Tingzhang WANG, Qiquan QUAN, Hongshuai GAO, Mengxue LI,Dewei TANG, Zongquan DENG

State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China

KEYWORDS Anchoring force;Asteroid exploration;Discrete element method;Force closure anchoring;Simulation verification

Abstract Asteroid exploration is significant for studying the origin of the solar system,establishing planetary defenses,and alleviating the resource crisis of the Earth.Asteroid anchoring is the basis of in-situ exploration and resource development and utilization.Therefore, the performance of asteroid force-closure anchoring is investigated using the discrete element method.The micro parameters of the simulated materials are calibrated with angle of repose and uniaxial compression experiments,based on which the regional modeling method is adopted to establish the anchoring discrete element model.Asteroid anchoring experiments are conducted on a self-developed microgravity simulation platform to verify the accuracy of the simulation model.The asteroid anchoring simulations are performed to investigate the influence of external force on the anchoring performance.The analysis of anchoring force varying with time and the interaction between the anchor and regolith particles reveals the influence mechanism of external force direction on the anchoring performance.The external force direction affects the critical anchoring force by influencing the failure of the force-closure structure.The comprehensive analysis of simulation results clarifies the variation of the critical anchoring force with the external forces.Finally, a stable anchoring region is established, beneficial for asteroid anchoring device design.

1.Introduction

Asteroids are known as the living fossils of the solar system,and they retain relatively complete information about the early formation and evolution of the solar system.1Asteroid exploration can trace the evolution of the solar system and is of great significance for revealing the origin of life on Earth and planetary defense.2According to the different compositions, the millions of asteroids discovered can be divided into several types and contain rich mineral resources, water, and organic matter.3,4Asteroid mining can help alleviate the earth’s resource crisis and provide a material basis for deep space exploration and space migration.5

Since the 1970s, ESA, America, Russia, Japan, and China have carried out 19 missions to explore asteroids and comets.6,7The exploration method shows a development trend from flying by,8flying around, impacting,9and sampling and returning10to in-situ exploration, and it often explores multiple targets in one mission.Fly-by exploration and fly-around exploration mainly use space-borne X-ray,radar,optical imaging, and other equipment to obtain information such as surface images, spectral characteristics, and rotation period.So far, there are only three asteroid sampling and returning missions, namely Hayabusa,11Hayabusa II,12,13and OSIRISREx,14,15all of which adopted the touch-and-go sampling method and did not establish a reliable mechanical connection with the asteroids.Affected by the space environment,although the results of remote sensing observations and even close flyby observations reflect the material information of asteroids to a certain extent,it cannot obtain the exact molecular compositions.16Therefore, to obtain information on the composition and evolution of asteroids, sensitive in-situ scientific exploration is necessary.The Philae first attempted to land on a comet surface and use a harpoon to anchor itself on the comet to provide a solid platform for in-situ scientific exploration.17Unfortunately, the harpoon anchor did not successfully launch, and the anchoring process failed.Philae bounced on the comet surface several times and eventually landed in a slit on the comet surface, causing part of the insitu exploration mission to fail.18Due to the low escape velocity of asteroids, a problem faced by in-situ scientific exploration is the rebound of the probe on the asteroid surface, so in most cases anchoring the asteroid probe on the asteroid is necessary.19

For the development and utilization of asteroid resources,the raw materials can be transported back to the Earth and further extracted, the resources can be in-situ mined and directly used, or the entire asteroid can be transported back to the Earth-Moon orbit and mined.No matter how the resources are developed and utilized, it is necessary to anchor the probe to the asteroid surface.

