Elsevier

Mechatronics

Volume 61, August 2019, Pages 58-68
Mechatronics

A novel Lyapunov-based trajectory tracking controller for a quadrotor: Experimental analysis by using two motion tasks

https://doi.org/10.1016/j.mechatronics.2019.05.006Get rights and content

Abstract

A novel model-based controller for quadrotor trajectory tracking is presented in this paper. The closed-loop stability is studied by Lyapunov’s theory guaranteeing local asymptotical stability of the resulting equilibrium point. The performance of the proposed scheme is compared in real time with respect to three known trajectory tracking controllers having different structures. Besides, two desired trajectories encoding different motion tasks are specified. The gains of the tested controllers are selected so that the mean value of the total thrust is the same. An analysis by using different performance indexes is employed in order assess the tracking performance of each scheme. The proposed controller presents the best experimental execution for the two implemented motion tasks.

Introduction

Much research in recent years has been focused on unmanned aerial vehicles (UAVs). These vehicles have captured the interest of the scientific community because they can be employed in a wide range of applications. Some examples of such applications are search, rescue, survey, transport, and mapping [1], [2]. A particular aerial unmanned vehicle is the quadrotor which includes four rotors placed in an X-shaped structure. This configuration allows the quadrotor performing hovering and vertical take-off and landing. However, the design of control schemes that ensure stability is a challenge since they are under-actuated systems with a highly nonlinear and strongly coupled dynamics, while the environmental disturbances affect it considerably.

The importance of the control schemes in the autonomous flight of quadrotors is crucial. Different controllers have been developed and tested. Classical PD and PID control laws were proposed for posture regulation in [3] and [4], respectively. A PID/PD double loop controller is proposed in [5]. The controller regulates the quadrotor pose under system variations, such as the battery drainage and mass changes. In [6], an optimal PID controller based on a multi-objective particle swarm optimization was designed and implemented to perform navigation task. A PID-type controller was introduced in [7], where the robustness was validated experimentally when disturbances occur in one of the actuators. The model-based control is another strategy in which the knowledge of the dynamic model and the system parameters were assumed. In [8], parameter identification was performed for a quadrotor. Besides, a model-based controller was proposed and real time experiments were presented. In [9], the authors introduced an attitude model-based control on the exponential coordinates parametrization of rotation. A dynamic model of quadrotor obtained by Hamiltonian approach was presented in [10]. The controller based on this model was developed and validated theoretically and experimentally. In [11], the quadrotor dynamic model was obtained by probabilistic Gaussian process models. Subsequently, a model predictive control scheme was proposed to address the trajectory tracking problem. Feedback linearization has been applied to the development of quadrotor controllers. In [12], the quadrotor dynamics was simplified by feedback linearization to design a linear formation control law which allows achieving the desired formation and heading synchronization. A global tracking strategy based on the chartwise dynamic feedback linearization was presented in [13]. Additionally, the positive sign of the thrust signal was guaranteed by adding a dynamic extension of it. In [14], the author proposed a method to design controllers in Euclidean space for systems defined on manifolds. The application of this methodology in the quadrotor was presented together with the theoretical analysis which guarantees exponential tracking.

A popular methodology for the design of quadrotor controllers is the backstepping technique. In [15], the quadrotor was stabilized with a controller designed by using Lyapunov and backstepping theory. In [16], a backstepping controller was proposed, which guarantees the stability of the quadrotor affected by external disturbances. An original control approach using backstepping-like feedback linearization was presented in [17]. The theoretical analysis ensures asymptotically stability for the tracking of the position and stabilizing the attitude of the quadrotor. Other optimal control techniques have been used in the quadrotor control. An LQR controller was implemented in a quadrotor to achieve the consensus of a heterogeneous multiagent system in [18]. In [19], a model predictive control and a fuzzy logic scheme were combined to develop a quadrotor tracking controller.

