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