Elsevier

Mechatronics

Volume 72, December 2020, 102450
Mechatronics

An approximation-free simple controller for uncertain quadrotor systems in the presence of thrust saturation

https://doi.org/10.1016/j.mechatronics.2020.102450Get rights and content

Abstract

In this paper, a new simple tracking control scheme is presented for quadrotor systems with uncertain dynamics. It precludes the necessity for prohibitive analytic computation of the derivatives of the desired (virtual) attitude that is typically employed in controlling quadrotor systems. Moreover, this control scheme is approximation-free in the sense that it does not incorporate any adaptive laws, observers, or command filters to compensate for unknown parameters in the dynamics and the absence of the analytic differentiation, thus exhibiting remarkably low complexity levels and making its implementation straightforward. The thrust saturation is approached in the position control design, which also enables the singularity in desired attitude extraction to be avoided entirely. It is demonstrated that based on the proposed scheme, the tracking errors can be made arbitrarily small by appropriately selecting design parameters. Extensive simulations and experiments are performed to verify the effectiveness of our method.

Introduction

The advent of ultra-fast microprocessors together with advanced sensing devices has stimulated the evolution of autonomous quadrotors due to the numerous applications that can be addressed by such systems like surveillance, transportation, or mapping [1], [2], [3]. As for the autonomous flight problem, a fundamental task is to embed appropriate and stable control schemes. Nevertheless, from a control perspective, designing control schemes for the quadrotor system constitutes a quite challenging issue. The difficulty arises from the fact that these systems possess particular characteristics such as the strong coupling nonlinearity, higher-order dynamics, underactuation, and multi-variable nature [4], [5].

Attempts to approach the stabilization and trajectory tracking control of the quadrotor with uncertain dynamics have been presented by applying linear and nonlinear control methods. Classic linear control algorithms such as proportional–derivative control were initially employed for quadrotors to actualize the position control [6], [7]; however, as remarked in [8], [9], these algorithms require a linearization of the quadrotor system at specific operation points and can barely perform tracking missions nearby these points. Motivated by these facts, some effective nonlinear control approaches have been proposed to carry out full-envelope tracking assignments for the quadrotors. With the help of model-free control and sliding mode control (SMC), a tracking control strategy is presented to control the attitude and position of a quadrotor in [10]. Nevertheless, this eminent strategy is restricted to locally approximated simple quadrotor model. Regarding the complicated original nonlinear model of the quadrotor, Lyapunov-based trajectory tracking schemes and robust adaptive control schemes are provided in the significant work [11], [12], [13]. However, the analytic calculation of the second derivatives of the desired (virtual) attitude, that is prohibitively difficult to obtain as emphasized in [14], is required by these schemes. Instead of using these analytic implementations, command filters are introduced to approximate these derivatives online and an adaptive control algorithm capable of stabilizing attitude and tracking trajectory is then designed for quadrotor system in [15]. Unfortunately, this algorithm suffers from the singular problems while computing the desired attitude for the control torque design. In light of robust compensator technique, a trajectory tracking controller without requiring desired attitude extractions is designed for quadrotors, and perfect tracking performance is achieved and verified through experiments in [16]. Under the presence of time-varying aerodynamic effect and bounded external disturbance, a novel augmented L1 adaptive strategy is presented in order to achieve precise trajectory control of quadrotors in [4]. Notwithstanding these efforts, little attention has been paid to the control of quadrotors in the presence of thrust saturation. Certainly, actuator saturation degrades the expected performance of the closed-loop system, and in a severe circumstance, may cause loss of closed-loop stability. Hence, taking into consideration the thrust saturation when designing control inputs is very important. Dealing with the singular problems induced by desired attitude extractions while considering the thrust constraint for the control of the quadrotor, via a hierarchical control strategy, is recently reported in [9], [14].

