Polarisation diversity of conformal arrays based on geometric algebra via convex optimisation

A simple and general design procedure is presented for the polarisation diversity of arbitrary conformal arrays; this procedure is based on the mathematical framework of geometric algebra and can be solved optimally using convex optimisation. Aside from being simpler and more direct than other derivations in the literature, this derivation is also entirely general in that it expresses the transformations in terms of rotors in geometric algebra which can easily be formulated for any arbitrary conformal array geometry. Convex optimisation has a number of advantages; solvers are widespread and freely available, the process generally requires a small number of iterations and a wide variety of constraints can be readily incorporated. The study outlines a two-step approach for addressing polarisation diversity in arbitrary conformal arrays: first, the authors obtain the array polarisation patterns using geometric algebra and secondly use a convex optimisation approach to find the optimal weights for the polarisation diversity problem. The versatility of this approach is illustrated via simulations of a 7×10 cylindrical conformal array. © 2012 The Institution of Engineering and Technology.

Coordinated motion design on lie groups

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. Applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3), SE(2) and SE(3). © 2010 IEEE.

Optimal data fitting on lie groups: A coset approach

This work considers the problem of fitting data on a Lie group by a coset of a compact subgroup. This problem can be seen as an extension of the problem of fitting affine subspaces in n to data which can be solved using principal component analysis. We show how the fitting problem can be reduced for biinvariant distances to a generalized mean calculation on an homogeneous space. For biinvariant Riemannian distances we provide an algorithm based on the Karcher mean gradient algorithm. We illustrate our approach by some examples on SO(n). © 2010 Springer -Verlag Berlin Heidelberg.