A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition

Round 1

Reviewer 1 Report

This manuscript studies the robotic path planning in a dynamic environment where an algorithm is designed based on Proper Generalized Decomposition (PGD). The investigated topic is interesting and the paper has some contributions. The paper might be improved from the following parts.

1. The contributions of the paper are better to be clearly claimed in Introduction.

2. The motion dynamics of the robot are preferred to be shown or explained before introducing the path planning algorithm.

3. It would be good if the goal of the designed path planning algorithm is clarified. The classic path planning algorithms are designed to try to achieve the optimal feasible path for a robot to travel from a start location to a goal location while the path planning algorithm designed here seem to generate a set of potential feasible paths for the robot in the presence of a dynamic obstacle.

4. The computational running complexity/time of the designed path planning algorithm is suggested to be analyzed/given. Second, it is not clear the moving speed of the dynamic obstacle shown in Section 4 compared with the computational time of the path planning algorithm for calculating the feasible paths.

5. Does the designed algorithm handle multiple dynamic obstacles? How about the case when the obstacle is not convex?

6. The literature review on path planning in Introduction is not complete. First, more recent relevant reference papers on path planning are needed for a better introduction. For example, the grid-based path planning strategy is used for a robot to optimally move in a drift field with obstacles in “Distributed multi-vehicle task assignment in a time-invariant drift field with obstacles” (2019). Furthermore, apart from the path planning approaches distinguished in Introduction, the optimal control theory based algorithm is also widely used for path planning, see some references related to multi-population genetic algorithms or clustering-based algorithms to solve vehicle task assignment. These papers are closely related to the path planning problem studied in the manuscript.

7. Some typos need to be revised: 1) In line 9 of Page 1，“are” seems to be “is”; 2) In the first line of Page 4, “take” seems to be “contains”; 3) In line 83 of Page 4, “contain” seems to be “contains”; 4) In line 86 of Page 4, (??) seems missing something; 5) In line 96 of Page 5, “satisfy” should be “satisfy”, and so on. Please check the remaining parts.

Author Response

Response to Reviewers Comments

A path planning algorithm for a dynamic environment based on Proper Generalized Decomposition

Manuscript ID: mathematics-1003690

Thank you for taking the time to review our paper. We hope that this new version of the manuscript “A path planning algorithm for a dynamic environment based on Proper Generalized Decomposition.PDF” meets the necessary standards for publication in the Mathematics journal.

What follows is a record of changes made to the initial submission of the paper attending to the comments of the reviewers and the editor.

Changes to the paper from the initial submission are highlighted in red in the new PDF File.   Below, the suggestions of the reviewers are considered and answered in turn and a detailed account of the changes made is included in every answer. Every change can be identified by its location in the revised manuscript submitted for review.

Finally, other minor changes have been carried out in order to maintain the logic of the paper wording and to correct some typographic errors identified during the revision of the paper.

Response to Reviewer 1

Comments and Suggestions for Authors

This manuscript studies the robotic path planning in a dynamic environment where an algorithm is designed based on Proper Generalized Decomposition (PGD). The investigated topic is interesting and the paper has some contributions. The paper might be improved from the following parts.

COMMENT R1.1: The contributions of the paper are better to be clearly claimed in introduction.

ANSWER R1.1: The contributions of the paper have been clarified in the introduction. The following paragraph has been rewritten (pages 1-2, lines 74-80):

“Our previous work developed a PGD-based computational Vademecum (PGD-Vademecum) to solve the Laplace equation, allowing the use of the potential flow theory in RT applications when the robot is guided in a static environment [27]. In this paper, the formulation of the PGD-Vademecum for dynamic environments with dynamic obstacles is derived, the Progressive PGD-vademecum, where the obstacles are considered in the representation as extra parameters. This is modeled as a matrix modifying the properties of the initial domain and, in the context of the potential flow theory, the porosity of the medium.”

COMMENT R1.2: The motion dynamics of the robot are preferred to be shown or explained before introducing the path planning algorithm.

COMMENT R1.3: It would be good if the goal of the designed path planning algorithm is clarified. The classic path planning algorithms are designed to try to achieve the optimal feasible path for a robot to travel from a start location to a goal location while the path planning algorithm designed here seem to generate a set of potential feasible paths for the robot in the presence of a dynamic obstacle.

