Nonlinear complementary filters on the special linear group

Abstract

This article proposes a nonlinear complementary filter for the special linear Lie-group SL(3) that fuses low-frequency state measurements with partial velocity measurements and adaptive estimation of unmeasured slowly changing velocity components. The obtained results have direct application on the problem of filtering a sequence of image homographies acquired from low-quality video data. The considered application motivates us to derive results that provide adaptive estimation of the full group velocity or part of the group velocity that cannot be measured from sensors attached to the camera. We demonstrate the performance of the proposed filters on real world homography data.

Keywords:

• nonlinear observer
• complementary filter
• special linear group SL(3)

Acknowledgements

This research was supported by the Australian Research Council through discovery grant DP0987411 ‘State Observers for Control Systems with Symmetry’ and the ANR-PSIRob project SCUAV ‘Sensory Control of Unmanned Aerial Vehicles’.

Notes

Notes

1. Here we use the fact that v is an eigenvector of with eigenvalue 1/λ² and that Z v is orthogonal to v, and hence is also an eigenvector of but with eigenvalue λ.

2. Available for download at http://esm.gforge.inria.fr.

3. We reserve the upper left index to indicate what frame of reference a physical object is expressed with respect to. The right subscript refers to the object being measured, and the left subscript refers to the frame with respect to which the measurement is made. Thus the vector _A ξ _B representing the origin of {B} with respect to {A} can be expressed in {A} as or in {B} as , with the normal attitude matrix representing the orientation of {B} with respect to {A}.

4. Where the vec operation stacks the columns of a matrix one on top of the other to form a vector and ⊗ is the Kroneker matrix product (see, e.g. Helmke and Moore 1994). Define I _n  ∈ ℝ ^n×n to be an n-dimensional identity matrix, 0 _n×m  ∈ ℝ ^n×m the zero matrix and 𝕋 ∈ ℝ^9×9 to be the permutation matrix such that 𝕋 vec(X) = vec(X ^⊤).