## 1 Introduction  
The legged robots have superior mobility and adaptation on the rough road compared to the wheeled and tracked robots. This significant advantage has encouraged scholars to research the legged robot over the past decades. [PERSON] et al. developed one-legged, biped, and quadruped running robots which performed highly dynamic ability in the 1980s [1, 2, 3]. All these robots used a pneumatic cylinder as a telescopic leg. The cylinder worked as an air spring that could store and recover the kinetic energy during the landing phase. [PERSON] et al. developed a one-legged hopping robot ARL-Monopod, which was driven by an electric motor. The mechanical springs located at the telescopic leg and hip joint improved this robot's hopping efficiency [4, 5]. Besides the telescopic type of the leg, the articulated leg, which could mimic the locomotion of animal, was developed and used in the bionic legged robots [6, 7, 8]. [PERSON] et al. designed an articulated one-legged robot Monopod, which mimicked the segment proportion of the horse's hindlimb [9]. [PERSON] et al. developed a bionic one-legged hopping robot KenKen [10, 11]. Boston Dynamics designed the famous quadruped robot BigDog in 2008 [12]. This robot was able to run at about 1.6 m/s with a trotting gait and keep balancing after a strong lateral disturbance. The success of the BigDog encouraged many researchers to focus on the large quadruped robot research all over the world (e.g., HyQ [13, 14], HyQ2 Max [15], Scalf [16], Baby Elephant [17]). All of these quadruped robots used the double-acting hydraulic cylinder as the joint actuator due to the wide dynamic range and the high power-to-weight ratio of the cylinder. The limbs of reptiles like lizard, alligator and gecko are laterally to the body, which cause a different locomotion posture from mammals. Alligators use sprawling posture to slide across the muddy land and erect posture to walk on the dry land [18]. To mimic the particular locomotion of alligators, [PERSON] et al. developed an alligator-inspired robot [19]. Geckos have extraordinary climbing ability due to the high adhesion and friction force on their toes [20]. [PERSON] et al. developed Stickybot to mimic the reptilian locomotion of geckos on vertical surfaces [21]. The jumping mechanisms of small animals like frog [22; 23], galago [24; 25], locust [26; 27; 28; 29] and flea [30] are studied and used in the small legged robots. Due to its compactness, diversity and controllability, the electric motor is wildly used in small legged robots. Based on the hopping strategies, small-sized legged robots can be divided into two kinds. Some hop through the direct actuation of the electric motor [31; 32], while the others are actuated by the catapult mechanism [33; 34; 35; 36; 37; 38]. Most existed catapult mechanisms work with two stages. First, the mechanical spring stores the electric motor's energy during the slow contraction of the leg. Then, the mechanical spring quickly releases to accelerate the robot's body. Due to the low power-weight ratio of the electric motor, a gear reduction device that converts the high speed into the large torque is necessary for the electric motor actuated legged robots. The gear reduction device introduces extra friction and damping and reduces the efficiency of the electric motor. The reduction gear is also easy to damage with excessive impact. Therefore, compared to the electric motor, the hydraulic cylinder is more suitable for the large legged robot. In most hydraulic-actuated legged robots, the ends of the hydraulic cylinder always directly connect to the robot leg segments, which is named as the direct attachment geometry (DAG) in this paper. During the take-off phase, the fast extension of the tibia leads to a continuous increment of the cylinder velocity and flow requirement under the DAG. On the other side, the unavoidable pressure drop and energy consumption existing at the orifice of the control valve will deteriorate with the increment of the flow rate.