ANSWER R1.2 and R1.3: The motion dynamics along with the goal of the path planning algorithm have been explained in the numerical example section, renamed as “Navigation example”. In this sense, the following paragraphs have been introduced on pages 7-8, lines 131-142:

“Harmonic functions describe flow dynamics by means of the Laplace equation, where the potential field is free of local minima and produces a set of streamlines [26], [16], [15], [31], [32]). These streamlines are time-independent and describe the movement of a massless fluid element traveling from a start to a target position, following the velocity field derived from the potential field gradient as in (11).

The streamlines generated by the velocity field can be computed using any interpolation technique (linear, cubic, spline, etc).  Through the streamlines reconstruction, it is possible to calculate the optimal path to be followed by a robot using any optimization technique that selects the optimal streamline meeting a specific criterion (shortest path, smooth path, etc). For instance, in order to guarantee a continuous path planning, the robot can select the streamline aligned with its orientation. Therefore, the final robot orientation will be the one associated to the selected streamline. In addition, the robot orientation can be modified and a different streamline cam be selected.  ”

And also the following sentence in lines 148-149:

“In Figure 1 the black holes represent sources introduced using a parametrized matrix A (in particular the parameters are given by the position of the hole’s center and its diameter) and the streamlines resulting from a linear interpolation of the PGD reconstruction are depicted as blue lines.”

COMMENT R1.4: The computational running complexity/time of the designed path planning algorithm is suggested to be analyzed/given. Second, it is not clear the moving speed of the dynamic obstacle shown in Section 4 compared with the computational time of the path planning algorithm for calculating the feasible paths.

ANSWER R1.4: Thanks for your comment. We have introduce specific details regarding the computational cost of the construction of the variational vademecum, which must be computed in advance (off-line) and that contains all the possible solutions for all the possible configurations of the robot and the obstacles. We have also remarked that the on-line stage does not have computational load, as only a set of sums and multiplications are needed for the PGD reconstruction.

The following sentences have been included in Section 4, page 9, lines 164-179:

“Due to the use of harmonic functions, it is assured that the robot always finds the target configuration, thus the solution is deterministic. Specific details related to the off-line stage about the convergence, complexity, computational time, etc., can be found in the previous authors’ work [1], where numerical examples are provided that describe the relationship between the PGD (or Greedy Rank-One Update algorithm) and the Finite-Element Method used for solving High-Dimensional PDEs based on the tensor product of one-dimensional base.

 The computational cost of the on-line stage is negligible. The recalculation of a new trajectory after the obstacle movement only needs two steps:  

1. Evaluate the Abacus at the point i defined by the current configuration given by the parameters P_i. This evaluation will give the solution of the Laplace’s equation for any position X of the domain and for the current set of parameters: u(X; P_i).
2. Evaluate the gradient of the solution (X; P_i) in order to define the streamline.

The evaluation of the solution (PGD reconstruction) is carried out at every time cycle within a region of interest (ROI). A ROI is a portion of the path composed by the surrounding nodes of the robot position that provides enough information to compute the velocity and the robot orientation in the next time step. Once the robot moves, the new robot position is obtained with the current robot and obstacles position by means of the reconstruction of the potential field in the ROI.”

COMMENT R1.5: Does the designed algorithm handle multiple dynamic obstacles? How about the case when the obstacle is not convex?

ANSWER R1.5: Many thanks for this question since it would help us to explain the potential of the proposed technique. Thanks to the parametric definition of obstacles, one can define any desired parametric shape of an obstacle (including convex ones) since it is only a matter of parameters. The only difference will be the offline computational cost due to the increase of the number of dimensions. A paragraph making emphasis in this aspect has been added to section 4, page 9, lines 180-192:

“The proposed algorithm can deal with several obstacles since, as mentioned before, each obstacle is defined by a set of parameters. So the addition of more obstacles can be performed by adding more parameters. Note that the easiest case is an obstacle with circular shape. However more complex, even parametric geometries of the holes can be considered just by adding more parameters. This would increase the dimensionality of the problem, i.e. each parameter represents an additional dimension, but the proposed solver can easily deal with high dimensional problems. Of course, the increase of the number of dimensions will increase the computational cost for calculating the Abacus, but the Abacus is calculated offline thus not affecting the online path planning. In any case, the computational cost of the PGD increases linearly with the number of parameters [13].