The design of a controller which provides robustness to the quadrotors flight also has been addressed considerably. For example, a robust global path following control law was introduced and experimentally validated in [20]. The controller rejects constant force disturbances, as is the case of constant wind or model uncertainties, while the path following error converges to zero globally. In [21], a robust control scheme based on a predictor was developed for time-delay systems, and its application in quadrotors was presented. In [22], a robust cascade controller was proposed to ensure the position and attitude tracking errors convergence to a neighborhood of the origin despite the disturbances and state time delays. In [23], a controller and an observer to estimate force and torque disturbances were combined to guarantee asymptotic stability of the overall closed-loop system. This scheme was experimentally validated in environments with obstacles and disturbances. A multi-time-scale controller was presented in [24]. The system uncertainties were estimated by a finite-time extended state observer while the controller guaranteed stability. In [25], the authors introduced a nonlinear internal model control (NLIMC) approach, which increases the robustness of the system changing the classic dynamic inversion technique by the flatness property. The sliding mode approach has been a common solution in the quadrotor control. In [26], a double-loop control system was proposed. An integral sliding mode controller for roll and pitch angles subsystem was applied while the position and yaw angle were controlled by classical techniques. In [27], the authors presented a controller based on sliding mode theory using an input-dependent sliding surface, which vanished the steady-state errors. Vision systems have also been used in control strategies to improve the performance of the quadrotors. A visual servoing scheme which estimates the linear velocity with an observer based on image features was presented in [28]. In [29], an image-based visual servo scheme with bounded-input was designed for the vertical take-off and landing task. A controller to track the feature trajectory which is defined directly in the image space was introduced in [30].

As can be observed in the bibliography review, a wide variety of linear-type and nonlinear controllers have been proposed to achieve the desired performance for quadrotors. Besides, multiple studies have presented numerical and experimental comparative analyses between control schemes. In [31], the authors presented the results of two control schemes, classical PID and LQ approach, applied to a quadrotor. Similarly, in [32], a feedback linearization controller was compared with an adaptive sliding mode controller, which is robust to sensor noise, and some uncertainty in the dynamic model. The performance of five nonlinear feedback control laws with saturation elements was evaluated numerically in [33]. The experimental validation was done for only one scheme and included only the hovering task. Hovering was established in [34] in order to evaluate the performance of three control strategies. The results showed that nested saturation control approach is the best option to stabilize the vehicle position with respect to an artificial visual landmark on the ground. In [35], different control schemes with quaternion representation were analyzed using numerical simulation. The analyzed schemes resulted from the combination of known control techniques.

It should be noticed, that different performance comparisons of linear-type and nonlinear controllers have been presented in literature. However, the real-time evaluation of control schemes with different structure in distinct motion tasks is a subject that deserves deeper study.

The contributions of this paper are as follows:

  • A novel controller, which is based on the dynamic model of the quadrotor.

  • A stability analysis based on Lyapunov’s theory which guarantees local asymptotically stability of the closed-loop system.

  • Experimental validation of the proposed controller for two different trajectory motion tasks. One corresponds to a trajectory drawing a circular path and another to a lemniscate path.

  • Performance analysis in real-time tasks for three control schemes taken from the literature and the proposed controller, which can be classified as linear-type and nonlinear.

  • The gains of the all tested controllers were selected so that the mean value of the total thrust produced by a controller was within 5% those of the others, which is a relevant consideration, since to the best of our knowledge no energetic criteria have been used to select gains when comparing quadrotor controllers.

While in our previous work [36] was devoted to study the robustness of a motion control scheme for a quadrotor, in this document a new controller is given for which the closed-loop system trajectories are rigorously studied by using the framework of the Lyapunov theory. Besides, the experimental tests presented here allow drawing important conclusions about the performance of the proposed controller and the structure of the studied schemes.

The remaining of this paper is organized as follows. Section 2 presents the quadrotor dynamics. The control objective, proposed controller, and stability analysis are given in Section 3. Section 4 presents the results obtained from the real-time experimental with the proposed controller and the already reported schemes. Besides, different performance indexes are discussed there. Finally, some conclusions derived from the experimental validation are expressed in Section 5.

