Dissipative Avoidance Feedback for Reactive Navigation Under Second-Order Dynamics

TL;DR

The proposed continuously differentiable controller solves the motion-to-goal problem while guaranteeing collision-free navigation by using the robot’s state and local obstacle distance information and achieves Almost Global Asymptotic Stability (AGAS) under certain curvature conditions.

Abstract

This paper addresses the problem of autonomous robot navigation in unknown, obstacle-filled environments with second-order dynamics by proposing a Dissipative Avoidance Feedback (DAF). Compared to the Artificial Potential Field (APF), which primarily uses repulsive forces based on position, DAF employs a dissipative feedback mechanism that accounts for both position and velocity, contributing to smoother and more natural obstacle avoidance. The proposed continuously differentiable controller solves the motion-to-goal problem while guaranteeing collision-free navigation by using the robot's state and local obstacle distance information. We show that the controller guarantees safe navigation in generic $n$-dimensional environments and that all undesired $ω$-limit points are unstable under certain controlled curvature conditions. Designed for real-time implementation, DAF requires only locally measured data from limited-range sensors (e.g., LiDAR, depth cameras), making it particularly effective for robots navigating unknown workspaces. Simulations in 2D and 3D environments are conducted to validate the theoretical results and to showcase the effectiveness of our approach.

Figures (6)

• Figure 1: Illustration of the different variables and parameters used in the proposed control approach \ref{['eq:controller']}.
• Figure 2: In 3D, the curvature condition \ref{['eq:conditionCompactForm']} of Theorem \ref{['theorem:theorem1']} can be satisfied for convex or saddle-shaped obstacles provided that one principal direction satisfies the curvature condition \ref{['eq:conditionCompactForm']} (e.g., strongly convex).
• Figure 3: On the left, $k_2/k_1=0.93$ violates the curvature condition \ref{['eq:conditionCompactForm']} making $p^*$ a local minimum. On the right, $k_2/k_1=0.68$ satisfies \ref{['eq:conditionCompactForm']}, making $p^*$ a saddle point.
• Figure 4: The trajectories of the robot in a 2D environment start from a set of initial positions (blue) away from the goal (red) while avoiding the obstacles (dark gray). The (light gray) regions are the dilation of the obstacles (dark gray) by the parameter $\epsilon$. The (black) and the (orange) dashed lines are a dilation by the parameters $\epsilon_1$ and $\epsilon_2$, respectively. The (blue) lines represent the trajectories under our approach, while the (green) lines represent the trajectories resulting from the APF approach. The animation is given at https://youtu.be/7TfEdOQXw7Y.
• Figure 5: Trajectories of the robot, in a 3D environment filled with convex and non-convex obstacles, using our approach (blue) and the APF approach (green) starting at a set of initial positions (blue) away from the goal (red) while avoiding the obstacles (gray). Animated visualization can be found at https://youtu.be/YqxY2tq8xTw.

Theorems & Definitions (4)

• Lemma 1
• Theorem 1
• proof
• Lemma 2: ms93rap