Robust Adaptive Geometric Tracking Controls on SO(3) with an Application to the Attitude Dynamics of a Quadrotor UAV Taeyoung Lee ∗ Abstract— This paper provides new results for a robust adaptive tracking control of the attitude dynamics of a rigid body. Both of the attitude dynamics and the proposed control system are globally expressed on the special orthogonal group, to avoid complexities and ambiguities associated with other attitude representations such as Euler angles or quaternions. By designing an adaptive law for the inertia matrix of a rigid body, the proposed control system can asymptotically follow an attitude command without the knowledge of the inertia matrix, and it is extended to guarantee boundedness of tracking errors in the presence of unstructured disturbances. These are illustrated by numerical examples and experiments for the attitude dynamics of a quadrotor UAV. I. I NTRODUCTION The attitude dynamics of a rigid body appears in various engineering applications, such as aerial and underwater vehi- cles, robotics, and spacecraft, and the attitude control prob- lem has been extensively studied under various assumptions (see, for example, [1], [2], [3], [4]). One of the distinct features of the attitude dynamics is that its configuration manifold is not linear: it evolves on a nonlinear manifold, referred as the special orthogonal group, SO(3). This yields important and unique properties that cannot be observed from dynamic systems evolving on a linear space. For example, it has been shown that there exists no continuous feedback control system that asymptotically stabilizes an attitude globally on SO(3) [?], [5]. However, most of the prior work on the attitude con- trol is based on minimal representations of an attitude, or quaternions. It is well known that any minimal attitude representations are defined only locally, and they exhibit kinematic singularities for large angle rotational maneuvers. Quaternions do not have singularities, but they have ambigui- ties in representing an attitude, as the three-sphere S 3 double- covers SO(3). As a result, in a quaternion-based attitude control system, convergence to a single attitude implies convergence to either of the two disconnected, antipodal points on S 3 [6]. Therefore, depending on the particular choice of control inputs, a quaternion-based control sys- tem may become discontinuous when applied to an actual attitude dynamics [7], and it may also exhibit unwinding behavior, where the controller unnecessarily rotates a rigid body through large angles [5], [8]. Geometric control is concerned with the development of control systems for dynamic systems evolving on nonlinear Taeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute of Technology, Melbourne, FL 39201 taeyoung@fit.edu ∗ This research has been supported in part by NSF under grants CMMI- 1029551. manifolds that cannot be globally identified with Euclidean spaces [9], [10], [11]. By characterizing geometric properties of nonlinear manifolds intrinsically, geometric control tech- niques completely avoids singularities and ambiguities that are associated with local coordinates or improper characteri- zations of a configuration manifold. This approach has been applied to fully actuated rigid body dynamics on Lie groups to achieve almost global asymptotic stability [11], [12], [13], [14], [15], [16]. In this paper, we develop a geometric adaptive controller on SO(3) to track an attitude and angular velocity command without the knowledge of the inertia matrix of a rigid body. An estimate of the inertia matrix is updated online to provide an asymptotic tracking property. It is also extended to a robust adaptive attitude tracking control system. Stable adaptive control schemes designed without consideration of uncertainties may become unstable in the presence of small disturbances [17]. The presented robust adaptive scheme guarantees the boundedness of the attitude tracking error and the inertia matrix estimation error even if there exist modeling errors or disturbances. Compared with a prior work in [15], the proposed adaptive tracking control system has simpler structures, and the proposed robust adaptive track- ing control system can be applied to unstructured or non- harmonic uncertainties without need for their frequencies. This paper is organized as follows. We present a global attitude dynamics model in Section II. Adaptive attitude tracking control systems on SO(3) are developed in Section III, followed by numerical and experimental results. II. ATTITUDE DYNAMICS OF A RIGID BODY We consider the rotational attitude dynamics of a fully- actuated rigid body. We define an inertial reference frame and a body fixed frame whose origin is located at the mass center of the rigid body. The configuration of the rigid body is the orientation of the body fixed frame with respect to the inertial frame, and it is represented by a rotation matrix R ∈ SO(3), where the special orthogonal group SO(3) is the group of 3 × 3 orthogonal matrices with determinant of one, i.e., SO(3)= {R ∈ R 3×3 | R T R = I, det R =1}. The equations of motion are given by J ˙ Ω+Ω × J Ω= u +∆, (1) ˙ R = R ˆ Ω, (2) where J ∈ R 3×3 is the inertia matrix in the body fixed frame, and Ω ∈ R 3 and u ∈ R 3 are the angular velocity of the rigid body and the control moment, represented with respect arXiv:1108.6031v1 [math.OC] 30 Aug 2011