# The Fast Generation of the Reachable Domain for Collision-Free Asteroid Landing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Trajectory Optimization for Collision-Free Landing

#### 2.1. Dynamic Models and Constraints

**r**= [x, y, z]

^{T}is the position vector,

**v**= [v

_{x}, v

_{y}, v

_{z}]

^{T}is the velocity vector, and m is the mass of the spacecraft system, respectively. Vector

**ω**= [0, 0, ω]

^{T}is the angular velocity of the asteroid,

**g**(

**r**) is the gravitational acceleration, T

_{max}represents the maximum magnitude of the thrust, u ∈ [0, 1] is the thrust ratio,

**α**is the direction of the thrust, I

_{sp}is the specific impulse, and g

_{0}= 9.80665 m/s

^{2}is the standard gravity.

**r**

_{1},

**r**

_{2,}and

**r**

_{3}are three vertexes of a triangle, i stands for the i-th triangle, and N

_{f}represents the total number of surface triangles of the selected polyhedral model. Then, for the soft landing problem with a given flight time t

_{f}, the final state vector of the spacecraft is known and written as

**R**means diagonal matrix, with ${\mathit{R}}_{e}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\frac{1}{{\mathrm{a}}^{2}},\frac{1}{{\mathrm{b}}^{2}},\frac{1}{{\mathrm{c}}^{2}})$, and a, b and c are the semiaxis of the ellipsoid. Vector

_{e}**n**is the outer normal vector of the triangle on the asteroid surface, and θ is the conical angle of the glide-slope constraint.

#### 2.2. Two-Stage Simplified Method for Collision-Free Trajectory Optimization

_{1}and J

_{2}are the performance index of the energy-optimal control problem in the descent stage and the gravity-free energy-optimal control problem [24] in the final landing stage, respectively, λ

_{0}= T

_{max}

^{2/}2m

_{0}

^{2}is a positive numerical factor that does not inherently change the optimal problem [25], m

_{0}is the initial mass of the spacecraft, t

_{p}is the flight time of the descent stage, and

**a**is the control acceleration of the final landing stage.

**λ**(t) satisfies

**λ**(t) = [

**λ**,

_{r}**λ**, λ

_{v}_{m}]

^{T}correspond to the state

**X**(t) = [

**r**,

**v**, m]

^{T}. The optimal thrust direction

**α**and the optimal thrust magnitude u are

**λ**(t

_{0}) = [

**λ**

_{r}_{0},

**λ**

_{v}_{0}, λ

_{m}

_{0}] can be solved by

**a**

^{*}is obtained as

_{pf}is the flight time of the final landing stage,

**r**

_{p}and

**v**

_{p}are the final position and velocity of the descent stage, respectively. When

**r**

_{p}and

**v**

_{p}are determined, by solving Equations (13) and (14), the energy consumption J

_{2}and initial costate of the final landing stage can be analytically obtained, while the trajectory is required to satisfy the glide-slope constraint given by Equation (6). Since the landing state and flight time are fixed, the trajectory only changes with

**r**

_{p}and

**v**

_{p}, i.e., the trajectory satisfying the glide-slope constraint can be obtained by changing

**r**

_{p}and

**v**

_{p}. In the descent stage, known the initial state in Equation (2) and flight time t

_{p}, if the initial costates of this stage are given, the trajectory and

**r**

_{p}and

**v**

_{p}can be obtained by integrating Equations (1) and (8), thereby generating the trajectory of the final landing stage by

**r**

_{p}and

**v**

_{p}. Under the conditions that the trajectory satisfies the path constraint of the descent stage, and

**r**

_{p}and

**v**

_{p}make the final landing stage satisfy the anti-collision path constraint, the initial costate that minimizes the energy consumption J of the two stages is found. Then, the problem can be solved.

## 3. Generation Method of the Reachable Domain

_{c}(t

_{f}). The obtained U

_{c}(t

_{f}) is a set of terminal positions r(t

_{f}) satisfying constraints described in the previous section.

