*5.1. Quasi-Incentre Definition*

Here, we define the target center as a quasi-incentre that satisfies the following two/s: firstly, the sum value of distance (*di*) between this point and any sides of the supporting polygon is the maximum; secondly, all the distances *di* should be as consistent as possible. Specifically, for regular polygons, the quasi-incentre is the center of the inscribed circle. As for the regular triangle, the quasi-incentre is just the incentre.

According to the definition of quasi-incentre, we can search the target point *q* ∗ by the following function:

$$\min f(q) = w\_1 \cdot \sum\_{i=1}^{n} \frac{1}{d\_i} + w\_2 \cdot \sum\_{i=1}^{n} \frac{|d\_i - d\_{\max}|}{d\_{\max}} \tag{17}$$

where *n* is the number of normal supporting legs, *dmax* is the maximum distance between all *di*, *w*<sup>1</sup> and *w*<sup>2</sup> is the weight coefficients, ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> <sup>1</sup> *di* illustrates the first demand while ∑*n i*=1 |*di*−*dmax*| *dmax* denotes the second demand.

#### *5.2. Quasi-Incentre Searching*

Particle Swarm Optimization (PSO) algorithm [32,33] is widely used in function optimization because it is simple to implement and has fewer adjusting parameters. In this algorithm, each particle possesses attributes of position and velocity. They are updated according to the following equation:

$$\upsilon\_{i}(k+1) = \upsilon\_{i}(k) + c\_{1} \cdot rand(\ ) \cdot (Popt\_{i}(k) - \mathbf{x}\_{i}(k)) + c\_{2} \cdot rand(\ ) \cdot (\mathsf{Wopt}\ (k) - \mathbf{x}\_{i}(k)) \tag{18}$$

$$
\pi\_i(k+1) = \pi\_i(k) + \upsilon\_i(k+1) \tag{19}
$$

where *xi*(*k*) and *xi*(*k* + 1) are the position of the i-th particle at the k-th and (k + 1)-th iteration, *vi*(*k*) and *vi*(*k* + 1) are the corresponding velocity, *rand*( ) is the random number between zero and one, *Popti*(*k*) is the best position of i-th particle, *Wopt* (*k*) is the best position of the swarm, *c*<sup>1</sup> and *c*<sup>2</sup> are the learning factors. The actual particle velocity satisfies the following limits:

$$v\_i(k+1) = \begin{cases} \begin{array}{c} v\_{\max\prime} \text{ if } v\_i(k+1) \ge v\_{\max\prime} \\ -v\_{\max\prime} \text{ if } v\_i(k+1) \le v\_{\max\prime} \end{array} \\ \begin{array}{c} \text{( $v\_i(k+1)$ )} \end{array} \end{cases} \tag{20}$$

where *vmax* is the maximum of the searching velocity.

#### 5.2.1. Searching in V Configuration

During five-legged landing, V-1 and V-2 are stable configurations. Here, we chose the V-2 configuration as the searching example. As shown in Figure 8, the numbers 1, 2, 3, 4, 5 and 6 denote the foothold positions of normal landing. All the particle positions at k-th iteration are expressed by the blue circles. The red star is the center of the hexagon that is constructed by all leg footholds and represented by the red dotted line. The new supporting polygon constructed by normal legs is expressed by the solid green line. Each subgraph is arranged in an increment of 5 iterations. At the first iteration, all the particles are scattered in the supporting plane. After 21 iterations, they are quickly concentrated into a small circular area. Then the particles slowly converge to the optimal position.

A similar process of particle convergence can also be found in the curve of target function value vs. iterations in V-2 configuration (Figure 9). At the first iteration, the function value is a larger number of 2.856. After a quick descent, it becomes a relatively stable value at the thirteenth iteration. Lastly, the value converges to a stable value of 1.692.

**Figure 8.** Particle evolution process in V-2 configuration (the horizontal axis denotes the forward/backward direction illustrated in Figure 2 while the vertical axis represents the leftward/rightward direction, respectively).

**Figure 9.** Target function value vs. iterations in V-2 configuration.

5.2.2. Searching in IV Configuration

During four-legged landing, the stable configurations are IV-2, IV-3, IV-4 and IV-6. Here, we take the IV-2 configuration as an example to illustrate the converge process. Figure 10 shows the particle evolution process in IV-2 configuration. All particles are distributed in the supporting plane at the first iteration. After a quick evolution process, they concentrate on a small circle district, then converge to the optimal position with slow velocity.

**Figure 10.** Particle evolution process in IV-2 configuration (the definition of vertical or horizontal axis is same as the one in Figure 8).

At the same time, the converged process is detailly illustrated in the curve of target function value vs. iterations in IV-2 configuration (Figure 11). The initial target function value is 2.124 at the first iteration. After 23 iterations, it reduces to 1.839 rapidly. Eventually, the value converges to the stable value of 1.834 at a slow velocity.

