## 1 Introduction  
Humanity has found the most effective way to move on land with the invention of the wheel. However, the need to move in different environments where wheeled vehicles cannot move forces scientists to develop mechanisms that mimic living things. Although a very small part of the Earth's surface is suitable for the locomotion of wheeled vehicles, living things can reach almost all surfaces of the world due to their unique physiology. Today, robots can perform different locomotion movements, such as walking, running, jumping, and swimming, that mimic living things. However, today's robots can perform these movements by consuming a large amount of energy, while living things can perform more complex and stable locomotion using much less energy. Therefore, it is important to understand the principles of animal locomotion to develop robots with low energy consumption. Research on this subject has focused on the energy consumed during locomotion [1, 2, 3, 4, 5]. The studies on the locomotion of legged robots have been mainly in the form of detailed examination of the movements of animals and the physical models related to these movements [6]. Legged robots can be divided into three main families. There are robots equipped with rigid limbs, whose locomotion is essentially based on rigid body dynamics. They have been extensively studied since the last decades of the last century [7, 8, 9, 10]. There are hopping robots that mimic hopping locomotion used by many animals, especially at high speeds. Hopping locomotion requires the presence of elastic limbs able to store elastic energy [11, 12, 10]. Finally, there are soft robots, which are the most recent family. They have some characteristics of the hopping robots, but they also mimic the behavior of many animals that adjust the stiffness and shape of their limbs in order to increase their mobility [13, 14, 15, 16]. The first issue of this research is vibration-based locomotion. Since this kind of locomotion shares many features and problems with hopping locomotion, a brief literature review on this topic is presented. The first research on hopping robots was carried out in the 1980s at the Leg Lab of the MIT [10]. As a result of these studies, robot mechanisms capable of two-dimensional and three-dimensional locomotion have been developed, but these mechanisms had complex structures and showed excessive energy consumption. In contrast, appropriately designed elastic limb mechanisms allow the development of legged robots with lower energy consumption by exploiting a natural vibration, and the energy efficiency of these systems may approach one of biological organisms [17]. The use of elastic limbs in robots brings energy efficiency together with a range of control difficulties. This difficulty can be eliminated by overlapping frequencies of the free vibrations of the elastic limbs and the periodic actuation of the robot. The identification of the stiffness properties of hopping robots is very important in order to generate accurate dynamic models, and, for this purpose, Experimental Modal Analysis (EMA) can be used [18; 19]. In recent years, many robotic systems with elastic limbs have been developed [20; 21; 22; 23; 24; 25]. For example, [PERSON] et al. have developed a simple but effective control unit for a single-legged hydraulic robot, which can change the jump height, step length, and, thus, the flight time [21]. [PERSON] et al. presented a soft-bodied jumping robot called JelloCube, which can be used for applications that require locomotion on uneven terrain [22]. Using a free vibration mode, [PERSON] et al. have developed an elastic mechanism that can perform locomotion with low energy consumption [23; 24; 25]. The second main issue of this research is amphibious locomotion [26; 27; 28; 29; 30]. The locomotion mechanism of many amphibious robots has been developed by taking inspiration from living things.