Spatial Path Smoothing for Car-like Robots Using Corridor-Based Quadratic Optimization

Round 1

Reviewer 1 Report

The authors consider path smoothing using a quadratic optimization procedure. The paper has a clear problem statement, the proposed methods look scientifically sound, and the extensive experimental section verifies the theoretical background. I find the paper easy to follow, and I have only a few comments to be addressed.

Major comments:

1.       Page 3, algorithm 1:

1.1.      What are D₁ and D₂? Did the authors mean d^L and d^R?

1.2.      Do the parameters above represent vector variables and how can we determine them? The authors write V^ref is a rough initial path, but I found no information about d^L and d^R.

1.3.      Parameters D₁ and D₂ (or d^L and d^R) are not mentioned explicitly in the algorithm.

1.4.      I recommend the authors provide more details about the path interpolation step.

1.5.      Line 2. The authors write, “i = 2, …,”—there is no final value of index i.

1.6.      L. 4. Why do we compute k_i’? The paper says nothing about this parameter.

1.7.      L. 5. What is K? Is it an array? If this array is empty before the for-loop, the authors should mention this explicitly.

1.8.      L. 8. Is it appropriate to use same notation k_i’ in this line? (Parameters K and K_smooth contain different variables.)

1.9.      L. 8. The authors use n to notate the final value of index i; the authors also use this notation in the text to indicate the dimension of V^ref. Array K_smooth, however, can have another dimension because of the interpolation step (l. 1). In addition, the authors should give an explicit definition of each variable in the algorithm: parameter n was not mentioned before l. 8.

1.10.   L. 11–12. Arrays V^L and V^R are not initialized: are they empty before the for-loop? In view of the previous comment, these arrays can have dimensions other than n, but the text (p. 3, l. 118+2) shows each array has n elements.

2.       In view of comment no. 1.9, the authors should check the ranges of index i in all the equations and the text.

3.       P. 9, linearization procedure:

3.1.      The authors perform linearization to deal with the nonlinear constraint (Eq. (27)). It is unclear, however, why this constraint is non-convex (at least for me). The authors should provide more details, because nonlinear constraints can be convex too. In this case, of course, we cannot consider the optimization problem as QP, but it can still be convex and solvable by other solvers.

3.2.      Eq. (30). This equation includes several unexplained parameters, e.g., F, F’, and P_i⁰. The bar and hat notations are unexplained either.

3.3.      Fig. 3. Shouldn’t it be μ instead of P_j? (The text says nothing about P_j.)

3.4.      Eq. (34). It is unclear (at least for me) how the authors got this equation. In addition, what are parameters H and G_i and shouldn’t it be H_i instead of H? The sentence below the equation does not provide explanations and seems incomplete.

3.5.      Eq. (35). What is A and shouldn’t it be A_i instead of A?

3.6.      According to Fig. 3 and Eq. (22), Δs is equal to the distance between points P_i and P_i+1. The coordinates of these points depend on optimization variables ρ_i (Eq. (4)). The authors, however, do not linearize Δs in Eq. (35). Do the authors consider this parameter constant? If so, why? I recommend the authors give some explanations.

4.       P. 10, algorithm 2:

4.1.      Shouldn’t the input include V^inter instead of V^ref? The authors should check other lines of the algorithm too.

4.2.      Why are there no κ_max and N_max in the list of input variables?

4.3.      L. 3. The authors perform a warm start, but the paper says nothing about it. In particular, why do we need to perform this warm start? If the solver can deal with the infeasible initial point, there is no need for this start because the convex optimization problem should be solved from any initial point.

4.4.      L. 6. There are no explanations about this step (GetPathMaxCurvature).

4.5.      L. 10. Shouldn’t “Flag” and “False” be typed in italic?

4.6.      L. 11–19. It is unclear why the authors implement this while-loop: shouldn’t the QP problem give the unique solution? I mean, why can’t we get the solution if we perform L. 12–13 once? If the optimization problem is convex, the optimization step in L. 12 should always give the same solution.

5.       I recommend the authors check the figure layout because some figures appear before they are cited in the text. For example, the authors refer to Fig. 13 on p. 14 (l. 249), but there were no references to Figs. 11 and 12.

6.       There are a few misspellings, e.g., “an non-linear,” “to constraint,” “a unconventional.”

Minor comments:

7.       P. 2, l. 56. Although the authors spell out the acronym QP in the introduction, I recommend them spell it out here too.

8.       P. 2, l. 69. I recommend the authors spell out the acronym RRT.

9.       P. 3, l. 118+3. Shouldn’t it be P_i^j instead of P_jⁱ in accordance with Eq. (1)? The authors should check notations x_jⁱ and y_jⁱ in this line too.

