Real-Time Sensor-Based Feedback Control for Obstacle Avoidance in Unknown Environments

TL;DR

This work revisits the Safety Velocity Cones (SVCs) obsta-cle avoidance approach for real-time autonomous navigation in an unknown n-dimensional environment and proposes a locally Lipschitz continuous implementation of the SVC controller using the distance-to-the-obstacle function and its gradient.

Abstract

We revisit the Safety Velocity Cones (SVCs) obstacle avoidance approach for real-time autonomous navigation in an unknown $n$-dimensional environment. We propose a locally Lipschitz continuous implementation of the SVC controller using the distance-to-the-obstacle function and its gradient. We then show that the proposed implementation guarantees safe navigation in generic environments and almost globally asymptotic stability (AGAS) of the desired destination when the workspace contains strongly convex obstacles. The proposed computationally efficient control algorithm can be implemented onboard vehicles equipped with limited range sensors (e.g., LiDAR, depth camera), allowing the controller to be locally evaluated without requiring prior knowledge of the environment.

Figures (6)

• Figure 1: Obstacle avoidance of a ball-shaped robot (blue). The (green) region around the obstacle (gray) is a dilation for the latest by the parameter $\epsilon$ while the (black) dashed line is a dilation by the parameter $\epsilon'$. The (orange) trajectory results from the application of the nominal controller which brings the robot's position to the desired goal $x_d$ (red) in the absence of the obstacle. The (blue) trajectory results from applying the smooth controller which brings the robot's position to the desired goal while avoiding the dilated obstacle.
• Figure 2: Different obstacle's topology affecting the nature of the equilibrium point:(left) a non-convex obstacle for which the trajectory of the robot converges to the undesired equilibrium, and (right) a convex obstacle for which the trajectory converges to the desired goal $x_d$.
• Figure 3: Two convex obstacles where the curvature affects the nature of the equilibrium point:(left) a flat obstacle, as viewed from the position of the vehicle, for which its trajectory converges to the undesired equilibrium, and (right) a strongly convex obstacle, as viewed from the position of the vehicle, for which its trajectory converges to the desired goal $x_d$.
• Figure 4: (Top) Example of a 2D LIDAR reading, which represents a polar curve. (Bottom) A presentation of the 3D LIDAR reading in a 2D grayscale map with the angles $\theta$ and $\phi$ as the axis and the color of a point $(\theta,\phi)$ is attributed according to the value of $\rho$, (black) when $\rho=0$, (white) when $\rho=R_s$ and shades of (gray) in between.
• Figure 5: the resulting navigation trajectories, in a 2D environment, of the smooth control law (\ref{['eq:smoothControl']}) starting at a set of initial positions (blue) away from the goal (red) while avoiding the obstacles (gray). The (green) region around the obstacle (gray) is a dilation for the latest by the parameter $\epsilon$ while the (black) dashed line is a dilation by the parameter $\epsilon'$. The (magenta) area represents the sensor range for the actual position of the robot (yellow).

Theorems & Definitions (10)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 2
• proof
• Theorem 3
• proof