Section snippets

Quadrotor dynamic model

In this Section, the motion equations of the quadrotor are presented. Fig. 1 shows the quadrotor in the fixed inertial and body reference frame.

In this work, we assume that the body reference frame coincides with the center of mass of the quadrotor. As result, the translational and the attitude dynamics can be separated into two equations as follows [37], [38]:mp¨+mgez=f,Ho(η)η¨+Co(η,η˙)η˙=τ,where p=[xyz]TR3 is the position with respect to the inertial frame, η=[ϕθψ]TR3 is the vector of Euler

Model-based control scheme

In this Section, we present a novel control scheme which is inspired in passivity-based control and the compensation of the nonlinear terms of the quadrotor dynamics. The trajectory tracking control goal is established, and the stability of the closed-loop system is proven using Lyapunov theory.

Experimental results

Real-time experiments have been conducted in the Qball 2 quadrotor. The attitude η(t) and the angular velocity ω(t) are obtained by an inertial measurements unit, while a sonar sensor provides the quadrotor height z(t). Additionally, the Optitrack motion capture system estimates position measurements. The values of the dynamic and kinematic parameters used in the controller implementations were obtained from the QBall 2 quadrotor user manual [41], and are shown in Table 1.

For the experimental

Conclusions

In this paper, the problem of position and orientation trajectory tracking control for a quadrotor was addressed. A model-compensation control scheme was designed for the trajectory tracking of a quadrotor. The closed-loop system was analyzed and its stability was proven by Lyapunov’s theory. The performance of linear and nonlinear control schemes having different structures was analyzed in two different trajectory tracking motion tasks. By using gains ensuring all the controllers have the same

Declaration of interests

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.

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Ricardo Párez-Alcocer was born in Mérida, México, in 1981. He received his B.Sc. degree in Computer Sciences from the University of Yucatán, México, in 2004, the M.Sc. in Mathematics from the University of Yucatán, México, in 2007, and the Ph.D. degree in Robotics and Advanced Manufacturing from CINVESTAV Research Center, Saltillo, México, in 2013. He is currently a CONACYT Research Fellow and is with Instituto Politécnico Nacional-CITEDI. His research interests include unmanned vehicles

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      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 [9,11–13]. Besides, no command filters [15,17] are employed to obtain such knowledge or compensate for their absence.

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    Ricardo Párez-Alcocer was born in Mérida, México, in 1981. He received his B.Sc. degree in Computer Sciences from the University of Yucatán, México, in 2004, the M.Sc. in Mathematics from the University of Yucatán, México, in 2007, and the Ph.D. degree in Robotics and Advanced Manufacturing from CINVESTAV Research Center, Saltillo, México, in 2013. He is currently a CONACYT Research Fellow and is with Instituto Politécnico Nacional-CITEDI. His research interests include unmanned vehicles (aerial, aquatics and wheeled), linear and nonlinear control, multi-agent systems, computer vision and intelligent systems.

    Javier Moreno-Valenzuela received the Ph.D. degree in Automatic Control from CICESE Research Center, Ensenada, México, in 2002. He was a Postdoctoral Fellow at the Université de Liége, Belgium, from 2004 to 2005. He is with the Instituto Politécnico Nacional-CITEDI, Tijuana, México. He is the author of many peer reviewed journal and international conference papers and the book entitled: Motion Control of Underactuated Mechanical Systems (Springer-Verlag, 2018). He has served as reviewer of a number of prestigious scientific journals. Actually, he is associate editor of IEEE Latin America Transactions and Mathematical Problems in Engineering. His research interests include nonlinear systems, mechatronics and intelligent systems.

    This work was supported by CONACYT Project Cátedras 1537, Project A1-S-24762, and by Secretaría de Investigación y Posgrado del IPN, México.

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