Asteroid anchoring is the basis for in-situ scientific exploration of asteroids and the development and utilization of asteroid resources.It is currently in the stage of concept proposal and principle device development.According to the anchoring mechanism, the existing anchoring methods mainly include telescopic anchor, glued anchor, magnetic anchor,envelope anchor, harpoon anchor, and force closure anchor.The American Jet Pushing Laboratory(JPL)developed a telescopic anchor, which principle is that high-energy gas drives the spikes to make the sleeves move and penetrate the asteroid to achieve anchoring.20The glued anchor is that the adhesive is injected into the asteroid surface from the landing pad to glue the probe on the asteroid.For metallic asteroids with magnetism,scholars have proposed a magnetic anchoring method,which has limited application,but it can assist other anchoring methods.Harpoon anchor is the anchoring method used by Philae,21and other researchers have also studied the anchoring process.22,23When anchoring, the harpoon shoots into the asteroid under fire work, and then the rope on the harpoon is tightened to anchor the probe.Harpoon anchors are suitable for the surface of softy asteroids.For the surface with higher hardness, it is easy to cause the anchor body to eject and cannot invade,which affects the anchoring performance.The spatial geometric force closure formed by the multi-point anchoring force is the basic principle of force closure anchoring.Force closure anchors include the micro-spine anchor,24,25drilling-based force closure anchor,26,27and disc-cut force closure anchor.28The force closure anchor has good performance, such as geological adaptability, anchoring force direction, anchoring duration, repeatability, and reliability,making it the ideal anchoring method.Therefore, this work is based on the proposed ultrasonic drilling-based force closer anchoring method with low drilling pressure and high drilling efficiency to carry out anchoring performance research.

Asteroid exploration has differences from Mars or Moon exploration, such as low gravity environment, vacuum environment,and vague knowledge of surface properties.29Before close exploration,the surface properties of asteroids have huge uncertainties, and the local topography is basically unknown,causing asteroid anchoring to be a challenging technical problem.30It is necessary to study the adaptability of the anchoring method to the complex asteroid surface environment to improve the actual asteroid anchoring performance and anchoring reliability.The anchoring simulation method can not only reduce the time and cost of experiments in a simulated weak gravitational environment of asteroids but also in-depth theoretically research the anchoring mechanism.The discrete element method (DEM) is the most attractive calculation method for designing, analyzing, and optimizing granular materials and related devices31and is widely used in mining,32agriculture,33milling,34and geotechnical applications.35In the field of rock and soil mechanics, DEM can simulate crack propagation and rock fragmentation processes.36In recent years, DEM is also been the simulation method in the exploration of extraterrestrial objects,37such as simulating the interaction process between the wheels of a Mars rover and the soil,38,39simulating the lunar soil,40simulating the interaction between the lunar soil and the sampling tool,41and even simulating the evolution of asteroids.42Here, DEM is adopted to investigate the influence of external force on the time response of anchoring force, anchoring failure process, and stable anchoring region in the process of asteroid force closure anchoring.

Henceforth, this work is organized to investigate the influence of external force on anchoring performance.A discrete element simulation model of force closure anchoring is established based on the discrete element micro parameter calibrating.Anchoring experiments are carried out on the developed asteroid microgravity environment simulation experimental platform, and the accuracy of the simulation model is verified by comparing the experimental results with the simulation results.After analyzing the probe structure and the external force characteristics, a simulation strategy is designed to study the influence of external forces on the anchoring performance.Finally, the simulation investigates the anchoring force with time, analyzes the influence of different external force directions on the anchoring forces and the anchoring failure process, and forms the boundary of the external forces in different directions for stable anchoring.

2.Drilling-based force closure anchoring of asteroids

The gravity of asteroids is weak,generally lower than 2×10-3g (e.g., the gravity of the Itokawa asteroid is only 10-6–10-5g11).Therefore, the stable anchoring of the probe becomes a prerequisite for in-situ exploration.The topography of asteroids is complex, showing gravel piles, loose weathered layers,ravines,and bulges formed by meteorite impact.The temperature on the asteroid surface changes widely, ranging from-150 ℃to 100 ℃.4These require the anchoring device to have good terrain adaptability and temperature adaptability.

