Covariant Differentiation of a Map in the Context of Geometric Optimal Control

This paper provides a detailed discussion of the second covariant derivative of a map and its role in the Lie group projection operator approach (a direct method for solving continuous time optimal control problems). We begin by briefly describing the iterative geometric optimal control algorithm and summarize the general expressions involved. Particular emphasis is placed on the expressions related to the search direction subproblem, writing them in a new compact form by using a new operator notation. Next, we show that the covariant derivative of a map between manifolds endowed with affine connections plays a key role in obtaining the required local quadratic approximations for the Lie group projection operator approach. We present a new result for computing an approximation of the parallel displacement associated with an affine connection which is an affine combination of two (or more) connections. As a corollary, an extremely useful approximation of the parallel displacement relative to the Cartan-Schouten (0) connection on Lie groups is obtained.