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Timed-Elastic-Band Based Variable Splitting for Autonomous Trajectory Planning

Hao Zhu, Kefan Jin, Rui Gao, Jialin Wang, C. -J. Richard Shi

TL;DR

The paper tackles the instability and endpoint errors impeding autonomous trajectory planning with traditional Timed-Elastic-Band (TEB) methods. It introduces TEB-VS, a variable-splitting framework that reformulates the global constrained optimization into smaller, tractable subproblems solved via augmented TEB on a G2O graph, with convergence guarantees under mild assumptions. The authors provide a convergence analysis and validate the approach through extensive TurtleBot2 experiments in both simulation and real environments, showing superior speed stability and trajectory fidelity while maintaining similar computation to TEB. These results offer a robust, real-time capable trajectory planner for autonomous systems operating in complex, dynamic settings and pave the way for further enhancements such as fastest-path optimization.

Abstract

Existing trajectory planning methods are struggling to handle the issue of autonomous track swinging during navigation, resulting in significant errors when reaching the destination. In this article, we address autonomous trajectory planning problems, which aims at developing innovative solutions to enhance the adaptability and robustness of unmanned systems in navigating complex and dynamic environments. We first introduce the variable splitting (VS) method as a constrained optimization method to reimagine the renowned Timed-Elastic-Band (TEB) algorithm, resulting in a novel collision avoidance approach named Timed-Elastic-Band based variable splitting (TEB-VS). The proposed TEB-VS demonstrates superior navigation stability, while maintaining nearly identical resource consumption to TEB. We then analyze the convergence of the proposed TEB-VS method. To evaluate the effectiveness and efficiency of TEB-VS, extensive experiments have been conducted using TurtleBot2 in both simulated environments and real-world datasets.

Timed-Elastic-Band Based Variable Splitting for Autonomous Trajectory Planning

TL;DR

The paper tackles the instability and endpoint errors impeding autonomous trajectory planning with traditional Timed-Elastic-Band (TEB) methods. It introduces TEB-VS, a variable-splitting framework that reformulates the global constrained optimization into smaller, tractable subproblems solved via augmented TEB on a G2O graph, with convergence guarantees under mild assumptions. The authors provide a convergence analysis and validate the approach through extensive TurtleBot2 experiments in both simulation and real environments, showing superior speed stability and trajectory fidelity while maintaining similar computation to TEB. These results offer a robust, real-time capable trajectory planner for autonomous systems operating in complex, dynamic settings and pave the way for further enhancements such as fastest-path optimization.

Abstract

Existing trajectory planning methods are struggling to handle the issue of autonomous track swinging during navigation, resulting in significant errors when reaching the destination. In this article, we address autonomous trajectory planning problems, which aims at developing innovative solutions to enhance the adaptability and robustness of unmanned systems in navigating complex and dynamic environments. We first introduce the variable splitting (VS) method as a constrained optimization method to reimagine the renowned Timed-Elastic-Band (TEB) algorithm, resulting in a novel collision avoidance approach named Timed-Elastic-Band based variable splitting (TEB-VS). The proposed TEB-VS demonstrates superior navigation stability, while maintaining nearly identical resource consumption to TEB. We then analyze the convergence of the proposed TEB-VS method. To evaluate the effectiveness and efficiency of TEB-VS, extensive experiments have been conducted using TurtleBot2 in both simulated environments and real-world datasets.
Paper Structure (11 sections, 2 theorems, 13 equations, 3 figures, 2 tables)

This paper contains 11 sections, 2 theorems, 13 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Methodology
  4. The General Framework
  5. Augmented TEB for Solving $\mathbf{x}$-subproblem
  6. Convergence Analysis
  7. Experiment Results
  8. Experimental Settings
  9. Simulation Results
  10. Real-Life Robot Results
  11. Conclusion

Key Result

Lemma 1

Let $\{\mathbf{x}^{(n)},\mathbf{v}^{(n)}, \bm{\eta}^{(n)},\bm{\zeta}^{(n)}\}$ be the sequence. If $\rho e_c^2 + \gamma e_p^2 >e_x$, and the Jacobian $\mathbf{J}_c$, and $\mathbf{J}_p$ have full-column rank with Then, the sequence $\{\mathbf{x}^{(n)},\mathbf{v}^{(n)}, \bm{\eta}^{(n)},\bm{\zeta}^{(n)}\}$ is nonincreasing with the iteration number $n$.

Figures (3)

  • Figure 1: Our architecture: i) Real-time dynamic robot state information $Q_i$ is acquired from sensors, encompassing variables such as position, orientation, and other pertinent data. ii) In the method, the objective function is introduced as a new edge in the graph, replacing the original constraint. This step is pivotal for assimilating the results of the optimization process back into the overall trajectory planning framework. Steps 1-3 correspond to formulas \ref{['eq:admm_x_linear']}, \ref{['eq:admm_iteration_v']}, and \ref{['eq:admm_iteration_eta']}). iii) General Graph Optimization (G2O) 2011G2o is employed to solve the subproblem arising in VS framework, where each vertex in the graph model denotes as a state, and edges represent objective functions. iv) The output of TEB-VS is the optimized trajectory for the robot.
  • Figure 2: Gazebo world, the right part of this figure shows the simulation room by gazebo and the left part is OctoMap 2013OctoMap based on this room. The black point is robot Turtlebot2. The details will also be updated on the map in real time when robot navigation in this room.
  • Figure 3: Autonomous trajectory planning generated by different methods. The starting point is at coordinates (2,5) with a specific orientation, and the goal is to navigate coordinates (9,5) with a different orientation in the OctoMap. There are two phases of autonomous trajectory planning motion: rotation and linear motion. The dotted lines are the ground truth. When comparing velocity stability, especially in the context of navigating the robot, it is important to distinguish between the rotational and linear processes. Special attention should be given to the left and right swing of the robot during the linear motion phase, as this significantly impact the accuracy of reaching the endpoint.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1: Convergence of TEB-VS
  • proof