By considering the low drilling pressure,compact structure,lightweight, wide temperature adaptability range, and better adaptation to the working conditions on the asteroid surface of ultrasonic drills, a force closure anchoring device is proposed based on the ultrasonic drill.The probe is supported with three landing legs, and the ultrasonic drill is coaxially arranged in the landing leg via a flexible connection to save space,as shown in Fig.1.The footpad connects to the landing leg with a spherical hinge to adapt to the different surface topography of asteroids.After the probe lands on the asteroid with the aid of the buffer device, the ultrasonic drill impacts and breaks the asteroid soil to make the anchor stick into the asteroid.Anchoring forces (F1, F2, and F3) of the three anchors form a spatial force closure to anchor the probe on the asteroid surface and resist the external force (Fe).When the probe needs to fly away from the asteroid or re-anchor,the anchor retracts into the landing leg to release the anchoring state.

3.Discrete element modeling of asteroid anchoring

3.1.Calibration of simulated asteroid soil micro parameters

The surface strength of asteroids distributes in a wide range,which is a significant factor affecting the actual anchoring performance.The asteroid surface presents obvious loose regolith,and the compressive strength of regolith is low, reducing the anchoring force.Here 100 kPa is adopted as the compressive strength of the virtual regolith on the asteroid surface to investigate the force closure anchoring performance, and the asteroid soil is assumed to be dry.Due to the lack of asteroid soil physical parameters, the virtual asteroid soil parameters refer to the related parameters of the lunar soil43,44.The 10 MPa gypsum, which is easy to prepare, is used as the anchoring medium in both the anchoring process simulations and experiments to compare the simulation and experimental anchoring forces and verify the accuracy of the simulation model.After validating the accuracy of the model, anchoring process simulations are performed on the virtual asteroid soil to investigate the influence of external force on the anchoring performance.The physical properties of the virtual regolith and gypsum material are shown in Table 1.45,46.

In the DEM model,the micro parameters of materials need to be determined so that the combination of numerical particles will show the same bulk behavior as the material.Calibrating the micro parameters of the virtual regolith and gypsum material is necessary to improve the calculation accuracy of the anchoring DEM model.There are two methods for calibration of the micro parameters, namely bulk calibration method and direct measuring method.47The direct measuring method measure the micro parameters directly and has a high requirement for materials and experimental equipment, so the bulk calibration method is adopted here.In the bulk calibration process, adjusting the micro parameters makes the DEM model output the same macro properties as the desired bulk, then the calibration is completed.48The micro parameters include particle contact parameters(i.e.,restitution coefficient,sliding friction coefficient and rolling friction coefficient)and particle bonding parameters(i.e.,normal stiffness,tangential stiffness,critical normal stress,critical tangential stress and contact radius).As the particle contact parameters affect the angle of repose (θ) and the particle bonding parameters are closely related to the uniaxial compression properties, the material accumulation experiments and uniaxial compression experiments are used to calibrate the particle contact parameters and particle bonding parameters, as shown in Fig.2.

Fig.1 Schematic diagram of drill-based force closure anchoring mechanism of asteroids.

Table 1 Physical properties of gypsum and virtual asteroid soil.45,46.

There exist many methods for measuring the angle of repose in literature,49and the fixed funnel is adopted here.During the measurement, the funnel bottom opens and allow the particles to flow and fall onto the table.When reaching the predetermined height or width, the funnel stops flowing out to obtain the experimental angle of repose by image processing.The particle contact parameters in the DEM simulation are adjusted repeatedly to make the angle of repose obtained by the simulation consistent with the experimental angle of repose to complete the calibration of the particle contact parameters.Nakashima et al.50found that the influence of gravity on the angle of repose is negligible, so the angle of repose can calibrate the particle contact parameters in the microgravity environment.In the calibration, the actual angle of repose is 40.02°,the simulated angle of repose is 39.45°,the error is within 1.5%, and the calibration completes.The contact parameters calibration results of the virtual regolith and gypsum are listed in Table 2.

