digging and pushing lunar regolith: classical soil mechanics and the forces needed for excavation...

Journal

www.elsevier.com/locate/jterra

Journal of Terramechanics 44 (2007) 133–152

ofTerramechanics

Review

Digging and pushing lunar regolith: Classical soil mechanicsand the forces needed for excavation and traction

Allen Wilkinson *, Alfred DeGennaro

Fluid Physics and Transport Branch, NASA Glenn Research Center, M.S. 110-3, 21000 Brookpark Road, Cleveland, OH 44135, United States

Received 11 January 2006; received in revised form 6 September 2006; accepted 7 September 2006Available online 13 November 2006

Abstract

There are many notional systems for excavating lunar regolith in NASA’s Exploration Vision. Quantitative system performance com-parisons are scarce in the literature. This paper focuses on the required forces for excavation and traction as quantitative predictors ofsystem feasibility. The rich history of terrestrial soil mechanics is adapted to extant lunar regolith parameters to calculate the forces. Thesoil mechanics literature often acknowledges the approximate results from the numerous excavation force models in use. An intent of thispaper is to examine their variations in the lunar context. Six excavation models and one traction model are presented. The effects of soilproperties are explored for each excavation model, for example, soil cohesion and friction, tool–soil adhesion, and soil density. Excava-tion operational parameters like digging depth, rake angle, gravity, and surcharge are examined. For the traction model, soil, opera-tional, and machine design parameters are varied to probe choices. Mathematical anomalies are noted for several models. Oneconclusion is that the excavation models yield such disparate results that lunar-field testing is prudent. All the equations and graphs pre-sented have been programmed for design use. Parameter ranges and units are included.Published by Elsevier Ltd on behalf of ISTVS.

1. Introduction

The parameters described in Fig. 1 are used throughoutSections 2 and 3 of this paper. It will become clear that thetool digging depth, d, soil internal cohesion forces, c (soil–soil sliding along the shear failure surface plane), rakeangle, b, and external friction forces (soil sliding on theblade) related to the external friction angle, d, and tool–soiladhesion, Ca and/or l, are the most prominent variables.Gravity manifests itself in this paper only by way of the soilweight. A likely overconsolidation ratio greater than onefor virgin regolith due to meteorite impact, tidal and ther-mal lunar quakes, and gravity settlement will induce mem-ory effects on soil cohesion and tool–soil adhesion. Thevacuum and plasma active lunar ambient environment alsowill induce particle surface activity that affects adhesion

0022-4898/$20.00 Published by Elsevier Ltd on behalf of ISTVS.

doi:10.1016/j.jterra.2006.09.001

* Corresponding author. Tel.: +1 216 433 2075; fax: +1 216 433 3793.E-mail addresses: [email protected], [email protected]

(A. Wilkinson).

and cohesion. Some models include an inertial forcerequired to give the soil its kinetic energy (v2) for movingwith the blade. Some models include surcharge pressure,q, due to soil mounding above the reference level of the ori-ginal soil surface. These models allow the machine force, T,to be applied to the excavator blade at an arbitrary angle,d, with horizontal, H, and vertical, V, forces resolved. Allmodels here include soil density and some cohesion effectsas a minimum. The contributions of each of the physicalsources in the models, like cohesion, are shown separatelyin the figures, except for the Osman model. Appendix Agives the exact expressions used for each physical sourcein each model plot.

Table 1 gives the parameter settings used in Sections 2and 3, except when a parameter is varied in the plots ofFigs. 2–6, 8 and 9. This parameter set represents Apollolunar soil parameters to the extent they exist [1]. AppendixB provides more definition and plausible ranges for theseparameters from terrestrial and lunar sources. No singlemodel uses all of these parameters.

mailto:[email protected]

mailto:[email protected]

Table 1Parameter set (SI units) used for Balovnev, Gill, McKyes, Osman, Swickand Viking models

Values Units

Tool width, w 1 mTool length, l 0.7 mTool depth, d 0.5 mSide length, ls 0.56 mSide thickness, s 0.02 mBlunt edge angle, ab 40 degBlunt edge thickness, eb 0.05 mMoon gravity, gM 1.63 m/s2

Earth gravity, gE 9.81 m/s2

Soil specific mass, c 1680 kg/m3

Surcharge mass, q 1 kg/m2

Rake angle, b 45 degShear plane failure angle, q 30 degTool speed, v 0.1 m/sGill’s cut resistance index, K 1000 N/mCohesion, c 170 N/m2

Internal friction angle, / 35 degSoil–tool adhesion, Ca 1930 N/m2

Soil–tool normal force, N0 100 NExternal friction angle, d 10 degOsman-d1 1Osman-d2 1Osman-d3 1Osman-d4 1Osman-d5 1Osman-d6 1Osman-d7 1Log radius, w 0 1Base log radius, r0 1 mCalculated log radius, r1 3.17 mRankine passive depth, t 0.5 m

Lunar soil parameters are used where available.

Fig. 1. Definition of angles and parameters [2].

134 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007) 133–152

When using excavation and traction results here, onewould balance H with the drawbar pull while V contrib-utes to the effective weight of the vehicle, either heavieror lighter. Using known soil parameters, one adjustsmachine parameters until H and drawbar pull match,with some established drawbar pull over-design factor.Given that we do not know the soil parameters precisely,one has to do soil parametric studies to be sure thedesign is tolerant.

Beyond the scope of this work is the customary fieldtesting required to validate design with reality. Field testingis a very expensive proposition for lunar exploration thatneeds careful consideration in order to avoid robotic orhuman support failures.

2. 2-D models of excavation

2.1. Osman [2,3]

This model uniquely allows for a curved soil shear fail-ure surface. A curved failure surface is more realistic inexcavating than the flat surface other models assume, butis determined by some authors as not a significant enoughquantitative contribution to warrant the mathematical dif-ficulty. The Osman model contains surcharge and tool–soileffects, while it lacks the inertial forces that bring the soilfrom rest up to the speed of the tool. This model also lacksexplicit inclusion of the soil internal friction angle.

The excavation force equation is

T ¼ w 0:5gct2ftan2ð45þ 0:5qÞgd1 þe2w0 tan q � 1

4 tan qr2

0gcd2

��

þ gqtftan2ð45þ 0:5qÞgd4

�d�1

3

�

þ w 0:5ðr21 � r2

0Þc

tan qþ 2ctftanð45þ 0:5qÞgd4

��

þ gqtsinð45� 0:5qÞ d5 þ Cald7

�d�1

6

�ð1Þ

and the horizontal and vertical components of the totalforce, T, are

H ¼ T sinðbþ dÞ;V ¼ T cosðbþ dÞ:

The authors here do not develop the use of this modelsince the di’s are indeterminate in the original Osmanpaper. To get the di’s an optimizing iteration is requiredon a parameter k, related to the location of the center ofrotation of the log-radius shear surface, that is not mathe-matically explicit in the formulation by either Osman orBlouin. As a result no quantitative conclusions are meritedhere. Fig. 2 shows the behavior of H for several parameterswhile holding all other parameters constant as in Table 1.The di’s used in this figure are arbitrary. In this model cer-tain values of q will cause the tan(f(q)) and 1/tan(q) func-tions to become divergent.

