  The motion of a mechanical system can be defined as a path through its
configuration space. Computing such a path has a computational complexity
scaling exponentially with the dimensionality of the configuration space. We
propose to reduce the dimensionality of the configuration space by introducing
the irreducible path --- a path having a minimal swept volume. The paper
consists of three parts: In part I, we define the space of all irreducible
paths and show that planning a path in the irreducible path space preserves
completeness of any motion planning algorithm. In part II, we construct an
approximation to the irreducible path space of a serial kinematic chain under
certain assumptions. In part III, we conduct motion planning using the
irreducible path space for a mechanical snake in a turbine environment, for a
mechanical octopus with eight arms in a pipe system and for the sideways motion
of a humanoid robot moving through a room with doors and through a hole in a
wall. We demonstrate that the concept of an irreducible path can be applied to
any motion planning algorithm taking curvature constraints into account.
