Adaptive Fast Smooth Second-Order Sliding Mode Control for Attitude Tracking of a 3-DOF Helicopter Xidong Wang, Zhan Li, Zhen He, Huijun Gao Abstract—This paper presents a novel adaptive fast smooth second-order sliding mode control for the atti- tude tracking of the three degree-of-freedom (3-DOF) heli- copter system with lumped disturbances. Combining with a non-singular integral sliding mode surface, we propose a novel adaptive fast smooth second-order sliding mode control method to enable elevation and pitch angles to track given desired trajectories respectively with the features of non-singularity, adaptation to disturbances, chattering suppression and fast finite-time convergence. In addition, a novel adaptive-gain smooth second-order sliding mode observer is proposed to compensate time-varying lumped disturbances with the smoother output compared with the adaptive-gain second-order sliding mode observer. The fast finite-time convergence of the closed-loop system with constant disturbances and the fast finite-time uniformly ultimately boundedness of the closed-loop system with the time-varying lumped disturbances are proved with the finite-time Lyapunov stability theory. Finally, the effective- Fig. 1. Structure of 3-DOF helicopter system with ADS ness and superiority of the proposed control methods are verified by comparative simulation experiments. Index Terms—Adaptive fast smooth second-order sliding mode control (AFSSOSMC), adaptive-gain smooth second- [4], [5], nonlinear control [2], [6]–[9] and intelligent control order sliding mode observer (ASSOSMO), 3-DOF helicopter. [10]–[12]. Among all the above-mentioned control methods, sliding mode control has attracted much attention because of its I. INTRODUCTION insensitivity and strong robustness to the disturbances. In the present sliding mode algorithms, the super-twisting algorithm UE to the merits of vertical take-off and landing, air is very popular and owns practical application value due to D hovering as well as aggressive maneuver, the small un- the features of finite-time convergence, strong robustness and manned helicopter has an extremely broad application prospect solely requiring the information of sliding mode variables in military and civil fields [1]. However, small unmanned [13]. In [14], a fast super-twisting algorithm is proposed to helicopters have the characteristics of high nonlinearity, strong solve the problem that the convergence speed becomes slow arXiv:2008.10817v2 [eess.SY] 27 Sep 2020 coupling, under-actuated and extremely vulnerable to lumped when the system states are far away from their equilibrium disturbances during flight, which make it a challenge to design points existing in [13]. However, the bound of disturbance the high-performance attitude tracking controller [2]. needing to be known in advance restricts the applications of Because of the similar dynamics with the real helicopter these super-twisting algorithms. To deal with this problem, an system, the 3-DOF laboratory helicopter, as shown in Figure adaptive super-twisting algorithm is proposed in [15], which 1, can act as an ideal experimental platform for testing various can adapt to the disturbance of unknown boundary. Combining advanced control methods of the helicopter [3]. In recent the merits of [14] and [15], an adaptive fast second-order years, researchers have proposed numerous methods to achieve sliding mode control method is proposed in [16]. An observer the attitude tracking object of 3-DOF helicopter and verified is also designed in the light of this method, but the output these methods by the helicopter experimental platform. These of the proposed observer is not smooth. In [17], a smooth methods can be mainly divided into three parts: linear control second-order sliding mode control is proposed to further alleviate the chattering effect existing in [13]. In [18], a fast smooth second-order sliding mode method is present based Xidong Wang is with the Research Institute of Intelligent Control and Systems, School of Astronautics, Harbin Institute of Technology, Harbin on the method of [17]. However, this method cannot adapt 150001, China (e-mail: [email protected]). to the unknown boundary disturbances. Considering all the above-mentioned modified super-twisting methods, there is TABLE I no method that can simultaneously achieve fast finite-time THE PARAMETERS OF THE 3-DOF HELICOPTER SYSTEM convergence, adaptation to the disturbances and smooth as of now. Symbol Definition Value 2 Jα Moment of inertia of elevation axis 1:0348kg · m In this paper, inspired by [16], [18], we propose a class 2 Jβ Moment of inertia of pitch axis 0:0451kg · m of smooth adaptive fast second-order sliding mode control La Distance from elevation axis to the center 0:6600m (SAFSOSMC) method and the method in [16] can be regarded of helicopter body as a special case of our proposed method to some extent. Lh Distance from pitch axis to either motor 0:1780m m Effective mass of the helicopter 0:094kg The new proposed