2.2. Gill and Vanden Berg [2,4]

This model, like Osman’s, is of older origins. It adds aninertial force contribution with the tool velocity term. Sur-charge is missing from this model, but simplified tool–soil

0.2 0.4 0.6 0.8 1.0

5000

6000

7000

8000

Horizontal Force vsRankine Passive Depth

OsmanRankine Passive Depth, t (m)

For

ce (

N)

0 50 100 150

5400

5450

5500

5550

5600

5650

5700

Horizontal Force vsSurcharge

Surcharge, q (N/m2)

For

ce (

N)

1500 2000 2500 3000 3500

5000

6000

7000

8000

Horizontal Force vsSoil Density

Soil Mass Density, γ (kg/m ) 3

For

ce (

N)

0 1000 2000 3000 4000

5000

1500

025

000

3500

0

Horizontal Force vsCohesion Stress

Cohesion, c (N/m2)

For

ce (

N)

0.0 0.5 1.0 1.5

4000

5000

6000

7000

8000

Horizontal Force vsLog Spiral Angle

Log Spiral Angle, w’ (radians)

For

ce (

N)

1 2 3 4 5

1000

020

000

3000

040

000

5000

0

Horizontal Force vsInitial Log Spiral Radius

Initial Log Spiral Radius, r0 (m)

For

ce (

N)

Fig. 2. Osman’s required drawbar force variation for significant parameters.

A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007) 133–152 135

friction, soil–soil cohesion, and inertial forces are included.In the simplest form, the model for the horizontal force canbe written as

H ¼ No sin bþ dNo cos bþ K � wwith an ideal penetration cutting force term contained inK Æ w. Blouin re-writes the force equation to allow leavingout that force as

H � ¼ H � K � w ¼ No sin bþ dNo cos b:

Eq. (2) breaks out the full complexity of No

H � ¼ gcsinðbþ qÞ

sin qlþ d cosðbþ qÞ

2 sin qþ d sinðbþ qÞ tan b

2 sin q

� ��

þ csin qðsin qþ / cos qÞ þ

cv2 sin bsinðbþ qÞðsin qþ / cos qÞ

�

� wdðsin bþ d cos bÞðsin qþ / cos qÞsinðqþ bÞð1� /dÞ þ cosðqþ bÞð/� dÞ : ð2Þ

0.2 0.4 0.6 0.8 1.0

020

0040

0060

0080

0010

000

Horizontal Force vsTool Depth

GillDepth, d (m)

For

ce (

N)

total

cut

tool_soil

depth

cohesion

kinetic

20 40 60 80

020

0060

0010

000

1400

0

Horizontal Force vsRake Angle

Rake Angle, , β (deg)

For

ce (

N)

20 25 30 35 40 45 50

010

0020

0030

0040

00

Horizontal Force vsSoil Internal Friction Angle

Internal Friction Angle, φ (deg)

For

ce (

N)

0 1000 2000 3000 4000

020

0040

0060

0080

00


Cohesion, c (N/m2)

For

ce (

N)

2 4 6 8 10

050

0010

000

1500

020

000

Horizontal Force vsGravity

Acceleration of Gravity, g (m/s2)

For

ce (

N)

1500 2000 2500 3000 3500

020

0040

0060

00


Soil Mass Density, γ (kg/m3)

For

ce (

N)

Fig. 3. Gill’s drawbar force variation for significant parameters.


The penetration K Æ w term is kept in results here as it hassignificant magnitude. The total and vertical componentsof this force are

T ¼ H cscðbþ dÞ;V ¼ H cotðbþ dÞ:From this point forward plot results will show the individ-ual contributions of physical sources of the forces in order

to gauge their relative contributions. Appendix A givesthe exact expressions used for each physical source notedby the legends. See Fig. 3 for the predicted drawbar forcerequirement under the variation of several parameters whileholding all other parameters constant as in Table 1. Thismodel indicates that depth and tool–soil friction are thedominant contributors to the drawbar pull required forexcavation. Depth, gravity, and rake angle, b, present the

0.2 0.4 0.6 0.8 1.0

010

0020

0030

0040

00


SwickDepth, d (m)

For

ce (

N)

totalsurchargetool_soildepthcohesionkinetic

20 40 60 80

–100

00

1000

2000

3000


Rake Angle, β (deg)

For

ce (

N)

20 25 30 35 40 45 50

050

010

0015

0020

00



For

ce (

N)

0 1000 2000 3000 4000

010

0020

0030

0040

00


Cohesion, c (N/m2)

For

ce (

N)

2 4 6 8 10

010

0020

0030

0040

0050

00



For

ce (

N)

1500 2000 2500 3000 3500

050

010

0015

0020

00



For

ce (

N)

Fig. 4. Swick and Perumpral drawbar force variation for the dominant parameters.


strongest functional dependence. The lunar soil cohesionused here makes cohesion a small contribution. Cohesionas a parameter projects roughly a 100% increase over therange, while the internal friction angle, /, projects a 34% in-crease. Over a plausible range of soil density the requireddrawbar force increases by 120% in a linear way, althoughno effect of density on K was allowed here. This model pre-sents mathematical problems as the rake angle, b, ap-

proaches 90�, as it might for a backhoe, due to the tanbfactor in the first line of Eq. (2).

2.3. Swick and Perumpral [2,5]

The Swick and Perumpral model includes a moresophisticated tool–soil friction term with the adhesionparameter Ca. All the physical effects of a simple blade

0.2 0.4 0.6 0.8 1.0

010

0020

0030

0040

00

McKyesDepth, d (m)

For

ce (

N)

totalsurchargetool_soildepthcohesionkinetic

20 40 60 80–100

00

1000

2000

3000


For

ce (

N)

20 25 30 35 40 45 50

050

010

0015

0020

00

Horizontal Force vs



Soil Internal Friction Angle


For

ce (

N)

0 1000 2000 3000 4000

010

0020

0030

0040

00


Cohesion, c (N/m2)

For

ce (

N)

2 4 6 8 10

010

0020

0030

0040

0050

00



For

ce (

N)

1500 2000 2500 3000 3500

050

010

0015

0020

00



For

ce (

N)

Fig. 5. McKyes’ drawbar force variation for significant parameters.


without end effects are included in this model. The magni-tude of the total force is written as follows:

T ¼ �Ca cosðbþ /þ qÞsin b

þ gcd2ðcot bþ cot qÞ sinð/þ qÞ

�

þgqðcot bþ cot qÞ sinð/þ qÞ þ c cos /sin q

þ cv2 sin b cos /sinðbþ qÞ

�

� wdsinðbþ /þ qþ dÞ

ð3Þ



See Fig. 4 for the variation of H for several parameters. Thismodel is problematic for certain angle ranges. If b + / +q + d P 180�, then the multiplier on the last line of Eq. (3)passes through a singularity and jumps negative. If b + /+ q < 90�, then the tool–soil force is non-physical for smallrake angles, b, which comes from the �Ca term in Eq. (3).This paper considered using the rule that q = 45� + //2.However, that produced the singular multiplier using lunarsoil parameters at larger rake angles. As a result a fixed valueof q is chosen in Table 1, that of a sand [6]. This still produced

0.2 0.4 0.6 0.8 1.0

020

0040

0060

0080

00


MuffDepth, d (m)

For

ce (

N)

totalfrictioncohesionkinetic

20 40 60 80

010

0030

0050

0070

00



For

ce (

N)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

050

015

0025

0035

00

Horizontal Force vsTool Velocity

Tool Velocity, v (m/s)

For

ce (

N)

0 1000 2000 3000 4000

020

0040

0060

0080

00


Cohesion, c (N/m2)

For

ce (

N)

2 4 6 8 10

050

0010

000

1500

020

000



For

ce (

N)

1500 2000 2500 3000 3500

010

0030

0050

0070

00



For

ce (

N)

Fig. 6. Viking model’s drawbar force variation for significant parameters under lunar conditions.


the negative tool–soil force contribution as seen in the ‘‘hor-izontal force vs. rake angle’’ plot in Fig. 4.