In the process of particle bonding parameter calibration,the material is made into a ?50× 100 mm standard cylinder,and the uniaxial compression experiment is performed at a speed of 0.3 mm/min to obtain the experimental stress–strain curve.The uniaxial compression process is repeated under the same conditions in the DEM simulation.The simulation results of the uniaxial compression stress–strain curve and compressive strength are compared with the experimental results to calibrate the particle bonding parameters.32The calibration results are shown in Table 3.

3.2.Asteroid anchoring model

Fig.3 depicts the three-dimensional simulation model of the asteroid probe.Under the premise of not affecting the simulation accuracy,the probe model is equivalent simplified.Under the condition that the overall dimensions, mass, and centroid of the probe remain unchanged, the probe body is replacedwith an equivalent mass block by ignoring the internal structure,and the result is shown in Fig.3(a).The size of the probe simulation model is the same as the actual probe size (Fig.3(b)).The working conditions of the anchoring discrete element simulation are initially set as that the length of the anchor deep into the asteroid surface is 100 mm, and the inclination angle of the anchor is 60°, as shown in Fig.3(c).

Table 2 Particle contact parameters of gypsum and virtual asteroid soil.

The discrete element simulation of force closure anchoring is conducted in the EDEM@ software.The contact mode between particles is set as the Hertz-Mindlin with bonding,which is good at simulating the fracture of rock and soil and has been used to simulate extraterrestrial celestial soil.The calibrated virtual asteroid regolith is adopted to be the anchoring object.In the discrete element simulation, the results will be affected by the particle size effect.Due to the large envelope size of the probe model, the simulation time will be longer if small particles with guaranteed simulation accuracy are used.Therefore, the regional modeling method is adopted, that is,the region close to the anchor is filled with smaller particles,and the remaining area is filled with large particles to ensure the simulation accuracy and reduce the simulation time.In previous studies,it was found that as the ratio of the minimum size of the particle bed to the particle radius is higher than 125,the particle size has little effect on the anchoring simulation results.Therefore, set the radius of coarse particles to be 10 mm and the radius of fine particles to be 2 mm.The established anchoring discrete element simulation model of the probe is shown in Fig.4.

Limited by the three degrees of freedom of the developed asteroid weak gravitational environment simulator, only the anchoring device with two landing legs can be used for anchoring experiments in a low gravitational environment.In order to verify the accuracy of the simulation method, an anchoring discrete element simulation model of the probe with two landing legs is established based on the calibrated gypsum, as shown in Fig.5.

Fig.2 Calibration method of discrete element simulation parameters.

Table 3 Particle bonding parameters of gypsum and virtual asteroid regolith.

Fig.3 Three-dimensional simulation model of asteroid probe.

Fig.4 Discrete element simulation model for anchoring of asteroids.

4.Experimental verification of simulation model

4.1.Experimental facility

Fig.5 Anchoring simulation model of probe with two landing legs.

At present,the commonly used microgravity environment simulation methods include the air-floating platform method,falling tower method, pendulum method, and the aircraft diving method, among which the air-floating platform method has the advantages of low cost, high accuracy, and long maintenance time of microgravity.The developed air-floating platform asteroid microgravity environment simulator and asteroid surface terrain simulation wall are shown in Fig.6.The air-floating platform uses the micro-level thickness air film formed between the air bearings arranged in a triangle and the marble plane to float the simulator to simulate a microgravity environment.The air-floating platform has three degrees of freedom: x-y plane movement and rotation around the z axis.The angle between the landing legs can be adjusted to realize the anchoring experiments with different anchor inclination angles.The Laval nozzle equipped on the probe can provide the drilling pressure for the ultrasonic drill to make the anchor into the asteroid.The loading device applies a vertical load to the probe simulator until the anchor body is pulled out,and a force sensor connected to the loading rope measures the anchoring force.The anchoring conditions in the experiments are as follows: the anchoring medium is gypsum, the inclination angle of the angle is 45°,the anchoring depth varies from 10 mm to 40 mm with an interval of 10 mm, and the anchor body is a cylinder with a diameter of 4 mm.