The proposed method shares the benefits of the harmonic functions for this type of problems, thus finding trajectories even with convex obstacles. Remember that the proposed methodology solves the Laplace problem for all possible obstacles configurations at once, so keeping the same properties of the initial problem.”

COMMENT R1.6: The literature review on path planning in Introduction is not complete. First, more recent relevant reference papers on path planning are needed for a better introduction. For example, the grid-based path planning strategy is used for a robot to optimally move in a drift field with obstacles in “Distributed multi-vehicle task assignment in a time-invariant drift field with obstacles” (2019). Furthermore, apart from the path planning approaches distinguished in Introduction, the optimal control theory based algorithm is also widely used for path planning, see some references related to multi-population genetic algorithms or clustering-based algorithms to solve vehicle task assignment. These papers are closely related to the path planning problem studied in the manuscript.

ANSWER R1.6: The literature review has been updated as suggested by the reviewer. In particular, the following paragraphs and references have been included in the introduction:

• Page 2, lines 48-49:

“These functions cannot be computed in closed form. Therefore, solutions can only be obtained using discrete approximations and the computational burden of these methods is really high.”

• Page 2, lines 54-62:

“In spite of this, some techniques have sped up this computation ([31], [32]) but the computational burden is still high for RT path planning, with 646s for an environment with 512x512 nodes using the EGSOR algorithm in [32].

Therefore, during the last years researchers have almost discarded these type of functions for RT path planning and have focused in similar approaches. For instance, those developed for underwater robots that account for ocean currents and obstacles by means of the definition of a drift fields and the use of grid-based planning strategies ([33], [3]). Also, the optimal control theory is also very popular for path planning in this context, particularly multi-population genetic algorithms or clustering-based algorithms to solve vehicle task assignments [2].”

• References:

2. Bai, X., Yan, W., Ge, S.S., et al. (2018) An integrated multi-population genetic algorithm for multi-vehicle task assignment in a drift field, Inf. Sci., 2018, 453, pp. 227–238.

3. Bai X, YanW, Cao M and Xue D (2019) Distributed multi-vehicle task assignment in a time-invariant drift field with obstacles IET Control Theory Appl., Vol. 13 Is 17, pp. 2886-2893.

27. Montés N, Chinesta F, Falcó A, Mora MC, Hilario L, Nadal E, Duval JL (2019) A PGD- based Method for Robot Global Path Planning: A Primer, in Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pp. 31-39.

31. Saudi A and Sulaiman J (2012) Path Planing for mobile robots using 4EGSOR via Nine-Point Laplacian (4EGSOR9L) Iterative method, International Journal of Computer Applications 53(16): 38-42.

32. Saudi A, Sulaiman J and Ahmad Hijazi MH (2014) Robot Path Planing with EGSOR Iterative Method using Laplacian Behaviour-Based Control (LBBC), 5th International Conference on Intelligent Systems, Modelling and Simulation, pp. 87-91.

33. Yan W, Bai X, Peng X, et al. (2014) The routing problem of autonomous underwater vehicles in ocean currents. MTS/IEEE Conf. OCEANS’14, Taipei, pp. 1–6.

COMMENT R1.7: Some typos need to be revised: 1) In line 9 of Page 1，“are” seems to be “is”; 2) In the first line of Page 4, “take” seems to be “contains”; 3) In line 83 of Page 4, “contain” seems to be “contains”; 4) In line 86 of Page 4, (??) seems missing something; 5) In line 96 of Page 5, “satisfy” should be “satisfy”, and so on. Please check the remaining parts.

ANSWER R1.7: The paper has been carefully revised in order to correct all the typos and errors present in the text. In addition, the English writing has been enhanced. Consequently, some sentences have been slightly modified in order to maintain the coherence of the text.

