Optimal Path Planning on the Three-input Six-dimensional Brockett’s Canonical System

This paper addresses an optimal path planning problem on the three-input six-dimensional Brockett’s canonical system. class of systems can be applied to control position and orientation a rigid body in SE(3), such as spacecraft, aerial vehicles, or underwater using only three inputs. Since number inputs is less than total degrees freedom, it raises non-trivial technical issues finding actual time sequence Here we show that shortest paths connecting two points are parametrized helix by introducing input quadratic norm Riemannian metric. In addition, present quasi-analytical procedure determine for any given target point. The characteristic feature our method explicit function state space, which enables solution derived without multidimensional iterations. approach was validated numerical computations aspects: matching arbitrary covering known special cases. result represented helices may also help guidance solving more general problems numerically, example, initial measure space.