**Figure 11.** Target function value vs. iterations in IV-2 configuration.

#### 5.2.3. Searching in III Configuration

In a three-legged landing, the stable landing configuration is III-4. As is well-known, the triangle must exist in the center. In the III-4 configuration, its incentre coincides with the old center of the normal supporting hexagon and is set as the origin of the world coordinate frame. It is not necessary to search the quasi-incentre by the PSO algorithm. However, we can verify the target position obtained in the PSO algorithm by comparing it with the incentre. Here, we chose the III-3 configuration to check the searching result. The particle evolution process is shown in Figure 12. The green star is the theoretical incentre of the supporting triangle. After 31 iterations, the scattered particles concentrate to the incentre quickly and are limited in a small circle. Then they converge to the optimal position with a slow velocity.

**Figure 12.** Particle evolution process in III-4 configuration (the definition of vertical or horizontal axis is same as the one in Figure 8 or Figure 10).

As shown in Figure 13, the curve of the target function value vs. iterations in the III-4 configuration can be divided into three terraces. The first terrace is the period from the first iteration to the fourth iteration; the second one is from the fifth iteration to the nineteenth iteration, while the third one is from the twentieth iteration to the final iteration. The corresponding function value reduces from initial 0.1302 to 0.0055, then to zero. The target function value of zero means that the distances between optimal position and any sides of the supporting triangle are equal, and the quasi-incentre is the incentre. Noticeably, we only use the second part in Equation (17) to achieve the searching in a supporting triangle while the weight coefficient of the first part is zero (*w*<sup>1</sup> = 0).

**Figure 13.** Target function value vs. iterations in III-4 configuration.

#### **6. Experiments**

In order to verify the fault-tolerant landing, we design a 5-Dof lunar gravity ground testing platform to provide the experiment scene. There were a series of experiments for stable configurations, including the five-legged, four-legged and three-legged soft-landing experiment conducted to verify the fault-tolerant landing capacity on the platform.

#### *6.1. Experiment Platform*

Figure 14a shows the detailed components, while Figure 14b expresses the construction of the counterweight system. The simulation capacity of the platform includes vertical movement *z*, horizontal *x*, and 3-Dof spatial rotation of *Rx*, *Ry* and *Rz*. The design principle of counterweight is to make that the resultant force for the lander system in the vertical direction is equal to the one on the lunar surface by trickily dividing the load weight into two parts of counterweight one and counterweight two. In the experiments, we would simulate the soft-landing with a load of 140 kg on the Moon, which means the total mass (*mt*) including lander and load is 180 kg, the counterweight one mass is 75 kg while the counterweight two mass is 65 kg.

**Figure 14.** 5-Dof lunar gravity ground testing platform. (**a**) components and Degrees of freedom; (**b**) components of counterweight system.

#### *6.2. Five-Legged Landing*

As shown in Figure 15, the lander adopts V-2 configuration to finish soft-landing while the other configuration can bring out similar results. No. 01 is the initial position. No. 02 and No. 03 are declining in the air. No. 04 is the moment of full-touching with the ground. No. 05 is compressing to the lowest position (No. 06). After a damped vibration, the body reaches the stable position No. 10.

**Figure 15.** Keyframe snapshots in five-legged fault-tolerant landing.

Owing to the non-centrosymmetric pentagon constructed by the normal supporting leg, the difference of joint torques is great. As illustrated in Figure 16, the maximum of peak joint torques in each leg occurs in leg 1 while the minimum occurs in leg 2. The torque changes dramatically at the moment of touching the ground. The thigh peak torque is 203.7 Nm, while the shank peak torque is −84.14 Nm in leg 1. As for leg 2, the thigh peak torque is 115.1 Nm, while the shank peak torque is −72.47 Nm. The side joint torques are always small. After abouta2s fluctuation, all joint torques reach a low, stable value. The curves of body angles are shown in Figure 17a. The fluctuation ranges are −1.8~0◦ and −7.98~1.13◦ for roll and pitch angles, respectively. The curves of position and velocity of the body are seen in Figure 17b. The lander touches the ground with a maximum velocity of −1.9 m/s in the z-direction. Next, the body continues to fall with a deceleration. After 0.36 s, the body velocity reduces to zero, and the body reaches the lowest position. Owing to the overturning force/torque caused by eccentricity, extra maximum velocities of 0.355 m/s and 0.04 m/s in the *x* and *y* direction are generated. Eventually, the body reaches a stable state after a damping vibration of 1.8 s.

**Figure 16.** Joint torques in five-legged fault-tolerant landing.

**Figure 17.** Body states in five-legged fault-tolerant landing.
