Trajectory tracking for a quadrotor under wind perturbations: sliding mode control with state-dependent gains

Abstract

The problem of position tracking of a mini drone subject to wind perturbations is investigated. The solution is based on a detailed unmanned aerial vehicle (UAV) model, with aerodynamic coefficients and external disturbance components, which is introduced in order to better represent the impact of the wind field. Then, upper bounds of wind-induced disturbances are characterized, which allow a sliding mode control (SMC) technique to be applied with guaranteed convergence properties. The peculiarity of the considered case is that the disturbance upper bounds depend on the control amplitude itself (i.e. the system is nonlinear in control), which leads to a new procedure for the control tuning presented in the paper. The last part of the paper is dedicated to the analysis and reduction of chattering effects, as well as investigation of rotor dynamics issues. Conventional SMC with constant gains, proposed first order SMC, and proposed quasi-continuous SMC are compared. Nonlinear UAV simulator, validated through in-door experiments, is used to demonstrate the effectiveness of the proposed controls.

Introduction

In the last decades there has been a growing interest for small unmanned aerial vehicles both for military and commercial applications. They are useful in different domains such as monitoring, inspection, and other actions, especially in urban areas or nearby buildings and dangerous interiors, for natural calamities such as earthquakes. These machines are often required to move in unfamiliar environments in terms of geography and in terms of aerological conditions. In addition, the low mass of such units (comparing to the forces generated by the air disturbances) reduces significantly the domain of stable flight, then additional constraints have to be considered in the control design. Thus, it appears inevitable that, if we want to let UAVs operate in urban environments, inside turbulent air flow patterns for which accurate prediction is not possible a priori with limited resources, we need to focus on detailed aerodynamic models and sophisticated control laws. This paper is part of the four years project “Small drones in the wind” at ONERA Lille, which aims to use the drone itself as a wind sensor. First year was dedicated to the identification of the UAV model affected by wind velocity, which is used in this article and explained in [1]. This control design is carried out in parallel with development of a wind estimation tool in [2].

The considered problem consists in design of a nonlinear robust control law, which ensures a stable and efficient navigation of a small UAV under unpredictable wind perturbations. The model described in [1] can also be used to estimate wind velocity, in this way the estimated values can themselves be used as inputs to the control to properly adapt the gains on-line and use this estimation in path planning and trajectory control to smartly avoid collisions.

There exist many control design techniques to counteract the effects of wind perturbations on flight of small UAVs, among which SMC plays a keyrole. Many methods have been proposed in the literature, for instance, some principal SMCs with their relative sliding surfaces and Lyapunov functions are illustrated in [3], [4]. Its insensitivity to the model errors, parametric uncertainties and other disturbances and its ability to globally stabilize the underactuated systems are two advantages of the sliding mode controller [5]. Sliding mode algorithms are extensively applied to dynamic systems and optimal algorithms are discussed in [6].

Dozens of articles have applied SMC methodology to UAVs in order to solve the position and the attitude tracking problems ensuring robustness against external disturbances. They have been compared extensively with respect to other controls in in-door experiments such as in [7], where SMC is compared with backstepping control for micro quadrotor. Chattering-free sliding mode controller is proposed in [8], by replacing a sign function with a high-slope saturation function, in [9], the energy saving effect because of chattering reduction is also evaluated. Second order sliding mode is used in [10], where two different sliding manifolds are defined for fully actuated and underactuated subsystems. In the paper [11], a robust backstepping-based controller is proposed that induces integral sliding modes for the Newton–Euler underactuated dynamic model of a quadrotor subject to smooth bounded disturbances. The trajectory tracking of uncertain underactuated nonlinear dynamic systems is tackled by an adaptive fuzzy hierarchical sliding-mode control in [12]. In the work [13], the controller of the fully actuated subsystem using a robust terminal sliding mode control algorithm is designed. A controller to provide robust position and attitude of the vehicle while relying only on knowledge of the limits of the disturbances is proposed in [14]. Sliding mode techniques are used also as observers estimators of the effect of the external disturbances such as wind and noise, and the whole observer-estimator-control law approach is presented in [15]. A SMC is proposed to stabilize a class of cascaded under-actuated systems, in which the UAV system is divided, in [16]. Famous super-twisting algorithm, which is able to ensure robustness with respect to bounded external disturbances, is discussed in [17]. In [18], the authors proposed control algorithms similar to the ones described in this note, however they do not consider the rotors dynamics, the 3D trajectory, the first and second derivative of the UAV position and angles, making their approach interesting in a mathematical point of view but impracticable in 3D trajectory tracking flight and far from the hover condition.

The quadrotor model itself is well established in the literature under various simplifications, in the present note we investigate the influence of the wind taking into account an identification work for the aerodynamic coefficients, which is useful for the control design. Two robust nonlinear SMC law design are described, which consider realistic assumptions on external disturbances of quadrotors. In the considered case the upper bound of matched disturbances depends nonlinearly on the control itself, the system state vector and wind disturbances. The closed-loop system stability is ensured for a selected maximum admissible value of the wind speed. The control strategy proposed in this article can be equipped with an additional wind estimator algorithm, as in [2], reducing automatically the control effort on the rotors when it is not necessary. However the peculiarity of the proposed SMC approach design is that the control allows the UAV to remain stable even without any coupled external disturbance observer.

The paper outline is as follows. In Section 2 the considered UAV drone is described and the flight dynamics model is derived. The control and disturbance bounds are calculated in Section 3. The two control designs are presented in Section 4, where the rotors issue, the chattering, the physical constraints and tools used for the implementation are also illustrated. The results of numerical experiments are shown in Section 5. The remarks and discussion conclude the paper in Section 6.