A Lyapunov-Based Switching Scheme for Selecting the Stable Closed-Loop Fixed Attitude-Error Quaternion During Flight

Abstract

We present a switching scheme, which uses both the attitude-error quaternion (AEQ) and the angular-velocity error, for controlling the rotational degrees of freedom of an uncrewed aerial vehicle (UAV) during flight. In this approach, the proposed controller continually selects the stable closed-loop (CL) equilibrium AEQ corresponding to the smallest cost between those computed with two energy-based Lyapunov functions. To analyze and enforce the stability of the CL switching dynamics, we use basic nonlinear theory. This research problem is relevant because the selection of the stable CL equilibrium AEQ directly determines the power and energy requirements of the controlled UAV during flight. To test and demonstrate the implementation, suitability, functionality, and performance of the proposed approach, we present experimental results obtained using a 31-gram quadrotor, which was controlled to execute high-speed yaw maneuvers in flight. These flight tests show that the proposed switching controller can respectively reduce the control effort and rotational power by as much as 49.75 % and 28.14 %, on average, compared to those corresponding to an often-used benchmark controller.

Figures (5)

• Figure 1: Photograph of the UAV used in the flight tests, the Crazyflie $\boldsymbol{2.1}$, with reflective markers attached to its body. Here, $\boldsymbol{\mathcal{B}} = \left\{\boldsymbol{b}_1,\boldsymbol{b}_2,\boldsymbol{b}_3 \right\}$, with its origin coinciding with the UAV's center of mass, is the body-fixed frame of reference; $\boldsymbol{\mathcal{N}} = \left\{\boldsymbol{n}_1,\boldsymbol{n}_2,\boldsymbol{n}_3 \right\}$ is the inertial frame of reference.
• Figure 2: Block diagram of the proposed switching controller. The switching law receives the desired and measured attitude quaternions, $\boldsymbol{ \mathrlap{\raisebox{-5.2pt}{$\space{\mathchar'26\mkern-9mu}$}} q}_\text{d}$ and $\boldsymbol{ \mathrlap{\raisebox{-5.2pt}{$\space{\mathchar'26\mkern-9mu}$}} q}$, and the desired and measured angular velocities, $\boldsymbol{\omega}_\text{d}$ and $\boldsymbol{\omega}$. Then, it computes the switching signal $\sigma^+ \in \left\{+1,-1\right\}$ and sends it to the attitude controller, which computes the control torque, $\boldsymbol{\tau}_{\space\sigma}$. Last, as explained in Ying_ICRA_2016, the actuator mapping receives $\boldsymbol{\tau}_{\space\sigma}$ and generates the pulse-width modulation (PWM) voltage signals, $e_i$, $i \in \left\{1,2,3,4 \right\}$, that excite the four DC motors of the controlled UAV.
• Figure 3: Experimental data corresponding to two sets of flight tests. (a) Yaw-angle reference, $\psi_{\text{d}}$, and measured yaw angle, $\psi$, for the pair $\left\{ \boldsymbol{\omega}_0, \psi_0 \right\} = \left\{3\cdot\boldsymbol{b}_3\,\text{rad}\cdot\text{s}^{-1},120^\circ \right\}$, obtained with the benchmark controller. (b) Yaw-angle reference, $\psi_{\text{d}}$, and measured yaw angle, $\psi$, for the pair $\left\{ \boldsymbol{\omega}_0, \psi_0 \right\} = \left\{3\cdot\boldsymbol{b}_3\,\text{rad}\cdot\text{s}^{-1},120^\circ \right\}$, obtained with the switching controller. (c) Instantaneous values of $\Lambda$ and $\sigma$ corresponding to the data shown in (b). (d) Yaw-angle reference, $\psi_{\text{d}}$, and measured yaw angle, $\psi$, for the pair $\left\{ \boldsymbol{\omega}_0, \psi_0 \right\} = \left\{4\cdot\boldsymbol{b}_3\,\text{rad}\cdot\text{s}^{-1},90^\circ \right\}$, obtained with the benchmark controller. (e) Yaw-angle reference, $\psi_{\text{d}}$, and measured yaw angle, $\psi$, for the pair $\left\{ \boldsymbol{\omega}_0, \psi_0 \right\} = \left\{4\cdot\boldsymbol{b}_3\,\text{rad}\cdot\text{s}^{-1},90^\circ \right\}$, obtained with the switching controller. (f) Instantaneous values of $\Lambda$ and $\sigma$ corresponding to the data shown in (e).
• Figure 4: Photographic sequences corresponding to two flight tests. (a) Sequence corresponding to a flight test with the initial-condition pair $\left\{ \boldsymbol{\omega}_0, \psi_0 \right\} = \left\{3\cdot\boldsymbol{b}_3\,\text{rad}\cdot\text{s}^{-1},120^\circ \right\}$ using the benchmark controller. (b) Sequence corresponding to a flight test with the initial-condition pair $\left\{ \boldsymbol{\omega}_0, \psi_0 \right\} = \left\{3\cdot\boldsymbol{b}_3\,\text{rad}\cdot\text{s}^{-1},120^\circ \right\}$ using the switching controller. Video footage of these and other flight tests can be viewed in the accompanying supplementary movie. This movie is also available at https://wsuamsl.com/resources/ROBOT2024movie.mp4.
• Figure 5: Comparison of flight performances. (a) Means and ESDs of $\Gamma_{\boldsymbol{\tau}}$, for five different pairs of initial conditions. (b) Means and ESDs of $\Gamma_{\text{p}}$, for five different pairs of initial conditions. Each data point in these plots was computed from ten back-to-back experiments.