a penetrating-anchoring mathematical model for the soft

Research ArticleA Penetrating-Anchoring Mathematical Model for the SoftAsteroid Anchoring System

Zhijun Zhao ,1 Shuang Wang,2 and Jingdong Zhao 3

1Beijing Key Laboratory of Intelligent Space Robotic Systems Technology and Applications, Beijing Institute of SpacecraftSystem Engineering, Beijing 100092, China2Space Star Technology Co., Ltd, China3State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China

Correspondence should be addressed to Jingdong Zhao; [email protected]

Received 11 January 2020; Revised 2 December 2020; Accepted 3 February 2021; Published 20 February 2021

Academic Editor: Jeremy Straub

Copyright © 2021 Zhijun Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The asteroid landing mechanism is necessary to be anchored to avoid flowing away. At present, the study on the anchoring systemis mainly focused on the mechanical design, but there are few researches on the penetrating or anchoring mathematical model, andthe researches on combining two models with each other are even more lacking. In the paper, based on the characteristics of Mohr-Coulomb material, a penetrating mathematical model of the anchoring system is established. This penetrating mathematical modelcan be used to calculate the penetrating depth of the anchor body according to the penetrating speed and the medium properties.Secondly, an anchoring mathematical model is established, which shows the relationships among the anchoring force, mediumproperties, and penetrating depth. Finally, a penetrating-anchoring mathematical model is built with the penetrating depth asthe link. The model establishes a relationship between the anchoring force and the initial penetrating conditions.

1. Introduction

Asteroid exploration has high scientific and economic values[1–3], such as finding evolution history of the planetary sys-tem, finding the origin of the solar system and the life, andexploring the asteroid resources. Moreover, asteroid explora-tion opens up a new field of deep space exploring [4, 5].

The landing mechanism must be fixed on the asteroidsurface by the anchoring system to prevent flowing awaydue to the microgravity on the asteroid. The greater theanchoring force, the easier it is for the landing mechanismto be anchored on the asteroid surface. The anchoring forceis closely related to the penetrating depth and the mediumcharacteristics, and the penetrating depth is related to themass of the anchor body, the shape of the anchor tip, thecharacteristics of the medium, and so on. Therefore, it is sig-nificant to analyze the penetrating-anchoring theory of theanchoring system on the asteroid surface. Penetrating analy-sis can predict penetrating depth of the anchoring system onthe asteroid surface, and anchoring analysis can estimate the

anchoring force that can be generated by the anchoring sys-tem at a certain penetration depth. Both of penetrating andanchoring analysis can guide the anchoring of the landingmechanism on the asteroid surface.

At present, the anchoring theory research on the asteroidlanding is still in the starting state, and the relevant theoreti-cal model has not been reported. Currently, anchoring theoryis mainly about the pile foundation engineering. Ilamparuthiand Dickin summarized some anchoring models establishedby the pioneers [6] (Majer et al. (1955) established a verticalslip surface model. Downs et al. (1966), Murray et al. (1987),and Clemence et al. (1977) established an inverted truncatedcone model. Balla (1961) and Meyerhof et al. (1968) estab-lished a curved slip surface model.). Sutherland studied theanchoring force of pile-based anchors in clay and analyzedvarious models [7]. Dickin studied the anchoring of anchorsin sand media from 1988 to 1992 [8]. In 1994, Rao andKumar studied the anchoring capacity of pile-based anchorsand analyzed various models [9]. Ilamparuthi and Dickinstudied the anchorage effects of different ends in the sand

HindawiInternational Journal of Aerospace EngineeringVolume 2021, Article ID 8649172, 10 pageshttps://doi.org/10.1155/2021/8649172

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https://doi.org/10.1155/2021/8649172

medium in 2001 [6]. Merifield et al. studied the anchoring ofpile anchors in clay [10]. Wu et al. studied the anchoring ofpile-based anchors in plaster-cement mixed media [11].Shanker et al. studied the anchoring of the light rod [12]. Nir-oumand et al. studied the anchoring of the late expanded pilefoundation in the sand medium [13]. Liu et al. studied thedeformation of the medium during the drawing process ofthe anchor [14]. Consoli et al. studied the anchoring forceof the pile body in cement [15]. The anchorage of the pilefoundation is larger (about several meters in length), andthe ground buried depth is deep (tens of meters). It is dis-tinctly different from the anchoring system in the volumeand penetration depth. Pile anchorage is different from aster-oid anchoring to some extent, but it can be taken as guidanceand reference for the study of the asteroid anchoring.

