Virtual holonomic constraints (VHCs) framework is a recent control paradigm for systematic design of motion controllers for wheel-less biologically inspired snake robots. Despite recent developments for VHC-based control systems for ground and underwater robotic snakes, they employ only two families of propulsive virtual holonomic constraints, i.e., lateral undulatory and eel-like virtual constraints. In this paper we extend the family of propulsive virtual constraints that can be used with VHC-based controllers by presenting a VHC analysis and synthesis methodology for planar snake robots that are subject to ground friction forces. In particular, we present a nonlinear differential inequality that guarantees forward motion of planar snake robots under the influence of VHCs. Furthermore, we provide a family of hyperbolic partial differential equations that can be employed to generate propulsive virtual holonomic constraints for these biologically inspired robots. Simulations are presented to verify the proposed analysis/synthesis methodology.

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