Author Response File: Author Response.pdf

Reviewer 2 Report

This is an interesting paper dealing with the path planning problem of a point-sized mobile robot in an environemnt containig dynamic obstacles. 

Considering some parts of the paper are modified, and additional details provided, the quality of the paper will be high, and the paper will be  interesting both for mathematicians, and for roboticists.

First of all - "taking envoronment into account" as you state in the line 1, is not one of the most important tools, it is a necessity. Withouth the environment, there is nothing to plan. So please rephrase that.

Line 9 - can be blocked - usually the phrase used here is can be trapped in local optima or similar.

Line 45 - uncertainties of positions are mentioned here. Please elaborate on which uncertainties you think. This certainly would play a role in a paper dealing with real life application for robot path planner, but here, a mathematical procedure, or a vademecum, is elaborated on a highly theoretical, or abstract level. So I am not sure where these uncertainties come from.

Line 73 - to illustrate the definition together a real life application - please reword (maybe with/along a real life application)

Generally,  I think that the quality of the paper would be significantly improved (especially from a roboticists perspective) if the authors would provide aditional details on:

• time required to find a potential flow of trajectories - it is an essential parameter, and at least some details regarding it should be provided
• then once you have all the trajectories, please explain how do you choose a prticular one the robot should execute
• I wolud like to have more information on the completness of the soultion found by the PGD in terms of number of obstacles, their behaviour etc. The exapmle privided in the Figure 1 which is the only figure in the paper) is a very basic one.
• is it possible to provide a statistical analysis of your approach which would provide comparison of a sucess rate of PGD in relation to complexity of the environment (number of obstacles, their positions and velocities), time for finding acceptable solutions etc?

Author Response

Response to Reviewer 2

Comments and Suggestions for Authors

This is an interesting paper dealing with the path planning problem of a point-sized mobile robot in an environment containing dynamic obstacles. 

Considering some parts of the paper are modified, and additional details provided, the quality of the paper will be high, and the paper will be interesting both for mathematicians, and for roboticists.

COMMENT R2.1: First of all - "taking environment into account" as you state in the line 1, is not one of the most important tools, it is a necessity. Without the environment, there is nothing to plan. So please rephrase that.

ANSWER R2.1: Thanks for the observation. The following modifications have been included in the abstract, lines 1 to 3:

“A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance.”

COMMENT R2.2: Line 9 - can be blocked - usually the phrase used here is can be trapped in local optima or similar.

ANSWER R2.2: The word “blocked” has been modified by “trapped” in line 9 according to this comment.

COMMENT R2.3: Line 45 - uncertainties of positions are mentioned here. Please elaborate on which uncertainties you think. This certainly would play a role in a paper dealing with real life application for robot path planner, but here, a mathematical procedure, or a Vademecum, is elaborated on a highly theoretical, or abstract level. So I am not sure where these uncertainties come from.

ANSWER R2.3: The reviewer is completely right regarding the uncertainty concept, which is not considered in the mathematical procedure presented in this paper. As a consequence, this sentence has been removed from the paper.

COMMENT R2.4: Line 73 - to illustrate the definition together a real life application - please reword (maybe with/along a real life application)

ANSWER R2.4: The sentence has been rewritten, line 87:

“In this section we introduce the notion of variational vademecum using the potential flow theory in robotics to illustrate the definition along with a real-life application}.”

COMMENT R2.5: Generally, I think that the quality of the paper would be significantly improved (especially from a roboticist’s perspective) if the authors would provide additional details on:

a. time required to find a potential flow of trajectories - it is an essential parameter, and at least some details regarding it should be provided

ANSWER R2.5a: In Section 4, we have introduce specific details regarding the computational cost of the construction of the variational vademecum, which must be computed in advance (off-line) and that contains all the possible solutions for all the possible configurations of the robot and the obstacles. We have also remarked that the on-line stage does not have computational load, as only a set of sums and multiplications are needed for the PGD reconstruction. The following sentences have been included in Section 4, page 9, lines 164-179:

“Due to the use of harmonic functions, it is assured that the robot always finds the target configuration, thus the solution is deterministic. Specific details related to the off-line stage about the convergence, complexity, computational time, etc., can be found in the previous authors’ work [1], where numerical examples are provided that describe the relationship between the PGD (or Greedy Rank-One Update algorithm) and the Finite-Element Method used for solving High-Dimensional PDEs based on the tensor product of one-dimensional base.

