# Leg Trajectory Planning for Quadruped Robots with High-Speed Trot Gait
## 2 Materials and Methods
### Leg Design of the Quadruped Robot
#### 2.1.2 Detailed Leg Design with Three-joint Leg Structure  
The leg was designed for high-speed locomotion of quadrupeds and inspired by the design of MIT Cheetah [9]. The physical leg parameters are shown in Table 1. The total mass of the leg is \(\sim\)5.5 kg, where the swing part (1.284 kg) only occupies 23% of total mass. Moreover, the center of leg mass is only 84 mm away from the rotation center of the shoulder module, which helps reduce leg inertia during the swing. Shoulder and knee actuators with planetary gears are coaxially located at the shoulder module part for achieving a coupled motion of two joints, as shown in Figure 6, thus that the mass and leg inertia can be decreased. In detail, the knee actuator drives the knee joint through a parallel linkage, while the shoulder actuator directly drives the shoulder joint. The leg consists of the thigh, calf, and foot, where a parallel linkage connects the calf and foot. Therefore, two parallel linkages can drive the motion of the foot and thigh in parallel without an extra actuator on foot.  
The detailed design of the shoulder module is shown in Figure 7. There are two identical motors, including a knee motor and a shoulder motor in the shoulder module. The shoulder motor drives the knee motor part and thigh part to rotate through the hollow motor shaft and a planetary gear reducer. The shoulder module is designed as a thin-walled structure and is very compact in order to reduce its mass. The hollow motor shafts and gears are made of T4 titanium alloy to ensure their strength. On the contrary, the elements bearing less stress are made of 7075 aluminum alloy to ensure their light weight.  
Actuators for quadruped robots with high-speed locomotion should provide high torque density which manages the dynamic physical interactions well. The proprioceptive actuator paradigm achieves a combination of high-bandwidth force control, high torque density, as well as impact mitigation [9]. For example, NABi-V2 utilizes six-back drivable high-power density electromagnetic actuators with a low gear ratio single-phase planetary gearbox to realize proprioceptive, force-controlled dynamic locomotion [12]. Similarly, a single-stage planetary gear reduction (main material is titanium alloy,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Units** \\ \hline Leg Mass & m & 5.500 & kg \\ Thigh Mass & m\({}_{1}\) & 0.670 & kg \\ Thigh Length & l\({}_{1}\) & 0.245 & m \\ Calf Mass & m\({}_{2}\) & 0.446 & kg \\ Calf Length & l\({}_{2}\) & 0.220 & m \\ Foot Mass & m\({}_{3}\) & 0.168 & kg \\ Foot Length & l\({}_{3}\) & 0.201 & m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical leg parameters.  
Figure 6: Structural design of one leg of the quadruped.  
and weight is 0.113 kg) with low gear ratio (6:1) combined with high torque density frameless motors was applied during our leg design.  
Moreover, during the high-speed locomotion, when the leg touched the ground, both linkages were pulled by two opposite forces, and neither bending moment nor torque is applied on the linkages. Therefore, these two linkages can be very thin. Hollow shafts are used in the joints of the leg, and ribbed plates are applied on the legs, which also decrease the total mass. In particular, the joints of linkages are so small that standard bearings cannot be used, so wear washers and dry bushing were adopted, which can effectively reduce the friction of linkages' joint by the self-lubrication characteristics of the dry bushing.