5.2.2. Step Height

The energy consumption conditions with different step height are shown in Figure 14 and Table 5. The step heights *H* are chosen as 0.04, 0.08, 0.12, 0.16, and 0.20 m. According to Equation (37), when the step length *S* is set as a constant, the energy consumption is proportional to *Ecost*.

**Figure 14.** The energy consumptions with different step heights.

**Table 5.** *Ecost* with different step heights.


Based on the equations of the foot trajectory, the step height has no influence on the stance phase, which can be verified in Figure 14. With the augmentation of the step height, the movement distance of the leg endpoint as well as the joint energy consumptions will increase. Considering the obstacle crossing ability of the robot, a proper value of the step height needs to be chosen.

#### 5.2.3. Standing Height

Considering the mechanical limitation and the ranges of the joint position, the standing heights *H*<sup>0</sup> are chosen between 0.5 to 0.7 m with a step of 0.05 m. The values of *Ecost* are proportional to the energy consumptions.

The results are illustrated in Figure 15 and Table 6. When the distance between the trunk and the ground is small, the joints have to sustain more weight, which will increase the joint torques, and the energy consumption will increase. However, if the distance between the trunk and the ground is too large, the COM of the robot will rise, which will reduce the movement stability of the robot.

**Figure 15.** The energy consumptions with different standing heights.



## 5.2.4. Gait Cycle

The gait cycle is the reciprocal of the gait frequency (*f*). Here, we choose the gait cycles as 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 s to calculate the energy consumption. The results are listed in Figure 16 and Table 7.

**Figure 16.** The energy consumptions with different gait cycle.



With a large gait cycle, the velocity and acceleration of the joint angle will be decreased and the joint torque can be reduced according to Equation (5). From Figure 16, we can know that the energy consumption in the swing phase will decrease with the augmentation of the gait cycle but that the energy consumptions during the stance phase remain the same. For the motion of the SCalf, a gait with a large cycle can reduce the energy consumption but will make the motion unstable.

#### 5.2.5. Duty Cycle

When the duty cycle is less or greater than 0.5, there will be conditions like a four-foot touchdown or a four-foot flight. The energy consumptions with different duty cycles are shown in Figure 17, and the *Ecost* are listed in Table 8.

For different duty cycles, the different energy costs *Ecost* are proportional to the energy consumptions, too. From Figure 17, we can know that the energy consumptions during the stance

phases are nearly the same. When the duty cycle is large or small, the energy consumption during the swing phase will grow rapidly. The optimal duty cycle falls into the range of 0.3 to 0.5.

**Figure 17.** The energy consumptions with different duty cycles.

**Table 8.** *Ecost* with different duty cycles.


#### *5.3. Discussion*

In robot motions, the velocity of the robot is an important comprehensive index. According to Equation (38), the robot velocity is decided by the step length *S* and gait cycle *T*; both big *S* or short *T* can increase robot velocity. Based on the results in Figures 13 and 16, big *S* and short *T* will both increase the energy consumption. When choosing the values of *S* and *T*, the mechanical limitations and control stability should be considered. We notice that the growth rate of the energy consumption caused by *S* is higher than that caused by *T*. Therefore, when the robot velocity is determined, a gait with a small step length and a short gait cycle should be chosen.

For the step height *H* and standing height *H*0, there are conflicts between low energy consumption and control effectiveness. Therefore, the selection of these parameters should consider the actual terrain and control situation.

Most researchers only study foot trajectories with the duty cycle as 50%. From the results in Figure 17, we can know that the optimal duty cycle falls into the range of 0.3–0.5, which would make the common trot gait become a flying trot gait.

According to the above discussions, for a typical situation with the robot velocity as 1 m/s, the gait parameters are chosen as follows: step length *S* = 0.25 m, gait cycle *T* = 0.5 s, step height *H* = 0.08 m, standing height *H*<sup>0</sup> = 0.6 m, and gait cycle *β* = 0.4. The energy consumptions under these gait parameters can be calculated as 217.45 J, 102.39 J, and 319.84 J for the stance phase, swing phase, and total consumption. The energy cost of this condition is 639.68 J/m.

#### **6. Conclusions**

In this work, the energy consumption conditions of the quadruped robot SCalf are studied. The kinematics model of the robot was studied, and the dynamics model was established based on the foot force distribution analysis. A complete energy model including the mechanical power and heat rate was derived. In order to describe the motions of the robot, a universal foot trajectory based on cubic spline interpolation is proposed. Different gait parameters including step length, gait cycle, step height, standing height, and duty cycle are considered in the trajectory. After obtaining the equations of the foot trajectory, the influences of different gait parameters on energy consumption are studied through a simulation. According to the results of the simulation, during the motion, a trot gait with a short step length and a small gait cycle should be chosen. The obstacle crossing ability and

motion stability need to be considered in the selection of the standing height and step height. The duty cycle of the gait should be selected in the range of 0.3 to 0.5. For a typical situation with the robot velocity as 1 m/s, the gait parameters are chosen as follows: step length *S* = 0.25 m, gait cycle *T* = 0.5 s, step height *H* = 0.08 m, standing height *H*<sup>0</sup> = 0.6 m, and gait cycle *β* = 0.4.

**Author Contributions:** Conceptualization, K.Y. and L.Z.; software, K.Y.; writing, K.Y. and L.Z.; supervision, X.R. and Y.L.; funding acquisition, L.Z., X.R., and Y.L.

**Funding:** This work was supported by the National Key R&D Program of China (Grant No. 2017YFC0806505), the National Natural Science Foundation of China (Grant No. U1613223, 61603216, and 61773226), the National High-tech R&D Program of China (Grant No. 2015AA042201), and the Key R&D Program of Shandong (Grant No. 2017CXGC0901).

**Conflicts of Interest:** The authors declare no conflict of interest.