#### 3.1. Initial Boundary of the Reachable Domain

_{0}is calculated, and the reachable domain boundary is obtained by using the circle section method, which is a method of obtaining the asteroid surface points between large and small radii by increasing and decreasing the radius by the same size. The size of increase and decrease is generally taken as one half of h, that is, the value range of radius r obtained is $\left\{r|{r}_{0}-h/2\le r\le {r}_{0}+h/2\right\}$. This makes the number of points needed to solve the reachable domain boundary less and evenly distributed, and it greatly reduces the time consumption for solving the optimization problem. The obtained group of asteroid surface points is expressed as P(r), as shown by the green “Δ“ in Figure 2b. The reachable domain boundary can be obtained by taking out the points with feasible solutions.

#### 3.2. Continuous Boundary of the Reachable Domain

_{1}is obtained by reducing r by h, and the value range obtained by the circular section method is $\left\{{r}_{{r}_{1}}|{r}_{0}-3h/2\le {r}_{{r}_{1}}\le {r}_{0}-h/2\right\}$, so as to obtain a group of asteroid surface points after the first expansion, expressed as P(r

_{1}). Points between all discontinuous points on the initial boundary are extracted from P(r

_{1}) along the vector direction from the asteroid centroid to the initial position. Through trajectory optimization, the obtained points with feasible solutions and the initial boundary form the boundary after the first expansion. Judge whether the boundary is continuous or not. If it is continuous, the final continuous boundary is obtained; otherwise, it continues to expand. The radius r

_{2}is obtained by reducing r

_{1}by h, and the value range obtained by using the circular section method is $\left\{{r}_{{r}_{2}}|{r}_{0}-5h/2\le {r}_{{r}_{2}}\le {r}_{0}-3h/2\right\}$, so as to obtain a group of asteroid surface points P(r

_{2}) after the second expansion. The points between discontinuous points in the boundary are taken out along the vector direction. The obtained feasible solution and the boundary after the first expansion form the boundary after the second expansion. Repeat the above process until the obtained boundary is continuous. From the above expansion process, it can be seen that the height h of the triangle on the asteroid surface is taken as an expansion unit because this expansion unit can appropriately take out a layer of asteroid surface points, and its number and distribution state are the most suitable for solving the boundary.

_{1}) is obtained, the points between green planes perpendicular to the paper surface are taken out according to the direction indicated by the arrow, and the points with feasible solutions form the boundary of this part. The direction indicated by the arrow is the vector direction from the mass center of the asteroid to the initial position. After determining the discontinuity of the “b” part in the boundary according to the distance, it continues to expand to obtain P(r

_{2}), and then takes out the points between red planes perpendicular to the paper according to the direction indicated by the arrow. At this time, the points with feasible solutions are continuous, and we can know that the continuous boundary of the “a” part is obtained after expanding it twice. The red dot in Figure 3b represents the points with the feasible solution obtained by trajectory optimization of all points on the asteroid surface.

## 4. Simulation Results and Analysis

_{max}= 20 N, the engine specific impulse I

_{sp}= 400 s, and the scaling is used, where the length scaling factor LU is set to 246 m and 1 km, respectively, and the time scaling factor $TU=\sqrt{\frac{L{U}^{3}}{{\mu}_{0}}}$s [24].

#### 4.1. Applications to 101,955 Bennu

^{T}km, [0.8 0 0]

^{T}km and [0.6 0 0.5]

^{T}km respectively. The initial velocity is set to 0 m/s.

#### 4.1.1. Initial Boundary

_{f}= 1200 s and the TOF of the final landing stage is 200 s. It can be seen from the figure that at different initial positions, the velocity increment increases with the increase of the distance from the initial position. Therefore, the section method can be used to obtain a point on the initial boundary of the reachable domain.

