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The Bounded Acceleration Shortest Path problem: complexity and solution algorithms

Stefano Ardizzoni, Luca Consolini, Mattia Laurini, Marco Locatelli

Abstract

The purpose of this work is to introduce and characterize the Bounded Acceleration Shortest Path (BASP) problem, a generalization of the Shortest Path (SP) problem. This problem is associated to a graph: the nodes represent positions of a mobile vehicle and the arcs are associated to pre-assigned geometric paths that connect these positions. BASP consists in finding the minimum-time path between two nodes. Differently from SP, we require that the vehicle satisfy bounds on maximum and minimum acceleration and speed, that depend on the vehicle position on the currently traveled arc. We prove that BASP is NP-hard and define solution algorithm that achieves polynomial time-complexity under some additional hypotheses on problem data.

The Bounded Acceleration Shortest Path problem: complexity and solution algorithms

Abstract

The purpose of this work is to introduce and characterize the Bounded Acceleration Shortest Path (BASP) problem, a generalization of the Shortest Path (SP) problem. This problem is associated to a graph: the nodes represent positions of a mobile vehicle and the arcs are associated to pre-assigned geometric paths that connect these positions. BASP consists in finding the minimum-time path between two nodes. Differently from SP, we require that the vehicle satisfy bounds on maximum and minimum acceleration and speed, that depend on the vehicle position on the currently traveled arc. We prove that BASP is NP-hard and define solution algorithm that achieves polynomial time-complexity under some additional hypotheses on problem data.

Paper Structure

This paper contains 23 sections, 14 theorems, 52 equations, 27 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem motivation
  3. Related works
  4. Notation
  5. Problem formulation
  6. Speed planning along an assigned path
  7. Solution of Problem \ref{['eqn_problem_pr_wg']}
  8. Bounded Acceleration Shortest Path Problem
  9. Complexity
  10. The $k$-BASP problem
  11. Adaptive A$^*$ algorithm for $k$-BASP
  12. Upper bounds on $T_{\mathbb{B}}(p)$
  13. Adaptive search for $k$
  14. Numerical experiments
  15. Nodes associated to different orientations
  16. ...and 8 more sections

Key Result

Proposition 3.3

The following properties hold:

Figures (27)

  • Figure 1: Comparison of BASP and SP solutions.
  • Figure 2: Optimal speed profile on $ABCD$.
  • Figure 3: Optimal speed profile on $AD$.
  • Figure 4: An example scenario.
  • Figure 5: The black line represents the path, while the red circle and the black cross represent the starting point and the end point, respectively.
  • ...and 22 more figures

Theorems & Definitions (28)

  • Definition 3.1
  • Proposition 3.3
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • Proposition 5.3
  • ...and 18 more