PROGRESS TOWARDS ROBUST MOBILITY ANALYSIS FOR A LEGGED PLANETARYFETCH ROVER

Brian Yeomans and Chakravathini M. Saaj

Surrey Space Centre, University of Surrey, Guildford, United KingdomEmail: B.Yeomans@surrey.ac.uk, C.Saaj@surrey.ac.uk

ABSTRACT

Legged rovers offer a potentially superior alternative towheeled locomotion for the exploration of challengingplanetary terrains. Compared with a wheeled vehicle, awalking rover will have improved agility, slope climbingand rough terrain capability. However, when it comesto the analysis of the interaction with deformable terrain,surprisingly little study has been made of the terrame-chanics applicable to the type of micro - legged vehiclethat might be deployed on a future mission.

This paper describes progress towards a robust mobilitymodel of legged vehicle performance on sands and otherdeformable terrains, applicable to a diverse range of ve-hicle and soil combinations. Software tools are describedwhich aid analysis of terrain interaction, assist with opti-misation of the vehicle design, and generate vehicle con-troller inputs enabling more reliable go / no go decisionsto be made, optimisation of path planning, and manage-ment of vehicle gait and foot positioning.

Key words: walking rover, terrain, trafficability, terrame-chanics, control.

1. STATE OF THE ART

Whilst the NASA Mars Exploration Rover (MER) mis-sions [1] demonstrated that wheeled rovers can provevery successful for planetary exploration, wheels sufferfrom a number of potential problems which mean thatthey may not be the ideal locomotion system to reachsample or data collection sites in particularly challeng-ing locations. A wheeled rover cannot climb very steepslopes or traverse extremely rugged regions, and it mayencounter such severe problems traversing sandy terrainthat the mission is at risk of complete failure [2].

Walking rovers offer an alternative to wheels which maybe more effective across difficult landscapes. The capa-bilities of these vehicles on steep, rocky terrain has beenstudied on a number of occasions, and impressive resultshave been achieved using reflexive behaviours to nego-tiate unstructured environments - for example, the ‘Big

Figure 1. DFKI SpaceClimber [5] - image courtesyDFKI

Dog’ project [3]. Combining the long distance capabil-ities and energy efficiency of a wheeled rover with theagility of a walking rover has been explored in [4] whichutilised the German Research Center for Artificial Intel-ligence’s (DFKI) SCORPION eight legged vehicle as ascout adjunct to a wheeled rover. DFKI are also devel-oping ‘SpaceClimber’ [5], see Fig. 1, a lunar crater ex-ploration rover constructed from space qualified compo-nents.

Although a walking rover’s superior agility may not be indoubt, performance over non - rigid terrain, and particu-larly the soft deformable areas which have proved sucha hazard on Mars, has not been well studied. There areexamples of designs which have been shown to performparticularly well in soft terrain; the CESAR robot, alsodeveloped by DFKI, won the European Space AgencyLunar Robotics Challenge in October 2008 [6], with a hy-brid legged-wheel design with five feet per wheel, whichmanaged to climb into and out of a lunar-like crater onthe island of Tenerife, collecting and delivering a 100gsoil sample without the aid of external illumination.

The work presented in this paper extends the under-standing of how a walking vehicle will perform over de-formable terrain. The theoretical basis of a detailed an-

alytical model of leg / terrain interaction for a walkingvehicle is presented in Section 2, discussion and analy-sis of the Model Predictions are presented in Section 3,and Conclusions and Planned Future Work are set out inSection 4.

2. THE LPTPT TOOL

2.1. Introduction

The Legged Performance and Traction Prediction Tool(LPTPT) comprises a comprehensive model of the inter-action between deformable terrain and the legs of a walk-ing rover [7] [8].

LPTPT uses the MATLAB computation engine for itscaculations and can produce both numerical and graph-ical output as required. The model contains a database ofreference vehicles and physical data on a range of plane-tary soil types, both real and simulated; currently interac-tions between ten vehicle and twenty five soil types canbe assessed. The model analyses the forces arising be-tween the vehicle and soil, and predicts the trafficabilityperformance. Leg loading, and the effect of gait modifi-cation can be varied. The model’s force predictions havebeen validated by experiment using a test rig comprisinga manipulator arm moving a representative leg / foot as-sembly through simulated planetary soils.

