Page 122 - Textos de Matemática Vol. 43

110 A. SACCON, J. HAUSER AND A. P. AGUIAR
stabilization of the attitude of a rigid body controlled by means of independent in- put torques [22], [21], [17], [10, 7] and references therein. Note that the stabilization of rigid body dynamics is often addressed employing a set of local coordinates (such as the Cayley-Rodrigues parameters) or a covering map (such as the 2:1 mapping from unit quaternions to rotational matrices) to describe rotational dynamics. As discussed in [10], a proof of global convergence in these sets of coordinates results either in a feedback law which is only defined locally (in the case that a set of local coordinates is chosen) or exhibits unwinding phenomenon (when a cover map is chosen), where the initial attitude may start arbitrarily close to the desired final attitude and yet rotate through large angles before coming to rest. In this paper, we are interested in working explicitly with the group of rotational matrices SO(3), studying the properties of the value function (e.g., its continuity and di↵erentiabil- ity) as a mapping from SO(3) to R. The complete characterization of the solution of the optimal control problem that we propose in this work will serve as a validation test for a numerical method we are developing for solving continuous-time optimal control problem on Lie groups [1],[2]. We consider the dynamical system
ˆ
(1.3) g˙(t) = g(t)⇠(t), g(0) = g0 2 SO(3),
where g 2 SO(3) is the state (the attitude) and ⇠ 2 R3 is the control input (the body angular velocity) and propose a “quadratic function” on SO(3) which replace the term kxk2Q in the cost function of (1.1). For a particular class of cost functions that depend explicitly on the state and the input, we derive a closed-form solution of the optimal feedback and prove that the non-di↵erentiable value function is the viscosity solution of an appropriate Hamilton-Jacobi-Bellman (HJB) equation on SO(3).
The paper is organized as follow. Section 2 introduces the notation and definitions used throughout the paper. In Section 3 we state the optimal control problem and derive the necessary conditions for optimality. A discussion on the cost function is also provided. In Section 4, we derive the explicit solution of the optimal control problem for a particular cost function and show that the obtained value function is the viscosity solution of the HJB equation on SO(3). Conclusions and future work are discussed in Section 5.
2. Notation and definitions
The special orthogonal group SO(3) is the set {g 2 R3⇥3 : gT g = I, det(g) = 1} , with standard matrix multiplication as group operation. The group identity is the identity matrix, usually represented by e.
The group SO(3) is a Lie group, since it possesses a smooth manifold structure with smooth group operation [18, Chapter 9]. We denote by TgSO(3) the tangent space and by Tg⇤SO(3) the cotangent space of SO(3) at g. The disjoint union of these spaces form the tangent bundle TSO(3) and cotangent bundle T⇤SO(3) of the group. Given a vector space X and its dual X⇤, the bilinear operator h·, ·i : X⇤ ⇥ X ! R,