Attitude and angular velocity tracking for a rigid body using geometric methods on the two-sphere

Abstract

The control task of tracking a reference pointing direction (the attitude about the pointing direction is irrelevant) while obtaining a desired angular velocity (PDAV) around the pointing direction using geometric techniques is addressed here. Existing geometric controllers developed on the two-sphere only address the tracking of a reference pointing direction while driving the angular velocity about the pointing direction to zero. In this paper a tracking controller on the two-sphere, able to address the PDAV control task, is developed globally in a geometric frame work, to avoid problems related to other attitude representations such as unwinding (quaternions) or singularities (Euler angles). An attitude error function is constructed resulting in a control system with desired tracking performance for rotational maneuvers with large initial attitude/angular velocity errors and the ability to negotiate bounded modeling inaccuracies. The tracking ability of the developed control system is evaluated by comparing its performance with an existing geometric controller on the two-sphere and by numerical simulations, showing improved performance for large initial attitude errors, smooth transitions between desired angular velocities and the ability to negotiate bounded modeling inaccuracies.

Global closed-loop equilibrium properties of a geometric PDAV controller via a coordinate-free linearization

“…CONTROL SYSTEM Before experimentally applying the PDAV controller [1], deep understanding of its closed-loop properties must be obtained; to this end, the controller is summarized next. For a thorough derivation of the controller see [1], [4]. The error function, [9],…”

“…it was shown that the desired equilibrium (q d , b ω d ) is almost globally exponentially stable, [1], [4].…”

“…A singularity-free controller was developed, demonstrating improved performance for large initial attitude errors and the ability to negotiate bounded parametric uncertainties. The modeling and control of an aerial platform (a vectoring tricopter UAV actuated by three outrunner motors that are pointed in 3D space) was studied, with rampmich@mail.ntua.gr 2 E. Papadopoulos is with the Department of Mechanical Engineering, NTUA, 15780 Athens (tel: +30-210-772-1440; fax: +30-210-772-1455) egpapado@central.ntua.gr the UAV utilizing the PDAV controller from [4], along with core modifications to cope with the platforms high precision vectoring requirements [1].…”

Smooth Feedback Construction Over Spherical Polytopes

“…In these cases, the controlled two degrees of freedom are naturally identified as the 2−sphere. Other applications include the spherical pendulum [3] and the visual tracking task [4].…”

“…One well-known theoretical result in this field is that there exists no time-invariant continuous control law that can globally stabilize a state on the sphere [5]. Many control strategies [1], [4], [6] are proposed in the literature by inherently considering the nonlinear manifold characteristics to avoid singluaries and ambiguities of other representations, and almost globally asymptotic stability is generally achieved.…”