Lagrangian is defined as the difference between the systems’ kinetic and potential energy. The configuration variables of such systems evolve in such a way that the integral of the Lagrangian is minimized according to the principal of least action, thus defining a variational problem. For multi-rigid-bodies robot, configuration variables are naturally decomposed into position and shape variables. Nonholonomic constraints as well as external forces are easily incorporated into the above variational problem that depicts the evolution of a mechanical system.
Recent research in mechanics has been utilizing the geometric features of the configuration space as well as the symmetries in the laws of physics to simplify and reduce the dynamic equations of motion. This reduction process has been proven to be quite useful in approaching the motion planning problem.