Observers for Kinematic Systems with Symmetry

Adding an integrator for output regulation of systems with matrix Lie-group states

“…This geometry can be modeled by the special linear group SL(3) and has been used extensively in computer vision applications [16], [18]. These examples are all cases where the system of interest is a kinematic system on a matrix Lie-group [19]. This class of problems has come under considerable attention recently in the context of non-linear observer design [3], [4], [17], [14], [5], [15], [21].…”

An intrinsic PID controller for mechanical systems on Lie groups

“…We remark that the controller (5)-(6) is a full state feedback controller due to the presence of the Ad E −1 term and hence cannot be implemented with only the measurement of (y, ζ ). However, there exist observers such as the ones proposed in Bonnabel et al (2009) and Mahony et al (2008Mahony et al ( , 2013 that provide a locally exponentially convergent estimate g o of g using the measurement of an equivariant output y and the system velocities ζ . Thus if a separation principle exists it is possible to implement the PID controller (11)-(12) with only the measurement of (y, ζ ).…”

“…This class of outputs have a wide interest (Bonnabel, Martin, & Rouchon, 2009;Khosravian & Namvar, 2012;Mahony et al, 2008Mahony et al, , 2013 in output feedback control and observers for rigid body systems. It is not too hard to see that when the Lie group is SO(3), Y is equal to S 2 and the action φ is given by Ad action then the map defined by y(R) Ad R (y 0 ) = Ry 0 for some fixed y 0 ∈ S 2 also satisfies this condition and gives rise to the direction tracking problem discussed at the beginning of this section.…”

“…To the best of our knowledge, it is the first time that such a general result has been proven. Furthermore, we apply the separation principle proved in Maithripala et al (2005) to show that the implementation of the controller in conjunction with a configuration and velocity estimator with local exponential convergence, such as the ones proposed in Mahony, Hamel, and Pflimlin (2008) and Mahony, Trumpf, and Hamel (2013), preserves the convergence properties of the closed-loop error system.…”

Feature‐based recursive observer design for homography estimation and its application to image stabilization

“…A preliminary version of the theoretical results in this paper was presented at a conference (). The present version provides a more formal derivation of the observer based on a recent theory (). Moreover, the M‐estimator‐like observer design proposed in Subsection is a new result that improves significantly the robustness of the proposed approach with respect to unavoidable feature correspondence outliers encountered in practice.…”