  Affine connection control systems are mechanical control systems that model a
wide range of real systems such as robotic legs, hovercrafts, planar rigid
bodies, rolling pennies, snakeboards and so on. In 1997 the accessibility and a
particular notion of controllability was intrinsically described by A. D. Lewis
and R. Murray at points of zero velocity. Here, we present a novel
generalization of the description of accessibility algebra for those systems at
some points with nonzero velocity as long as the affine connection restricts to
the distribution given by the symmetric closure. The results are used to
describe the accessibility algebra of different mechanical control systems.
