A New Type of Axis-Angle Attitude Control Law for Rotational Systems: Synthesis, Analysis, and Experiments

TL;DR

This paper addresses the limitation of quaternion-based attitude control, which can yield non-unique CL equilibria and diminished proportional action for large attitude errors. It proposes two axis-angle control laws that use a scaled Euler-axis (SEA) vector to ensure a unique CL equilibrium AEQ and to maintain or increase proportional control as the error grows. By constructing strict Lyapunov functions, the authors prove uniform asymptotic stability of the CL equilibria for both laws and validate performance through extensive numerical simulations and real-time tumble-recovery experiments on a small quadrotor, demonstrating faster stabilization than a quaternion-based benchmark. The approach offers a practical, actuator-conscious framework suitable for integration into switching or angular-velocity-aware schemes, with potential applicability to various rigid-body platforms; future work will address actuator saturation and broader implementation aspects.

Abstract

Over the past few decades, continuous quaternion-based attitude control has been proven highly effective for driving rotational systems that can be modeled as rigid bodies, such as satellites and drones. However, methods rooted in this approach do not enforce the existence of a unique closed-loop (CL) equilibrium attitude-error quaternion (AEQ); and, for rotational errors about the attitude-error Euler axis larger than πrad, their proportional-control effect diminishes as the system state moves away from the stable equilibrium of the CL rotational dynamics. In this paper, we introduce a new type of attitude control law that more effectively leverages the attitude-error Euler axis-angle information to guarantee a unique CL equilibrium AEQ and to provide greater flexibility in the use of proportional-control efforts. Furthermore, using two different control laws as examples-through the construction of a strict Lyapunov function for the CL dynamics-we demonstrate that the resulting unique equilibrium of the CL rotational system can be enforced to be uniformly asymptotically stable. To assess and demonstrate the functionality and performance of the proposed approach, we performed numerical simulations and executed dozens of real-time tumble-recovery maneuvers using a small quadrotor. These simulations and flight tests compellingly demonstrate that the proposed axis-angle-based method achieves superior flight performance-compared with that obtained using a high-performance quaternion-based controller-in terms of stabilization time.

Figures (5)

• Figure 1: Photograph of the quadrotor---a Crazyflie $\boldsymbol{2.1}$---used in the flight tests performed to assess and demonstrate the functionality and performance of the proposed attitude control approach. Here, $\boldsymbol{\mathcal{N}} = \left\{\boldsymbol{n}_1, \boldsymbol{n}_2, \boldsymbol{n}_3\right\}$ and $\boldsymbol{\mathcal{B}} = \left\{\boldsymbol{b}_1, \boldsymbol{b}_2, \boldsymbol{b}_3\right\}$ denote the inertial and body-fixed frames used for kinematic description and dynamic modeling. As customary, $\boldsymbol{\mathcal{N}}$ is fixed to the planet Earth and the origin of $\boldsymbol{\mathcal{B}}$ coincides with the CoM of the controlled rotational system.
• Figure 2: Block diagram of the upper-level attitude control scheme con-sidered in this paper. In this scheme, the attitude controller receives as inputs the desired attitude quaternion, $\boldsymbol{ \mathrlap{\raisebox{-5.2pt}{$\space{\mathchar'26\mkern-9mu}$}} q}_{\text{d}}$; the desired angular velocity, $\boldsymbol{\omega}_{\text{d}}$; the measured attitude quaternion, $\boldsymbol{ \mathrlap{\raisebox{-5.2pt}{$\space{\mathchar'26\mkern-9mu}$}} q}$; and, the measured angular velocity, $\boldsymbol{\omega}$. Using these inputs, at any given instant, it computes the control torque, $\boldsymbol{\tau}\in\{\boldsymbol{\tau}_{\text{b}}, \boldsymbol{\tau}_{1}, \boldsymbol{\tau}_{2}\}$. The actuator mapping receives as its input $\boldsymbol{\tau}$ and generates as its output the vector signal $\boldsymbol{a}$ that excites the actuators of the open-loop dynamics of the controlled rotational system.
• Figure 3: Simulation results.(a) Stabilization times versus the initial Euler-axis rotational error, $\Theta_{\text{e},0}$, obtained using the three tested attitude control laws, $\boldsymbol{\tau}_{\text{b}}$, $\boldsymbol{\tau}_1$, and $\boldsymbol{\tau}_2$. (b) Regulation responses of the Euler-axis rotational errors, $\Theta_{\text{e}}(t)$, for $t \in \left[0,2\right]$ s and initial condition $\Theta_{\text{e},0} = 300$ °, obtained using the three tested control laws, $\boldsymbol{\tau}_{\text{b}}$, $\boldsymbol{\tau}_1$, and $\boldsymbol{\tau}_2$. (c) SEA vector norms---$\|\boldsymbol{n}_{\text{e}}\|_2$, $\|\boldsymbol{p}_{\text{e},1}\|_2$, and $\|\boldsymbol{p}_{\text{e},2}\|_2$---corresponding to the flight cases in (b), evolving over time within the range $\left[0,2\right]$ s.
• Figure 4: Experimental setup used during the performance of flight control tests. All the experiments were performed outdoor using a ground computer equipped with a Crazyradio $2.0$ dongle, which sends the initialization and stop commands for the execution of controlled flight maneuvers. Also, during flight, the flier---a Crazyflie $2.1$---sends its instantaneous state to the ground computer for data collection.
• Figure 5: Experimental results.(a) Time evolutions of the Euler-axis rotational errors, $\Theta_{\text{e}}(t)$, for $t \in \left[0,1.5\right]$ s, obtained using the benchmark attitude control law, $\boldsymbol{\tau}_{\text{b}}$, and the new attitude control law, $\boldsymbol{\tau}_2$. (b) SEA vector norms---$\|\boldsymbol{n}_{\text{e}}\|_2$, $\|\boldsymbol{p}_{\text{e},1}\|_2$, and $\|\boldsymbol{p}_{\text{e},2}\|_2$---corresponding to the flight cases in (a), evolving over time within the range $\left[0,1.5\right]$ s.~(c) Mean and SEM of the stabilization times corresponding to $25$ back-to-back experiments respectively performed using $\boldsymbol{\tau}_{\text{b}}$ and $\boldsymbol{\tau}_2$.