_{0}. Taking Figure 5a as an example, the radius r

_{0}= 625 m, and the height h of the triangle on the asteroid surface is taken as 25 m. Using the circular section method, the surface points of the asteroid with a radius of more than 612.5 m but less than 637.5 m are taken out, and the points with feasible solutions form the initial boundary, as shown in green “o”. Figure 5a–f display the reachable domains of the asteroid surface at different initial positions when t

_{f}= 700 s and t

_{f}= 900 s. Among them, the purple dots indicate the points with feasible solutions obtained after trajectory optimization of all points on the asteroid surface, which are used as the references of the reachable domain to verify the effectiveness and accuracy of the reachable domain boundary solution.

_{f}= 900 s, histories of the position, velocity, mass, and descent trajectory of the energy optimal control problem landing on the asteroid surface from above the North Pole are shown in Figure 6. These results show that the two-stage simplified solution method based on the anti-collision path constraint enables the spacecraft to land safely at the target landing site. In the figure, the red “Δ“ and “●“ respectively indicate the transition position and landing position. In addition, the time consumption for solving an optimal landing trajectory is about 60 s.

#### 4.1.2. Continuous Boundary

_{0}= 625 m at the initial boundary is obtained, and the radius r

_{1}= r

_{0}− h = 600 m after the first expansion. The surface point P(r

_{1}) of the asteroid after the first expansion is obtained using the circular section method. Along the vector direction from the asteroid centroid to the initial position, the points between all discontinuous points on the initial boundary are taken from P(r

_{1}) for trajectory optimization, and the obtained points with feasible solutions are connected with the initial boundary to become the boundary after the first expansion. Judge the distance of the boundary, repeat the above expansion process until the distance between all adjacent points is less than 75 m, and then obtain the final continuous boundary to determine the reachable domain.

_{f}= 700 s, where the green connecting line represents the initial boundary, and the purple connecting line represents the continuous boundary. It can be seen that most of the initial boundary and the continuous boundary are coincident, and only when the distance between adjacent points is too large to describe the boundary is expansion required. In Figure 5, the initial boundary can basically surround the actual reachable domain, while the continuous boundary further improves the accuracy of the reachable domain boundary on the premise of a small increase in calculation.

_{f}on the asteroid surface, and their ratio, i.e., the time-consuming ratio. In the table, P(d) represents the number of points needed for trajectory optimization when solving a point on the boundary; P(r) represents the number of points needed for trajectory optimization when solving the initial boundary; P(e) represents the number of points needed for trajectory optimization to obtain a continuous boundary by expanding the initial boundary and n represents the number of extension times.

_{f}is not more than 6%, that is, the time-consuming ratio of the proposed method and the traditional method for solving the reachable domain is less than 6%. Therefore, the reachable domain solution method based on the section and expansion method proposed in this study can reduce the solution time by more than 94%, and greatly improve the solution efficiency. It should be pointed out that the use of the parfor greatly improves the efficiency of trajectory optimization.

#### 4.2. Applications to 2063 Bacchus

^{T}km, [1 0 0]

^{T}km and [0.4 0 0.6]

^{T}km, respectively. The initial velocity was set to 0 m/s.

#### 4.2.1. Initial Boundary

_{f}= 1600 s and the TOF of the final landing stage is 300 s is shown in Figure 1. It can be clearly seen that the velocity increment increases with the increase of the distance from the initial position. The process of obtaining a point on the initial boundary by using the section method is shown by the yellow connecting line in Figure 8, Figure 9 and Figure 10. The yellow “o” represents the point with the feasible solution in P®. After ignoring the scattered points, the yellow point “o” farthest from the initial position is the point on the initial boundary. Use this point to determine the radius to obtain the initial boundary, as shown by the green connecting line.

_{0}and the height h of the triangle on the asteroid surface are taken as 620 m and 40 m, respectively. Using the circular section method, the asteroid surface points with radius r greater than 600 m but less than 640 m are extracted, and the initial boundary is obtained by trajectory optimization of these points. Figure 8, Figure 9 and Figure 10 show the initial boundaries of the reachable domain with different TOF when the initial positions of the spacecraft are over the North Pole, the Equator, and the mid-latitude region.

