Conference Paper

Output-feedback control for almost global stabilization of fully-acuated rigid bodies

Dept. of Electr. Eng. & Comput. Sci., Inst. Super. Tecnico, Lisboa, Portugal
DOI: 10.1109/CDC.2008.4738956 Conference: Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008, December 9-11, 2008, Cancún, México
Source: IEEE Xplore

ABSTRACT

This paper addresses the problem of stabilizing a fully-actuated rigid body. The problem is formulated considering the natural space for rigid body configurations, the Special Euclidean group SE(3). The proposed solution consists of an output-feedback controller for force and torque actuation that guarantees almost global asymptotic stability of the desired equilibrium point. As such the equilibrium point is a stable attractor for all initial conditions except for those in a nowhere dense set of measure zero. As an additional feature, the controller is required to verify prescribed bounds on the actuation.

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    • "The development of combined position and attitude controllers from existing attitude controllers has some advantages over techniques based on the special Euclidean group SE(3), where rotations are represented directly by rotation matrices [27] [28] [29]. In the latter, asymptotically stability of the combined rotational and translational motion is proven by either defining two different error functions for the position and attitude error [29] or, in two steps, by first proving the asymptotical stability of the rotational motion before the asymptotical stability of the translational motion can be proven [27] (recall that the translational motion depends on the rotational motion). The use of dual quaternions to describe the kinematics allows the use of a single error function, the " error dual quaternion " (defined by analogy to the classical rotation error quaternion) to represent the combined position and attitude error. "
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    • "The technique proposed in this paper for developing combined position and attitude controllers from existing attitude controllers has some advantages over techniques based on the special Euclidean group SE(3), where rotations are represented directly by rotation matrices [17], [18], [19]. In the latter, asymptotically stability of the combined rotational and translational motion is proven by either defining two different error functions for the position and attitude error [19] or, in two steps, by first proving the asymptotical stability of the rotational motion before the asymptotical stability of the translational motion can be proven [17] (note that the translational motion depends on the rotational motion). In our approach, a single error function, the error dual quaternion (defined by analogy to the classical rotation error quaternion), is used to represent the combined position and attitude error. "
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    ABSTRACT: In this paper, we suggest a new representation for the combined translational and rotational dynamic equations of motion of a rigid body in terms of dual quaternions. We show that with this representation it is relatively straightforward to extend existing attitude controllers based on quaternions to combined position and attitude controllers based on dual quaternions. We show this by developing setpoint nonlinear controllers for the position and attitude of a rigid body with and without linear and angular velocity feedback based on existing attitude-only controllers with and without angular velocity feedback. The combined position and attitude velocity-free controller exploits the passivity of the rigid body dynamics and can be used when no linear and angular velocity measurements are available.
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    • "As we saw in Section 3.2, asymptotic stability analysis for external actuation depended crucially on linearization. However, it is not possible to linearize the present system as done in [6] [7], since it is not clear how the closed loop system can be derived from a Riemannian structure. We present the linearization relevant to the present context in the following subsection. "
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