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The use of free vibration in elastic structure can lead to energy efficient robot locomotion, since it significantly reduces the energy expenditure if properly designed and controlled. However, it is not well understood how to harness the dynamics of free vibration for the robot locomotion, because of the complex dynamics originated in discrete events and energy dissipation during locomotion. From this perspective, the goal of this paper is to propose a design strategy of hopping robot based on elastic curved beams and actuated rotating masses, and identify the minimalistic model that can characterize the basic principle of robot locomotion. Since the robot mainly exhibits vertical hopping, three one-dimensional models are examined that contain different configurations of simple spring-damper-mass components. The real-world and simulation experiments show that one of the models best characterizes the robot hopping, through analyzing the basic kinematics and negative works in actuation. Based on this model, the self-stability of hopping motion under disturbances is investigated and design and control parameters are analyzed for the energy efficient hopping. Additionally, further analyses show that this robot can achieve the energy efficient hopping with the variation in payload, and the source of energy dissipation of the robot hopping is investigated.