  Controlling large-scale particle or robot systems is challenging because of
their high dimensionality. We use a centralized stochastic approach that allows
for optimal control at the cost of a central element instead of a decentralized
approach. Previous works are often restricted to the assumption of fully
actuated robots. Here we propose an approach for underactuated robots that
allows for energy-efficient control of the robot system. We consider a simple
task of gathering the robots (minimizing positional variance) and steering them
towards a goal point within a bounded area without obstacles. We make two main
contributions. First, we present a generalized coordinate transformation for
underactuated robots, whose physical properties should be considered. We choose
Euler- Lagrange systems that describe a large class of robot systems. Second,
we propose an optimal control mechanism with the prime objective of energy
efficiency. We show the feasibility of our approach in numerical simulations
and robot simulations.