If one accepts that the failure plane angle in triaxialshear cell tests is the same angle as q, then using the Mohrstress circle defined by triaxial test results, one getsq = 45� + //2 [7, p. 14–15]. Hoek also suggests that qdepends on b as well [8]. Some forms of dependence of qon b could either eliminate or exacerbate the singular mul-tiplier problem, depending on the form.

Overall, depth remains the largest contributing term as itwas for Gill. Depth, gravity, and rake angle show strongeffects. However, soil–soil cohesion has a strong depen-dence at values larger than the lunar value here with

a 270% increase over the range plotted. Soil internal frictionangle, /, projects a 150% increase over the range. Soil den-sity shows a 90% increase over the range plotted. Surchargeremains small as its chosen mass per area here is small.

2.4. McKyes [2,7]

Like Swick and Perumpral, the McKyes model includesall the physical effects of a two-dimensional blade withoutend effects. This model is an enhancement of early work byReece [9]. In the simplest form this model shows a heritagefrom the Terzaghi coefficients, Nx [10]. The total forceequation is


T ¼wðcgd2N cþ cdN cþCadN caþ gqdN qþ cm2dN aÞ;

T ¼ cgdðcotbþ cotqÞ2

þ gqðcotbþ cotqÞ�þ cð1þ cotqcotðqþ/ÞÞþCað1� cotbcotðqþ/ÞÞ

þcv2ðtanqþ cotðqþ/ÞÞ1þ tanqcotb

�� wdcosðbþ dÞþ sinðbþ dÞcotðqþ/Þ

ð4Þ



A feature of the results in Fig. 5 is the quantitativeequivalence to Swick and Perumpral, even with a differ-ent mathematical description. However, the chief differ-ence between Eqs. (3) and (4) is in the handling of thetrigonometric factors. This model shows the same math-ematical problems for the angle sums as noted in Section2.3; negative tool–soil adhesion force for some rake an-gles depending on the choices of q and / and a vanishingdivisor on the last line of Eq. (4) for other rake angles.Together these cause concern for the generality of thesemodels.

2.5. Lockheed-Martin/Viking [11–13]

This model was used by Martin-Marietta Corp. in coop-eration with the Colorado School of Mines for design ofthe Mars Viking lander robotic excavation arm. The origi-nal forms come from Luth and Wismer testing sands forfriction terms and clays for cohesive terms [12,13]. Theseequations are also in active application for characterizationof a bucket-wheel excavator for lunar and Martian use.The model includes tool–soil friction, soil–soil shear resis-tance due to cohesion, and inertial effects. Velocity hassome non-quadratic contribution in the cohesion equa-tions; v0.121 in Hcohesion and v0.041 in Vcohesion. Surcharge,shear plane failure angle and soil–tool adhesion are not fac-tors in this model. The use of assorted exponents and con-stant terms make it hard to reflect on the physical details ofthe model.

H and V stand for horizontal and vertical force compo-nents respectively in Eqs. (5)

H friction ¼ cgwl1:5b1:73ffiffiffidp d

l sin b

� �0:77

� 1:05dw

� �1:1

þ 1:26v2

glþ 3:91

( );

V friction ¼ cgwl1:5ffiffiffidpf0:193� ðb� 0:714Þ2g d

l sin b

� �0:777

� 1:31dw

� �0:966

þ 1:43v2

glþ 5:60

( );

H cohesion ¼ cgwl1:5b1:15ffiffiffidp d

l sin b

� �1:21

� 11:5ccgd

� �1:212v3w

� �0:121

0:055dw

� �0:78

þ 0:065

!(

þ 0:64v2

gl

�;

V cohesion ¼ cgwl1:5ffiffiffidpf0:48� ðb� 0:70Þ3g d

l sin b

� �

� 11:5ccgd

� �0:412v3w

� �0:041

9:2dw

� �0:225

� 5:0

!(

þ 0:24v2

gl

�: ð5Þ

Examining Fig. 6, friction contributions far outweigh othercontributions under low cohesion lunar conditions andmodest excavation speed. Inertial (kinetic) contributionsremain small as they have in other models in this paper.Depth, gravity, and rake angle dependence remain strong.Cohesion effects have a weakly non-linear character as dis-tinct from the other models and projects a 140% increaseover the plotted range. This model, unique from the othersin this paper, almost triples the required drawbar pull overthe density range considered. The angles d, /, and q do notfigure into this model, and the rake angle does not createthe mathematical problems of the Gill, Swick and Perump-ral, and McKyes models.

2.6. 2-D section summary

In this section the Gill, Swick and Perumpral, along withthe McKyes and Viking models, predict in the plots factor-of-2-like differences between the models in the expecteddrawbar forces required under identical lunar conditions.Cohesion causes drawbar pull requirements to vary morethan soil internal friction for the range of each parameterconsidered plausible here. It is beyond the scope of this paperto survey the comparison of these predictions with terrestrialfield test data. The authors suggest that the level of quantita-

tive variance in these models requires field tests in situ on the

lunar surface to validate their design dependability for long-

term space hardware procurement. Pending validation theauthors cannot recommend one model over another.

3. 3-D models for excavation

Most excavator buckets have sides that cut through soiland confine material loss from the bucket. This sectionadds the cutting and tool–soil adhesion effects from thesesides. Some Russian workers laid the foundations for thisthinking [14–16].

3.1. Hemami [2,17]

Hemami has partitioned the excavation forces withattention to all the physical actions as identified below.

Table 2Balovnev data set: Cartesian forces for various excavation depths

Depth (m) Horizontal force (N) Vertical force (N)

1 0.05 74.84 52.402 0.10 156.71 109.733 0.15 262.87 184.074 0.20 393.34 275.425 0.25 548.11 383.796 0.30 727.17 509.177 0.35 930.53 651.578 0.40 1158.19 810.98


Fig. 7 illustrates these forces. Eq. (6) adds up all the hori-zontal (x) components

F x ¼ f1x þ f2x þ f3x þ f4x þ f5x þ f6x; ð6Þwhere f1 is the weight of the accumulating material in thebucket, f2 is the resistance from compacting the material,f3 is all the friction forces of material sliding on bucket sur-faces, f4 is the penetration or cutting resistance, f5 is theinertial force from accelerating the material to the toolvelocity much like in Section 2. Finally, f6 is the inertialforce to move the empty bucket.

3.2. Balovnev [2,14]

The non-inertial components of Hemami’s model can befurther defined by associating the fi to the horizontal forcesaccording to Balovnev:

f4x ¼ P 1 þ P 2 þ P 3 and f 3x ¼ P 4;

where P1 is the cutting and surface friction resistance of aflat trenching blade with a sharp edge; P2 is the additionalcutting resistance due to resistance from a blunt edge; P3 isthe resistance offered by cutting from the two confiningsides of the bucket; and P4 is the resistance due to frictionon those sides. Interestingly f1, f2, f5, f6 are not included inthis picture. The Russian literature considers these second-ary and small [16,15].

The horizontal component of the total force is now writ-ten as

H ¼ f4x þ f3x ¼ ðP 1 þ P 2 þ P 3Þ þ P 4

¼ wdð1þ cot b cot dÞA1

dgc2þ c cot /þ gqþ BURIED

�

�ðd � l sin bÞ gc1� sin /1þ sin /

� ��þ webð1þ tan d cot abÞ

� A2

ebgc2þ c cot /þ gqþ dgc

1� sin /1þ sin /

� ��

þ 2sdA3

dgc2þ c cot /þ gqþ BURIED

�

�ðd � ls sin bÞ gc1� sin /1þ sin /

� ��þ 4 tan dA4lsd

� dgc2þ c cot /þ gqþ BURIED � ðd � ls sin bÞ

�

� gc1� sin /1þ sin /

� ��; ð7Þ

Fig. 7. Definition of forces used in this section [2].

where BURIED = TRUE or FALSE is a 1 or 0 dependingon whether the whole bucket is below the soil surface ornot.

