Integration Constants as State Variables for Optimal Path Planning

This paper explores the advantages of a state parameterization in integration constants to optimally path plan multi-dimensional systems governed by ordinary differential equations. The idea is inherited from astrodynamics, where the integration constants of the unperturbed equations of orbital motion - orbital elements - are used to path plan optimally artificial satellites in space. This paper shows that the advantages gained from this state parameterization can be extended to other applications. In order to do so, firstly, it presents a general methodology to model the governing equations-in general non-integrable and nonlinear- using the integration constants of a reduced integrable part. Subsequently, it formalizes the optimal path planning problem in integration constant space, where a geometric interpretation of the minimum input cost path is provided together with an algorithmic procedure to identify it. It is shown that working in integration constant space simplifies the path planning. In particular, the integrable dynamics can be exploited intrinsically, while the part of the dynamics which is not integrable can be isolated and compensated.