Taking account of the quite limited onboard computation power, a new simple control scheme is designed to realize trajectory tracking control for the quadrotors with unknown moment of inertia and known mass in this paper. It precludes the necessity for involved analytic computation of the derivatives of the desired (virtual) attitude that is commonly required in controlling quadrotors. Moreover, this control scheme is approximation-free in the sense that it does not include any adaptive laws, observers, and command filters to compensate for unknown parameters in the dynamics and the absence of the analytic differentiation. Owing to the aforementioned attributes, it needs quite a few and manageable calculations to generate the control input, thus rendering its implementation straightforward. Additionally, the thrust saturation is considered in the position control design by using differentiable saturated functions, which also enables the singularity in desired attitude extraction to be avoided entirely.

Compared with the existing related results, our contribution is mainly reflected in the following three aspects. First, different from the control algorithm [10], our developed scheme does not require the local simplified quadrotor model. Consequently, our results include a broader class of quadrotor systems and are more applicable to real-world applications than this published result. Second, our control scheme obviates the need for analytic calculation of the derivatives of the desired attitude that is essential in controlling quadrotor systems [9], [11], [12], [13]. Besides, no command filters [15], [17] are employed to obtain such knowledge or compensate for their absence. Therefore, the developed control scheme represents a computational simplicity solution and is more easily derived and executed. Third, in contrast to ultimate boundedness results in the context of the thrust saturation [14], the tracking errors can be adjusted arbitrarily small by the proposed control scheme via simply choosing the appropriate parameter in the angular velocity controller.

The rest of this paper is organized as follows. The control problem and its technical assumptions are formally represented in Section 2. The approximation-free simple control scheme is proposed in Section 3, and the theorem summarizing its main result is given in Section 4. Further, in Section 5, simulation and experimental studies are conducted to demonstrate the performance of the developed controller. Concluding notes are provided in Section 6.

Section snippets

Problem statement

The configuration of the quadrotor including the free body diagram and coordinates system is depicted in Fig. 1. Let B={BxByBz} represent the body fixed frame and its origin coincide with the mass center of the quadrotor. Let p=[x,y,z]TR3 denote the position of the mass center of the quadrotor expressed in the fixed inertial frame E={ExEyEz}, where xR, yR, and zR are, respectively, the longitudinal, latitudinal, and vertical positions. γ=[ϕ,θ,ψ]TR3 is the attitude of the quadrotor, where ϕ

Control scheme design

Since the uncertain quadrotor modeled by (1), (2) possesses the higher-order nonlinear dynamics with the mismatched condition, the control design will be proceeded by adopting a step-by-step procedure known as the backstepping technique [20]. Nevertheless, distinct from the standard backstepping technique, further efforts are needed to address the thrust saturation, the simple control design, and the strong nonlinear coupling between the position dynamics and the attitude angles. The design

Main result

So far, we have completed the control scheme design procedure for the quadrotor with unknown dynamics. The overall trajectory tracking control structure is displayed in Fig. 2. The main result of our proposed control scheme in Section 3 is summarized in the following theorem.

Theorem 1

Consider the closed-loop system consisting of the quadrotor (1))–(2), the desired trajectory pd and ψd satisfying Assumptions 12, the translational thrust (5), and the control torque (17). The position and attitude

Simulation study

In this section, simulations are carried out on a quadrotor to verify the effectiveness of the proposed control scheme. We consider the quadrotor with m=1.336 kg, l=0.8 m, Ix=0.0259 kg m2, Iy=0.0260 kg m2, Iz=0.0397 kg m2, Jr=0.0231 kg m2, and ω=0.12 rads. The gravity acceleration is set as g=9.8 ms2. The quadrotor is required to track the desired trajectory pd(t) and time-varying yaw ψd(t), which are described as pd(t)=[2cos(t3),2sin(t3),8+t10]T m and ψd=πsin(t3)6 rad.