_{f}= 1000 s, histories of the position, velocity, mass, and descent trajectory of the energy optimal control problem landing on the asteroid surface from above the North Pole are shown in Figure 11. These results show that the two-stage simplified solution method enables the spacecraft to land on the target point without collision. Similarly, the time consumption for solving an optimal landing trajectory is about 60 s.

#### 4.2.2. Continuous Boundary

_{0}= 620 m, and the radius r

_{1}= r

_{0}− h = 580 m after the first expansion. Based on r

_{1}, asteroid surface points P(r

_{1}) after the first expansion is obtained by using the circular section method. Along the vector direction from the asteroid centroid to the initial position, the points between all discontinuous points on the initial boundary are taken from P(r

_{1}) for trajectory optimization, and the obtained points with feasible solutions are connected with the initial boundary to become the boundary after the first expansion. Judge the distance of the boundary, repeat the above expansion process until the distance between all adjacent points is less than 120 m, and then obtain the final continuous boundary to determine the reachable domain.

_{f}= 1000 s, where the green connecting line represents the initial boundary and the purple connecting line represents the continuous boundary. It can be seen that most of the initial boundary and the continuous boundary are coincident, and only when the distance between adjacent points is too large to describe the boundary, expansion is required. From Figure 8, Figure 9 and Figure 10, it can be seen that the initial boundary can basically surround the actual reachable domain, while the continuous boundary further improves the accuracy of the reachable domain boundary on the premise of increasing a small amount of calculation. The expansion indicated by the red arrows in Figure 12d, f clearly illustrates the necessity of solving the continuous boundary.

_{f}on the asteroid surface, and their ratio, i.e., time-consuming ratio. It can be seen from Table 2 that the ratio of N to N

