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Equivariant Systems Theory and Observer Design

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Document pages: 19 pages

Abstract: A wide range of system models in modern robotics and avionics applicationsadmit natural symmetries. Such systems are termed equivariant and the structureprovided by the symmetry is a powerful tool in the design of observers.Significant progress has been made in the last ten years in the design offilters and observers for attitude and pose estimation, tracking ofhomographies, and velocity aided attitude estimation, by exploiting theirinherent Lie-group state-space structure. However, little work has been donefor systems on homogeneous spaces, that is systems on manifolds on which aLie-group acts rather than systems on the Lie-group itself. Recent research inrobotic vision has discovered symmetries and equivariant structure onhomogeneous spaces for a host of problems including the key problems of visualodometry and visual simultaneous localisation and mapping. These discoveriesmotivate a deeper look at the structure of equivariant systems on homogeneousspaces. This paper provides a comprehensive development of the foundationtheory required to undertake observer and filter design for such systems.

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