The control

Conclusions

In this paper, a novel approximation-free simple control scheme for achieving trajectory tracking is proposed for quadrotor systems with uncertain parameters in the dynamics, which prevents the necessity for involved analytic computation of the derivatives of the desired (virtual) attitude. The thrust saturation capable of bypassing the singularity in desired attitude extraction is also approached in the position control design. Compared with the relevant results in the literature, the proposed

CRediT authorship contribution statement

Gang Wang: Conceptualization, Methodology, Writing - original draft, Simulation. Weixin Yang: Experimental validation, Formal analysis. Na Zhao: Experimental Validation. Yantao Shen: Writing - review & editing, Project administration. Chaoli Wang: Methodology.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Gang Wang received the B.Sc. degree in Information and Computing Science and the Ph.D. degree in Systems Analysis and Integration from the University of Shanghai for Science and Technology, Shanghai, China, in 2012 and 2017, respectively. He is currently a Research Associate in the Department of Electrical and Biomedical Engineering, University of Nevada, Reno, NV, USA. His research interests include distributed control of nonlinear systems, adaptive control, and robotics. He was a finalist for

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    Gang Wang received the B.Sc. degree in Information and Computing Science and the Ph.D. degree in Systems Analysis and Integration from the University of Shanghai for Science and Technology, Shanghai, China, in 2012 and 2017, respectively. He is currently a Research Associate in the Department of Electrical and Biomedical Engineering, University of Nevada, Reno, NV, USA. His research interests include distributed control of nonlinear systems, adaptive control, and robotics. He was a finalist for the Best Paper Award at the 2019 IEEE/ASME International Conference on Advanced Intelligent Mechatronics.

    Weixin Yang was born in Yunnan China, in 1991. He received his M.S degree in Electrical Engineer in 2016, and a Ph.D. in Electrical Engineering in 2019, from the University of Nevada, Reno, United State. His research interests include bio-inspired robotics design and control, focus on robotic motion control and path planning.

    Na Zhao received her B.S. and M.S. from Northeastern University in 2012 and 2014, respectively, and a Ph.D in Beijing Institute of Technology in 2019. Her research interests include coaxial rotors, morphing quadrotors and bio-mimetic robots.

    Yantao Shen received his Ph.D. degree in Mechanical and Automation Engineering from The Chinese University of Hong Kong in 2002. He is currently an associate professor with the Department of Electrical and Biomedical Engineering at the University of Nevada, Reno. Dr. Shen’s research is in the areas of Bio-instrumentation & Automation, Biomechatronics/robotics, Sensors and Actuators, and Tactile/Haptic Interfaces. His research has been supported by federal agencies such as the National Science Foundation (NSF), National Institute of Health (NIH), as well as the state’s and local agencies. Dr. Shen has published more than 120 research papers in the fields. Several publications have been nominated for or have won the Best Paper Awards, including in the IEEE international conferences ICRA, IROS, ROBIO, AIM and ROMAN. In addition, Dr. Shen is a recipient of NSF CAREER Award.

    Chaoli Wang is a Professor of School of Optical–Electrical and Computer Engineering at the University of Shanghai for Science and Technology. In 1999, he received the Ph.D. degree in Control Theory and Engineering from Beijing University of Aeronautics and Astronautics, China. He received his M.Sc. degree in 1992 and B.S. degree in 1986, both from Mathematics Department at Lanzhou University, China.

    From 1999 to 2000, he worked as a Postdoctoral Research Fellow with Robotics Laboratory of Chinese Academy of Sciences, China. From 2001 to 2002, he was a Research Associate at Department of Automation and Computer-Aided Engineering of The Chinese University of Hong Kong, Hong Kong. Since 2003, he has been with the Department of Electrical Engineering at the University of Shanghai for Science and Technology. Currently, Wang’s research interests are in the areas of nonlinear control, robust control, robot dynamic and control, visual servoing feedback control and pattern identification.

    This paper was recommended for publication by Associate Editor Chun-Yi Su.

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