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Learning Input Constrained Control Barrier Functions for Guaranteed Safety of Car-Like Robots

Sven Brüggemann, Dominic Nightingale, Jack Silberman, Maurício de Oliveira

TL;DR

A robust ICCBF that can be efficiently implemented is obtained by learning a smooth function of the environment using Support Vector Machine regression, and takes into account steering constraints.

Abstract

We propose a design method for a robust safety filter based on Input Constrained Control Barrier Functions (ICCBF) for car-like robots moving in complex environments. A robust ICCBF that can be efficiently implemented is obtained by learning a smooth function of the environment using Support Vector Machine regression. The method takes into account steering constraints and is validated in simulation and a real experiment.

Learning Input Constrained Control Barrier Functions for Guaranteed Safety of Car-Like Robots

TL;DR

A robust ICCBF that can be efficiently implemented is obtained by learning a smooth function of the environment using Support Vector Machine regression, and takes into account steering constraints.

Abstract

We propose a design method for a robust safety filter based on Input Constrained Control Barrier Functions (ICCBF) for car-like robots moving in complex environments. A robust ICCBF that can be efficiently implemented is obtained by learning a smooth function of the environment using Support Vector Machine regression. The method takes into account steering constraints and is validated in simulation and a real experiment.
Paper Structure (7 sections, 2 theorems, 19 equations, 4 figures)

This paper contains 7 sections, 2 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Safety design for car-like robots
  4. Learning a robust safety filter
  5. Simulation & experimental results
  6. Limitations
  7. Conclusion & future direction

Key Result

Lemma 1

Consider system eq:sys_generic with input constraints $\mathcal{U}$ and $u_{\rm nom}\in\mathcal{U}$. Define If $h_N(x)$ is an ICCBF and $L_g h_{N}(x)$ is full rank, then renders eq:sys_generic safe. If $L_g h_{N}(x)$ is not full rank but $L_fh_{N}(x)+ \alpha_N( h_{N}(x))\geq0$ then eq:sys_generic is still safe.

Figures (4)

  • Figure 1: Overview of approach and photos of experiment.
  • Figure 2: Preprocessing for offline learning for closed track we wish to follow. From the image we generate data $\mathcal{D}$.
  • Figure 3: Simulation results. By observing the simulated travelled path in closed loop we notice that the safety filter keeps the robot safe even though nominal control steers directly off track towards the goal $z_{\rm goal}$. The resulting steering angle and rate remain within bounds. The safety filter overrides the nominal control whenever the ICCBF $h_2 (x)$ approaches zero (the robot approaches the road edges), see Figures \ref{['fig:sim_u']} and \ref{['fig:sim_h']}. The safety filter is mostly engaged, except in the three periods in which the nominal steering command drives the robot away from the edges (see Figure \ref{['fig:learned_cbf']}). Importantly, the track itself represents a simply connected set.
  • Figure 4: Experimental results. In Fig. \ref{['fig:exp_delta']}: the learned safety filter keeps the car on the road while avoiding obstacles and allowing nominal action, $u_{\rm nom}(x)$, (zero steering angle) if safe. Fig. \ref{['fig:exp_u']} shows that $\hat{h}_0\geq0$ at all time, despite input constraints and uncertainties which include: measurement noise from the LiDAR sensor used for map building and localization; limited map resolution (here $5$cm) measurement noise related to the car's velocity; process noise due to discretization of continuous-time model dynamics, and approximated dynamics in general (wheel slip is neglected); input disturbance for the steering angle; limited real-time capabilities of used hardware and ROS.

Theorems & Definitions (5)

  • Definition 1: Forward Invariance & Safety
  • Definition 2: agrawal2021safe
  • Lemma 1
  • Theorem 1
  • proof