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Quaternion-based Adaptive Backstepping Fast Terminal Sliding Mode Control for Quadrotor UAVs with Finite Time Convergence

Arezo Shevidi, Hashim A. Hashim

TL;DR

This work addresses reliable 6-DOF pose control for underactuated quadrotor UAVs by introducing a quaternion-based cascaded control framework that couples Adaptive Backstepping for translation with Adaptive Fast Terminal Sliding Mode Control for attitude. The unit-quaternion representation avoids attitude singularities and, together with adaptive laws, mitigates unknown time-varying uncertainties and initialization errors while reducing control chattering. The authors provide Lyapunov-based proofs of asymptotic stability for translation and finite-time convergence for attitude, and validate the approach via simulations that demonstrate robust tracking and improved performance over Euler-angle-based methods.

Abstract

This paper proposes a novel quaternion-based approach for tracking the translation (position and linear velocity) and rotation (attitude and angular velocity) trajectories of underactuated Unmanned Aerial Vehicles (UAVs). Quadrotor UAVs are challenging regarding accuracy, singularity, and uncertainties issues. Controllers designed based on unit-quaternion are singularity-free for attitude representation compared to other methods (e.g., Euler angles), which fail to represent the vehicle's attitude at multiple orientations. Quaternion-based Adaptive Backstepping Control (ABC) and Adaptive Fast Terminal Sliding Mode Control (AFTSMC) are proposed to address a set of challenging problems. A quaternion-based ABC, a superior recursive approach, is proposed to generate the necessary thrust handling unknown uncertainties and UAV translation trajectory tracking. Next, a quaternion-based AFTSMC is developed to overcome parametric uncertainties, avoid singularity, and ensure fast convergence in a finite time. Moreover, the proposed AFTSMC is able to significantly minimize control signal chattering, which is the main reason for actuator failure and provide smooth and accurate rotational control input. To ensure the robustness of the proposed approach, the designed control algorithms have been validated considering unknown time-variant parametric uncertainties and significant initialization errors. The proposed techniques has been compared to state-of-the-art control technique. Keywords: Adaptive Backstepping Control (ABC), Adaptive Fast Terminal Sliding Mode Control (AFTSMC), Unit-quaternion, Unmanned Aerial Vehicles, Singularity Free, Pose Control

Quaternion-based Adaptive Backstepping Fast Terminal Sliding Mode Control for Quadrotor UAVs with Finite Time Convergence

TL;DR

This work addresses reliable 6-DOF pose control for underactuated quadrotor UAVs by introducing a quaternion-based cascaded control framework that couples Adaptive Backstepping for translation with Adaptive Fast Terminal Sliding Mode Control for attitude. The unit-quaternion representation avoids attitude singularities and, together with adaptive laws, mitigates unknown time-varying uncertainties and initialization errors while reducing control chattering. The authors provide Lyapunov-based proofs of asymptotic stability for translation and finite-time convergence for attitude, and validate the approach via simulations that demonstrate robust tracking and improved performance over Euler-angle-based methods.

Abstract

This paper proposes a novel quaternion-based approach for tracking the translation (position and linear velocity) and rotation (attitude and angular velocity) trajectories of underactuated Unmanned Aerial Vehicles (UAVs). Quadrotor UAVs are challenging regarding accuracy, singularity, and uncertainties issues. Controllers designed based on unit-quaternion are singularity-free for attitude representation compared to other methods (e.g., Euler angles), which fail to represent the vehicle's attitude at multiple orientations. Quaternion-based Adaptive Backstepping Control (ABC) and Adaptive Fast Terminal Sliding Mode Control (AFTSMC) are proposed to address a set of challenging problems. A quaternion-based ABC, a superior recursive approach, is proposed to generate the necessary thrust handling unknown uncertainties and UAV translation trajectory tracking. Next, a quaternion-based AFTSMC is developed to overcome parametric uncertainties, avoid singularity, and ensure fast convergence in a finite time. Moreover, the proposed AFTSMC is able to significantly minimize control signal chattering, which is the main reason for actuator failure and provide smooth and accurate rotational control input. To ensure the robustness of the proposed approach, the designed control algorithms have been validated considering unknown time-variant parametric uncertainties and significant initialization errors. The proposed techniques has been compared to state-of-the-art control technique. Keywords: Adaptive Backstepping Control (ABC), Adaptive Fast Terminal Sliding Mode Control (AFTSMC), Unit-quaternion, Unmanned Aerial Vehicles, Singularity Free, Pose Control
Paper Structure (22 sections, 4 theorems, 71 equations, 7 figures, 2 tables)

This paper contains 22 sections, 4 theorems, 71 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Work and Challenges
  4. Contributions
  5. Structure
  6. Preliminaries
  7. Math Notation
  8. Unit-quaternion
  9. UAV Dynamical Model and Quaternion Representation
  10. UAV Translation and Rotational Dynamics
  11. Expanded Vehicle Nonlinear Dynamics
  12. Problem Formulation and Controller Design
  13. Adaptive Backstepping for Position Control
  14. Fast Terminal Sliding Mode Control Attitude Control
  15. Attitude Control and Finite Time Convergence
  16. ...and 7 more sections

Key Result

Lemma 1

labbadi2019robust Based on Barbalet Lemma, if $f(s)$ is a uniformly bounded continuous function and $\lim_{s\to+\infty}\int_{0}^{\top}f(s)\,ds$ exists, $f(s)$ converges to the origin asymptotically.

Figures (7)

  • Figure 1: Quadrotor UAV configuration
  • Figure 2: Illustrative diagram of the proposed control system for UAV
  • Figure 3: UAV Attitude and Position Trajectory Tracking: True value represented in the blue solid-line, while red dash-lines plotted references
  • Figure 4: UAV Rotational Control (plotted in red) and Total Thrust Input (represented in blue).
  • Figure 5: Bounded and smooth trajectory of the adaptive parameters of controllers.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2