It should be noted that in very loosely packed granu-lar materials and / or with increase in limb velocities,“walking” behaviours can very rapidly transition to acompletely different form of locomotion resembling slowswimming through the granular material [9]. The anal-ysis presented here is confined purely to walking be-haviours.

2.2. LPTPT Model - Primary Components

The diagram in Fig. 2 illustrates the forces arising on aleg stepping into soil with a stepping angle α. Four forcetypes are described and quantified by LPTPT:

(i) Soil Force Ho, is the shear force acting on thefoot / soil interface, providing forward traction.The soil force at the foot / soil shear interface isbased on the Mohr - Coulomb equation [10]. Themaximum shear stress τmax arising is:

τmax = Co + σtanφ, (1)

where Co and φ are the soil physical properties ofcohesion and friction angle, and σ is the normalstress on the soil / foot interface.

(ii) Draught Force Fd, is the force between the soil andthe leg / foot assembly cutting through the soil. This

Figure 2. Leg / Soil Forces - vehicle body and upper leglinks omitted for clarity

force provides additional traction for the vehicle un-less the leg is stepping “forward” into the soil, inwhich case it will act to resist forward motion.

The draught force derivation in LPTPT is based onthe application of tillage theory, the study of the me-chanics of tool / soil interaction [11]. Terzaghi’sUniversal Earthmoving Equation [12] as further de-veloped by Reece [13] is used, and is expressed asfollows:

F = γgz3Nγ +Coz2Nc + qz2Nq +Caz

2Nca (2)

where γ = unit weight of soil, g is acceleration dueto gravity, z is the sinkage, Nc,q,a,ca are Terzaghi’sfour dimensionless soil bearing capacity factors, q isthe soil surcharge pressure, and Ca is soil adhesion.

(iii) Active Force Fa, which arises as soil falls back intothe trench created by the leg as it moves through thesoil. This force acts to assist the leg moving throughthe soil and so reduces traction.

The active force derivation is also based on Terza-ghi’s analysis [12]. It is not described in detail hereas at low to moderate levels of sinkage, active forcehas a negligible effect on the total forces arising.

(iv) Frictional Force Ff , the effect of friction betweenthe soil and the foot / leg. Where the stepping angleα is high (α ≈ 90o) , this force will provide furtherresistance to the leg moving back through the soiland so provides additional forward traction.

Frictional force is modelled in LPTPT following thesame principles applied to determining the shearforce at the foot / soil interface. At high valuesof stepping angle (α ≈ 90o), friction derives prin-cipally from the sides of the leg / foot assemblyand depends on soil cohesion, the geometry of theleg / foot, and the sinkage depth.

(v) Other Factors The model also addresses other rel-evant factors such as the effect of gravity, and

Figure 3. Predicted Rover soil /leg interaction.

changes to leg loading that can arise during the step-ping cycle in more complex gait patterns.

The forces derived by the model are illustrated in Fig. 3.This scenario is based on a 20kg hexapod walking in alow density lunar - type granular soil. This is similar tothe DFKI SpaceClimber vehicle shown in Fig. 1, and rep-resents the type of walking rover which could well be de-ployed on a future Lunar mission, for example to explorethe polar regions.

The plot shows the horizontal components of each of theforces arising, plotted against sinkage. The figure high-lights a number of significant features:

(i) All forces other than the Active Force increase thrustper leg as they increase.

(ii) All forces other than the soil Thrust increase assinkage increases; in the case of Draught Force,markedly so.

It can be seen that a degree of sinkage, provided this is notso large as to overwhelm the vehicle, is of assistance to alegged vehicle as the force available to generate forwardmovement is increased. This is the opposite of the posi-tion for a wheeled rover, where sinkage is a negative fac-tor as it increases compaction and other resistances [14].

2.3. LPTPT Model - Further Development

The model described above derives the basic force analy-sis and computes the maximum horizontal force availablefrom all sources at the soil interface, given a known levelof sinkage. Whilst this information enables a view to beformed of the potential of the rover / terrain combinationto deliver thrust, it does not describe the impact of slip

at the soil interface, and the consequent effect on sinkageand the forces generated.