method integrates the advantages of all the g Gravitational acceleration constant 9:81m=s2 modified super-twisting algorithms mentioned before, includ- K Propeller force-thrust constant 0:1188N=V ing fast finite-time convergence, adaptation to the disturbances f V Front motor voltage input [−24; 24]V and smooth. To the best of the authors’ knowledge, this is f V Back motor voltage input [−24; 24]V the first time that a modified super-twisting algorithm can b simultaneously own the above three merits. Based on this new method, a novel adaptive-gain smooth second-order sliding II. PROBLEM FORMULATION AND PRELIMINARIES mode observer (ASSOSMO) is also proposed to alleviate chat- tering effect and adjust the parameters automatically without A. The dynamics of the 3-DOF Helicopter knowing the boundary of the lumped disturbances derivative. As shown in Fig.1, the 3-DOF helicopter system studied The main contributions of this paper can be summarized as in this paper has elevation, pitch and travel motions, which follows: are driven by two DC motors called the front motor and back motor. A positive voltage applied to each motor can 1) A novel SAFSOSMC method is proposed to enable ele- generate the elevation motion, and a higher voltage applied on vation and pitch angles to track given desired trajectories the front motor can produce the positive pitch motion . The respectively, which not only maintains the characteristics travel motion can be generated by thrust vectors when the of fast finite-time convergence and adaptation to the distur- helicopter body is pitching. Moreover, an active disturbance bance of unknown boundary, but also enormously reduces system (ADS) serving as the lumped disturbances is installed the chattering effect. Combined with a non-singular integral on the arm. sliding mode surface [19], this method can ensure the fast Due to the under-actuated mechanism of the 3-DOF heli- finite-time convergence of the tracking errors with constant copter system, only two of the three degree of freedoms can disturbances and the fast finite-time uniformly ultimately be controlled to track arbitrary trajectories in the operating boundedness with the time-varying lumped disturbances. domain. In this work, we investigate the elevation and pitch 2) A novel ASSOSMO is proposed to estimate the time- motions, while the travel motion is set to move freely. The varying lumped disturbances of helicopter system control models of elevation and pitch channels can be expressed as channels with smoother output than that of the adaptive- follows [2] gain second-order sliding mode observer (ASOSMO) pro- posed in [16]. Jαα¨ = Kf La cos(β)(Vf + Vb) − mgLa cos(α) ¨ (1) Jββ = Kf Lh(Vf − Vb) The fast finite-time convergence of the closed-loop system where α and β represent the elevation and pitch angle re- with constant disturbances and the fast finite-time uniformly spectively. Taking into account the mechanical constraints, the ultimately boundedness of the closed-loop system with time- operating domain of the helicopter system is defined as follows varying lumped disturbances will be proved with the cor- o o responding finite-time Lyapunov stability theory. The effec- −27:5 ≤ α ≤ +30 (2) tiveness and superiority of the proposed control methods are −45o ≤ β ≤ +45o verified by comparative experiments. The definitions and values of the relevant parameters are _ The remainder of this paper is organized as follows. In shown in Table I. Denote x1 = α; x2 =α; _ x3 = β; x4 = β. In Section 2, the dynamic model and control objective of the view of the external disturbance and system uncertainty, the 3-DOF helicopter system, some essential lemmas and the model of helicopter system can be rewritten as new proposition are given. The controller design process and x_ = x stability analysis of the closed-loop system are presented in 1 2 L g Section 3. Section 4 provides contrastive experiment results x_ = a cos(x )u − mL cos(x ) + d (x) 2 J 3 1 J a 1 1 and discussion. Section 5 concludes this paper. α α (3) x_ 3 = x4 Notation: In this paper, we use k·k to denote the Euclidean Lh norm of vectors and sign(·) to denote the standard signum x_ 4 = u2 + d2(x) Jβ function. Moreover, λmax (·) and λmin (·) are used to denote T the maximum and minimum eigenvalues of a matrix, respec- where x = [x1; x2; x3; x4] denotes the state vector of system tively. (1), which is assumed to be measurable in this study. d1 (x) and d2 (x) represent the lumped disturbances existing in the where θ1 and θ2 are arbitrary positive constants holding corresponding control channels. In addition, u1and u2 are θ1 2 (0; c1); θ2 2 (0; c2), then x(t; x0) can converge to a defined by neighborhood of origin in a finite time T . In addition, the u1 = Kf (Vf + Vb) convergent region can be given by: (4) u2 = Kf (Vf − Vb) n p1−p2 1−p2 o D = x : θ1V (x) + θ2V (x) < c3 (10) The control objective is to design the controllers such that the elevation and pitch angles can track the given desired Define an auxiliary variable θ3 2 (0; 1).