A1 ¼ AðbÞ; A2 ¼ AðabÞ; A3 ¼ A4 ¼ A p2

� are geometric

factors depending on the angle of a surface with respectto a reference plane. To calculate a particular Ai replaceb with the appropriate argument in the following equation:

AðbÞ ¼ 1� sin / cosð2bÞ1� sin /

if b < 0:5 sin�1 sin dsin /

� �� d

� �;

cos d cos dþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2 /� sin2 d

q� �1� sin /

e

2b�pþdþsin�1 sin dsin /ð Þ�

tan /

if b P 0:5 sin�1 sin dsin /

� �� d

� �:

The total and vertical components of this force are

T ¼ H cscðbþ dÞ;V ¼ H cotðbþ dÞ:

Table 2 provides horizontal and vertical forces as a func-tion of cutting depth. These numbers are between thoseof the Gill and McKyes models. As a compromise they willbe used in the discussions of Section 4.

Fig. 8 shows the dependence on digging depth, soildensity, surcharge, soil cohesion, and gravity. The dis-continuity in the Depth plot is caused by the bucketentering the buried condition. The drawbar force showslinear dependence on soil cohesion and gravity like allthe other models except the Viking model. Surcharge,as in all the other models, has little effect unless it isgreatly increased. Soil density causes a 133% increaseof the drawbar force over the range plotted. Cohesion

9 0.45 1410.15 987.4010 0.50 1776.48 1243.9111 0.55 2158.48 1511.3812 0.60 2577.94 1805.0913 0.65 3034.87 2125.0414 0.70 3529.27 2471.2215 0.75 4061.13 2843.6416 0.80 4630.47 3242.2917 0.85 5237.27 3667.1718 0.90 5881.54 4118.3019 0.95 6563.27 4595.6520 1.00 7282.48 5099.25


projects a 740% increase over the plotted range, muchgreater than any other model. Depth remains the stron-gest contributor to the drawbar pull, as it was in theother models.

Fig. 9 shows the dependence on various angles. The dis-continuity in the rake angle plot is due to the bucket exitingthe buried condition as the rake angle increases. The shearplane failure angle, q, does not enter this model. The rakeangle in various angle summations does not create theproblem noted for the Gill, Swick and Perumpral, andMcKyes models. However, if d is greater than /, thenA(b) misbehaves in this model. Overall though, the qualita-tive shapes of the curves are similar to the other models.

The Balovnev model shows that the blunt edge has littleeffect. Soil–tool adhesion caused by the sides is a prominentcontributor. Side effects are ignored in all Section 2 models.The soil internal friction angle, /, projects a 130% increasein horizontal force over the plot range.

3.3. 3-D section summary

In all the excavation models of this paper soil propertiesare considered homogeneous. Chapter 9 and Section 1.4 of

0.2 0.4 0.6 0.8 1.0

020

0040

0060

00


Balovnev.1Depth, d (m)

For

ce (

N)

totalsurchargedepthcohesionsharp_bladeblunt_bladesides_cutsides_friction

0 1000 2000 3000 4000

020

0040

0060

0080

0010

000


Cohesion, c (N/m2)

For

ce (

N)

Fig. 8. Balovnev’s drawbar force variation for some significant parameters. NoAlso the sum of sharp_blade (P1), blunt_blade (P2), sides_cut (P3), and sides_intended to help comparison to previous models as well as examine the new f

the Lunar Sourcebook gives estimated fits of density withdepth [1]. The interdependence of parameters like (b,/,q)and (g,/,c), which seem physically plausible, are neitherdealt with here nor often in the literature. The Balovnevmodel includes some larger contributors to excavationforces not seen in Section 2. Cohesion causes requireddrawbar pull to vary more than it does with soil internalfriction and by a wider margin than in the 2-D models.Table 3 provides a list of excavation depths for a 1 m widebucket that requires a drawbar pull expected for the Apollorover as estimated in Section 4 for the moon. Lunar-fieldtest data is needed to sort which depth is valid. Pendingvalidation the authors cannot recommend one model overanother.

4. Traction [18–20]

Three notable methods of performing traction calcula-tions are: (1) agricultural engineering’s mobility index-based [21] method, (2) the NATO Reference MobilityModel [22,23, p. 120] method, and (3) the normal and shearstress-based method of Bekker [20]. The first two use datafrom cone-penetrometer measurements as the single source

1500 2000 2500 3000 3500

050

010

0015

0020

0025

0030

00



For

ce (

N)

2 4 6 8 10

020

0040

0060

0080

00



For

ce (

N)

te that the sum of surcharge, cohesion, and depth add up to the total force.friction (P4) add up to the total force in these plots. This break-out was

eatures of this model.

20 40 60 80

050

010

0015

0020

0025

00


Balovnev.2Rake Angle, β (deg)

For

ce (

N)

0 5 10 15 20 25 30 35

020

0040

0060

0080

00

Horizontal Force vsExternal Friction Angle

External Friction Angle, δ (deg)

For

ce (

N)

totalsurchargedepthcohesionsharp_bladeblunt_bladesides_cutsides_friction

15 20 25 30 35 40 45

050

010

0015

0020

0025

00



For

ce (

N)

0.01 0.02 0.03 0.04 0.05

050

010

0015

00

Horizontal Force vsBlunt Edge Thickness

Blunt Edge Thickness, eb (m)

For

ce (

N)

Fig. 9. Balovnev’s required drawbar force variation for additional significant parameters.

Table 3Spread of digging depth for roughly the same drawbar pull on the moon

Model Depth (m) Required drawbarpull (N)

Gill <0.05 239McKyes and Swick

and Perumpral0.15 246

Viking 0.062 239Balovnev 0.14 239


of soil information. The third uses bevameter measure-ments along with soil cohesion, internal friction angle,and slippage information for calculations. NASA has his-torically worked with the Bekker approach (3). Theauthors employ the Bekker approach here, while quantita-tively examining the effects of several parts of the modelusing Apollo lunar rover [24] and soil parameters wherepossible.

This paper does not deal with the effects of the runninggear. It focuses on the soil mechanics.

Notation for this section is defined in Fig. 10 andAppendix B. Further definition is available in Refs.[18,20]. This model starts from the linear Mohr–Coulombrelation between normal and shear stresses, s = c +r tan/.

4.1. Soil thrust and slip

Slip directly affects the traction, or drawbar pull. Thispaper does not include effects from the inertial forces ofmoving the soil during slip. The effect is related to whatwe saw for excavation in Section 2. There, the inertial con-tribution was small.

