On the Design of Hybrid Pose and Velocity-bias Observers on Lie Group SE(3)

Abstract

This paper deals with the design of globally exponentially stable invariant observers on the Special Euclidian group SE(3). First, we propose a generic hybrid observer scheme (depending on a generic potential function) evolving on SE(3) × R 6 for pose (orientation and position) and velocity-bias estimation. Thereafter, the proposed observer is formulated explicitly in terms of inertial vectors and landmark measurements. Interestingly, the proposed observer leads to a decoupled rotational error dynamics from the translational dynamics, which is an interesting feature in practical applications with noisy measurements and disturbances.Index terms-Nonlinear Observer, Special Euclidean group SE(3), Hybrid dynamics, Inertialvision systems.

Design of Globally Exponentially Convergent Continuous Observers for Velocity Bias and State for Systems on Real Matrix Groups

“…There have been papers on estimation of pose and velocity measurement bias for SE(3), e.g. [7], [12], [13] and references therein. A globally exponentially convergent hybrid (not continuous) observer is proposed in [12], and a non-global exponentially convergent observer is proposed in [13].…”

“…[7], [12], [13] and references therein. A globally exponentially convergent hybrid (not continuous) observer is proposed in [12], and a non-global exponentially convergent observer is proposed in [13]. A gradient-like observer design on SE(3) with system outputs on the real projective space was proposed in [7].…”

“…E A , e b ) +ǫ E A e b , V 3 ( E A , e b ) = (k P − (2B ξ + B b ) − ǫk I ) E A ǫ(k P + B b + 2B ξ ) E A e b .Hence, there are numbers α > 0 and β > 0 such that (24) holds. LetV (E A , e b ) ǫ E A , e b ,which satisfies (25) for all (E A , e b ) ∈ R n×n × g. It is easy to show that along any trajectory of the composite system consisting of the rigid body (1) and the observer(12),V ≤ −V 3 ≤ −βV 2 ≤ −βV.As in the proof of Theorem II.4, it is east to show that there are numbers C > 0 and a > 0 such thatE A (t) + e b (t) ≤ C( E A (0) + e b (0) )e −at (26) for all ( Ā(0), b(0)) ∈ R n×n × g and all t ≥ 0. Since E A = I − A −1 Ā = A −1 E A or E A = AE A = F gE A , we have σ min (F )L g E A ≤ E A ≤ σ max (F )U g E A .…”