Lyapunov-like functions for attitude control via feedback integrators Tejaswi K. C. 3 , Srikant Sukumar 1 and Ravi Banavar 2 Abstract—The notion of feedback integrators permits Eu- clidean integration schemes for dynamical systems evolving on manifolds. Here, a constructive Lyapunov function for the attitude dynamics embedded in an ambient Euclidean space has been proposed. We then combine the notion of feedback integrators with the proposed Lyapunov function to obtain a feedback law for the attitude control system. The combination of the two techniques yields a domain of attraction for the closed loop dynamics, where earlier contributions were based on linearization ideas. Further, the analysis and synthesis of the feedback scheme is carried out entirely in Euclidean space. The proposed scheme is also shown to be robust to numerical errors. I. I NTRODUCTION There are many established techniques for attitude control design employing parametrization of the set of rotational matrices [1]. A brief summary of the representations involved in description of the kinematics of motion is given in [2]. Simple control laws in terms of Euler parameters [3], Cayley- Rodrigues parameters [4] have been formulated. However, using such parametrization and hence local charts could cause undesirable unwinding behavior [5] and require switching between these local coordinate systems for control design. On the other hand, in the recent past, coordinate-free techniques using geometric ideas have been used to design rigid body attitude controllers [6], [7], [8]. However, imple- menting such feedback laws from geometric control theory [9], [10] requires special variants of numerical integrators (e.g. variational integrator) to preserve the geometric structure of the manifold and yield reliable results. R M x 0 Fig. 1. Stabilization in Euclidean space While simple geometric PD controllers can be used to stabilize a rigid body [7], numerical integration errors quickly creep into the digital implementations of these schemes, thus resulting in the states not lying on the SO(3) manifold, and 1 Associate Professor, Systems and Control Engineering, Indian Institute of Technology, Bombay srikant@sc.iitb.ac.in 2 Professor, Systems and Control Engineering, Indian Institute of Technol- ogy, Bombay ravi.banavar@iitb.ac.in 3 Undergraduate Student, Aerospace Engineering, Indian Institute of Tech- nology, Bombay kctejaswi999@gmail.com being pushed into the ambient space of 3 × 3 real-matrices. In such situations can we still guarantee that these numerical schemes will recover and converge to the manifold? Feedback integrators [11] provide a positive answer to this question. Figure 1 illustrates this scenario with R being the set of tuples of 3 × 3 matrices and angular velocity vectors, while M is the set of tuples of valid rotation matrices and angular velocities. In [11] the authors have shown that if the rigid body dynamics is seen as the restriction of a special vector field in an ambient Euclidean space, then Euclidean numerical integration schemes also lead to convergence of states to the manifold. Further, for the case when trajectories starting from an ambient space converge to an embedded submanifold, [12] shows that the omega limit set lies in a unique connected component of the level sets corresponding to a Lyapunov- like function. Our work builds on these two techniques to design Euclidean controllers which guarantee that the rigid body converges to an equilibrium point x 0 on M, even if at some instants the states do not lie on M. More recent work by [13] has addressed this problem by linearizing the ambient dynamics, thus is only valid in a small neighborhood around the desired set-point. We briefly introduce the same in section II. In this article (primarily in section III), we have developed a new procedure for nonlinear design using Lyapunov-like functions on the ambient system to guarantee asymptotic convergence to an equilibrium point in M. Finally, to demonstrate the performance of the controller, numerical simulations are presented in section IV. Notation: • Euclidean inner product is used in this paper: 〈A, B〉 =  i,j A i,j B i,j = tr(A T B) for matrices of identical dimensions. The norm induced by this inner product is used for vectors and matrices. • SO(3) = {R ∈ R 3×3 | R T R = I, det(R)=1} is the Lie group of all rotations and so(3) = {A ∈ R 3×3 | A = −A T } is the corresponding Lie algebra. • Hat map ∧ : R 3 → s0(3), ˆ Ω=   0 −Ω 3 Ω 2 Ω 3 0 −Ω 1 −Ω 2 Ω 1 0   for Ω ∈ R 3 . The inverse map is the vee map, ∨, such that ( ˆ Ω) ∨ =Ω for all Ω ∈ R 3 and  (A ∨ )= A for all A ∈ so(3). • For a square matrix A, A s := (A + A T )/2 is the symmetric part and A k := (A − A T )/2 is the skew- symmetric part. arXiv:1903.09841v1 [cs.SY] 23 Mar 2019