Currently, there are few researches on the anchoringmodel for the asteroid anchoring, let alone the penetrating-anchoring model. However, the mathematical modeldescribing the relationship between the penetrating and theanchoring is necessary in asteroid anchoring. In the paper,the vertical penetration equation of the anchoring systemon the asteroid surface is derived. This equation can describethe mathematical relationship among the penetration speed,penetration depth, and the medium properties. Thus, thisequation can predict the penetration depth of the anchorbody if the medium property is known. Then, the anchoringmodel of the anchoring system is established. The model canbe used to estimate the anchoring force generated by theanchoring system. Finally, the penetrating-anchoring modelis established, which displays the relationship between thepenetration velocity, the medium characteristics, and theanchoring force.

2. Anchoring System

The landing mechanism anchors on the asteroid via theanchoring system. The anchoring system contains penetrat-ing unit, advancing unit, winding unit, damping unit, andso on. Its schematics and performances are shown separatelyin Figure 1. The mass of the system is about 1.5 kg. The pen-etrating unit drafting a wire of about 2m in length can pen-etrate the soft asteroid surface. Then, a wire connection isestablished between the landing mechanism and the asteroid.

Penetrating unit in the anchoring system is active anddeformable, and its mechanical structure is shown in thetop graph of Figure 2. The penetrating unit is mainly com-posed of anchor tip, anchor claw, barb, handspike, piston,pyrotechnics, anchor body, etc. As the penetrating unit pen-etrated the asteroid surface, the control system ignites thepyrotechnics and produces high-pressure gas. The gas pushesthe handspike via the piston, inducing the splay of the clawand the barbs. The deployed penetrating unit is shown inthe lower graph of Figure 2. In soft media, the claws andbarbs are fully opened, and the open barbs increase the con-tact area between the anchor tip and the media, increasingthe anchoring force. In hard media, the claws and barbs willgenerate irregular and plastic deformation when opening dueto the great thrust generated by the pyrotechnic. This phe-

nomenon is similar to the effect of expansion bolt, and itcan enhance the anchoring force.

The anchoring force is a key index of the anchoring sys-tem, directly related to its penetrating depth of the penetrat-ing unit. However, this penetrating depth depends on theinitial conditions such as the medium properties, the shapeof the anchor tip, the mass, and penetrating velocity. Thus,it is necessary to establish the relationship between theanchoring force and the initial condition, which can providetheoretical foundation for reliable anchorage of the anchor-ing system.

3. Penetrating Model

3.1. Penetrating Resistance Force Analysis. The anchoringforce is directly related to the penetrating depth. In order toestablish the relationship between the anchoring force andthe initial condition, it is firstly necessary to establish the rela-tionship between the penetrating depth and the initialcondition.

The forces on arbitrary shape anchor tip are shown inFigure 3 in penetrating.

Based on the cavity expansion model [16–18] and citingthe deduction process of the paper [19], we can get the axialforce on the anchor tip [18].

Fz1 = 2πðr0

Aτ0 + Bρ0Vz2 1

1 + φ′ rð Þh i2

0B@

1CArdr +

2πμðz0

Aτ0 + Bρ0Vz2

f ′ zð Þh i2

1 + f ′ zð Þh i2

0B@

1CAf zð Þdz0 ≤ z ≤ p

Fz2 = 2πðR0

Aτ0 + Bρ0Vz2 1

1 + φ′ rð Þh i2

0B@

1CArdr +

2πμðp0

Aτ0 + Bρ0Vz2

f ′ zð Þh i2

1 + f ′ zð Þh i2

0B@

1CAf zð Þdzz > p,

ð1Þ

Advancing unit Winding unit

Anchor tip

Damping unit

Penetrating unit

Figure 1: Schematics of anchoring system.