4.2.Analysis of simulation and experimental results

The anchoring experiment and discrete element simulation for the gypsum medium are performed under the same conditions.The simulation process gradually approaches the theoretical anchoring force by changing the applied external force method.The final simulation results use a force range to represent the critical anchoring force to reduce the calculation time.Fig.7 depicts the comparison between the simulation results and the experimental results of the anchoring force.

Fig.7 Comparison of simulation results with experimental results of anchoring force.

The experimental critical anchoring forces are all located in the anchoring force range obtained by simulation.The difference between the simulation results and experimental results of the anchoring force shows an increasing trend with the increase of the anchoring depth, and the maximum error is 15%.In most anchoring conditions, the simulation results of the critical anchoring force are greater than the experimental results.The reason is that the deformation of the anchor body and the rigidity of the anchoring device in the experiments make the anchor easy to slide out along the anchoring hole,which damages the force closure structure and reduces the anchoring force.In addition, the difference in the strength of the anchoring medium between the two anchors makes the anchoring force different.When the anchor with weak anchoring ability fails, the anchoring of the probe fails, and the anchoring force reduces.

The simulation results and experimental results of the failure surface when the anchoring fails are compared and analyzed in Fig.8.With the increase of the anchoring depth, the number of stressed particles around the anchor increases,and the damaged area and damage depth of the anchoring medium increase.The simulation results and experimental results of the failure surface show a consistent trend of change.By comparing the simulation results and experimental results of the anchoring force and the failure surface,it can be considered that the anchoring discrete element simulation method is accurate.

Fig.6 Comprehensive simulator of asteroid microgravity environment.

Fig.8 Comparative analysis of gypsum failure surface form in experiment and simulation.

5.Simulation analysis of asteroid anchoring performance

After anchoring,the probe is subjected to external forces such as sampling operation force, radiation force, electromagnetic force, and the asteroid probe can maintain a stable anchoring state only when the external force is within the reliable range of the anchoring forces.The external force acting on the probe is complex, of which the magnitude, direction, and location are unknown.As different external forces will have a different effect on the anchoring of the probe, the external force states are necessary to consider in the discrete element simulation of the asteroid probe anchoring.Therefore, the anchoring discrete element simulation is conducted on the virtual asteroid soil to investigate the influence of magnitude and direction of the external force on the anchoring state of the probe, to analyze the anchoring force variation during the anchoring process by comparing the anchoring medium damage under different external forces, and to obtain the critical anchoring ability in different external force directions.

5.1.Asteroid anchoring simulation strategy

In simulations, the probe is regarded as a rigid body, and the complex external force acting on the probe is equivalent to the force in the corresponding direction at the center of mass of the probe.The angle between the external force horizontal projection and the landing leg and the angle between the external force vertical projection and the z-axis direction determine the external force direction.By considering the spatial symmetry structure of the asteroid detector (Fig.9(a)), the angle between the external force horizontal projection direction and the landing leg is selected to vary from 0° to 60° with an interval of 30° to explore the influence of the external force horizontal projection direction on the anchoring force, as shown in Fig.9(b).The angle between the external force vertical projection direction and the positive direction of the z-axis is selected to change from 0° to 90° with an interval of 30° to investigate the influence of the external force vertical projection direction on the anchoring force, as shown in Fig.9(c).

Using the calibrated 100 kPa compressive strength of the virtual asteroid regolith as the anchoring medium,the anchoring discrete element simulation experiments are carried out with the anchoring conditions of the anchor inclination angle of 60°and anchoring depth of 100 mm.The simulation process is divided into three groups according the horizontal projection of the external force.The combination of the horizontal direction and the vertical direction of the external force and the corresponding simulation results are listed in Table 4.

