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Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control

Armin Lederer, Jonas Umlauft, Sandra Hirche

TL;DR

This paper employs the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions and demonstrates how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of this bound.

Abstract

Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on these models. However, existing error bounds require restrictive assumptions. In this paper, we employ the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions. Furthermore, we demonstrate how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of our bound. Finally, we derive safety conditions for the control of unknown dynamical systems based on Gaussian process models and evaluate them in simulations of a robotic manipulator.

Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control

TL;DR

This paper employs the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions and demonstrates how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of this bound.

Abstract

Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on these models. However, existing error bounds require restrictive assumptions. In this paper, we employ the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions. Furthermore, we demonstrate how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of our bound. Finally, we derive safety conditions for the control of unknown dynamical systems based on Gaussian process models and evaluate them in simulations of a robotic manipulator.

Paper Structure

This paper contains 20 sections, 7 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Gaussian Process Regression and Uniform Error Bounds
  4. Related Work
  5. Probabilistic Uniform Error Bound
  6. Exploiting Lipschitz Continuity of the Unknown Function
  7. Probabilistic Lipschitz Constants for Gaussian Processes
  8. Analysis of Asymptotic Behavior
  9. Safety Guarantees for Control of Unknown Dynamical Systems
  10. Tracking Control Design
  11. Stability Analysis
  12. Numerical Evaluation
  13. Synthetic System with Unknown Lipschitz Constant $L_f$
  14. Robotic Manipulator with 2 Degrees of Freedom
  15. Conclusion
  16. ...and 5 more sections

Key Result

Theorem 3.1

Consider a zero mean Gaussian process defined through the continuous covariance kernel $k(\cdot,\cdot)$ with Lipschitz constant $L_k$ on the compact set $\mathbb{X}$. Furthermore, consider a continuous unknown function $f:\mathbb{X}\rightarrow \mathbb{R}$ with Lipschitz constant $L_f$ and $N\in\math Moreover, pick $\delta\in (0,1)$, $\tau\in\mathbb{R}_+$ and set Then, it holds that

Figures (3)

  • Figure 1: Snapshots of the state trajectory (blue) as it approaches the desired trajectory (green). In low uncertainty areas (yellow background), the set $\mathbb{B}$ (red) is significantly smaller then in high uncertainty areas (blue background).
  • Figure 2: When the ultimate bound (red) is large, the tracking error (blue) increases due to the less precise model.
  • Figure 3: The task space of the robot (left) shows the robot is guaranteed to remain in $\mathbb{B}$ (red) after a transient phase. Hence, the remaining state space $\mathbb{X}\setminus\mathbb{B}$ (green) can be considered as safe. The joint angles and velocities (right) converge to the desired trajectories (dashed lines) over time.

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Theorem 3.3
  • Definition 4.1: Ultimate Boundedness
  • Theorem 4.1
  • proof : Proof of Theorem 3.1
  • Lemma B.1
  • proof
  • ...and 6 more