Motion planning for a planar mechanical system with dissipative forces

In this paper, the geometric motion planning problem is addressed for an under-actuated mechanical system with dynamic non-holonomic constraints. Such constraints are the result of conservation of momentum that limits the mobility of the system in ambient space. However, dissipation forces due to interaction with the environment play a role enabling the system to move in constrained directions. Geometric mechanics tools are used to represent system dynamics in a structured form, which help better understand the motion planning problem. The geometric structure can be utilized to choose appropriate gaits intuitively by considering the properties of functions involved in the system dynamics. In a similar manner, dissipation forces also show the same type of geometric properties in terms of Stokes' connection and Stokes' Gamma functions. We can choose a gait intuitively without the need for integrating the system dynamics to generate motion in ambient space. We achieve this by exploiting the geometric properties of the friction model along with the natural dynamics of the system. By the proposed gait selection methodology, gaits are devised to move the system along a fiber direction. The simulation results are consistent with the results predicted by the proposed motion planning method. The proposed methodology is validated using experimental demonstration which also supports the simulation results. The proposed Stokes' Height functions and Stokes' Gamma functions can help to better understand the contribution of the dissipative forces and their anisotropy in motion of biological snakes and their robotic counterparts.