2 International Journal of Aerospace Engineering

where

A = 1α

1 + τ0/2Eγ

� �2α−1λ, ð2Þ

B = ð3/ð1 − η ∗Þð1 − 2αÞð2 − αÞÞ + ð1/γ2Þð1 + τ0/2E/γÞ2αfð3τ0/EÞ + η ∗ ð1 − 3τ0/2EÞ2−γ3½2ð1 − η ∗Þð2 − αÞ + 3γ3�/ð1− η ∗Þð1 − 2αÞð2 − αÞð1 + τ0/2EÞ4g:

α = 3λ/3 + 2λ, where λ = tan φ and φ is the internal fric-tion angle.

η ∗ = 1 − ρ0/ρ ∗, where ρ0 is the volume density beforedistortion and ρ ∗ is the volume density after distortion.

γ = Vc= 1 + τ0

2E� �

− 1 − η ∗ð Þh i1/3

, ð3Þ

where E is the Young’s modulus and τ0 is the cohesion of themedium.

3.2. Penetrating Depth. The generatrix equation of the conicanchor tip is as follows:

rz= R

p= tan θ, ð4Þ

where θ is the semivertical angle of the anchor tip and θ =9:46.

Obtaining

r = f zð Þ = z tan θ,z = φ rð Þ = r cot θ,

(

f ′ zð Þ = tan θ,φ′ rð Þ = cot θ:

( ð5Þ

Integrating the above equation with equation (4) willobtain

Fz1 = πr2 Aτ0 + Bρ0Vz2 ⋅ sin2θ

� �+

2πμðz0Aτ0 + Bρ0Vz

2 ⋅ sin2θ� �

z tan θdz0 ≤ z ≤ p

Fz2 = πR2 Aτ0 + Bρ0Vz2 ⋅ sin2θ

� �+

2πμðp0Aτ0 + Bρ0Vz

2 ⋅ sin2θ� �

z tan θdzz > p:

ð6Þ

(1) When penetrating depth 0 ≤ z ≤ p

To the conic anchor tip, according to equation (6),

Fz1 = πr2 Aτ0 + Bρ0Vz2 ⋅ sin2θ

� �1 + μ cot θð Þ0 ≤ z ≤ p: ð7Þ

It can be written as follows:

Fz1 = πr2Aτ0 1 + μ cot θð Þ + πr2Bρ0 sin2θ 1 + μ cot θð ÞVz2:

ð8Þ

Substituting r = z tan θ into the equation above, it isobtained that

Fz1 = α1z2 + β1z

2Vz2, ð9Þ

where

α1 = π tan2θAτ0 1 + μ cot θð Þ,

β1 = π tan2θBρ0 sin2θ 1 + μ cot θð Þ:ð10Þ

According to Newton second law, we can get

mVzdVz

dz= − α1 + β1Vz

2� �z2,

m2

dVz2

α1 + β1Vz2� � = −z2dz:

ð11Þ

Integrating

−m2β1

ln α1 + β1Vz2� �

+ C1 =13 z

3: ð12Þ

Anchor tip Claw Barb Handspike

Piston Pyrotechnics Anchor body

Figure 2: Schematic of the penetrating unit.

R

z

Vz

dr

rdlp

𝜎n

𝜎𝜏

dz

z

r

Figure 3: Force on arbitrary anchor tip.

3International Journal of Aerospace Engineering

When t = 0, Vz =V0, and z = 0, we find

C1 =m2β1

ln α1 + β1V02� �: ð13Þ

Substituting C1 into equation (12), we can get

z = 3m2β1

ln α1 + β1V02� �

α1 + β1Vz2� �

" #1/3: ð14Þ

When z = p,

Vzp =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiα1β1

+V02

� �e−2β1p3/3m −

α1β1

s: ð15Þ

When maximum penetrating depth h1 < p, setting Vz = 0, we can get

h1 =3m2β1

ln 1 + β1α1

V02

� � 1/3: ð16Þ

By the way, according to the Newton second law, we canget

md2z

dt2= − α1 + β1Vz

2� �z2: ð17Þ

As Vz = dz/dt, the above equation can be written as

md2z

dt2= − α1 + β1

dzdt

� �2" #

z2: ð18Þ

Integrating the above equation, we can get the relationbetween the depth z and the time t. And then differentiating,we can get the relation between the acceleration and the time.In the paper, we are only interested in the relationshipbetween the initial penetration velocity and the final penetra-tion depth. Thus, the solution to the equation is not discussedin detail.

