This letter proposes a real-time optimal robust path following control scheme for planar snake robots without sideslip constraints using model predictive control (MPC). One of the features is that a linear double-integrator model rather than the complex dynamic model of snake robots is used for the MPC design to improve calculation efficiency. Moreover, in addition to constraints on joint angles, velocity, and acceleration for security, constraints on the joint offset and velocity are also considered to keep the snake robot moving forward efficiently. And the multiple inequality constraints are handled by a novel constraints translator. Furthermore, to achieve robust path following control subject to unknown and varied friction forces and disturbances, two reduced-order structure-improved extended state observers are designed to avoid complex environmental modeling. Extensive comparative experiments were conducted to verify the constraints design and the effectiveness of the proposed control scheme.