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Risk-aware Control for Robots with Non-Gaussian Belief Spaces

Matti Vahs, Jana Tumova

TL;DR

This work defines the belief state and belief dynamics for continuous-discrete PFs and construct safe sets in the underlying belief space and designs a controller that provably keeps the unknown robot’s belief state within this safe set.

Abstract

This paper addresses the problem of safety-critical control of autonomous robots, considering the ubiquitous uncertainties arising from unmodeled dynamics and noisy sensors. To take into account these uncertainties, probabilistic state estimators are often deployed to obtain a belief over possible states. Namely, Particle Filters (PFs) can handle arbitrary non-Gaussian distributions in the robot's state. In this work, we define the belief state and belief dynamics for continuous-discrete PFs and construct safe sets in the underlying belief space. We design a controller that provably keeps the robot's belief state within this safe set. As a result, we ensure that the risk of the unknown robot's state violating a safety specification, such as avoiding a dangerous area, is bounded. We provide an open-source implementation as a ROS2 package and evaluate the solution in simulations and hardware experiments involving high-dimensional belief spaces.

Risk-aware Control for Robots with Non-Gaussian Belief Spaces

TL;DR

This work defines the belief state and belief dynamics for continuous-discrete PFs and construct safe sets in the underlying belief space and designs a controller that provably keeps the unknown robot’s belief state within this safe set.

Abstract

This paper addresses the problem of safety-critical control of autonomous robots, considering the ubiquitous uncertainties arising from unmodeled dynamics and noisy sensors. To take into account these uncertainties, probabilistic state estimators are often deployed to obtain a belief over possible states. Namely, Particle Filters (PFs) can handle arbitrary non-Gaussian distributions in the robot's state. In this work, we define the belief state and belief dynamics for continuous-discrete PFs and construct safe sets in the underlying belief space. We design a controller that provably keeps the robot's belief state within this safe set. As a result, we ensure that the risk of the unknown robot's state violating a safety specification, such as avoiding a dangerous area, is bounded. We provide an open-source implementation as a ROS2 package and evaluate the solution in simulations and hardware experiments involving high-dimensional belief spaces.
Paper Structure (23 sections, 3 theorems, 15 equations, 3 figures, 2 tables)

This paper contains 23 sections, 3 theorems, 15 equations, 3 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. Related Work
  3. Preliminaries
  4. Continuous-discrete Particle Filters
  5. Risk Measures
  6. Stochastic Control Barrier Functions
  7. Problem Setting
  8. Risk-Aware Control
  9. Particle Filter Belief Dynamics
  10. Construction of Safe Sets
  11. Control Synthesis
  12. Experiments
  13. Motivating Example Continued
  14. Setup
  15. Discussion
  16. ...and 8 more sections

Key Result

Theorem 1

If a locally Lipschitz control input $\bm{u}(t)$ satisfies Def. def:NSCBF for a given safe set, then $\mathrm{Pr}\left[\bm{x}(t) \in \mathcal{C}_x, \space \forall t\geq t_0\right]=1$, provided that $\bm{x}(t_0) \in \mathcal{C}_x$.

Figures (3)

  • Figure 1: An illustration of our experiments with the ABB Mobile YuMi Research Platform operating in an uncertain environment. The robot is equipped with a LiDAR sensor used for localization. Due to uncertainties in motions and observations, we only have access to a belief which is provided by a PF shown as red dots. The robot is supposed to avoid a dangerous area that cannot be detected by local sensing such as e.g. an untraversable area.
  • Figure 2: Left: A drone with uncertain position $x$ and its pdf $p(x)$ is moving in one dimension. A safety specification is defined as not colliding with the wall, i.e. $h_x = 2 - x$. The true pdf is a Gaussian shown in red and the PF is shown in blue. Right: The distributions over $h_x$ are shown on the right with their corresponding risk measures visualized.
  • Figure 3: (a) Values of the true $\mathrm{CVaR}$, the empirical $\widehat{\mathrm{CVaR}}$ and our lower bound $\overline{\mathrm{CVaR}}$ over time for a single simulation of Example \ref{['exmpl:drone']}. (b) Illustration of the PF belief over time for a 2D avoidance task where the initial belief follows a mixture of Gaussians. The ground truth trajectory is shown in blue and the trajectory for the reference controller is shown in green. (c) Illustration of the PF belief and the AMCL estimate for our hardware experiment.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Example 1
  • Theorem 2: thomas2019concentration, Thm. 3
  • Theorem 3