(2) When the penetrating depth z > p

When z > p, we can get the antiforce of the anchor tip byequation (6).

Fz2 = πR2Aτ0 1 + μ cot θð Þ + πR2Bρ0 sin2θ 1 + μ cot θð ÞVz2z > p:

ð19Þ

It can be written as follows:

Fz2 = α2 + β2Vz2 ð20Þ

where

α2 = πR2Aτ0 1 + μ cot θð Þ,

β2 = πR2Bρ0 sin2θ 1 + μ cot θð Þ:ð21Þ

According to the Newton second law, we can get

mVzdVz

dz= − α2 + β2Vz

2� �: ð22Þ

Obtaining

m2

dVz2

α2 + β2Vz2� � = −dz: ð23Þ

Integrating

−m2β2

ln α2 + β2Vz2� �

+ C2 = z: ð24Þ

When z = p, Vz = Vzp, getting

C2 =m2β2

ln α2 + β2Vzp2� �

+ p: ð25Þ

Substituting C2 into equation (26), we can get

z = m2β2

lnα2 + β2Vzp

2

α2 + β2Vz2

!+ p: ð26Þ

Setting Vz = 0, we can get the maximum penetratingdepth h2 ðh2 > pÞ.

h2 =m2β2

ln 1 + β2α2

Vzp2

� �+ p: ð27Þ

By the way, according to the Newton second law, we canget

mdVz

dt= − α2 + β2Vz

2� �: ð28Þ

Integrating the above equation, we can get

Vz =ffiffiffiffiffiα2β2

rtan −

ffiffiffiffiffiffiffiffiffiα2β2

pm

t + C3

" #: ð29Þ

Taking the initial condition VzðtpÞ =Vzp (tp is the time ofz = p) into the above equation, we can obtain the relationshipbetween Vz and the time t. Then, the relationship betweenthe acceleration and the time can be known.

3.3. Penetrating Test. In order to verify the abovementionedpenetration equation, the relationship between the penetrat-ing depth and the penetrating speed of the anchor body in


hard clay and fine sand media is tested experimentally. Test-ing platform of the anchoring system is shown in Figure 4.The anchoring element is fired by the powder, which isignited via the trigger, and the trigger is controlled by theelectric cable. When the anchor body is fired by the anchor-ing element, the firing speed of the anchor body is measuredby the speedometer, and the penetrating depth is measuredby the thread on the rear of the anchor body. Then, theanchor body is pulled out of the medium by the rocker-armvia the pulley. There is a force gauge between the rocker-arm and the pulley; thus, the anchoring force of the anchor-ing element will be measured. The diameter of anchor body(penetrating unit) used in the test is about 20mm, and themass is about 160 g.

The parameters of the media are shown in Table 1.Hard clay and fine sand are used in penetrating tests,and soft clay is used in anchoring tests in Section 4.2.Figure 5 shows the prediction of the penetration equationand the corresponding experimental results. It can be seenthat the penetration equation can logically describe thecorrespondence between the penetrating depth and thepenetrating speed. As the depth of the experimental mediais about 0.5m, the experiment cannot penetrate the depththat exceeds 0.5m. Experimental result errors are mainlyfrom the media parameters error and equipment measure-ment error. Simultaneously, it can be found that the depthincrease in fine sand is less than that in hard clay as thepenetrating speed increases. This phenomenon is relatedto the fluidity (characterized by the cohesion) of the finesand. The smaller the cohesion, the easier it is to flow.Thus, the cavity formed in fine sand as the anchor bodypenetrating is difficult to maintain. Compared with hardclay, flowing fine sand is easier to consume the penetra-tion energy of the anchor body.

4. Anchoring Model

4.1. Anchoring Force Analysis. After the anchor body haspenetrated, the barbs on the anchor tip are opened to pro-duce a greater anchoring force. The shape of the open barbsformed can be approximately regarded to be round and likea disc. There will be shear damage between the round discanchor tip and the medium. It is found that a circular trun-cated cone concave was formed on the surface of the mediumafter the anchor body is pulled out. Thus, an anchoringmodel shown in Figure 6 is established. The maximum prin-cipal stress of the element in medium is σ1, which is the pas-sive stress of the Rankine. The minimum principal stress σ3 isthe compressive stress caused by the gravity of the upperlayer. The depth of the round disc on the anchor tip is H.