10.    P. 4, Fig. 1. The authors write in the figure caption, “the black lines represent the road boundaries,” but the black line represents the reference path too.

11.    P. 4, Eq. (1). Shouldn’t it be k_i instead of k in the third line of the equation?

12.    P. 4, l. 120–121; p. 6, the top line. The authors notate superscripts in an italic style, but they notated them in an upright style in the preceding equations.

13.    P. 5, l. 136–137. Shouldn’t there be P_i^L and P_i^R instead of P_1,i and P_2,i?

14.    P. 6, Eqs. (7)–(9). Shouldn’t there be any subscripts or superscripts to indicate that matrices H and B depend on index i too? The authors should also check other cost functions.

15.    P. 6, l. 149+2. The authors mention B-splines, but the preceding text said nothing about these splines.

16.    P. 7, a line below Eq. (13). Vectors ρ should have subscript i in accordance with the subsequent equations.

17.    P. 7, Eqs. (14)–(15). The authors introduce parameter N, but the paper says nothing about it. (See also comment no. 2.)

18.    P. 7, l. 152. It should be H_d,y instead of H_d,x.

19.    P. 8, Eq. (25). The equation shows parameter κ, but there are no explanations in the text.

20.    P. 13, Table 1; p. 14, Table 2. The tables have no units.

21.    P. 14, Fig. 9. I recommend the authors add some explanations about the green and the red paths in the figure.

22.    P. 19, Ref. 31. The authors should check the reference: “Boyd” appears twice.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The manuscript proposes a corridor-based method for spatial path optimization. The spatial corridor construction module is designed to extract an optimization domain for path optimization. The two-step path optimization approach is designed to generate a smooth path in real-time while ensuring the feasibility of path curvature. The experimental results verify the validity of the proposed method. In general, the paper is standard and has certain innovation. However, there some points listed below are not clear. The authors should consider the following comments:

 

1.    There are some grammar errors in the manuscript. On page 9 “where H and Gi are the Combining with Eq.(33) and Eq.(34).” is incomplete. The authors should carefully check the manuscript to ensure and improve its presentation quality.

2.      The formulation in Eq. (36) seems not self-explanatory. We do not know what is H, ρ, and q in your case either. Please explain them with the proposed path problem problem.

3.      We cannot really know the behavior of the obstacle in Fig. 8. Are they static obstacles? Please explain.

4.      The section 5.4 gives the experiment of on road path optimization. Is the safe distance from obstacles taken into account? Please given some discussions.

5.      Figure3 illustrates the procedure of linearization of curvature constraint. The description of figure3 and this procedure is not enough for explaining this procedure.

 

Author Response

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Reviewer 3 Report

This paper proposed an efficient spatial path smoothness approach, which is capable to optimize a rough spatial path to be a high quality one under corridor and curvature constraints. The proposed approach is viewed as an independent path optimization module, achieving spatial path optimization efficiently. The topic is interesting and worth researching. Overall, the paper is technically correct and well written. Some minor suggestions/comments for improvement of the work are listed below.

1.    The readability and presentation of the work should be improved.

2.    The related work should have improvement with benefits of quadratic equations and comparisons with many other standards techniques and what are the role and specifications of quadratic programming in the real field. The introduction section does not clearly describe the motivation of this paper.

3.    This paper formulates its problem as a quadratic programming (QP) problem. Why not formulate it as other nonlinear optimization problems? What are the adventages of a quadratic programming (QP) problem?

4.     “Eq.(27) is a nonlinear inequality, which is viewed as a non-convex constraint.”  The following section has discussed how to linearize Eq.(27). What is the purpose of linearizing it? the If not linearize it, what problems will occur? Please state.

5.    We cannot really know the behavior of the obstacle in Fig. 8. Are they static obstacles? Please explain.

Author Response

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Round 2

Reviewer 1 Report

The authors have addressed all the comments and improved the paper quality significantly. There are just a few minor remarks provided below:

1.       P. 4, Algorithm 1. Is it correct that line 8 of the algorithm repeats line 6?

2.       P. 6, a line below Eq. (7). Subscripts of the matrices differ from the ones used in Eq. (7).

3.       P. 9, a line above Eq. (26). The authors refer to Eq. (5) to compute parameter ||μ||. Is this reference correct? Eq. (5) actually shows a computation of a cost function.

4.       P. 10, Algorithm 2. Lines 3 and 4 of the algorithm use different notations ρ^init and ρ^start, which seem to designate the same variables.

Author Response

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