of the proposed methodologies with respect to existing numerical techniques. The validity of the new techniques developed by the authors is demonstrated, and although a single planar parallel manipulator is used as an example of application, this manipulator has an associated kinematic analysis that, indeed, is not simple, so it can be assumed that these methodologies are also valid for more complex manipulators. The authors indicate that this will be precisely the line of future work.

The English writing is adequate.

Author Response

Thank you very much for the detailed reading of the article and the help in locating errors in the article. We also appreciate the words about the formulation and organisation of the paper, which has undergone several restructurings in the writing process.

Comment 1: Typos in line 45: “…the have become research tools, awith…”. Typo in line 137, coma instead of dot point. In line 187, it should be “offers”.

Response: A typos revision has been carried.

Comment 2: When explaining Fig. 1, the passive angles of the joints are mentioned, hovewer, in that figure only the first passive angle psi_1 is depicted. Although is quite obvious which are the remaining passive angles psi_2 and psi_3, I recommend to depict them also in the figure so that all variables and parameters appear.

Response: Angles psi_2 and psi_3 have been included. In the same line, in Figure 2, angles gamma_2 and gamma_3, which will be mentioned in Equation (6), are now depicted.

Comment 3: Please note that according to Eq. (5), alpha_1 is the sum of phi and gamma_1, but this does not fit with the angle depicted in Figure 2.

Response: A typo had remained as a result of the different formulations of this first part of kinematics that we have made. Equation (6) was also incorrect and has been corrected.

Comment 4: In line 359, the second referred Eq. should be (23).

Response: It has been corrected. 

Reviewer 3 Report

I am impressed by the results of these two AI-based methodologies for FK modeling of parallel manipulators. They have the potential to overcome the limitations of existing techniques and enable new applications. I am particularly interested in the potential of NN to be trained experimentally, without even needing to use the IK model. This could make it much easier to use these techniques for robots with complex kinematics. The vulnerability of the method represents that requires a bigger dataset for the training process.

The graphic processing of the article is at a good level.

I recommend publishing the article.

Author Response

We thank you for reviewing the article and hope you enjoyed the results. We are currently working on using NN to kinematically model more types of robots.

As you comment, we have tried to take care of the figures in the article, as we felt they were essential for a good understanding of the equations.