 The computational cost of the on-line stage is negligible. The recalculation of a new trajectory after the obstacle movement only needs two steps:

1. Evaluate the Abacus at the point i defined by the current configuration given by the parameters P_i. This evaluation will give the solution of the Laplace’s equation for any position X of the domain and for the current set of parameters: u(X; P_i).
2. Evaluate the gradient of the solution (X; P_i) in order to define the streamline.

The evaluation of the solution (PGD reconstruction) is carried out at every time cycle within a region of interest (ROI). A ROI is a portion of the path composed by the surrounding nodes of the robot position that provides enough information to compute the velocity and the robot orientation in the next time step. Once the robot moves, the new robot position is obtained with the current robot and obstacles position by means of the reconstruction of the potential field in the ROI.”

b. then once you have all the trajectories, please explain how do you choose a particular one the robot should execute

ANSWER R2.5b: A paragraph has been introduced regarding the trajectory selection in the numerical example section, renamed as “Navigation example”. Pages 7-8, lines 137-142:

“Through the streamlines reconstruction, it is possible to calculate the optimal path to be followed by a robot using any optimization technique that selects the optimal streamline meeting a specific criterion (shortest path, smooth path, etc). For instance, in order to guarantee a continuous path planning, the robot can select the streamline aligned with its orientation. Therefore, the final robot orientation will be the one associated to the selected streamline. In addition, the robot orientation can be modified and a different streamline cam be selected.

c. I would like to have more information on the completeness of the solution found by the PGD in terms of number of obstacles, their behaviour etc. The example provided in the Figure 1 which is the only figure in the paper is a very basic one.

ANSWER R2.5c: Regarding the PGD, each obstacle is understood as a set of parameters, thus the increase in the number of obstacles will also increase the number of parameters and additional computational cost in the offline calculation, but not affecting to the online vehicle path planning.

Two paragraphs making emphasis in this aspect has been added to section 4.

The first one on page 8, lines 166-173:

“Because of the use of harmonic functions, it is assured that the vehicle always finds the target point, thus the solution is deterministic. The computational cost of recalculating the new trajectory after the obstacle movement is small since it only needs two steps:

1. Evaluate the Abacus at the point i defined by the current configuration given by the parameters Pi. This evaluation will give the solution of the Laplace’s equation for any position X of the domain and for the current set of parameters: u(X; Pi).
2. Evaluate the gradient of the solution (X; Pi) in order to define the streamline.”

The first second one on page 9, and lines 180-192:

 “The proposed algorithm can deal with several obstacles since, as mentioned before, each obstacle is defined by a set of parameters. So the addition of more obstacles can be performed by adding more parameters. Note that the easiest case is an obstacle with circular shape. However more complex, even parametric geometries of the holes can be considered just by adding more parameters. This would increase the dimensionality of the problem, i.e. each parameter represents an additional dimension, but the proposed solver can easily deal with high dimensional problems. Of course, the increase of the number of dimensions will increase the computational cost for calculating the Abacus, but the Abacus is calculated offline thus not affecting the online path planning. In any case, the computational cost of the PGD increases linearly with the number of parameters [13].”

d. is it possible to provide a statistical analysis of your approach which would provide comparison of a success rate of PGD in relation to complexity of the environment (number of obstacles, their positions and velocities), time for finding acceptable solutions etc?

ANSWER R2.5d: The proposed algorithm solves the Laplace’s problem in the whole domain, thus the solution is deterministic and it always provides a deterministic trajectory given a configuration of obstacles. The PGD computes the solution in the whole domain for every position of obstacles at once, thus inheriting the same properties of the former Laplace’s problem

A paragraph clarifying this aspect has been added at the end of section 4, page 9, lines 189-192.

“The proposed method shares the benefits of the harmonic functions for this type of problems, thus finding trajectories even with convex obstacles. Remember that the proposed methodology solves the Laplace problem for all possible obstacles configurations at once, so keeping the same properties of the initial problem.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

 Recommendation: Accept