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Multi-Agent Motion Planning with Bézier Curve Optimization under Kinodynamic Constraints

Jingtian Yan, Jiaoyang Li

TL;DR

This paper presents a three-level MAPF-based planner called PSB, which fully considers the kinodynamic capability of the agents and produces solutions with smooth speed profiles and shows up to 49.79% improvements in solution cost compared to existing methods while achieving significant improvement in scalability.

Abstract

Multi-Agent Motion Planning (MAMP) is a problem that seeks collision-free dynamically-feasible trajectories for multiple moving agents in a known environment while minimizing their travel time. MAMP is closely related to the well-studied Multi-Agent Path-Finding (MAPF) problem. Recently, MAPF methods have achieved great success in finding collision-free paths for a substantial number of agents. However, those methods often overlook the kinodynamic constraints of the agents, assuming instantaneous movement, which limits their practicality and realism. In this paper, we present a three-level MAPF-based planner called PSB to address the challenges posed by MAMP. PSB fully considers the kinodynamic capability of the agents and produces solutions with smooth speed profiles that can be directly executed by the controller. Empirically, we evaluate PSB within the domains of traffic intersection coordination for autonomous vehicles and obstacle-rich grid map navigation for mobile robots. PSB shows up to 49.79% improvements in solution cost compared to existing methods.

Multi-Agent Motion Planning with Bézier Curve Optimization under Kinodynamic Constraints

TL;DR

This paper presents a three-level MAPF-based planner called PSB, which fully considers the kinodynamic capability of the agents and produces solutions with smooth speed profiles and shows up to 49.79% improvements in solution cost compared to existing methods while achieving significant improvement in scalability.

Abstract

Multi-Agent Motion Planning (MAMP) is a problem that seeks collision-free dynamically-feasible trajectories for multiple moving agents in a known environment while minimizing their travel time. MAMP is closely related to the well-studied Multi-Agent Path-Finding (MAPF) problem. Recently, MAPF methods have achieved great success in finding collision-free paths for a substantial number of agents. However, those methods often overlook the kinodynamic constraints of the agents, assuming instantaneous movement, which limits their practicality and realism. In this paper, we present a three-level MAPF-based planner called PSB to address the challenges posed by MAMP. PSB fully considers the kinodynamic capability of the agents and produces solutions with smooth speed profiles that can be directly executed by the controller. Empirically, we evaluate PSB within the domains of traffic intersection coordination for autonomous vehicles and obstacle-rich grid map navigation for mobile robots. PSB shows up to 49.79% improvements in solution cost compared to existing methods.
Paper Structure (31 sections, 4 theorems, 6 equations, 7 figures, 1 algorithm)

This paper contains 31 sections, 4 theorems, 6 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Intersection Model
  5. Grid Model
  6. MAMP on Intersection
  7. System Overview
  8. PBS-based planner
  9. Safe Interval Planner
  10. Bézier-curve-based Planner
  11. Background on Bézier Curves
  12. Bézier-curve-based Planner (BCP)
  13. Find Bézier curve given arrival time
  14. Search for optimal arrival time
  15. Pseudocode for BCP
  16. ...and 16 more sections

Key Result

Lemma 1

$B^{T}(t)$ exists iff a single time range $[T_{min}, T_{max}]$ exists such that $\delta_T^* = 0$ for $T \in [T_{min}, T_{max}]$ and $\delta_T^* > 0$ for $T \notin [T_{min}, T_{max}]$, making $T_{min}$ the optimal travel time $T^*$. Moreover, $\nabla \delta_T^* < 0$ for $T \in [0, T_{min})$, $\nabla

Figures (7)

  • Figure 1: Illustration of the intersection model and the grid model we use. (a) Traffic intersection coordination model. (b) Grid map navigation model. Collision points are marked as black dots. The black cells in (b) are static obstacles.
  • Figure 2: Overview of PSB. The shadowed strips denote time intervals occupied by other agents, the green segments denote safe intervals. The dark green segments represent the intervals in the open list.
  • Figure 3: Illustration of the search for optimal arrival time. The agent moves along the path from $c_0$ to $c_3$ with associated safe intervals represented by green segments. The optimal spatio-temporal profiles found by Eq. \ref{['eq:bezier']}-\ref{['eqn:5']} for the four different arrival time $T_x$ shown on the right are represented by four dashed curves. Since the two gray ones do not intersect with all green segments, they are infeasible, and $\delta^*_{T_{x}}$ indicates how far away they are from feasible profiles. As the figure shows, minimizing $\delta^*_{T_{x}}$ can bring these profiles closer to feasible ones.
  • Figure 4: Delay and average runtime (in log scale) for the intersection model. Error bars show the standard deviations.
  • Figure 5: Success rate, solution cost, and runtime of PSB, PSL, and SIPP-IP across all maps for the grid model. The solution cost and runtime are averaged only over scenarios where the planner successfully generates a solution. Note that PSL uses a relaxed agent model.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1: Completeness and optimality of BCP
  • proof
  • Theorem 2: Completeness and suboptimality of PSL
  • proof