5.2.Influence of the external force direction on anchoring performance

Since the simulation process is grouped according to the external force horizontal projection direction, and the simulation analysis process of each group is similar, the angle between the external force horizontal projection and the landing leg of 0° is adopted as an example for analysis.From the simulation results of the critical anchoring force(F)in the four vertical projection directions of the external force,when the vertical projection angles are 30°and 60°,the critical anchoring force is the smallest (80–90 N).Therefore, 80 N and 90 N external forces with a vertical projection angle of 60° are selected to investigate the anchoring force variation with time and the interaction between the anchor and the regolith particles during the non-failure and failure anchoring process,respectively.

When subjected to an external force of 80 N(i.e.,anchoring without failure), the anchoring force variation of the probe and each anchor with time is shown in Fig.10.As the probe is impacted by an external force, the anchoring force acting on the probe increases sharply and exceeds the external force in a short time and then stabilizes at a level equal to the external force,achieving stable anchoring(Fig.10(a)).The anchoring force of the probe in the y direction is significantly greater than the anchoring forces in the x and z directions, and the anchoring forces in the x and z directions are similar.The reason is that the angle between the projection direction of the external fore in the horizontal plane and the landing leg of 0° means that as the external force is in the y plane, it has a greater impact on the anchoring in the y direction.Fig.10(b)describes the anchoring force variation of each anchor with time, and the changing trend is consistent with the changing trend of the anchoring force of the probe.Affected by the overlap between the external force horizontal projection and Anchor 2 horizontal projection,at the initial stage of suffering from the external force,the anchoring force of Anchor 2 is less than the anchoring forces of the other two anchors.At t = 0.2 s, the anchoring force of the Anchor 2 starts to be greater than the anchoring force of the other two anchors.

Table 4 Critical anchoring force range of probe under external force in different directions.

In order to investigate the anchoring mechanism from the perspective of the interaction between the anchor and the regolith particles, three representative moments are selected according to the anchoring force curve of the probe, namely,the moment when the asteroid probe begins to suffer external force (t = 0.00 s), the moment when the anchoring force is maximum (t = 0.06 s), and the moment when the anchoring is stable (t = 0.80 s), to obtain the deformation cloud and the probe movement, as shown in Fig.11.When the probe is not subjected to the external force, the anchor does not produce anchoring force, manifested as the non-existence of stressed particles around the anchor body.Since the external force causes the probe to rotate clockwise around the footpad of Anchor 2, the squeezed particles around Anchor 1 and Anchor 3 appear above the anchor.The squeezed particles around Anchor 2 are mainly concentrated under the footpad and at the tip of the anchor.The rotating motion of the probe causes Anchor 1 and Anchor 3 to pull out upward.After t = 0.2 s, the anchoring force of the probe is balanced with the external force,and the regolith particles above the anchors are not further damaged.

Fig.10 Anchoring force of asteroid probe when subjected to an external force of 80 N.

Fig.11 Deformation cloud of virtual asteroid soil at different times under action of external force.

Since the existence of external force will cause the probe to rotate, the movement speed and displacement of the probe after being subjected to the external force are extracted as shown in Fig.12.When the probe is impacted by an external force, its moving speed undergoes a process of rapid increase and decrease, then gradually tends to 0 after t = 0.2 s, and finally reaches a stable anchoring (Fig.12(a)).As the anchoring force is generated by the mutual extrusion of the anchor and the regolith particles, the anchor will generate displacement in the virtual regolith to generate the anchoring force that balances with the external force.In this simulation condition,the maximum displacement of the probe after equilibrium is 4.45 mm, as shown in Fig.12(b).