_{f}is not more than 10%, that is, the time-consuming ratio of the proposed method and the traditional method to solve the reachable domain is less than 10%. Therefore, the reachable domain solution method based on the section and expansion method proposed in this study can reduce the solution time by more than 90%, and greatly improve the solution efficiency.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zhang, Y.; Michel, P. Shapes, structures, and evolution of small bodies. Astrodynamics
**2021**, 5, 293–329. [Google Scholar] [CrossRef] - Zhang, Y.; Yu, Y.; Baoyin, H.X. Dynamical behavior of flexible net spacecraft for landing on asteroid. Astrodynamics
**2021**, 5, 249–261. [Google Scholar] [CrossRef] - Zhang, X.; Luo, Y.; Xiao, Y.; Liu, D.; Guo, F.; Guo, Q. Developing Prototype Simulants for Surface Materials and Morphology of Near Earth Asteroid 2016 HO3. Space Sci. Technol.
**2021**, 2021, 9874929. [Google Scholar] [CrossRef] - Rozitis, B.; Ryan, A.J.; Emery, J.P.; Nolan, M.C.; Green, S.F.; Christensen, P.R.; Lauretta, D.S. High-Resolution Thermophysical Analysis of the OSIRIS-REx Sample Site and Three Other Regions of Interest on Bennu. J. Geophys. Res. Planets
**2022**, 127, e2021JE007153. [Google Scholar] [CrossRef] - Yada, T.; Abe, M.; Okada, T.; Nakato, A.; Yogata, K.; Miyazaki, A.; Hatakeda, K.; Kumagai, K.; Nishimura, M.; Hitomi, Y.; et al. Preliminary analysis of the Hayabusa2 samples returned from C-type asteroid Ryugu. Nat. Astron.
**2021**, 6, 214–220. [Google Scholar] [CrossRef] - Ploen, S.R.; Seraji, H.; Kinney, C.E. Determination of spacecraft landing footprint for safe planetary landing. IEEE Trans. Aerosp. Electron. Syst.
**2009**, 45, 3–16. [Google Scholar] [CrossRef] - Cui, P.; Ge, D.; Zhu, S.; Zhao, D. Research progress of autonomous planetary landing site assessment and selection. Sci. Sin. Technol.
**2021**, 51, 1315–1325. (In Chinese) [Google Scholar] [CrossRef] - Huang, M.; Liang, Z.; Cui, P. Reachable zone generation for irregularly shaped asteroid landing. J. Astronaut.
**2021**, 42, 1550–1558. (In Chinese) [Google Scholar] - Benito, J.; Mease, K.D. Reachable and controllable sets for planetary entry and landing. J. Guid. Control Dyn.
**2010**, 33, 641–654. [Google Scholar] [CrossRef] - Arslantaş, Y.E.; Oehlschlägel, T.; Sagliano, M. Safe landing area determination for a moon lander by reachability analysis. Acta Astronaut.
**2016**, 128, 607–615. [Google Scholar] [CrossRef] - Chen, Q.; Qiao, D.; Wen, C. Reachable domain of spacecraft after a gravity-assist flyby. J. Guid. Control Dyn.
**2019**, 42, 931–940. [Google Scholar] [CrossRef] - Lee, S.; Hwang, I. Reachable set computation for spacecraft relative motion with energy-limited low-thrust. Aerosp. Sci. Technol.
**2018**, 77, 180–188. [Google Scholar] [CrossRef] - Kulumani, S.; Lee, T. Low-thrust trajectory design using reachability sets near asteroid 4769 castalia. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Long Beach, CA, USA, 13–16 September 2016. [Google Scholar] [CrossRef] [Green Version]
- Wen, T.; Zeng, X.; Circi, C.; Gao, Y. Hop reachable domain on irregularly shaped asteroids. J. Guid. Control Dyn.
**2020**, 43, 1269–1283. [Google Scholar] [CrossRef] - Kim, H.; Kim, B. Energy-optimal transport trajectory planning and online trajectory modification for holonomic robots. Asian J. Control
**2021**, 23, 2185–2200. [Google Scholar] [CrossRef] - Zhang, Y.; Wang, J.; Xu, Y.; Yang, D. Energy-optimal problem of multiple nonholonomic wheeled mobile robots via distributed event-triggered optimization algorithm. Chin. Phys. B
**2019**, 28, 030501. [Google Scholar] [CrossRef] - Neely, M. Energy optimal control for time-varying wireless networks. IEEE Trans. Inf. Theory
**2006**, 52, 2915–2934. [Google Scholar] [CrossRef] - Yang, H.; Bai, X.; Baoyin, H. Rapid generation of time-optimal trajectories for asteroid landing via convex optimization. J. Guid. Control Dyn.
**2017**, 40, 628–664. [Google Scholar] [CrossRef] - Liu, X.; Yang, H.; Li, S. Collision-free trajectory design for long distance hopping transfer on asteroid surface using convex optimization. IEEE Trans. Aerosp. Electron. Syst.
**2021**, 57, 3071–3083. [Google Scholar] [CrossRef] - Zhang, Y.; Huang, J.; Cui, H. Trajectory optimization for asteroid landing with two-phase free final time. Adv. Space Res.
**2020**, 65, 1210–1224. [Google Scholar] [CrossRef] - Zhang, B.; Zhang, Y.; Bai, J. Twistor-Based Adaptive Pose Control of Spacecraft for Landing on an Asteroid with Collision Avoidance. IEEE Trans. Aerosp. Electron. Syst.
**2021**, 58, 152–167. [Google Scholar] [CrossRef] - Zhu, S.; Yang, H.; Cui, P.; Xu, R.; Liang, Z. Anti-collision zone division based hazard avoidance guidance for asteroid landing with constant thrust. Acta Astronaut.
**2022**, 190, 377–387. [Google Scholar] [CrossRef] - Zhao, Y.; Yang, H.; Li, S. Real-time trajectory optimization for collision-free asteroid landing based on deep neural networks. Adv. Space Res.
**2022**, 70, 112–124. [Google Scholar] [CrossRef] - Yang, H.; Li, S.; Bai, X. Fast homotopy method for asteroid landing trajectory optimization using approximate initial costates. J. Guid. Control Dyn.
**2019**, 42, 585–597. [Google Scholar] [CrossRef] - Jiang, F.; Baoyin, H.; Li, J. Practical techniques for low-thrust trajectory optimization with homotopic approach. J. Guid. Control Dyn.
**2012**, 35, 245–258. [Google Scholar] [CrossRef] - Ma, H.; Xu, S. Optimization of bounded low-thrust rendezvous with terminal constraints by interval analysis. Aerosp. Sci. Technol.
**2018**, 79, 58–69. [Google Scholar] [CrossRef] - Zeng, X.; Jiang, F.; Li, J.; Baoyin, H. Study on the connection between the rotating mass dipole and natural elongated bodies. Astrophys. Space Sci.
**2015**, 356, 29–42. [Google Scholar] [CrossRef]

