## 1 Introduction  
Compared with wheeled or tracked robots, legged robots have certain advantages in the unstructured environment and have become one of the hotspots in the field of robotics. For legged robots, quadruped robots have simple structure and are easy to control; therefore, they're more widely used than biped and hexapod robots. Foot trajectory planning is important for the study of quadruped robots. [PERSON] et al. [1] proposed a trot gait trajectory based on a sinusoidal curve and realized by [PERSON] et al. [2]. This trajectory was simple and easy to be realized, but when the robot touched the ground, the bottom of the foot would slide and move. [PERSON] et al. [3] used the cycloid foot trajectory to plan motions of HyQ--a hydraulically and electrically actuated quadruped robot. However, only a set of typical trot gait parameters including a 50% duty cycle was used. [PERSON] et al. [4] used a virtual model to study the torso motion control of a quadruped robot, where a flight toe trajectory generator was established. In the flight phase, a rectangular trajectory was used. However, the foot endpoint would have a big velocity before landing, which could cause an impact and could affect the control system. To deal with the above problem, a new foot trajectory needs to be studied. The foot trajectory can solve problems like the landing impact and more gait parameters need to be included in the trajectory. Despite the fact that legged robots promise great mobility in rough and unstructured terrains, they still have some deficiencies in energy consumption when compared with tracked or wheeled robots. In order to improve the energy efficiency of the legged robots, many researchers studied the energy consumption of legged robots. In the last paragraph, the foot trajectories were studied in the view of motions not energy. In this part, energy models proposed by different researchers are introduced. [PERSON] et al. [5] analyzed the energy flow of a quadruped robot with a flexible trunk joint. The energy consumption conditions were studied under different gaits. However, this paper did not establish any specific energy model of the robot. It only divided the energy flow into the energy input, the friction loss, and the collision loss. It lacked an analysis of the mechanical power change. [PERSON] et al. [6] and [PERSON] et al. [7] both established their own energy model and studied the energy consumption by using different gait, and several criteria were proposed to evaluate the performances of different conditions. These two works focused on body forward velocity; therefore, only the step length and gait cycle were taken into consideration. Moreover, the energy model of References [6] and [7] only calculated the mechanical power, and the heat rate was ignored, which is a big part of the loss in practical application. [PERSON] et al. [8] analyzed the energy consumption of a quadruped robot with three foot trajectories based on their trot gait. The energy model only took the mechanical power into account, and the influence on energy consumption caused by a duty cycle was not studied. [PERSON] et al. [9] established the kinematics and dynamics models of a quadruped robot and studied the influences of different parameters on energy consumption. In this work, the feet forces were regarded as half of the robot total mass, which was not rigorous. Moreover, the heat rate was ignored in the energy model and only the situations with a 50% duty cycle were studied. [PERSON] et al. [10] studied the effects of turning gait parameters on energy consumption of an electric actuated six-legged robot. In the paper, the joint positions were programmed by quintic polynomial. Moreover, the foot force distribution problem was discussed and a relative complete energy model was established. However, the heat rate of the hydraulic cylinder was different from the electric motor. In this work, a complete energy model that is suitable for a hydraulic actuated quadruped robot is studied.