Optimal path planning on matrix Lie groups

The problem is to plan an optimal trajectory for an airplane moving at a constant velocity from some starting point and orientation to some final point and orientation. The problem is formulated as an optimal control problem of a left invariant control system on the Lie group SE(3). This paper considers the problem of optimal control on Lie groups in general, and relates the control tower problem to classic work on elastica. Through the use of Pontryagin´s maximum principle and the techniques of numerical optimization, a solution to the problem is presented. Several examples are worked out in detail, including not only the model airplane on SE(3) but also similar systems on SE(2) and SO(3)