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Finding globally shortest paths through a sequence of adjacent triangles by the method of orienting curves

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Abstract

In this paper, an exact algorithm based on the method of orienting curves is developed for solving the convex non-differentiable optimization problem on the closed unit cube in a finite dimensional space: finding the shortest path joining two points going through a sequence of adjacent triangles in 3D. As a result, the global solution of the problem is determined successively by some orienting curves and final curve, which can be exactly constructed with ruler and compass. A detailed numerical example is presented.

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Authors and Affiliations

  1. Institute of Mathematical and Computational Sciences (IMACS) and Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam

    Phan Thanh An

  2. Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam

    Phan Thanh An

  3. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road, Cau Giay District, Hanoi, Vietnam

    Hoang Xuan Phu

Authors
  1. Phan Thanh An
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  2. Hoang Xuan Phu
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Correspondence to Phan Thanh An.

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The research of the first author was funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under Grant Number T2022 - 20 -01. The second author was supported by the Vietnam Academy of Science and Technology under Grant Number NVCC01.02/22-23. They thank the reviewers for their value comments.

Appendix

Appendix

Connection of the results to the field of global optimization

In [22], to get a shortest path joining two points on a smooth surface (or “the shortest path problem on smooth surfaces", for short), Pham-Trong at al. trangulate the surface then solve the auxiliary problems of finding the globally shortest path joining two these points going through a sequence of selected adjacent triangles, in which the next sequence of selected adjacent triangles is chosen by a so-called pivot vertex (see Algorithm 4.3 [22]). Finally, the limit of solutions of such auxiliary problems is a locally shortest path joining two points on the smooth surface. Thus, although a local solution of an auxiliary problem is a global one (Corollary 2), for the shortest path problem on smooth surfaces, there can be many local optimal solutions which are not global.

Fig. 6
figure 6

Four sequences of adjacent triangles joining \(p_0\) and \(p_5\): (0, 1, 2, 3, 4, 5), \((0, 0', 3, 4, 5), (0, 1, 2, 5', 5)\), and \((0, 0', 3, 2, 5', 5)\)

Full size image

To avoid this issue, we determine the number of all such auxiliary problems and then the globally shortest path on a triangulated surface is obtained from their minimum. For example, to get a globally shortest path joining two points \(p_0\) and \(p_5\) on the triangulated surface consisting of triangles 0, 0’, 1, 2, 3, 4, 5 and 5’ (see Fig. 6), we solve 4 following auxiliary problems: finding the globally shortest path joining two points \(p_0\) and \(p_5\) going through the sequence of triangles \((0, 1, 2, 3, 4, 5), ((0, 0', 3, 4, 5), (0, 1, 2, 5', 5)\), and \((0, 0', 3, 2, 5', 5)\), respectively).

Recall that the number of all such auxiliary problems can be reduced by pivot vertex technique mentioned above.

A real life application of the problem of finding the globally shortest path joining two points going through a sequence of selected adjacent triangles

The problem of finding the globally shortest path joining two points going through a sequence of selected adjacent triangles has been applied successfully in the following autonomous robot problem: A robot in with a limited vision range to find a path to a goal in the unknown environment of polygonal obstacles. To solve this problem, suitable sequences of adjacent triangles of the explored map are constructed then the robot finds the shortest paths between two locations on.

We experienced our approximate algorithms [2] and [4] on the autonomous robot TurtleBot3 Burger [32], which has the exact vision range of 0.8 m, in real environment [33]. During operations of the robot, suitable sequences of adjacent triangles of the explored map are constructed (this is displayed on the window on the upper right corner in the video [33]). The full video [33] that captured this event was recorded at our institute in April 2022, in which the robot finds a path avoiding obstacles to the goal that is a red wine box located in another room that is 14m far from the robot.

Our exact Algorithms 1–5 can be implemented in Python and integrated in autonomous robots moving on terrains. This will be the subject of another paper.

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An, P.T., Phu, H.X. Finding globally shortest paths through a sequence of adjacent triangles by the method of orienting curves. J Glob Optim 85, 1037–1063 (2023). https://doi.org/10.1007/s10898-022-01244-x

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