1. Introduction

Quadrotor helicopters, also known as "quadrotors," are currently employed in diverse scenarios, which range from search and rescue missions to infrastructure inspection, precision agriculture, and wildlife monitoring ([1, Ch. 1], [2, 3]). Employing quadrotors in enclosed industrial environments or in proximity of untrained personnel is still considered as a challenge for the high reliability

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

required to these aircraft. Additional complexity in the use of quadrotors for commercial applications, such as parcel delivery, is that users demand satisfactory trajectory following capabilities without tuning the controller's gains prior to each mission, whenever the payload is changed.

properties, such as mass, moment of inertia, and location of the center of mass are unknown. Moreover, we assume that the quadrotor's reference frame is centered at an arbitrary point, which does not necessarily coincide with the vehicle's center of mass. In addition, we suppose that the coefficients characterizing the aerodynamic force and moment are unknown. In order to reduce the rotational kinematic and dynamic equations to a form that is suitable for MRAC, we employ an output-feedback linearization approach so that the controlled rotational dynamics is captured by a linear dynamical system, whose virtual input is designed using the

Robust Adaptive Output Tracking for Quadrotor Helicopters

http://dx.doi.org/10.5772/intechopen.70723

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We design the autopilot's outer loop so that it regulates the vehicle's position in the inertial frame and the inner loop so that it regulates the vehicle's attitude. In conventional autopilots for quadrotors, the outer loop regulates the vehicle's position in the horizontal plane and the inner loop controls the quadrotor's altitude and orientation. As in conventional architectures, our outer loop defines the reference pitch and roll angles for the inner loop to track. However, our control architecture allows us to verify a priori that the reference pitch and roll angles meet the sufficient conditions for strong accessibility of the quadrotor's altitude and rotational dynamics. Conventional autopilots' outer loop may generate large reference pitch and roll angles that are

The conventional MRAC architecture ([17], Ch. 9) is designed to regulate time-invariant dynamical systems and, for this reason, autopilots for quadrotors based on the classical MRAC are unable to account for time-varying terms in the vehicle's dynamics, such as the inertial counter-torque. Moreover, autopilots for quadrotors based on the classical MRAC architecture are robust to both matched and parametric uncertainties, but not unmatched uncertainties, such as aerodynamic forces. The autopilots presented in this chapter, instead, are robust to unmatched uncertainties as well and account for the fact that quadrotors are inherently time-varying dynamical systems.

A numerical example illustrates our theoretical framework by designing a control law that allows a quadrotor to follow a circular trajectory, although the vehicle's inertial properties are unknown, one of the motors is suddenly turned off, the payload is dropped over the course of the mission, and the wind blows at 16 m/s, which is usually considered as a prohibitive velocity for conventional quadrotors. In this example, we clearly show how the proposed robust control algorithm outperforms both the classical proportional-derivative (PD) control and the conventional MRAC. Indeed, it is shown that quadrotors implementing autopilots based on the PD framework crash as soon as one motor is turned off. Moreover, we verify that quadrotors implementing autopilots based on the classical MRAC framework [6] are unable to fly in the presence of wind gusts faster than 6 m/s. Lastly, we show that, to fly in a wind blowing at 16 m/s, our autopilot requires a control effort that is smaller than the one required

In this section, we establish the notation and the definitions used in this chapter. Let R denote the set of real numbers, <sup>R</sup><sup>n</sup> the set of <sup>n</sup> 1 real column vectors, and <sup>R</sup><sup>n</sup> <sup>m</sup> the set of <sup>n</sup> <sup>m</sup> real

proposed robust MRAC architecture.

not guaranteed to lay in the vehicle's reachable set.

by a conventional MRAC-based autopilot to fly in a 6 m/s wind.

2. Notation and definitions

Autopilots for commercial-off-the-shelf quadrotors are currently designed assuming that the vehicle's and the payload's inertial properties are known and constant in time. Moreover, it is assumed that the propulsion system is able to deliver maximum thrust whenever needed. These assumptions considerably simplify the design of control algorithms for quadrotor helicopters, but also undermine these vehicles' reliability in challenging work conditions, such as in case the propulsion system is partly damaged or the payload is not rigidly attached to the vehicle. For instance, the authors in [4] show that if the payload's mass and matrix of inertia vary in time, then autopilots for quadrotors designed using classical control techniques, such as the proportional-derivative control, are inadequate to guarantee satisfactory trajectory tracking.