Patel offers S ¼ xWktL2 [23] to calculate slip. The authors use

this form for calculations here. The result of this predictionfor lunar soil is given in Table 4. However, more commonlyit is an experimental observable that is plugged into a cal-culation. Once the ideal drawbar pull, H0, is calculated, asseen later in this subsection, there are several forms to cal-culate the effective drawbar pull before various resistancesare subtracted. At low velocity for plastic soils one com-putes the effective drawbar pull most simply as [20]

H ¼ H 0ð1� e�S�L=jÞ: ð8Þ

Another form used for Apollo rovers, assuming plasticsoils, is [1,6]

H ¼ H 0 1� jS � L ð1� e�S�L=jÞ

h i: ð9Þ

This form is used for the plots shown here. However, theuse of these forms for lunar soils is unclear, as lunar soil

D

zε

l

soil surface

α

Fig. 10. Simple definition of geometrical parameters for traction.


may well behave as brittle dry granular soils. A form forsoils in general (including brittle soils) is [20, p. 8]

H ¼ H 0 exp �K2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2

2 � 1q� �

K1S � L� ��

� exp �K2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2

2 � 1q� �

K1S � L� ��

: ð10Þ

Table 4Parameter set base: (SI units) used for traction calculations

Values Units

Calculated slippage, S 8.30 %Calculated sinkage, z 0.00863 mTire squat, e 0.0081 mSoil rupture angle, q 62.5 radSoil specific mass, c 1680 kg/m3

Vehicle mass, W 698.5 kgGravity level, gM 1.63 m/s2

Terrain slope angle, h 0 degInternal friction angle, / 35 degSoil cohesion, c 170 N/m2

Calculated wheel/track contact area, A 0.0400 m2

Wheel nominal width, B 0.27 mWheel diameter, D 0.81 mCalculated wheel contact length, L 0.188 mNumber of wheels/tracks, n 4Contact grousers per wheel/track, Ng 4Grouser height, h 0.0015875 mCalculated k 825,185 Pa=mn0

kc 1400 Pa=mn0�1

k/ 820,000 Pa=mn0

Calculated Kc 48.6Nc 25Calculated Kc 0.537Nc 1.5Calculated l0 0.00234 mkt 50Degree of brittleness K1 5.56 m�1

Slip strength K2 2.5Coefficient of surface adhesion, x 0.000518 m2/NShear deformation slip modulus, j 0.018 mSoil deformation exponent, n0 1Tracked OR not 0 1 = TRUE,

0 = FALSECalculated net drawbar pull, DP 239 N

In Fig. 11 all three of these forms for drawbar pull are com-pared using values for K1 and K2 that are selected to trackEq. (9) as well as possible. K1 and K2 are called slip coeffi-cients, and values do not exist for lunar soils. Larger valuesof K1 cause steeper initial slope and quicker fall-off at largerslip in plots of H = f(S). K1 is the degree of brittleness, acohesion effect. Larger values of K2 cause an increase inthe magnitude of H = f(S) for all slippage values. K2 isthe degree of slip strength, a friction effect. Today onewould measure the drawbar pull versus slippage and fitthe curve for the best values of K1 and K2.

Appendix B gives a range for K1 and K2 that span fromloose sand (low cohesion and high friction angle) to brittle/compact (high cohesion and low friction angle) soils in ter-restrial experience [19, p. 266–7]. Notice K1 and K2 usedhere are of lower cohesion and higher friction angle soilthan a slightly moist sandy loam that Bekker describes.

To get H0, the ideal tractive thrust available, for a vehi-cle with a soil contact area of A = bL per wheel, track, orleg, and W is the total vehicle weight, one uses the equation[18]

H 0 ¼ Acþ Wn

l ¼ Acþ Wn

tan /:

For an n-wheeled or n-legged powered vehicle, total soilthrust is given by

H 0 ¼ nbLcþ W tan /:

Different weight loading per wheel, track, or leg is not con-sidered here for simplicity. Likewise the effects of pre-con-solidation or confining pressure from neighboring wheelsor legs on improved soil shear strength and the resultant ef-fects on H0 and slippage is not considered.

For tracked vehicles, the soil thrust increases due to thegreatly increased ground contact area. For two tracks, soilthrust is given by

H 0 ¼ 2bLcþ W tan /:

4.2. Grousers

Grousers, or cleats, are an additional way to engage moresoil (excavation-like) for greater traction. This paperincludes the number of grousers interacting with the soil asthe wheel diameter changes, given a fixed grouser spacing.However, it does not include the soil shear strength decreasewhen the grousers are tall enough and close enough to dis-rupt the soil of neighboring grousers. The Apollo lunar roverhad 54 titanium alloy chevrons, 1.6 mm high, covering 50%of the wheel surface [24]. This leads to a 2.4 cm spacingbetween grousers. For a grousered, n-wheeled/legged vehi-cle, the maximum soil thrust is given by [19, p. 226–7]

H 0 ¼ nT 0 þ W tan /;

where T 0 ¼ sN gð h�bcos q�sin qþ h2

cos qÞ.

For a double tracked vehicle with Ng grousers of thick-ness, or height h, in contact with the soil, the maximum soilthrust per track is

2 4 6 8 10

050

010

0015

0020

0025

00

Draw Bar Pull vsGravity


Net

Dra

w B

ar P

ull (

N)

TotalCompressiveBulldozing

0 20 40 60 80 100

020

040

060

080

0

Draw Bar Pull vsSlippage Alone

Slippage, s (%)

Net

Dra

w B

ar P

ull (

N)

Simple PlasticLSB PlasticGeneral & Brittle

0.00 0.04 0.08 0.12–100

00–5

000

050

0010

000

Draw Bar Pull vsSinkage Alone

Sinkage, z (m)

Net

Dra

w B

ar P

ull (

N)


0e+00 2e+05 4e+05 6e+05 8e+05

–300

–100

010

020

030

0

Draw Bar Pull vsModulii of Soil Deformation

kφ or 100 X kc

Net

Dra

w B

ar P

ull (

N)

kφkc

0 1000 2000 3000 4000

050

100

150

200

250

Draw Bar Pull vsSoil Cohesion

Soil Cohesion, c (N/m2)

Net

Dra

w B

ar P

ull (

N)


20 25 30 35 40 45 50

010

020

030

040

0

Draw Bar Pull vsSoil Internal Friction Angle


Net

Dra

w B

ar P

ull (

N)


Fig. 11. Bekker’s predicted drawbar pull as a function of gravity, slippage, sinkage, and soil strength.


H 0 ¼ 2N gbLc 1þ 2hb

� �þ W tan /

hb

cot�1 hb

� �:

4.3. Motion resistances

There are several forces that subtract from the effectivedrawbar pull after slippage. These forces are sinkage, bull-dozing, and hill climbing, along with others relevant to aspecific context like wheel flexure elastic losses or soil iner-tia with slippage of grousers or transmission and runninggear losses. This paper allows for the first three. Inertialosses may be small at low speeds as we saw with the exca-

vation equations earlier. Sinkage and bulldozing depend onsoil behavior.

4.3.1. Soil compaction due to sinkage

Energy is lost in packing down the soil by a wheel, leg, ortrack. The modulus of soil deformation, k = kc/b + k/ [20, p.240, 313–40 and 447], provides a cohesional, kc, and fric-tional, k/, measure of the resistance to compaction due tosinkage. One should find kc and K1, as well as k/ and K2,coupled by their cohesive and frictional physical origins,respectively. The computer code supporting this paper usesthe general form of sinkage for n wheels or tracks as follows:


z ¼ Wn � A � k

� � 1n0

¼ Pk

� � 1n0

; ð11Þ

where the total vehicle weight, W, is rationalized by the totalground contact area, n Æ A, for the graphs and tables here.

Slippage induced sinkage [20, p. 139] is not covered inthis paper. Multiple passes in the same track by a wheelis not covered by Eq. (11), unless one has new measure-ments of k and A.

Another form for sinkage of a balanced n-wheeled vehi-cle is given by

z ¼ 3W =n

ð3� n0Þk � bffiffiffiffiDp

! 2ð2n0þ1Þ

:

This is not a special case of Eq. (11), but rather an empiricalform which Bekker wrote down that described three differentexperimental results for a rigid wheel if one uses different val-ues of n 0. See Ref. [20, p. 437–9] for more discussion.

