Covariant differentiation of a map in the context of geometric optimal control Alessandro Saccon ∗ John Hauser ∗∗ A. Pedro Aguiar ∗∗∗ ∗ Department of Mechanical Engineering, Eindhoven University of Technology, the Netherlands (e-mail: a.saccon@tue.nl) ∗∗ Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder (e-mail: hauser@colorado.edu) ∗∗∗ Faculty of Engineering, University of Porto (FEUP) (e-mail: pedro.aguiar@fe.up.pt) Abstract: 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. Keywords: Optimal control, Riccati equations, differential geometry, geometric approaches, Lie groups, Projection operator approach 1. INTRODUCTION In Saccon et al. (2013), we have proposed an algorithm for solving continuous time optimal control problems for systems evolving on (noncompact) Lie groups (including, as a particular case, the flat space R n ). The approach borrows from and expands the key results of the projec- tion operator approach for the optimization of trajectory functionals, developed in Hauser (2002). The algorithm can be viewed as a generalization of New- ton’s method to the infinite dimensional setting and ex- hibits a second order convergence rate to a local mini- mizer at which the second order sufficient condition for optimality holds. At each step, a quadratic model of a cost functional (given by the composition of the original cost functional with the projection operator) is constructed about the current trajectory iterate. This quadratic model is developed using the first and second derivatives of the incremental cost, terminal cost, and the control system vector field. To this end, the second covariant derivative of ⋆ This work was supported in part by projects CONAV/FCT- PT (PTDC/EEACRO/113820/2009), FCT (PEst- OE/EEI/LA0009/2011), and MORPH (EU FP7 under grant agreement No. 288704). The work of the second author was supported in part by AFOSR FA9550-09-1-0470 and by an invited scientist grant from the Foundation for Science and Technology (FCT), Portugal. The first author benefited from a postdoctoral fellowship from FCT. a map between two manifolds plays a key role in providing a chain rule for the required Lie group computations. Motivated by this, we provide technical details and a historical perspective of the second covariant derivative of a map between smooth manifolds endowed with affine connections. 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. Having at hand such an approximation of the parallel displacement is key for computing the second covariant derivatives of the cost function and system dynamics that are used in the projection operator approach. The paper is organized as follows. Section 2 introduces the notation used throughout the paper. The projection operator approach on Lie groups for the optimization of trajectory functionals is reviewed in Section 3 and rewrit- ten in a new compact form by using operator theory. In Section 4, historical development and important properties of the second covariant derivative of a map are detailed. Finally, in Section 5 an approximation of the parallel displacement relative to the (0) Cartan-Schouten connec- tion on Lie groups is introduced and justified. Concluding remarks are given in Section 6. 9th IFAC Symposium on Nonlinear Control Systems Toulouse, France, September 4-6, 2013 ThB3.2 Copyright © 2013 IFAC 499