In recent years, numerous authors, such as Bouadi et al. [5]; Dydek et al. [6]; Jafarnejadsani et al. [7]; Loukianov [8]; Mohammadi & Shahri [9]; Zheng et al. [10], to name a few, employed nonlinear robust control techniques, such as sliding mode control, model reference adaptive control (MRAC), adaptive sliding mode control, and L<sup>1</sup> adaptive control, to design autopilots for quadrotors that are able to account for inaccurate modeling assumptions and compensate failures in the propulsion system. These autopilots are generally designed assuming perfect knowledge of the location of the quadrotor's center of mass, supposing that the vehicle's Euler angles are small at all times, and neglecting the inertial counter-torque. Furthermore, in several cases also the aerodynamic force and the corresponding moment are omitted. Because of these simplifying assumptions, these autopilots are inadequate for aircraft performing aggressive maneuvers, flying in adverse weather conditions, and transporting payloads not rigidly connected to the vehicle's frame [11]. The vehicle's guidance system is usually delegated to avoiding obstacles detected by proximity sensors and cameras installed aboard. For details, see the recent works by Faust et al. [12]; Gao & Shen [13]; Lin & Saripalli [14].

In the first part of this chapter, we present the equations of motion of quadrotors and analyze those properties needed to design effective nonlinear robust controls that enable output tracking. Specifically, we present the equations of motion of quadrotors without assuming a priori that the Euler angles are small and without neglecting the inertial counter-torque and the gyroscopic effect. Since the inertial counter-torque cannot be expressed as an algebraic function of the quadrotor's state and control vectors, we account for this effect as an unmatched time-varying disturbance on the vehicle's dynamics and hence, we consider the equations of motion of a quadrotor as a nonlinear time-varying dynamical system. Successively, we verify for the first time sufficient conditions for the strong accessibility of quadrotors' altitude and rotational dynamics; strong accessibility [15] is a weak form of controllability for nonlinear time-varying dynamical systems. As a result of this analysis, we show that a conservative control law for quadrotors must prevent rotations of a π/2 angle about either of the two horizontal axes of the body reference frame; otherwise, the vehicle may be uncontrollable.

In the second part of this chapter, we present a robust autopilot for quadrotors, which is based on a version of the e-modification of the MRAC architecture [16]. This autopilot is characterized by numerous unique features. For instance, we assume that the quadrotor's inertial properties, such as mass, moment of inertia, and location of the center of mass are unknown. Moreover, we assume that the quadrotor's reference frame is centered at an arbitrary point, which does not necessarily coincide with the vehicle's center of mass. In addition, we suppose that the coefficients characterizing the aerodynamic force and moment are unknown. In order to reduce the rotational kinematic and dynamic equations to a form that is suitable for MRAC, we employ an output-feedback linearization approach so that the controlled rotational dynamics is captured by a linear dynamical system, whose virtual input is designed using the proposed robust MRAC architecture.

We design the autopilot's outer loop so that it regulates the vehicle's position in the inertial frame and the inner loop so that it regulates the vehicle's attitude. In conventional autopilots for quadrotors, the outer loop regulates the vehicle's position in the horizontal plane and the inner loop controls the quadrotor's altitude and orientation. As in conventional architectures, our outer loop defines the reference pitch and roll angles for the inner loop to track. However, our control architecture allows us to verify a priori that the reference pitch and roll angles meet the sufficient conditions for strong accessibility of the quadrotor's altitude and rotational dynamics. Conventional autopilots' outer loop may generate large reference pitch and roll angles that are not guaranteed to lay in the vehicle's reachable set.

The conventional MRAC architecture ([17], Ch. 9) is designed to regulate time-invariant dynamical systems and, for this reason, autopilots for quadrotors based on the classical MRAC are unable to account for time-varying terms in the vehicle's dynamics, such as the inertial counter-torque. Moreover, autopilots for quadrotors based on the classical MRAC architecture are robust to both matched and parametric uncertainties, but not unmatched uncertainties, such as aerodynamic forces. The autopilots presented in this chapter, instead, are robust to unmatched uncertainties as well and account for the fact that quadrotors are inherently time-varying dynamical systems.

A numerical example illustrates our theoretical framework by designing a control law that allows a quadrotor to follow a circular trajectory, although the vehicle's inertial properties are unknown, one of the motors is suddenly turned off, the payload is dropped over the course of the mission, and the wind blows at 16 m/s, which is usually considered as a prohibitive velocity for conventional quadrotors. In this example, we clearly show how the proposed robust control algorithm outperforms both the classical proportional-derivative (PD) control and the conventional MRAC. Indeed, it is shown that quadrotors implementing autopilots based on the PD framework crash as soon as one motor is turned off. Moreover, we verify that quadrotors implementing autopilots based on the classical MRAC framework [6] are unable to fly in the presence of wind gusts faster than 6 m/s. Lastly, we show that, to fly in a wind blowing at 16 m/s, our autopilot requires a control effort that is smaller than the one required by a conventional MRAC-based autopilot to fly in a 6 m/s wind.