Using z of Eq. (11), the soil compaction resistance for n

tracked, wheeled, and legged vehicles is determinedthrough the following equation [20, p. 458 and 484]:

Rc ¼ nb � k

n0 þ 1

� �zðn0þ1Þ: ð12Þ

Table 5Drawbar pull versus sinkage under lunar conditions

Sinkage (m) Drawbar pull (N) Compressiveresistance (N)

Bulldozingresistance (N)

1 0.0010 292.27 0.446 0.7242 0.0078 249.26 26.764 17.4203 0.0145 150.01 93.687 49.7434 0.0213 �2.37 201.216 94.5965 0.0280 �206.39 349.350 150.4756 0.0348 �461.08 538.090 216.433

4.3.2. Bulldozing

Bulldozing comes from the pushing of soil in front of awheel by the action of that wheel. For soft soils with largesinkage, this term can dominate over compaction losses.

A form for the bulldozing resistance per wheel or trackis [20, p. 453, Eqs. (2)–(25)]

Rb¼bsinðaþ/Þ2sinacos/

ð2zcKcþgcz2KcÞ

þ pgcl30ð90�/Þ540

þpcl20

180þcl2

0 tan 45þ/2

� �� : ð13Þ

Bulldozing for wheeled vehicles involves all the terms ofEq. (13). But for tracked vehicles, only the first term is rel-evant. The key parameters of Eq. (13) are [10]

Kc ¼ ðnc � tan /Þ cos2 /;

Kc ¼2nc

tan /þ 1

� �cos2 / and

l0 ¼ z tan2ð45� /=2Þ:
7 0.0415 �765.79 767.435 291.7938 0.0483 �1119.97 1037.385 376.0269 0.0550 �1523.20 1347.940 468.701
10 0.0618 �1975.11 1699.101 569.45111 0.0685 �2475.38 2090.867 677.95512 0.0753 �3023.72 2523.238 793.92513 0.0820 �3619.88 2996.214 917.10214 0.0888 �4263.60 3509.796 1047.245

4.3.3. Gravitational

This force is just the simple projection of the vehicleweight vector along an incline during ascent or descent, writ-ten here as

Rg ¼ W sin h: ð14Þ
15 0.0955 �4954.67 4063.983 1184.12816 0.1023 �5692.88 4658.776 1327.54217 0.1090 �6478.02 5294.174 1477.28618 0.1158 �7309.90 5970.177 1633.16719 0.1225 �8188.35 6686.785 1795.00220 0.1293 �9113.17 7443.999 1962.612
4.4. Drawbar pull

With all the forces in hand the net drawbar pull is writ-ten, assuming R is independent of slip, as follows:

DP ¼ H � R ¼ H � ðRc þ Rb þ Rg þ RotherÞ: ð15ÞTable 4 gives the parameters used for the calculations ofTable 5 and Figs. 11 and 12, except where they were specif-ically overridden to isolate a particular effect. The tablevalues reflect literature values for the moon and the 4wheel-drive Apollo lunar rover [1,24].

Table 5 presents the data behind the sinkage plot inFig. 11. This predicts that if the Apollo lunar rover were tosink about 2.1 cm into the regolith, then it would be stuck.Compressive resistance increases more rapidly with sinkagethan bulldozing resistance for this range of sinkage. Note,this paper does not develop the consequences of soil param-eter changes with depth, like density, as they are not certainenough in the literature. For example, the lunar surface pow-der layer that sometimes is crusty is not accounted for at all.

Fig. 11 maps the effects of gravity, slippage, sinkage, andmodulus of sinkage deformation, k, on traction. Realize thatslippage and sinkage, both proportional to vehicle weight inthese calculations, are increasing with increasing gravity.Also soil shear strength increases with gravity since the nor-mal stress from the vehicle weight increases. Bulldozing resis-tance increases more rapidly than compressive resistance asgravity increases. At about two-thirds of Earth’s gravitydrawbar pull has a maxima for this soil and vehicle.

Recall there are three slippage models above. One mightexpect that the lunar soil could be described as brittle ratherthan plastic with drawbar pull declining after ‘‘failure’’ ataround 60% slippage in Fig. 11. The coincidence of the‘‘General and Brittle’’ curve and the ‘‘Lunar SourceBook(LSB) Plastic’’ curve is arbitrary, as there is no lunar datato fit for K1 and K2. The K1 and K2 used for the plots arerelevant to a loose frictional sand. The ‘‘Simple Plastic’’and ‘‘LSB Plastic’’ curves are markedly different usingknown lunar parameters. The ‘‘LSB Plastic’’ equation isused here in all calculations.

0.1 0.2 0.3 0.4 0.5

050

100

150

200

250

300

Draw Bar Pull vsWheel Width

Wheel Width, B (m)

Net

Dra

w B

ar P

ull (

N)

1 2 3 4 5

050

100

150

200

250

300

Draw Bar Pull vsWheel Diameter

Wheel Diameter, D (m)

Net

Dra

w B

ar P

ull (

N)

1500 2000 2500 3000 3500

220

225

230

235

240

245

Draw Bar Pull vsSoil Density


Net

Dra

w B

ar P

ull (

N)

0 2000 4000 6000 8000

050

010

0020

0030

00

Draw Bar Pull vsVehicle Mass

Vehicle Mass, W (kg)

Net

Dra

w B

ar P

ull (

N)

Fig. 12. Bekker’s predicted drawbar pull for wheel width, diameter, soil density, and vehicle mass.


The reader should be aware that in the k dependence plot,when either kc (the cohesive contribution) or k/ (the fric-tional contribution) are varied, then the other one is set toits Lunar Sourcebook value. That is why the k/ curve inter-sects the kc curve when k/ reaches its lunar value. k/ has thedominant dependence for the soil strength. When k/ < 415kPa and z > 1.7 cm bulldozing resistance is predicted todominate compressive resistance. If that friction contribu-tion were less than roughly 225 kPa, then the rover wouldbe stuck. The sinkage at this point is about 3 cm, or approx-imately 4% of the wheel diameter. Perhaps something likethis happened during Apollo when the lunar rover (LRV)drove in some soft soil. Small k/ gives rise to large sinkage,that causes the ‘‘approach angle’’, a, to be greater than90�; the wheel has sunk over its axel. It is worth asking whatwould happen when driving on lose excavated regolith pilesin the future.

The k/ dependence has a mathematical pathology forthe smallest k/. The problem arises from the sin(a + /)/sina coefficient for the bulldozing resistance in Eq. 13.When a + / > 180�, then Rb is negative, adding to drawbarpull unphysically. This can be seen in the ‘‘Draw Bar Pullvs. k’’ plot. However, it is unrealistic for a or / to everbe greater than 90�, and this pathology should not be apractical problem.

Cohesion shows about a 20% decrease in drawbar pullover the same range considered for excavation forces ear-

lier. However, soil internal friction angle increases drawbarpull something more than 300% over the range used in theexcavation sections here.

Fig. 12 shows that traction is not strongly dependent onsoil density (11% decrease), assuming other parameters likek are constant. Not surprisingly narrow wheels get stuck.For a width less than 3.8 cm the loaded lunar rover is stuckin this calculation. On the other hand, changing the widthto greater than 30–40 cm does not improve traction much.Increasing the wheel diameter decreases traction in thismodel. That is a consequence of decreased soil shearstrength with wheel bearing pressure from increased con-tact area and the 1/L2 dependence of slippage used here.The exponential in Eq. (9) dominates the 1/(S Æ L) factorleading to a decreasing H with wheel diameter. If S Æ L werea constant, then increased wheel diameter would increasedrawbar pull as the increasing grouser number in soil con-tact would overcome decreased wheel bearing pressure, buta law of diminishing returns occurs. The number of grous-ers per unit circumferential length was held constant.Increasing vehicle mass shows the strongest effect in thisfigure. At about 209 kg, the mass of the Apollo rover with-out cargo, the drawbar pull is roughly 26 N. An optimummass for maximum drawbar pull is seen. It may pay to self-load an excavator or rover with regolith to greatly improveits traction without having to bring mass from the Earth.However, overloading beyond the optimum is detrimental.