Trigger Electric cableRocker-arm

Pulley

Force gauge

Speedometer

Anchor body

Powder

Bracket

Medium

Anchoringelement

Figure 4: Testing platform of the anchoring system.

Table 1: Parameters of the clay and fine sand.

Parameter Density ρ0(kg/m3)

Internal frictionalangle φ (°)

Cohesion τ0(kPa)

Elasticity module E(MPa)

Poissonratio

Watercontent

Voidratio

Locked volumetricstrain η ∗Medium

Hard clay 1:98 × 103 14 160 23.8 0.3 19.91% 0.709 0.15

Fine sand 1:71 × 103 35.5 0.2 45.0 0.35 0 0.585 0.1

Soft clay 1:85 × 103 14 5.0 — — — — —

800

600

400

200

Pene

trat

ing

dept

h (m

m)

Penetrating speed (m/s)

00 20 40 60 80 100

Hard clayHard clay test

Fine sandFine sand test

Figure 5: Results of penetration equation and experiments ofanchor penetration.


The medium is sheared layedly when the anchor is pulledout. When the round disc of the anchor tip is pulled out bythe distance of H1 and reaches A position, the anchoringforce F reaches the maximum value. Then, under the effectof this maximum force, the surface of the medium is sub-jected to the overall shear failure and forms the circular trun-cated cone concave. The height of the circular truncated coneconcave is H-H1. Its bottom radius is R, and its top radius isequal to the round disc radius on the anchor tip, which is r.The angle between medium surface normal and the concavegeneratrix is δ = π/4 + φ/2 (based on soil mechanics, theangle between the shear failure surface and the maximumprincipal stress surface is δ = π/4 + φ/2). Finally, the anchortip is separated from the medium by holding the circulartruncated cone.

The maximum principal stress σ1 and the minimumprincipal stress σ3 are

σ1 = ρg H − zð Þ tan2δ + 2τ0 tan δ,

σ3 = ρg H − zð Þ,ð30Þ

where ρ is the density of the medium and g is the gravitycoefficient.

From the Mohr-Coulomb yield criterion, the shearstrength of the medium is

τ = τ0 + σ tan φ: ð31Þ

The positive pressure σ of the medium in column shearfailure stage (0 ≤ z ≤H1) and circular truncated cone shearfailure stage (H1 < z ≤H), respectively, are

σ =σ1 0 ≤ z ≤H1,σ1 + σ3

2 + σ1 − σ32 cos 2δ H1 < z ≤H:

8<: ð32Þ

Thus, the shear strength is as follows:

τ1 = τ0 + σ1 tan φ 0 ≤ z ≤H1,

τ2 = τ0 +σ1 + σ3

2 −σ1 − σ3

2 sin φ� �

tan φ H1 < z ≤H:

ð33Þ

The anchoring force F1 generated in column shear failurestage (0 ≤ z ≤H1) can be expressed as follows:

F1 =ðz02πrτ1dz: ð34Þ

The anchoring force F2 generated in circular truncatedcone shear failure stage (H1 < z ≤H) can be expressed as fol-lows:

F2 =ðHH1

2π r + z −H1ð Þ tan δ½ �τ2dz

cos δ : ð35Þ

According to the assumption of the anchoring model,when z =H1, the anchoring force reaches the maximumvalue, and this force causes the shear failure of part (H-H1),resulting in a circular truncated cone concave. Thus,

F1 z =H1ð Þ = F2 cos δ: ð36Þ

Anchor body

Anchoringhole shape

Medium (soil)

Mediumshear shape

𝛿

Figure 8: Medium (soil) and its shape when the anchor body ispulled out.

Anchor body

Disc

Figure 7: 40mm and 90mm diameter disc mounted on the anchorbody.

R

Z

F 𝛿

𝜏2

𝜏1

H1

H

𝜎3

𝜎1

r

Z

L

A

O

dz

Figure 6: Anchoring model of the anchoring system after barbsstretching.


