  In this paper we use an affine connection formulation to study an optimal
control problem for a class of nonholonomic, under-actuated mechanical systems.
In particular, we aim at minimizing the norm-squared of the control input to
move the system from an initial to a terminal state. We consider systems
evolving on general manifolds. The class of nonholonomic systems we study in
this paper includes, in particular, wheeled-type vehicles, which are important
for many robotic locomotion systems. The two special aspects of this optimal
control problem are the nonholonomic constraints and under-actuation.
Nonholonomic constraints restrict the evolution of the system to a distribution
on the manifold. The nonholonomic connection is used to express the constrained
equations of motion. Furthermore, it is used to take variations of the cost
functional. Many robotic systems are under-actuated since control inputs are
usually applied through the robot's internal configuration space only. While we
do not consider symmetries with respect to group actions in this paper, the
fact that the system is under-actuated is taken into account in our problem
formulation. This allows one to compute reaction forces due to any inputs
applied in directions orthogonal to the constraint distribution. We illustrate
our ideas by considering a simple example on a three-dimensional manifold.
