Studying snake robot locomotion in a cluttered environment has been a complicated and computationally expensive task. This is because the motion model is discontinuous due to the physical contact with obstacles, and the contact force cannot be determined solely by contact positions. We present a unique mathematical model of the robot interacting with obstacles in which the contact forces are mapped on the basis of a viscous friction model. Also a motion planning strategy has been introduced which helps deriving the simplest path that ensures sufficient number of contacts of the robot with the obstacles required to reach a goal position. Numerical simulations and experimental results are presented to validate the theoretical approach.