# Leg Trajectory Planning for Quadruped Robots with High-Speed Trot Gait
## 1 Introduction  
Legged robots are more suitable for applications with rough terrain and complex cluttered environments compared to wheeled robots [1; 2; 3]. With respect to the leg length, quadruped robots can freely select contact points while making contact with the environment. Therefore, it is promising that they can be used for rescuing people in forests and mountains, climbing stairs, to carry payloads in construction sites, and so on [4]. Recently, several quadruped robots are being developed which are equipped with hydraulic actuators or electric actuators. For example, Boston Dynamics developed a series of robots equipped with hydraulic actuators with high energy density. Hydraulic drive quadruped robots, such as Big Dog [5] and HyQ [6], make use of the characteristics of the high power density of hydraulic pressure to realize a stronger carrying capacity and motion ability. However, their large noise and size have limited their applications to outdoor applications. Moreover, they require additional hydraulic source and oil circuit, which often makes the structure more complex. Different from that, quadruped robots equipped with electric actuators can even be used to indoor environments, such as SpotMini [7], ANYmal [8], and MIT Cheetah [9; 10; 11]. Amongst them, MIT Cheetah realized a fast, efficient design with advanced proprioceptive actuator design [9]. ANYmal applies a bioinspiredactuator design, making it robust against impact, while allowing accurate torque measurement at the joints. However, the complicated actuator design also increases cost and compromises the power output of the robot [4]. SpotMini did not publish their detailed researches; however, public media has released its capabilities, such as climbing stairs and stabilizing several gaits [7]. On the other hand, high-speed locomotion involves high acceleration and extensive loadings on the legs, which brings a trade-off between weight and strength of legs [12]. There are generally two main ways to apply the leg motors for quadruped robots: putting the motor at each joint and concentrating the motor at the shoulder. Quadruped robots, such as ANYmal, adopt the method of putting motors at each joint to achieve a fully actuated motion with high compliant series elastic actuation, suitable for highly dynamic motions. Although by using this method the quadruped robots are easier to control and can walk steadily, the swing speed of their legs is slow, which leads to low walking speed. Some quadruped robots, such as SpotMini and MIT Cheetah, adopt the method of putting motors at the shoulder. This strategy can make the leg mass and inertia of the swinging parts of the legs very low and achieve a high running locomotion. Moreover, as for the leg structure of quadruped robots, the main leg structure is a two-joint structure, which is used in most quadruped robots, such as ANYmal, Wildcat [13], SpotMini, etc. This structure is simple, intuitive, and easy to control. However, from a biological point of view, too animals, such as tigers, leopards, lions, and wolves, have a three-joint structure in their legs, which imparts great advantages in running speed and energy efficiency. MIT Cheetah adopted the three-part structure, which achieved a 6 m/s running speed, and the ability to cross obstacles with high energy efficiency [14]. Also, Cheetah-cub designed with pantograph leg configuration to simplify the control of three links with only two joints and reached a running trot with short flight phases [15]. Besides, a real \"cat-sized\" robot called Pneuupard also explored this same pantograph method [16]. The method of putting motors at the shoulder has been proved to be effective on the high running locomotion. However, there is still a lack of analysis with infinitesimal kinematics for a favorable leg structure of quadruped robots with high-speed locomotion. Except for the suitable leg design, the stable gait trajectory is the basis of movement stability for quadruped robots.