  Dynamical System (DS)-based closed-loop control is a simple and effective way
to generate reactive motion policies that well generalize to the robotic
workspace, while retaining stability guarantees. Lately the formalism has been
expanded in order to handle arbitrary geometry curved spaces, namely manifolds,
beyond the standard flat Euclidean space. Despite the many different ways
proposed to handle DS on manifolds, it is still unclear how to apply such
structures on real robotic systems. In this preliminary study, we propose a way
to combine modern optimal control techniques with a geometry-based formulation
of DS. The advantage of such approach is two fold. First, it yields a
torque-based control for compliant and adaptive motions; second, it generates
dynamical systems consistent with the controlled system's dynamics. The salient
point of the approach is that the complexity of designing a proper
constrained-based optimal control problem, to ensure that dynamics move on a
manifold while avoiding obstacles or self-collisions, is "outsourced" to the
geometric DS. Constraints are implicitly embedded into the structure of the
space in which the DS evolves. The optimal control, on the other hand, provides
a torque-based control interface, and ensures dynamical consistency of the
generated output. The whole can be achieved with minimal computational overhead
since most of the computational complexity is delegated to the closed-form
geometric DS.
