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Rapid Vector-based Any-angle Path Planning with Non-convex Obstacles

Yan Kai Lai

TL;DR

This work advances vector-based any-angle path planning in maps with non-convex obstacles by introducing the best-hull concept and progression-based navigation (source progression and target progression) to enable monotonically increasing cost estimates even when line-of-sight checks are delayed. It presents two novel algorithms, R2 and its successor R2+, which leverage delayed LOS checks, phantom points, and convex-hull inferences to find Euclidean shortest paths efficiently when the solution has few turning points. The methods combine new mechanisms (phantom points, best-hull, angular and occupied-sector rules) with rigorous completeness and optimality proofs, and demonstrate substantial speed-ups over state-of-the-art vector-based planners in sparse, non-convex environments. The work also contributes a versatile multi-dimensional ray tracer for occupancy grids and outlines future extensions, including a three-dimensional angular sector framework, with strong potential to guide robotics global planning and related decision-making systems.

Abstract

Vector-based algorithms are novel algorithms in optimal any-angle path planning that are motivated by bug algorithms, bypassing free space by directly conducting line-of-sight checks between two queried points, and searching along obstacle contours if a check collides with an obstacle. The algorithms outperform conventional free-space planners such as A* especially when the queried points are far apart. The thesis presents novel search methods to speed up vector-based algorithms in non-convex obstacles by delaying line-of-sight checks. The "best hull" is a notable method that allows for monotonically increasing path cost estimates even without verifying line-of-sight, utilizing "phantom points" placed on non-convex corners to mimic future turning points. Building upon the methods, the algorithms R2 and R2+ are formulated, which outperform other vector-based algorithms when the optimal path solution is expected to have few turning points. Other novel methods include a novel and versatile multi-dimensional ray tracer for occupancy grids, and a description of the three-dimensional angular sector for future works.

Rapid Vector-based Any-angle Path Planning with Non-convex Obstacles

TL;DR

This work advances vector-based any-angle path planning in maps with non-convex obstacles by introducing the best-hull concept and progression-based navigation (source progression and target progression) to enable monotonically increasing cost estimates even when line-of-sight checks are delayed. It presents two novel algorithms, R2 and its successor R2+, which leverage delayed LOS checks, phantom points, and convex-hull inferences to find Euclidean shortest paths efficiently when the solution has few turning points. The methods combine new mechanisms (phantom points, best-hull, angular and occupied-sector rules) with rigorous completeness and optimality proofs, and demonstrate substantial speed-ups over state-of-the-art vector-based planners in sparse, non-convex environments. The work also contributes a versatile multi-dimensional ray tracer for occupancy grids and outlines future extensions, including a three-dimensional angular sector framework, with strong potential to guide robotics global planning and related decision-making systems.

Abstract

Vector-based algorithms are novel algorithms in optimal any-angle path planning that are motivated by bug algorithms, bypassing free space by directly conducting line-of-sight checks between two queried points, and searching along obstacle contours if a check collides with an obstacle. The algorithms outperform conventional free-space planners such as A* especially when the queried points are far apart. The thesis presents novel search methods to speed up vector-based algorithms in non-convex obstacles by delaying line-of-sight checks. The "best hull" is a notable method that allows for monotonically increasing path cost estimates even without verifying line-of-sight, utilizing "phantom points" placed on non-convex corners to mimic future turning points. Building upon the methods, the algorithms R2 and R2+ are formulated, which outperform other vector-based algorithms when the optimal path solution is expected to have few turning points. Other novel methods include a novel and versatile multi-dimensional ray tracer for occupancy grids, and a description of the three-dimensional angular sector for future works.
Paper Structure (162 sections, 7 theorems, 90 equations, 59 figures, 13 tables, 9 algorithms)

This paper contains 162 sections, 7 theorems, 90 equations, 59 figures, 13 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Synopsis of the Thesis
  3. Contributions of the Thesis
  4. Applications of the Thesis
  5. Literature Review
  6. Path Planning Concepts
  7. Optimality and Completeness
  8. World Representations
  9. Curse of Dimensionality
  10. Types of Path Planners
  11. Grid-based Planners
  12. Topological Planners
  13. Sampling-based Planners
  14. Artificial Potential Fields
  15. Deep Reinforcement Learning
  16. ...and 147 more sections

Key Result

Theorem 1

In an unbounded map, or bounded map with a convex boundary, the target-pledge method can find a path to the target point if a path exists, provided that a cast occurs from a trace at the first corner that satisfies $\sigma\theta_T \le 0$. All traces and casts have to be simultaneously examined.

