Smart materials and soft robotics have been seen to be particularly well-suited for developing biomimetic devices and are active fields of research. In this study, the design and modeling of a new biomimetic soft robot is described. Initial work was made in the modeling of a biomimetic robot based on the locomotion and kinematics of jellyfish. Modifications were made to the governing equations for jellyfish locomotion that accounted for geometric differences between biology and the robotic design. In particular, the capability of the model to account for the mass and geometry of the robot design has been added for better flexibility in the model setup. A simple geometrically defined model is developed and used to show the feasibility of a proposed biomimetic robot under a prescribed geometric deformation to the robot structure. A more robust mechanics model is then developed which uses linear beam theory is coupled to an equivalent circuit model to simulate actuation of the robot with ionic polymer-metal composite (IPMC) actuators. The mechanics model of the soft robot is compared to that of the geometric model as well as biological jellyfish swimming to highlight its improved efficiency. The design models are characterized against a biological jellyfish model in terms of propulsive efficiency. Using the mechanics model, the locomotive energetics as modeled in literature on biological jellyfish are explored. Locomotive efficiency and cost as a function of swimming cycles are examined for various swimming modes developed, followed by an analysis of the initial transient and steady-state swimming velocities. Applications for fluid pumping or thrust vectoring utilizing the same basic robot design are also proposed. The new design shows a clear advantage over its purely biological counterpart for a soft-robot, with the newly proposed biomimetic swimming mode offering enhanced swimming efficiency and steady-state velocities for a given size and volume exchange.