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C-Uniform Trajectory Sampling For Fast Motion Planning

O. Goktug Poyrazoglu, Yukang Cao, Volkan Isler

TL;DR

This paper tackles biased trajectory sampling by introducing C-Uniformity, which enforces uniform sampling over level-sets $L(t)$ rather than uniform control inputs. It derives a closed-form solution for the 1-D case and a network-flow optimization for general systems to compute trajectory-probability distributions that achieve C-Uniformity, and then integrates this sampling into MPPI-style controllers. Empirical results show up to 40% improvement in coverage over the best baselines in simulations, plus successful real-world validation on a 1/10-scale racer and F1Tenth platform, including obstacle-rich and cluttered environments. The approach is computationally intensive but amenable to offline precomputation and parallelization, enabling practical real-time deployment with future work focusing on real-time performance and map-aware extensions.

Abstract

We study the problem of sampling robot trajectories and introduce the notion of C-Uniformity. As opposed to the standard method of uniformly sampling control inputs (which lead to biased samples of the configuration space), C-Uniform trajectories are generated by control actions which lead to uniform sampling of the configuration space. After presenting an intuitive closed-form solution to generate C-Uniform trajectories for the 1D random-walker, we present a network-flow based optimization method to precompute C-Uniform trajectories for general robot systems. We apply the notion of C-Uniformity to the design of Model Predictive Path Integral controllers. Through simulation experiments, we show that using C-Uniform trajectories significantly improves the performance of MPPI-style controllers, achieving up to 40% coverage performance gain compared to the best baseline. We demonstrate the practical applicability of our method with an implementation on a 1/10th scale racer.

C-Uniform Trajectory Sampling For Fast Motion Planning

TL;DR

This paper tackles biased trajectory sampling by introducing C-Uniformity, which enforces uniform sampling over level-sets rather than uniform control inputs. It derives a closed-form solution for the 1-D case and a network-flow optimization for general systems to compute trajectory-probability distributions that achieve C-Uniformity, and then integrates this sampling into MPPI-style controllers. Empirical results show up to 40% improvement in coverage over the best baselines in simulations, plus successful real-world validation on a 1/10-scale racer and F1Tenth platform, including obstacle-rich and cluttered environments. The approach is computationally intensive but amenable to offline precomputation and parallelization, enabling practical real-time deployment with future work focusing on real-time performance and map-aware extensions.

Abstract

We study the problem of sampling robot trajectories and introduce the notion of C-Uniformity. As opposed to the standard method of uniformly sampling control inputs (which lead to biased samples of the configuration space), C-Uniform trajectories are generated by control actions which lead to uniform sampling of the configuration space. After presenting an intuitive closed-form solution to generate C-Uniform trajectories for the 1D random-walker, we present a network-flow based optimization method to precompute C-Uniform trajectories for general robot systems. We apply the notion of C-Uniformity to the design of Model Predictive Path Integral controllers. Through simulation experiments, we show that using C-Uniform trajectories significantly improves the performance of MPPI-style controllers, achieving up to 40% coverage performance gain compared to the best baseline. We demonstrate the practical applicability of our method with an implementation on a 1/10th scale racer.
Paper Structure (14 sections, 1 theorem, 7 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 1 theorem, 7 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation and Preliminaries
  4. Method
  5. Solution for the one-dimensional case
  6. The general case
  7. Experiments
  8. Experiment setup
  9. Coverage Analysis
  10. Environment with an Obstacle
  11. Cluttered Environments
  12. Real Experiments
  13. Conclusion and Future Work
  14. Acknowledgement

Key Result

Lemma 1

There exist C-Uniform control inputs between levels $L$ and $L'$ if and only if $n \times m$ units of flow can be transported by the flow network described above.

Figures (6)

  • Figure 1: Comparison of uniformly sampling the actions versus C-Uniform trajectories for the 1-D random walk system (top figure) and the Dubins car (bottom left vs. right). For the Dubins car, note how long sharp turns are not generated by uniformly sampling the control inputs (bottom left).
  • Figure 2: C-Uniform control inputs the case of $n=5$ and $m=9$. Probabilities are obtained by dividing each entry by 9.
  • Figure 3: Overview of approach: We generate level sets by densely sampling control inputs and tiling $\delta$-measure regions over each level set. To compute the action probabilities, a flow network is generated whose maximum-flow yields C-Uniformity.
  • Figure 4: Comparison of the MPPI, log-MPPI and C-Uniform for the suddenly appearing obstacle experiment. The obstacle becomes visible after 1.0s of the robot movement (middle). The baselines can not avoid collisions because their samples lie inside the obstacle. In contrast, C-Uniform samples avoid the obstacle by achieving a wider coverage, which allows for successful re-planning (right).
  • Figure 5: Top: Experiment setup Bottom: Successful run trajectories for each method: MPPI, Log-MPPI, and C-Uniform
  • ...and 1 more figures

Theorems & Definitions (1)

  • Lemma 1