As the external force is 90 N, the anchoring fails, and the anchoring force variation of the probe and each anchor with time is shown in Fig.13.Under the external force acting, the anchoring force of the probe increases rapidly with time and then decreases and maintains stability for a short time.With the time further increasing, the anchoring force quickly decreases to about 0 N, then appears a small second peak,and finally decreases to 0 N,as shown in Fig.13(a).In the first half of the anchoring failure period (i.e., t < 0.64 s), the anchoring force in the y direction is greater than the anchoring force in the x and z directions, and the variation trend of the anchoring forces in the x and z directions is the same.In the latter half of the anchoring failure period, the anchoring force in the z direction is greater than the anchoring force in the x and y directions.Since Anchor 1 and Anchor 3 are symmetrical about the external force plane, their force state is the same,and the same force variation curve also verifies the accuracy of the simulation, as shown in Fig.13(b).During the anchoring failure process, the anchoring force of Anchor 2 is large, and the variation is also relatively drastic.

Fig.12 Motion state of asteroid probe under action of external force during anchoring.

Fig.13 Anchoring force of asteroid probe when external force is 90 N.

The key moments on the anchoring force curve of the probe are selected to extract the corresponding deformation cloud of the interaction between the anchor and the virtual regolith particles for analysis, as shown in Fig.14.Since the probe is impacted by the external force,at t=0.06 s,under the action of inertia,the anchoring force of the probe is 117.38 N greater than the external force 90 N, and a large number of stressed particles are distributed around the three anchors.Subsequently,the anchoring force is reduced to around 90 N to balance with the external force and remains stable for a short time.At this time,the probe begins to overturn in a clockwise direction with the footpad of Anchor 2 as the center, causing the anchor force of Anchor 1 and Anchor 3 to slowly decreasing, and the anchoring force of Anchor 2 to increase.At t = 0.56 s, part of the anchor is pulled out from the virtual regolith, as shown in Fig.14.The actual anchoring depth begins to affect the anchoring force that the virtual regolith can provide for Anchor 1 and Anchor 3.At t=0.64 s,Anchor 1 and Anchor 3 are separated from the virtual regolith completely, and the anchoring force is minimum.With time increasing, the resistance of Anchor 2 from leaving the virtual regolith particles increases, increasing the anchoring force.At 0.75 s, the anchoring force of Anchor 2 reaches the maximum value of 27.32 N.Finally,at t=0.86 s,Anchor 2 breaks away from the virtual regolith, and the anchoring fails.

Fig.14 Deformation cloud of virtual asteroid soil at different times during anchor failure.

5.3.Investigating stable anchoring region

Comprehensive analysis of the anchoring simulation results,the stable anchoring regions in the vertical plane where the angle between the external force horizontal projection and the landing leg is 0°, 30°, and 60° are obtained as shown in Fig.15.When the angle between the external force horizontal projection and the landing leg is 0°, the minimum anchoring force appears in the direction where the angle between the vertical projection and the z axis is 30°and 60°,and the minimum is 80–90 N,as shown in Fig.15(a).While the angle between the external force horizontal projection and the landing leg is 30°and 60°,the minimum anchoring force is in the direction where the angle between the vertical projection and the z axis is 60°(Fig.15(b) and (c)).For the three cases, when the external force is along the z direction,the anchoring force is the largest,between 120 and 130 N.

Fig.15(d) describes the stable anchoring region in threedimensional space.As the angle between the horizontal projection of the critical external force and the landing leg increases,the anchoring force tends to decrease.With the angle between the vertical projection of the external force and the z-axis increasing, the critical anchoring force shows a trend of first decreasing and then increasing.Therefore, when the external force with an angle between the horizontal projection and the landing leg of 60°and an angle between the vertical projection and the z axis of 60°acts on the probe,the critical anchoring force is minimum, between 40 N and 50 N.

The simulation results are extended to the entire threedimensional space, and the stable anchoring region is obtained, as shown in Fig.16.The distance from the point on the surface to the original (0, 0, 0) represents the critical anchoring force obtained by the simulation.The horizontal projection of the stable anchoring region is an equilateral triangle related to the landing leg distribution, as shown in Fig.16(b).Due to the support of the landing leg, the critical anchoring force is the largest when the external force is along the direction of the landing leg.With the angle between the horizontal projection of the external force and the landing leg increasing, the critical anchoring force decreases.The anchoring is the lowest when this angle increases to 60°.The projection of the stable anchoring region in the vertical plane is also a triangle,as shown in Fig.16(c).Therefore,as the angle between the vertical projection of the external force and the zaxis increases, the critical anchoring force first increases and then decreases.

