Controlling one-legged hopping locomotion is a challenging problem due to the hybrid dynamics of the hopper and the interaction with ground. The hybrid dynamics of the one-legged hopper consists of mainly two sub-dynamics, one when the hopper is in contact with ground, and the other when there is no contact. The ground model can effect the hopper behavior since the hopper interact with ground when the hopper in contact with ground. Here we investigate the locomotion behavior of the one-legged multi-link hopper hopes on three different ground models, namely, rigid, elastic, and viscoelastic ground. The rigid ground apply an impulsive force to the hopper when the hopper came in contact with ground resulting energy losses. A partial feedback linearization is used to control the internal dynamics of the hopper. A Poincar\\'e map is used to construct a discrete-time system and a controller with integral action is designed to achieve the control objectives. The elastic ground, the ground modeled as massless spring, the spring in the ground store some of the energy of the hopper during the contact. A continuous backstepping controller is designed to control the energy level and internal dynamics of the hopper. A Poincar\\'e map is used to construct a discrete-time system and a controller with integral action is designed to achieve the control objectives. The viscoelastic ground, the ground modeled as an under-damped mass-spring-damper system, the damper and the impact with ground mass resulting in energy losses and the ground spring store some of the energy of the hopper during the contact. A continuous backstepping controller is designed to control the energy level and internal dynamics of the hopper. A Poincar\\'e map is used to construct a discrete-time system and a controller with integral action is designed to achieve the control objectives. We considered multiple versions of one-legged hoppers, namely, two-DOF two-mass, two-DOF ankle-knee-hip, and four-link hopper. Simulation results are presented to demonstrate the efficacy of the controllers.
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