A Multiple Artificial Potential Functions Approach for Collision Avoidance in UAV Systems

TL;DR

The paper addresses UAV collision avoidance with static obstacles using a MAPOF framework based on $\ddot{\xi}=u$, where the goal is to reach $\eta$ while avoiding a detected obstacle. It proposes MAPOF, a switched-control scheme that combines four artificial potential forces into a control law $u_{\sigma(t)}=F_{\sigma(t)}(\xi)-k_d\dot{\xi}$, with a state- and time-dependent switching rule and dwell-time constraints to prevent chattering. The closed-loop dynamics are cast as a switched affine system $\dot{x}=A_{\sigma(t)}x+b_{\sigma(t)}$ with $x_1=\eta-\xi$, $x_2=-\dot{\xi}$, and the origin is shown to be the unique equilibrium under the MAPOF switching; stability conditions on dwell-times $T_{D1},T_{D2}$ are derived using hybrid-system theory. Simulations in two scenarios (one obstacle and multiple obstacles) demonstrate smooth, stable obstacle avoidance and convergence to the target, and the authors also propose dwell-time relaxations to handle high-frequency switching, indicating practical applicability and avenues for robustness and asymptotic stability in future work.

Abstract

Collision avoidance is a problem largely studied in robotics, particularly in unmanned aerial vehicle (UAV) applications. Among the main challenges in this area are hardware limitations, the need for rapid response, and the uncertainty associated with obstacle detection. Artificial potential functions (APOFs) are a prominent method to address these challenges. However, existing solutions lack assurances regarding closed-loop stability and may result in chattering effects. Motivated by this, we propose a control method for static obstacle avoidance based on multiple artificial potential functions (MAPOFs). We derive tuning conditions on the control parameters that ensure the stability of the final position. The stability proof is established by analyzing the closed-loop system using tools from hybrid systems theory. Furthermore, we validate the performance of the MAPOF control through simulations, showcasing its effectiveness in avoiding static obstacles.

Figures (7)

• Figure 1: Introduction of geometric variables for the state space partitioning proposed in Section \ref{['subsec:partitioning']}. The numbers indicate the respective subregion.
• Figure 2: Schematic representation of the MAPOF force map, with the four forces introduced in \ref{['eq:forces']}, relative to the obstacle and goal positions.
• Figure 3: Illustration of the scenario considered in Corollary \ref{['cor:1']}. The switching after the unstable mode $\sigma=4$ is restricted by the dwell-time $T_{D_2}$ in \ref{['eq:td3']}, so that the UAV maintains the avoidance force $\sigma=2$ until the point $g_{1}^{*}$.
• Figure 4: UAV trajectory with MAPOF strategy for collision avoidance in scenario 1: one obstacle.
• Figure 5: Switches in the different subsystems over time for scenario 1.

Theorems & Definitions (3)

• Lemma 1
• Theorem 1
• Corollary 1