Geometric Control of Quadrotor Attitude in Wind with Flow Sensing and Thrust Constraints

Quadrotor vehicles show great potential over a range of tasks, but effective control in windy environments continues to be a challenge. This paper develops a thrust-saturated controller on the Lie group SO(3) that uses flow sensing in order to reduce the effect of gusts on the vehicle. Designing the controller on SO(3) establishes almost-global exponential stability, and avoids the pitfalls of representing rigid-body kinematics using Euler angles. We prove that exponential stability is retained in the presence of thrust saturation. Aerodynamics are incorporated into the dynamics and control through a model of the blade-flapping phenomena experienced by rotorcraft. Numerical examples show that the system control remains effective despite thrust saturation, and that flow sensing improves both the initial response and steady-state error of the system in wind. INTRODUCTION Quadrotor helicopters have become popular in myriad tasks, ranging from entertainment to highly utilitarian work. Their ability to hover and maneuver with great agility provides advantages over fixed-wing aircraft, and their simplicity makes them preferable over traditional helicopters. A variety of control methods have been employed in order to stabilize quadrotors, including PID [1, 2], adaptive [3, 4], robust [5, 6], feedback linearization [7], and optimal [8] control. This paper employs a feedback linearization controller on the ∗Address all correspondence to this author. geometric Lie group SO(3) following [7], with the addition of saturation of the vehicle’s thrust inputs. Cao and Lynch [1], and Roza and Maggiore [9] approach thrust saturation using the nested saturation method from Teel [10], which is designed to address saturation in the case of a chain of integrators. Cao and Lynch [1] bound the roll and pitch angles of the system as well as the thrust by placing limits on system inputs, whereas Roza and Maggiore [9] place the bound on thrust only. Cutler and How [2] address saturation by choosing a trajectory that keeps the system states within the bounds required in order to avoid thrust saturation. This paper uses the method of Pappas et al. [11] to bound the thrust on the system in order to guarantee stability when the cost of feedback linearization does not saturate the thrust. Quadrotors are able to accomplish an array of tasks such as surveying farmland and aiding in natural disasters [12] that require multi-rotor aircraft to fly outdoors in potentially adverse weather. High winds pose a challenge to small UAS [13–15], and developing an understanding of how they respond to wind and the mechanics behind that response is key to compensation. This paper builds on previous work [16] in order to incorporate flow sensing in the attitude controller of a three-degree-of-freedom (DOF) quadrotor test stand. In [16], the blade-flapping response of a quadrotor propeller is analyzed in order to identify the aerodynamic moment acting on a propeller in the presence of a wind gust. This moment is described analytically and may be included in the model directly, rather than using an uncertainty block characteristic of a robust-control scheme. Using traditional inertial sensing along with flow sensing to predict the aerodynamic mo1 Copyright c © 2017 by ASME ment yields improved results as compared to using inertial feedback alone. Flow sensing will be performed with two-port, foreand aft-facing probes that measure flow using differential pressure measurements [17]. Flight-dynamics applications often apply Euler angles [18] due to the intuitive nature of measuring each angle directly. However, Euler angles kinematics suffer singularities at gimbal lock and each angle is not necessarily constrained to the unit circle, which allows for non-physical error representations in the case of deviations over one rotation. We therefore use a geometric approach on the Lie group SO(3), which is a compact set representing the configuration space of the orientation of a rigid body. Additionally, a geometric description of the rotational kinematics in SO(3) does not encounter singularities, allowing for potentially global solutions. The contributions of this paper are (1) the addition of propeller aerodynamics to the quadrotor dynamics, yielding a more accurate description in the presence of wind disturbances; (2) a nonlinear, feedback-linearizing controller on SO(3) with saturated thrust inputs; and (3) an assessment of the relative merits of adding flow sensing to the vehicle controller versus using inertial feedback alone. We show improved stabilization through the use of flow sensing, which promises to allow for controlled flight in unfavorable weather and improved safety when flight is required in spite of weather concerns. The outline of the paper is as follows. The first section describes the dynamics of the quadrotor vehicle. The second section describes the controller and proves stability under bounded thrust constraints. The third section shows numerical results comparing the controller with and without thrust saturation, and with and without flow sensing. The final section summarizes the paper and discusses ongoing work. QUADROTOR DYNAMICS This paper investigates a quadrotor constrained to operate on an attitude stand, such that the quadrotor is mounted on a ball joint at its center of mass, allowing for full attitude motion while constraining the translational degrees of freedom. Let rotation matrix R ∈ SO(3) represent the orientation of the vehicle’s body frame with respect to the inertial frame and employ rigid-body kinematics and Euler’s second law to describe the rotational dynamics. We have Ṙ = RΩ̂ JΩ̇ =−Ω× JΩ+Mthrust +Maero, (1) where Ω= [p,q,r]T is the angular velocity of the quadrotor in the body frame, Mthrust is the moment acting on the system due to the thrust input, Maero is the aerodynamic moment on the system due to the interaction between the rotors and the wind, and J is the moment of inertia of the quadrotor. Here, J is a diagonal matrix due to the symmetry of the quadrotor; specifically [J1,J2,J3] = [m``/12+2mm`,m``/12+2mm`,m``/6+4mm`] where m` is the mass of one cross-beam of the quadrotor, ` is the length of one cross-beam, and mm is the mass of each motor. The wedge operator ∧ converts a vector in R3 to a 3×3 skew symmetric matrix in so(3), which can also be used to represent a cross product, such that for any vectors x and y in R3, x̂y = x×y. The inverse of the wedge operator is the vee operator ∨, which transforms a matrix in so(3) to a vector in R3.