**Figure 1.**Velocity increment distributions at different initial positions. (

**a**) Over the North Pole. (

**b**) Over the Equator. (

**c**) Over the mid-latitude region.

**Figure 2.**Use of the section method. (

**a**) Solving a point on the initial boundary by the section method. (

**b**) A group of asteroid surface points used to solve the initial boundary. The yellow “o” stand for points with feasible solutions between parallel planes. The green “Δ“stand for points between parallel planes when solving initial boundary.

**Figure 3.**Use of the extension method. (

**a**) The extension from the initial boundary to the continuous boundary. (

**b**) The extension of the “a” part. “0” represents the initial boundary, and “1” and “2” represent the boundary obtained after the first and second expansion, respectively.

**Figure 4.**Distribution of velocity increment with different initial positions at t

_{f}= 1200 s. (

**a**) Over the North Pole. (

**b**) Over the Equator. (

**c**) Over the mid-latitude region.

**Figure 5.**Initial boundary of the reachable domain under different initial states. (

**a**) Initial boundary for landing from over the North Pole with t

_{f}= 700 s. (

**b**) Initial boundary for landing from over the North Pole with t

_{f}= 900 s. (

**c**) Initial boundary for landing from over the Equator with t

_{f}= 700 s. (

**d**) Initial boundary for landing from over the Equator with t

_{f}= 900 s. (

**e**) Initial boundary for landing from over the mid-latitude region with when t

_{f}= 700 s. (

**f**) Initial boundary for landing from over the mid-latitude region with when t

_{f}= 900 s. The yellow “o” stand for points with feasible solutions between parallel planes. The green “o” stand for points that make up the initial boundary. The purple dots indicate the points with feasible solutions on the asteroid surface.

**Figure 6.**Trajectory optimization results of landing on the asteroid Bennu from above the North Pole. (

**a**) Histories of the position. (

**b**) Histories of the velocity. (

**c**) Histories of the thrust. (

**d**) History of the descent trajectory. The red “Δ” and “●” indicate the transition position and landing position, respectively. “●” indicate the transition position and landing position, respectively.

**Figure 7.**Boundary of the reachable domain under different initial states with t

_{f}= 700 s. (

**a**) 3D view of the boundary when the initial position is over the North Pole. (

**b**) Supplementary view of the boundary when the initial position is over the North Pole. (

**c**) 3D view of the boundary when the initial position is over the Equator. (

**d**) Supplementary view of the boundary when the initial position is over the Equator. (

**e**) 3D view of the boundary when the initial position is over the mid-latitude region. (

**f**) Supplementary view of the boundary when the initial position is over the mid-latitude region. The green connecting line stands for the initial boundary. The purple connecting line stands for the continuous boundary.

**Figure 8.**Initial boundary when the initial position is above the North Pole. (

**a**) 3D view of the initial boundary with t

_{f}= 900 s. (

**b**) Supplementary view of the initial boundary with t

_{f}= 900 s. (

**c**) 3D view of the initial boundary with t

_{f}= 1000 s. (

**d**) Supplementary view of the initial boundary with t

_{f}= 1000 s. The yellow “o” stand for points with feasible solutions between parallel planes. The green “o” stand for points that make up the initial boundary. The purple dots indicate the points with feasible solutions on the asteroid surface.

