A Class of Axis-Angle Attitude Control Laws for Rotational Systems

TL;DR

This paper tackles global attitude stabilization for 3D rotational systems by moving beyond quaternion and Euler-angle controllers to a generalized axis-angle framework. It introduces a control law that uses a scaled Euler axis with an extended $K_{ obreak ext{o}}\infty$ function, and provides a Lyapunov-based proof of global asymptotic stability for a unique fixed attitude error point. The proposed design offers flexible choice of the proportional function and integrates smoothly with switching schemes that account for angular velocity. Through extensive simulations and outdoor quadrotor experiments, the method achieves shorter stabilization times and lower control effort than both quaternion-based and geometric benchmarks, especially in high-error tumble-recovery scenarios.

Abstract

We introduce a new class of attitude control laws for rotational systems, which generalizes the use of the Euler axis-angle representation beyond quaternion-based formulations. Using basic Lyapunov's stability theory and the notion of extended $K_{\infty}$ functions, we developed a method for determining and enforcing the global asymptotic stability of the single fixed point of the resulting closed-loop (CL) scheme. In contrast with traditional quaternion-based methods, the proposed generalized axis-angle approach enables greater flexibility in the design of the control law, which is of great utility when employed in combination with a switching scheme whose transition state depends on the angular velocity of the controlled rotational system. Through simulation and real-time experimental results, we demonstrate the effectiveness of the proposed approach. According to the recorded data, in the execution of high-speed tumble-recovery maneuvers, the new method consistently achieves shorter stabilization times and requires lower control effort relative to those corresponding to the quaternion-based and geometric-control methods used as benchmarks.

Figures (4)

• Figure 1: Picture of the UAV used in the flight control experiments. The Crazyflie 2.1 quadrotor and the definitions of the inertial frame of reference, $\boldsymbol{\mathcal{I}} = \left\{\boldsymbol{i}_1, \boldsymbol{i}_2, \boldsymbol{i}_3\right\}$, fixed to the planet and the body-fixed frame of reference, $\boldsymbol{\mathcal{B}} = \left\{\boldsymbol{b}_1, \boldsymbol{b}_2, \boldsymbol{b}_3\right\}$ (shifted for clarity), whose origin is attached to the center of mass of the robot.
• Figure 2: Numerical results with Crazyflie 2.1 parameters using quaternion-based, geometric, and axis--angle control laws.(a) Mean and SD of the stabilization time over different initial rotation errors. (b) Mean and SD of the control effort variation over different initial rotation errors. (c) Time evolution of the rotation error associated with the attitude-error Euler axis. (d) Time evolution of the SEAs magnitudes for the quaternion-based and proposed axis--angle law.
• Figure 3: Experimental setup used during the flight tests. The attitude flight control experiments were run outdoors using a ground computer equipped with the Crazyradio 2.0 to communicate with the Crazyflie 2.1.
• Figure 4: Experimental results.(a) Rotation errors corresponding to three tests employing $\boldsymbol{\tilde{\tau}}_{\text{b}}$, $\boldsymbol{\tau}_{\text{g}}$, and $\boldsymbol{\tilde{\tau}}_{\gamma}$. (b) Magnitudes of the SEAs $\|\boldsymbol{n}_{\text{e}}\|_2$ and $\|\boldsymbol{\alpha}_{\text{e}}\|_2$ corresponding to $\boldsymbol{\tilde{\tau}}_{\text{b}}$ and $\boldsymbol{\tilde{\tau}}_{\gamma}$, respectively. (c) Comparison of mean (circle) and SEM (vertical bars) of the stabilization times using $\boldsymbol{\tilde{\tau}}_{\text{b}}$, $\boldsymbol{\tau}_{\text{g}}$, and $\boldsymbol{\tilde{\tau}}_{\gamma}$ over 30 experiments. (d) Comparison of mean and SEM of the control effort using $\boldsymbol{\tilde{\tau}}_{\text{b}}$, $\boldsymbol{\tau}_{\text{g}}$, and $\boldsymbol{\tilde{\tau}}_{\gamma}$ over 30 experiments.