The Dynamics of a Chaplygin Sleigh with an Elastic Internal Rotor

In this paper the dynamics of a Chaplygin sleigh like system are investigated. The system consists a of a Chaplygin sleigh with an internal rotor connected by a torsional spring, which is possibly non-Hookean. The problem is motivated by applications in robotics, where the motion of a nonholonomic system is sought to be controlled by modifying or tuning the stiffness associated with some degrees of freedom of the system. The elastic potential modifies the dynamics of the system and produces two possible stable paths in the plane, a straight line and a circle, each of which corresponds to fixed points in a reduced velocity space. Two different elastic potentials are considered in this paper, a quadratic potential and a Duffing like quartic potential. The stiffness of the elastic element, the relative inertia of the main body and the internal rotor and the initial energy of the system are all bifurcation parameters. Through numerics, we investigate the codimension-one bifurcations of the fixed points while holding all the other bifurcation parameters fixed. The results show the possibility of controlling the dynamics of the sleigh and executing different maneuvers by tuning the stiffness of the spring.