*6.3. Four-Legged Landing*

Here, we chose the IV-6 configuration as the example to present the four-legged softlanding. As shown in Figure 18, the buffer process in four-legged landing is similar to the one in five-legged landing. No. 01, No. 04, No. 06 and No. 10 are the initial position, touching ground moment, lowest position, and stable position, respectively.

**Figure 18.** Keyframe snapshots in four-legged fault-tolerant landing.

Thanks to the centrosymmetric supporting rectangle, all curves of joint torque in each leg are basically the same. The maximum peak torque occurs in leg 4. As illustrated in Figure 19, the thigh peak torque is 175.2 Nm, while the shank peak torque is −98.39 Nm. At the moment of touching the ground, the torques of the thigh or shank and the angles of roll or pitch change greatly. As expressed in Figure 20, the maximum roll angle is −1.46◦ while the one of pitch angle is −0.93◦. The touching ground velocity is the same as the one in a five-legged landing, but the fluctuation time is shorter and lasts about 1.6 s. The extra horizontal velocities in the *x* and *y* direction are almost zero because of the excellent symmetry of the supporting polygon.

**Figure 19.** Joint torques in four-legged fault-tolerant landing.

**Figure 20.** Body states in four-legged fault-tolerant landing.

#### *6.4. Three-Legged Landing*

As for three-legged landing, there is only one stable configuration denoted by III-4. Figure 21 shows the fluctuation process with damping vibration. The curves of joint torque in each leg are almost consistent, and the maximum peak torque occurs in leg 1. As illustrated in Figure 22, the thigh peak torque is 184.7 Nm, while the shank peak torque is −78.57 Nm. The fluctuation ranges of the angles of roll and pitch are −1.26~0.53◦ and −1.16~1.08◦ in Figure 23a, respectively. The maximum velocity in the z-direction is −1.9 m/s at the moment of touching the ground. The body velocity reduces to zero and reaches the lowest position after 0.416 s. Lastly, the body keeps a stable height by a damping vibration of about 2 s.

**Figure 21.** Keyframe snapshots in three-legged fault-tolerant landing.

**Figure 22.** Joint torques in three-legged fault-tolerant landing.

**Figure 23.** Body states in three-legged fault-tolerant landing.

#### **7. Discussion**

As shown in Table 7, the soft-landing performances in different landing configurations with the same touch-ground conditions are obviously numerous. While the number of supporting legs has a great influence on landing performance, the spatial distribution of normal legs also plays an important role. In the four-legged and three-legged landing, all normal legs are evenly distributed, which generates almost no derivative velocity and results in a small angle derivation of roll and pitch (≈ ±1.5◦). Thanks to more normal legs, the peak torque is smaller, and the damping vibration duration is shorter in four-legged landing than the ones in three-legged landing. As for five-legged landing, this case has the most supporting legs than three/four-legged landing, but its landing performance is not very great due to the terrible non-centrosymmetric distribution of normal legs. The indexes of peak torque and angle derivation in five-legged landing are worse than the ones in the centrosymmetric configuration, like four/three-legged landing. Furthermore, a derivative velocity of 0.355 m/s and 0.04 m/s in the x and y direction is separately generated due to the large angle derivation. Thanks to more supporting legs, the damping vibration duration in a five-legged landing is longer than the one in a four-legged landing, but it is still shorter than the one in a three-legged landing.


**Table 7.** The comparison of the key index in different fault-tolerant landings.

#### **8. Conclusions**

To execute the tasks of landing and roving simultaneously, a six-legged movable repetitive lander is designed and manufactured. Instead of absorbing the landing impact energy by irreversible deformation of aluminum honeycomb material in the current legged lander, a new electric IDU with high power, low weight and small volume is utilized to dissipate the energy actively by simulating the dynamic characters of spring and damper based on impedance control. The leg structure is still intact rather than permanent deformation, so the HexaMRL can perform repetitive exploration like the lander and rover. Fault-tolerant landing capacity is important for adapting this repetitive work mode. The main contributions are as follows:


In future work, we will study the fault-tolerant walking capacity for HexaMRL to further improve the control theory under faults.

**Author Contributions:** Conceptualization, K.Y. and F.G.; methodology, K.Y.; software, K.Y. and Q.S.; validation, K.Y., S.Z. and Q.S.; formal analysis, K.Y.; investigation, K.Y.; resources, F.G.; data curation, K.Y.; writing—original draft preparation, K.Y.; visualization, K.Y. and S.Z.; supervision, F.G.; project administration, F.G.; funding acquisition, F.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the National Natural Science Foundation of China (No. U1613208), the National Key Research and Development Plan of China (No. 2017YFE0112200), the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement (No. 734575), the Research Program of National Major Research Instruments of China (No. 51927809).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declared no potential conflict of interest with respect to the research, authorship, and/or publication of this article.