4.5. Traction summary

Table 4 predicts that the Apollo lunar rover could pro-vide about 239 N of drawbar pull fully loaded on a levelsurface. Compressive resistance is the most significant lossfor traction, given lunar soil parameters. However, whenk/ is small and sinkage is large, or when cohesion or grav-ity are larger, then bulldozing resistance dominates. Brittlelunar soil slippage effects are unknown for lack of tractionmeasurements during Apollo. Sinkage gets one stuck rathereasily on the moon. Increased soil density and cohesionhave a modest decreasing effect on traction, while soil inter-nal friction has a large effect. This is the opposite of whatwas seen for cohesion and density effects on excavation.Not surprisingly wheel width and diameter play an impor-tant role for designers, and slippage constraints can makelarger a diameter helpful or a handicap. A band of vehiclemass offers a maxima for traction performance.

This work enables convenient calculations for a trackedvehicle as well, but is not presented here.

5. Conclusions

Besides base-lining calculations using Apollo soil sampleresults, this paper presents selected soil and machine para-metric studies to highlight the most significant parametersfor both the excavating bucket and the tractive vehicle driv-ing the bucket. If all were as modeled, this work suggests theApollo rover could excavate from roughly 1 to 15 cm depthdepending on which excavation model is most accurate.However, the lunar rover could get stuck if it sank nomi-nally more than 2.1 cm into virgin regolith (ignoring theloose powder on the surface). There is no available workon traction and sinkage into excavated lose lunar regolith,and getting stuck on such soil will be more likely.

Soil cohesion has a large increasing effect on excavationforces but a small decreasing effect on traction. Soil internalfriction has a relatively small increasing effect on excavationforces but a rather large increasing effect on traction. Soildensity produces a moderate to strong increase in excavationforces, while producing a small decrease in traction. Thecompeting effects of these soil parameters when balancingexcavation forces against traction forces suggests that versa-tility of equipment designs for diverse lunar soils is hard toaccomplish.

Some mathematical anomalies of the excavation andtraction models are noted.

A key capability of these results is that one could testmodels and lunar parameters with small scale excavationand traction devices on lunar robotic precursor missionsimmediately. All equations in this paper have been com-puter programmed for future calculations. These modelshave the better part of a century of application to terrestrialsoils and machines. In all such applications patient field test-ing was needed to establish the soil parameters to match themodels. However, the predictability of these equations forthe moon are unknown, without field testing, across the

widely variable soils of the lunar highlands, maria, and fro-zen poles. Excavation and traction tests during robotic lunarmissions is strongly recommended by this work.

Modern excavating companies like Caterpillar and JohnDeere are actively working on more granular physics-basedpredictive algorithms, knowing that soil mechanics predic-tions use many empirical parameters to fit observationsand that tight tolerance machine designs are not possiblewithout more fundamental physics understanding. As duringApollo, NASA could choose to advance the state-of-the-artin terramechanics, starting with robotic lunar missions.

Acknowledgements

This work was funded by the NASA Exploration Initia-tive transitional support under In-Situ Resource Utiliza-tion. We thank Chris Gallo and Dr. Juan Agui and Dr.Richard Rogers for editorial improvements to this manu-script. Edward Katich, a summer intern, assisted with liter-ature collection and database entry.

Appendix A. Expressions used to calculate curves for eachphysical force of each model

Gill and Vanden Berg model:

Cutting: Kw,Tool–soil:

lgcsinðbþ qÞ

sin q� wdðsin bþ d cos bÞðsin qþ / cos qÞsinðqþ bÞð1� /dÞ þ cosðqþ bÞð/� dÞ :

Depth:

gcsinðbþ qÞ

sin qd cosðbþ qÞ

2 sin qþ d sinðbþ qÞ tan b

2 sin q

� �

� wdðsin bþ d cos bÞðsin qþ / cos qÞsinðqþ bÞ ð1� /dÞ

þ cosðqþ bÞð/� dÞ:

Cohesion:

csinqðsinqþ/ cosqÞ �

wdðsinbþ d cosbÞðsinqþ/ cosqÞsinðqþ bÞð1�/dÞ þ cosðqþ bÞð/� dÞ :

Kinetic:

cv2 sin bsinðbþ qÞðsin qþ / cos qÞ

� wdðsin bþ d cos bÞðsin qþ / cos qÞsinðqþ bÞð1� /dÞ þ cosðqþ bÞð/� dÞ :

Swick and Perumpral model: (Each term multiplied bysin(b + d) for horizontal component)

Surcharge:

gqðcot bþ cot qÞ sinð/þ qÞ � wdsinðbþ /þ qþ dÞ :


Tool–soil:

�Ca

cosðbþ /þ qÞsin b

� wdsinðbþ /þ qþ dÞ :

Depth:

gcd2ðcot bþ cot qÞ sinð/þ qÞ � wd

sinðbþ /þ qþ dÞ :

Cohesion:

c � cos /sin q

� wdsinðbþ /þ qþ dÞ :

Kinetic:

cv2 sin b cos /sinðbþ qÞ �

wdsinðbþ /þ qþ dÞ :

McKyes model: (Each term multiplied by sin(b + d) forhorizontal component)

Surcharge:

gqðcot bþ cot qÞ � wdcosðbþ dÞ þ sinðbþ dÞ cotðqþ /Þ :

Tool–soil:

Cað1� cotb cotðqþ/ÞÞ � wdcosðbþ dÞ þ sinðbþ dÞ cotðqþ/Þ :

Depth:

cgdðcot bþ cot qÞ

2� wdcosðbþ dÞ þ sinðbþ dÞ cotðqþ /Þ :

Cohesion:

cð1þ cot q cotðqþ /ÞÞ � wdcosðbþ dÞ þ sinðbþ dÞ cotðqþ /Þ :

Kinetic:

cv2 tan qþ cotðqþ /Þ1þ tan q cot b

� wdcosðbþ dÞ þ sinðbþ dÞ cotðqþ /Þ :

Lockheed-Martin/Viking model:

Friction:

cgwl1:5b1:73ffiffiffidp d

l sin b

� �0:77

1:05dw

� �1:1

þ 1:26v2

glþ 3:91

( ):

Cohesion:


l sin b

� �1:21

ð11:5ccgd

Þ1:21 2v3w

� �0:121(

� 0:055dw

� �0:78

þ 0:065

!þ 0:64

v2

gl

):

Kinetic:


lsinb

� �0:77

1:26v2

glþ cgwl1:5b1:15

ffiffiffidp d

lsinb

� �1:21

� 11:5ccgd

� �1:212v3w

� �0:121

0:055dw

� �0:78

þ 0:065

!þ 0:64

v2

gl

( ):

Balovnev model:

Surcharge:

gqðwdð1þ cot b cot dÞA1 þ webð1þ tan d cot abÞA2 þ 2sdA3

þ 4 tan dA4lsdÞ:

Depth:

wdð1þ cotb cotdÞA1dgc2þBURIED � ðd � l sinbÞ

�

� gc1� sin/1þ sin/

� ��þwebð1þ tand cotabÞA2dgc

1� sin/1þ sin/

� �

þ 2sdA3

dgc2þBURIED � ðd � ls sinbÞ gc

1� sin/1þ sin/

� ��

þ 4 tandA4lsddgc2þBURIED � ðd � ls sinbÞ

�

� gc1� sin/1þ sin/

� ��:

