Bounded attitude stabilization: Application on four-rotor helicopter J. F. GUERRERO-CASTELLANOS, A. HABLY, N. MARCHAND, S. LESECQ Abstract— A quaternion based feedback is developed for attitude stabilization of rigid body. The control design takes into account the input bounds and is based on cascaded saturation approach. The global stability is guaranteed. A simulation study of the proposed scheme is illustrated for the four-rotor helicopter. I. I NTRODUCTION The problem of attitude control of a rigid body has attracted considerable amount of interest since the 1950’s within the scientific communities of aeronautics, aerospace, control and robotics. This is due to the fact that many systems such as spacecrafts, satellites, helicopters, tactical missiles, coordinated robot manipulators, underwater vehicles and others enter within the framework of rigid body with a need for attitude control. Several approaches are applied such as optimal time control [13], Lyapunov design procedures [20], quaternion feedback [4], [6] and [22], predictive control (applied to spacecraft in [23] and to micro satellite in [5]), backstepping (quaternion-based in [8] and nonlinear adaptive in [15]), and robust control applied to tactical missiles [16]. This list is not exhaustive. The above cited approaches do not consider the problem of attitude control which takes the input constraints into account. Few publications have treated this problem. In [21], the stabilization with non smooth control law of an under- actuated rigid spacecraft subject to input saturation is studied. In [1], a control law that drives a rigid underwater vehicle between arbitrary initial and final region of the state space while satisfying bounds on control and state is proposed. The authors in [2] have studied the robust sliding mode stabiliza- tion of the spacecraft attitude dynamics in presence of control input saturation based on the variable structure control (VSC) approach. Unfortunately, the stabilizing bounded control law that are applied are non smooth and this fact renders difficult the practical implementation. The approach proposed in the present paper is more in the spirit of the approach [24] where the problem of reorienting a rigid spacecraft within the physical limits of actuators has been investigated based on the cascaded saturation approach proposed by [19]. However, in [24] no formal stability proof is given. Although Teel’s results is nice and founding, its performance in term of convergence speed is very poor for system of dimension n ≥ 3 [9]. Nevertheless, as is mentioned in [12], for a double This work was partly supported by CONACYT-M´ exico and CNRS-Liban The authors are with Control System Department, GIPSA-lab, CNRS UMR 5216, CNRS-INPG-UJF, ENSIEG BP 46, 38402 Saint Martin d’H` eres Cedex, France fergue@lag.ensieg.inpg.fr, Ahmad.Hably@inpg.fr, Nicolas.Marchand@inpg.fr, Suzanne.Lesecq@inpg.fr integrator plant, such as the presented in this work, the Teel’s approach presents a good performance with regard to settling time stabilization. The orientation of a rigid body can be parameterized by several methods: a rotation matrix, a unit quaternion (i.e. Euler parameters) and Euler angles. The unit quaternion is a four-parameter representation and is considered as a globally nonsingular parametrization. For more details on attitude representations, the reader can refer to the survey written by Shuster [14]. In this paper, the bounded attitude control of a rigid body is studied. The control scheme is applied on a four-rotor helicopter. The complete model of this special type of mini helicopter, also known as X-4 flyer or even quad-rotor, is developed in [10], [11]. This four-rotor helicopter has some advantages over conventional helicopters: owing to symmetry, this vehicle is dynamically elegant, inexpensive, and simple to design and construct. To our knowledge, the attitude stabilization of the four-rotor helicopter using quaternion feedback was firstly studied in [17] and more recently in [18]. In these papers, a quaternion-based feedback control scheme for attitude stabilization is applied without considering the boundedness of the control inputs . The present paper is organized as follows. In section II, a rigid body quaternion-based orientation is given. The main problem is formulated in section III. The control law design is presented and its stability is proved in section IV. The application of this control law on a four-rotor helicopter is explained in section V. The simulation results are given in section VI. The paper ends with some conclusions given in section VII. II. MATHEMATICAL BACKGROUND As mentioned in the introduction, the attitude of a rigid body can be represented by a quaternion, consisting of a unit vector  e, known as the Euler axis, and a rotation angle β about this axis. The quaternion q is then defined as follows q =  cos β 2  e sin β 2  =  q 0  q  ∈ H (1) where H = {q | q 2 0 + q T  q = 1, q =[q 0  q] T , q 0 ∈ R, q ∈ R 3 } (2)  q =[q 1 q 2 q 3 ] T and q 0 are known as the vector and scalar parts of the quaternion respectively. In attitude control applications, the unit quaternion represents the rotation from an inertial coordinate system N(x n , y n , z n ) located at some point in the space (for instance, the earth NED frame), to 2007 IEEE International Conference on Robotics and Automation Roma, Italy, 10-14 April 2007 WeB12.2 1-4244-0602-1/07/$20.00 ©2007 IEEE. 730