Jointing equations (30) and (33)~(36), we can firstlyobtain H1. Then, the value of the anchoring force in 0 ≤ z ≤H1 interval can be obtained.

When the circular truncated cone is generated after shearfailure of the medium (z >H1), the anchoring force F3 is the

gravity of the circular truncated cone pulled out by theanchor tip. It can be expressed as follows:

F3 =ðHH1

ρgπ r + z −H1ð Þ tan δ½ �2dz: ð37Þ

0 0.1 0.2 0.30

100

200

300

400

Pull-out displacement (m)

Anc

hori

ng fo

rce

(N)

(a)

Pull-out displacement (m)

Anc

hori

ng fo

rce

(N)

0 0.05 0.1 0.15 0.20

50

100

150

(b)

Anc

hori

ng fo

rce

(N)

0

50

100

150

200

0 0.1 0.2 0.3Pull-out displacement (m)

(c)

Anc

hori

ng fo

rce

(N)

0

20

40

60

80

Pull-out displacement (m)0 0.05 0.1 0.15 0.2

(d)

Figure 9: Anchoring experiments in soft clay.

0

20

40

60

80

Anc

hori

ng fo

rce

(N)


(a)

0

10

20

30

Anc

hori

ng fo

rce

(N)


(b)

0

20

40

60

Anc

hori

ng fo

rce

(N)


(c)

0

5

10

15

20

Anc

hori

ng fo

rce

(N)


(d)

Figure 10: Anchoring experiments in fine sand (also called soft sand).


Thus, the anchoring force F in the whole pulling out pro-cess of the anchor body can be expressed as follows:

F =F1 0 ≤ z ≤H1,F3 z >H1:

(ð38Þ

4.2. Anchoring Test. The anchoring model is verified by test-ing the anchoring forces at different depths in fine sand andsoft clay media. After several times of penetratings, the stateof the hard clay is destroyed and its parameters are changed.We remeasure the parameters of the clay and call the clay inthis state as soft clay. Moreover, soft clay is convenient foroperating in test. The key parameters of the fine sand andthe soft clay are shown in Table 1. Discs with 90mm diame-ter and 40mm diameter are separately mounted on the top ofthe anchor body, which is shown in Figure 7. Then, theanchoring forces of the anchor body in two different pene-trating depths are tested. There are two penetrating depthvalues: one is 285mm which represents full penetrating ofthe anchor body, and the other is 165mm representing par-tial penetrating.

The medium on the anchor body after it is pulled outfrom the soft soil in 285mm penetrating depth is shown inFigure 8, where the anchoring hole shape is also displayed.As the medium on the anchor body would collapse underthe effect of the gravity, it is difficult to find the integral shearshape of the medium on the anchor body at the time of pull-ing out. Nonetheless, it is obvious that the medium shearshape on the left side of the anchor body in Figure 8 is verysimilar to that shown in Figure 6.

When the anchor body is pulled out from the soft clayand the fine sand, the relationship between the anchoringforce and the pullout displacement is shown in Figures 9and 10, and the maximum anchoring force can be found intwo figures. We can conclude from the testing that theanchoring force increases gradually during the pullout pro-cess, and the surface of the media is ruptured suddenly whenthe anchoring force is up to the maximum (due to strongliquidity of the sand, the phenomenon is not obvious). Afterthe anchoring force decreases to a constant value, this valueequals to the total weight of the anchor body, the disc, andthe medium on its surface. At the same time, it can be seenthat the cohesion of the medium, the internal friction angle,the penetration depth of the anchor body, and the diameterof the disc have great influence on the anchoring force.

The calculated maximum anchoring force can beobtained by substituting the parameters of the experimentalmedium and the penetrating depth of the anchor body into

the anchoring model. The calculated and the testing maxi-mum anchoring force are shown in Table 2. It can be foundthat the calculated maximum anchoring force is close to themaximum anchoring force obtained by the experiment, sothe anchoring model is considered as reasonable.

5. Penetrating-Anchoring Mathematical Model

At the time of the asteroid landing mechanism landing, theanchor body of the anchoring system is projected into theasteroid surface at a certain speed, which produces theanchoring force. The anchoring force is the key performanceindex for the asteroid landing mechanism. Establishing therelationship between the initial penetrating speed of theanchor body and the anchoring force is of great importance.