Figures (59)

  • Figure 1: Problems with symmetric ray tracing algorithm. (a) a line that passes through a corner may cause an extra cell to be identified (red bordered blue cell). (b) a line that travels along a grid line may miss out cells on one side of the line (red bordered cells) and only return the other (blue cells).
  • Figure 2: The contour assumption reduces ambiguity when a ray collides with an obstacle. A blue circle and arrow represents the left trace's starting position and direction respectively, and an orange circle and arrow represents the right trace. The black arrow is the bisecting vector $\mathbf{v}_\mathrm{crn}$. Under the contour assumption, the obstacle lies an infinitesimal distance away from the grid lines (exaggerated in illustration). A cast that collides with a corner, and that points slightly to the right of $\mathbf{v}_\mathrm{crn}$ (a, d), or to the left (b, e), will be treated to have collided at the right or left edge respectively. The collided corner will be the first corner for a left trace in (a, d), or for a right trace in (b, e). The collided corner will be the first corner for (c, f), and the trace directions depend on the implementation, (R2+ is shown). As there is no corner ambiguity for (g, h, i), the adjacent vertices can lie on any adjacent vertex. By eliminating the ambiguity, the sector rule for R2 and R2+ can infer that a ray has been crossed. A ray is crossed when the trace is located at the first corner found after the ray collides.
  • Figure 3: A greedy vector-based algorithm casts to the destination $\mathbf{x}_T$ as early as possible, and is susceptible to getting trapped in a highly non-convex obstacle such as a 'G'-shaped obstacle. A cast from collides at $\mathbf{x}_{\mathrm{col},1}$, resulting in a left trace (orange) that reaches $\mathbf{x}_1$ and a right trace that goes out of map. A second cast is performed from $\mathbf{x}_1$ to $\mathbf{x}_T$, colliding at $\mathbf{x}_{\mathrm{col},2}$. The left trace from $\mathbf{x}_{\mathrm{col},2}$ (blue) will cast again at $\mathbf{x}_1$, repeating the process, while the right trace goes out of map.
  • Figure 4: The figure illustrates a pruning method after $L$-sided turning points were placed. (a) $L$-sided source turning point at $\mathbf{x}_S$, placed by the same trace that reaches $\mathbf{x}$, is pruned. (b) $L$-sided target turning point at $\mathbf{x}_T$, placed by a prior $L$-side trace, is pruned once the current trace reaches $\mathbf{x}$. (c) The tautness check in Eq. (\ref{['ncv:eq:prune:istautsrc']}) and are only valid when $\mathbf{v}_S$ is not rotated by more than half a round from $\mathbf{v}_{SS}$ after the trace progresses, and if $\mathbf{x}_S$ is pruned immediately like in (a).
  • Figure 5: Cases for Theorem \ref{['ncv:thm:tpldg']}. The obstacle width is exaggerated for illustration. (a) Case 1.1, a zero-width obstacle containing two C2 corners (circle with '-') and no C4 (circle with '+') corners . (b) Non-convex extrusion in Case 1.2. (c,d) Cast collides with same obstacle in Case 1.3. (e,f) Cases 1.1 to 1.3 are repeated when casts reach disjoint obstacles in Case 2. (g) Trace will not cast if it walks the interior boundary of an obstacle enclosing the target point $\mathbf{x}_T$. (h) Trace may not cast if there is no path to $\mathbf{x}_T$.
  • ...and 54 more figures

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 4 more