5.4.Influence of external force direction on anchoring failure

Fig.15 Stable anchoring region under different horizontal directions of external forces.

Fig.16 Stable anchoring region and anchoring force changing with external force direction.

Fig.17 Influence of asteroid probe response to external force on anchoring stability during anchoring.

From the above simulation analysis, it is found that the external force direction has a significant impact on the magnitude of the critical anchoring force.The resistance of the regolith particles to the pull-out process of the anchor is the basic mechanism of the anchoring force.Therefore, the external force direction affects the pull-out sequence of the anchor and thus affects the magnitude of the critical anchoring force.There are usually-two peaks in the anchoring force versus time curve during the anchoring process, as shown in Fig.17.The first peak is due to the impact of external force, and the second peak is the resistance during the pulling out of the final anchor.When the angle between the horizontal projection of the external force and the landing leg is 0°,the order of pulling out the three anchors is two anchors first, then one anchor.For the cases where the angle between the horizontal projection of the external force and the landing leg is 30°, the three anchor bodies are pulled out in sequence.When the angle between the horizontal projection of the external force and the landing leg is 60°,the order in which the three anchors are pulled out is that the first anchor is pulled out,and the last two anchors are pulled out together.In the third case (i.e., one by two), since the resistance of the last two anchors needs to be resisted simultaneously, the second peak of the anchoring force is greater than the first peak.Although the second peak value of the anchoring force is relatively large, the force closure structure is destroyed, and the anchoring failure is inevitable.For the first case,it is necessary to resist the pull-out resistance of the two anchors first to destruct the force closure structure,so the critical anchoring fore is relatively large.The difference between the second and third cases is that the external force direction affects the force component in the first anchor to be pulled out,which affects the difficulty of pulling out the first anchor body.This is why the critical anchoring force is the smallest when the angle between the horizontal projection of the external force and the landing leg is 60°.

6.Conclusions

This work focuses on investigating the performance of asteroid force-closure anchoring using the discrete element method.The following conclusions are drawn:

(1) In simulations and experiments, the consistent trend of anchoring force with anchoring depth, the maximum error of the anchoring force less than 15%,and the similar failure surface indicate that the simulation method is accurate.

(2) Under the external force acting, the anchoring force curve changing with time has two peaks and a relatively steady stage approximately equal to the external force.The first peak is caused by the impact of the external force and the inertia of the probe,while the second peak relates to the anchoring force of the last pulled-out anchor.

(3) As the angle between the external force horizontal projection and the landing leg increases,the critical anchoring force decreases.With the angle between the external force vertical projection and the z-axis increasing, the critical anchoring force shows a trend of first decreasing and then increasing.

(4) A stable anchoring region of the anchoring device is obtained.The external force direction affects the critical anchoring force by affecting the pull-out sequence of the anchors and the failure of the closure force structure.When the external force is along the z-axis, it pulls out three anchors simultaneously, so the critical anchoring force is the largest.The second is that the external force horizontal projection is along the landing leg, which needs to pull out two anchors simultaneously.When the angle between the external force horizontal projection and landing leg and the angle between the vertical projection and the z-axis are both 60°, the anchors are pulled out sequentially, and the critical anchoring force is the smallest.

Due to the lack of physical and mechanical properties of actual asteroid soils, constructing simulated asteroid soils is difficult, which limits the development of research work to a certain extent.Future research will focus on constructing the simulated asteroid soil and studying the influences of soil property, anchoring condition parameters, and asteroid surface topography on the anchoring performance.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (Nos.52105012, 51975139 and 52111530038) and Support by Self-Planned Task of State Key Laboratory of Robotics and System (No.SKLRS202101C).