**Figure 9.**Initial boundary when the initial position is above the Equator. (

**a**) 3D view of the initial boundary with t

_{f}= 1000 s. (

**b**) Supplementary view of the initial boundary with t

_{f}= 1000 s. (

**c**) 3D view of the initial boundary with t

_{f}= 1100 s. (

**d**) Supplementary view of the initial boundary with t

_{f}= 1100 s.

**Figure 10.**Initial boundary when the initial position is above the mid-latitude region. (

**a**) 3D view of the initial boundary with t

_{f}= 900 s. (

**b**) Supplementary view of the initial boundary with t

_{f}= 900 s. (

**c**) 3D view of the initial boundary with t

_{f}= 1000 s. (

**d**) Supplementary view of the initial boundary with t

_{f}= 1000 s.

**Figure 11.**Trajectory optimization results of landing on the asteroid Bacchus from above the North Pole. (

**a**) Histories of the position. (

**b**) Histories of the velocity. (

**c**) Histories of the thrust. (

**d**) History of the descent trajectory. The red “Δ“ and “●” indicate the transition position and landing position, respectively.

**Figure 12.**Boundary of the reachable domain under different initial states with t

_{f}= 1000 s. (

**a**) 3D view of the boundary when the initial position is over the North Pole. (

**b**) Supplementary view of the boundary when the initial position is over the North Pole. (

**c**) 3D view of the boundary when the initial position is over the Equator. (

**d**) Supplementary view of the boundary when the initial position is over the Equator. (

**e**) 3D view of the boundary when the initial position is over the mid-latitude region. (

**f**) Supplementary view of the boundary when the initial position is over the mid-latitude region. The green connecting line stands for the initial boundary. The purple connecting line stands for the continuous boundary. The red arrow stands for the direction of expansion.

Initial Condition | Over the North Pole | Over the Equator | Over the Mid- Latitude Region | |||
---|---|---|---|---|---|---|

700 s | 900 s | 700 s | 900 s | 700 s | 900 s | |

P(d) | 39 | 33 | 34 | |||

P(r) | 97 | 77 | 49 | 78 | 61 | 115 |

P(e) | 22 | 39 | 51 | 40 | 40 | 0 |

n | 2 | 2 | 2 | 2 | 2 | 0 |

N | 158 | 155 | 133 | 151 | 135 | 149 |

N_{f} | 2692 | |||||

Time-consuming ratio (%) | 5.8692 | 5.7578 | 4.9406 | 5.6092 | 5.0149 | 5.5349 |

Initial Condition | Over the North Pole | Over the Equator | Over the Mid- Latitude Region | |||
---|---|---|---|---|---|---|

900 s | 1000 s | 1000 s | 1100 s | 900 s | 1000 s | |

P(d) | 49 | 53 | 29 | |||

P(r) | 139 | 126 | 84 | 54 | 102 | 157 |

P(e) | 65 | 94 | 76 | 53 | 56 | 153 |

n | 3 | 2 | 1 | 3 | 1 | 3 |

N | 254 | 269 | 213 | 160 | 187 | 339 |

N_{f} | 4092 | |||||

Time-consuming ratio (%) | 6.210 | 6.574 | 5.2053 | 3.9101 | 4.5699 | 8.2845 |

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**MDPI and ACS Style**

Zhao, Y.; Yang, H.; Hu, J.
The Fast Generation of the Reachable Domain for Collision-Free Asteroid Landing. *Mathematics* **2022**, *10*, 3763.
https://doi.org/10.3390/math10203763

**AMA Style**

Zhao Y, Yang H, Hu J.
The Fast Generation of the Reachable Domain for Collision-Free Asteroid Landing. *Mathematics*. 2022; 10(20):3763.
https://doi.org/10.3390/math10203763

**Chicago/Turabian Style**

Zhao, Yingjie, Hongwei Yang, and Jincheng Hu.
2022. "The Fast Generation of the Reachable Domain for Collision-Free Asteroid Landing" *Mathematics* 10, no. 20: 3763.
https://doi.org/10.3390/math10203763