Cohesion:

c � cot /ðwdð1þ cot b cot dÞA1 þ webð1þ tan d cot abÞA2

þ 2sdA3 þ 4 tan dA4lsdÞ:

Sharp blade:

wdð1þ cot b cot dÞA1

dgc2þ c cot /þ gq

�

þ BURIED � ðd � l sin bÞ gc1� sin /1þ sin /

� ��:

Blunt blade:

webð1þ tand cotabÞA2ebgc

2þ c cot/þ gqþ dgc

1� sin/1þ sin/

� �� :

Sides cut:

2sdA3dgc2þ c cot /þ gqþ BURIED � ðd � ls sin bÞ

�


� ��:

Sides friction:

4 tan dA4lsddgc2þ c cot /þ gqþ BURIED � ðd � ls sin bÞ

�


� ��

Appendix B. Define notation

Model symbol Coding parameter Units {Range} description

Tooleb eb m {0.005:0.10} Blunt edge thicknessl l m {0.1:1} Tool length (front to back)ls ls m {0.8 * l} Length of side plates s m {0.01:0.1} Side plate thicknessw w m {0.3:3} Tool widthab alphab deg {10:45} Blunt edge angle

Soil s = c + r(c) Æ tan/c c N/m2 {68:4500} Cohesion, Bekker p. 332 and 340 [20], LSB

p. 529 [1]c gamma kg/m3 {1,200:3,500} Specific mass {[1] p. 494 and 536},

Blouin mixed this up with specific weight betweenmodels

/ phi deg {20:50} Internal friction angle

Tool–soil Tool shear = Ca + No(c) tandCa Ca N/m2 {200:5000} Soil–tool adhesiond delta deg {0:50} External friction angle

Operation

d d m {0.05:1.0} Tool depthd1 d1 Dimensionless {unknown} Osman’s graphical distanced2 d2 Dimensionless {unknown} Graphical distanced3 d3 Dimensionless {unknown} Graphical distanced4 d4 Dimensionless {unknown} Graphical distanced5 d5 Dimensionless {unknown} Graphical distanced6 d6 Dimensionless {unknown} Graphical distanced7 d7 Dimensionless {unknown} Graphical distanceK K N/m {0:10} Cutting resistance indexq q kg/m2 {0.5:100} Surcharge * g = N/m2

r0 r0 m {0.5:5} Initial radius of log spiralr1 r1 m Calculated radius of log spiralt t m {0.05:1} Depth of Rankine passive zonev v m/s {0.01:0.3} Tool speedw 0 wprime deg {0:90} Polar angle in log spiralb beta deg {5:90} Rake or cutting angleq rho deg q = 45 + //2 OR {20:55} [6] soil rupture angle

(traction) OR shear plane failure angle (excavation),w.r.t. soil surface

r sigma N/m2 Normal stress ¼ Wn�A per wheel, for traction

s tau N/m2 Soil shear strength = c + rtan/

Gravity

g gM 1.63 m/s2 Gravitational acceleration (MOON)g gE 9.81 m/s2 Gravitational acceleration (EARTH)

Miscellaneous

Na m Inertia coefficientNc m Cohesion coefficientNca m Adhesion coefficientNo No N/m2 {0:1,000} Load normal to bladeNq m Surcharge coefficientNc m Weight coefficient


Appendix B (continued )


Forces

H H N Horizontal component of T OR Soil Thrust intraction

T T N Resultant vector cutting forceV V N Vertical component of T

Traction

A A m2 Ground contact area per wheel; leg; or track;p4 b � L for wheel OR B � L for track and leg

b b m {0.025:0.5} Effective wheel ground contact width¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðB� eÞ � e

p, NOT USED

B B m {0.025:1.0} Width of wheel/track/legD D m {0.175:5} Wheel diameterDP DP N Net drawbar pullh h m {0.001:0.10} Grouser heightH0 H0 N Ideal soil thrust without slipk k Pa

mðn0�1Þ, Patel; Pamn0 , Bekker Modulus of sinkage deformation = kc + b Æ k/ per

Patel [18] (‘‘soil consistency’’); Bekker otherwisedefines k = kc/b + k/, {[20] p. 240, 313–40 and 447}

kc kc Pamðn0�1Þ {0:1593, Terrestrial range} modulus of cohesion of

soil deformation, {[20] p. 240–3} & {[1] p. 529}kt kt Dimensionless Tangential stiffness of wheel or track, derived from

‘S Æ L’ physical observationk/ kphi Pa

mn0 {0:160,000, Terrestrial range} modulus of friction ofsoil deformation, {[20] p. 240–3} & {[1] p. 529}

K1 K1 1/m {3.9:39} Degree of brittleness or compactness orcoherence (cohesion effect) [19, p. 265 ff]

K2 K2 Dimensionless {3:1} Degree of slip strength (friction effect)[19, p. 265 ff]

Kc Kc Dimensionless {[20] p. 453} For bulldozing resistance, corrects Patel[18]

Kc Kgamma Dimensionless {[20] p. 453} For bulldozing resistance, corrects Patel[18]

l0 l0 m {[20] p. 453} Distance of rupture for bulldozingresistance, corrects Patel [18]

L L m {0.1:5} Ground contact length: wheel:L ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD� eÞe

p, NOT USED wheel or track.

n 0 nprime Dimensionless {0:1.2} Soil deformation exponent, {[20] p. 340}n n Dimensionless {1:6} Number of wheels/tracks/legsnc nc Dimensionless {0:3} Coefficient of passive earth pressure, {[20]

p. 139 and 142} & [10]Ng Ng Dimensionless {0:20} Number of grousers touching soil per track/

wheel/legnc ngamma Dimensionless {15:35} Coefficient of passive earth pressure, {[20]

p. 139} & [10]P P N/m2 Ground pressure = W/A = rR R N Total resistive forceRb Rb N Bulldozing resistanceRc Rc N Soil compaction resistanceRg Rg N Gravitational resistanceRr Rr N Rolling resistance of elastic tire, NOT USEDRother Rother N Other resistive forces

(continued on next page)


Appendix B (continued )


S S Dimensionless {(0.10:0.12)/L} Low speed slip, {[20] p. 138 and 447}say S Æ L = 10–12 cm, max. traction. S = x Æ W/kt Æ L2 per Patel.

T0 T0 N Available grousered wheel thrust ¼ s � Ngh�b

cos q�sin qþ

h2

cos qÞW W N = kg m/s2 {50:8000} * g Vehicle weight, WLRV = 460 lbs

vehicle + 1080 lbs cargo = 1540 lbs onEarth = 698.5 kg = 6852 N on Earth = 1138 N onthe moon

z z m Sinkage, not counting multiple passes in wheel tracks.Bekker definition is used in this paper,z ¼ W

n�A�k� 1

n0 ¼ Pk

� 1n0

a a deg Approach angle ¼ arccos 1� 2 zþeD

� . e is often

negligiblee E m {0.01:0.10} Wheel squat/deflection (<10% of wheel

dia., elasticity)h theta deg {0:10} Slope angle of the ground (w.r.t. horizontal)j kappa m {0.01:1.0} Shear deformation slip modulus, S Æ L/

j = {10:0.1} is observed physically {[1] p. 529}l mu Dimensionless Soil coefficient of friction = tan/m nu Dimensionless {1:5} Poisson’s ratio ¼ straintransv

strainlong

x omega m2/N, Pa�1 Coefficient of surface adhesion;reciprocal of soil–tool adhesion ¼ S�kt�L2

W ¼ 1=Ca


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digging and pushing lunar regolith: classical soil mechanics and the forces needed for excavation...

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