  The aim of this paper is to give some existence results of optimal control of
robotic systems with a Riemannian geometric view, and derive a formulation of
the PMP using the intrinsic geometry of the configuration space. Applying this
result to some special cases will give the results of avoidance problems on
Riemannian manifolds developed by A. Bloch et al. We derive a formulation of
the dynamic programming approach and apply it to the quadratic costs and extend
the linear quadratic regulator to robotic systems on Riemannian manifolds and
giving an equivalent Riccati equation. We give an optimisation aspect of the
Riemannian tracking regulator of F. Bullo and R.M. Murray. Finally, wee apply
the theoretical developments to the regulation and tracking of a rigid body
attitude.