The anchoring model mainly studies the relationshipbetween the penetrating depth H and the anchoring force F, and the penetration equation mainly describes the relation-ship between the penetration depth H and the initial pene-trating velocity V0. Therefore, on the base of the anchoringmodel and the penetration equation, the mathematical rela-tionship between the initial penetrating velocity V0 and theanchoring force F can be established with the penetrationdepth H as the link. Through this mathematical relationship,we can easily calculate the minimum initial penetrating speedneeded to obtain the required anchoring force.

The penetrating depth of the anchor body can beobtained by the penetrating equations (16) and (27), whichcan be expressed as follows:

H =h1 0 ≤ z ≤ p,h2 z > p,

(ð39Þ

written as

H = f1 m, V0, θ, μ, τ0, E, φ, ρ, ρ ∗ð Þ: ð40Þ

When the anchor body is anchored at the depth H, themaximum anchoring force occurrence point H1 can beobtained by combining equations (30) and (33)~(36). It iswritten as follows:

H1 = f2 H, τ0, φ, ρ, gð Þ: ð41Þ

And then substituting the z =H1 into equation (34), wecan obtain the maximum anchoring force Fmax.

Fmax = f3 H, τ0, φ, ρ, gð Þ: ð42Þ

Table 2: Comparison of anchoring model with experiments in soft clay and sand.

H = 285mm2r = 90mm

H = 165mm2r = 90mm

H = 285mm2r = 40mm

H = 165mm2r = 40mm

Soft clayMaximum anchoring force in the anchoring model 348.8N 131.5N 176.1N 67.0N

Maximum anchoring force in the tests 355.0N 133.6N 195.4N 70.1N

Fine sandMaximum anchoring force in the anchoring model 84.8N 20.9N 61.0N 13.7N

Maximum anchoring force in the tests 77.0N 27.1N 59.3N 17.5N


Substituting equation (40) into equation (42), we obtain

Fmax = f4 m, V0, θ, μ, τ0, E, φ, ρ, ρ∗,gð Þ: ð43Þ

Equation (43) is the penetrating-anchoring model. Thismodel incorporates the penetration equation and the anchor-ing model with the penetrating depth as a link, which com-bines two independent processes (penetrating andanchoring) as a whole. This model describes the mathemati-cal relationship among the maximum anchoring force Fmax,the initial penetrating velocity V0, and the medium proper-ties. The penetrating-anchoring model can easily predictthe required minimum initial penetrating speed that isadopted for generating the expected maximum anchoringforce.

6. Conclusions

In the paper, based on the cavity expansion model and theMohr-Coulomb yield criterion, the force on the anchor bodyduring the penetration is analyzed. A penetrating equation isintroduced for calculating the resistance force and penetrat-ing depth of the anchor body when it is penetrating. The pen-etrating depth estimated by the model is approximate to thetesting results in clay and sand media. Utilizing soil mechan-ics, the force on the deployed anchor body in the process ofpulling out is analyzed, and the anchoring model is built.This model describes the relationship between the penetrat-ing depth and the anchoring force. The anchoring force onthe anchor body equipped with 40mm disc and the 90mmdisc which represent deployed anchor tip are tested in theclay and fine sand. The maximum anchoring force obtainedby the tests is close to the calculated maximum anchoringforce. The maximum anchoring force in the test appears atthe time of media surface failure, and the residual medium’sshape on the anchor body after it is extracted is similar to themedium’s shear shape in the anchoring model. Thepenetrating-anchoring model is built by uniting the anchor-ing model and the penetration equation with the penetrationdepth as the link. This model establishes the relationshipamong the anchoring force, the initial penetrating velocity,and the media properties, and it provides a theoretical foun-dation for the design and performance estimation of theanchoring system in the asteroid or the comet landing explo-ration. Tests from different angles or close to the actual situ-ation are the follow-up plan, and we are planning to carry outsimilar tests in the future.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (No. 51775129 and No.51975139) and the State Key Laboratory of Robotics and Sys-tem (HIT).

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a penetrating-anchoring mathematical model for the soft

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