LPTPT has therefore been developed further to incorpo-rate slip and sinkage modelling, to enable it to directlyshow the effect on forces as these parameters vary. Addi-tionally, sinkage and slip are vehicle operational param-eters that can be measured or estimated, enabling terraininteraction predictions to be directly linked with vehicleperformance.

Sands deform under stress in a characteristic manner; thestress / shear relationship can typically be characterisedby one of two types of exponential function [15]. Theshear stress tends to a constant residual level, and in thecase of compacted rather than loose sands [15], the curveshows a pronounced peak. Given that the degree of com-paction of the sand may not be known, and to avoid over-stating the forces available at the soil interface, LPTPTmodels the stress / shear relationship using the curvewithout a peak, as follows [16]:

τ = τmax

(1− e(−j/K)

), (3)

where j is the shear displacement at the relevant pointin the interface, and K is the shear deformation param-eter. To determine K, LPTPT adopts a model proposedby Godbole and Alcock [17] to compute K derived fromknown laboratory reference values using the the relativecontact patch area, which was found to be well supportedby experimental results:

(K1/K2) =√(A1/A2), (4)

where the ratio of the value of K sought to the referencevalue equals the square root of the ratio of actual A1 andlaboratory A2 contact patch areas.

It can be seen from equation (3) that any stress appliedto the soil interface gives rise to some shear; the shear

results in movement of the soil material resulting in slipof the leg against the soil. Slip can be quantified by meansof the Slip Ratio; this measures the extent to which theforward traction theoretically available at the leg / soilinterface fails to be converted to actual forward motion.The slip ratio i can be defined as [15]:

i = 1− V

Vt,(5)

where V is the actual forward speed and Vt is the theo-retical forward speed with perfect traction.

The force developed at the foot / soil interface, and itsdependence on i is computed by substituting the Mohr -Coulomb relation from equation (1) for τmax and inte-grating the resulting stress values over the contact patcharea. In the simple case of a flat, rectangular foot, ofwidth b, length l and area A, the shear displacement j ata point under the foot is related to the slip ratio i and thedistance from the front of the foot x as j = ix [15], andthe force can be derived analytically as:

F = b

∫ l

0

(Co +

W

bltanφ

)(1− e(−ix/K)

)dx (6)

F = (ACo +Wtanφ)

[1− K

il

(1− e(−il/K)

)](7)

Static sinkage is modelled using the Bernstein-Bekkermethodology relating pressure p and sinkage z [10]:

p =

(kcb

+ kφ

)zn (8)

where kc is the cohesive modulus of soil deformation, kφthe frictional modulus of soil deformation, n is an experi-mentally determined exponent (typically between 0.7 and1.3), and b is the smaller dimension of the contact patch(the radius, in the case of a circular contact area).

With respect to slip sinkage, a number of methodologieshave been developed [10] [18] [19], which depend onslip,static sinkage zo and, in some cases, grouser height.LPTPT implements the following relationship,which hasbeen verified against experimental results [20].

ztotal = Kss.zo (9)

where

Kss =

(1 + i

1− 0.5i

)(10)

Each of the methodologies considered gives a similar re-sult at low to moderate slip levels, but results diverge athigh slip ratios. The approach chosen was selected be-cause:

(i) Sinkage does not trend to infinity at high slip levelsunlike some of the other available methodologies;

(ii) Grousers may or may not be fitted, depending onthe foot design, and this model does not depend ongrouser height;

(iii) It is non linear, in accordance with observed results,and accords well with observed data.

Figure 4. schematic of Single Leg Testbed

2.4. Model Testing

The basic force relations in LPTPT have been extensivelytested using a lower leg segment moved through soil bya robotic arm manipulator. The difficulty with this ar-rangement however when it comes to verifying the slipdependency relations is that the kinematics are substan-tially different from those applicable to the actual vehi-cle, principally because the manipulator base and the soilare fixed relative to each other. This makes it difficultto replicate certain of the kinematic features of a movingleg in soil, and in particular means that only zero or 100%slip ratios can be achieved.This issue is addressed using a specially developed singleleg testbed, comprising a suspended carriage to which thetest leg is attached. A schematic of the testbed is shownin Figure 4.

(i) The carriage height above the soil can be adjusted,to allow for a wide variety of leg sizes and designs.

(ii) The carriage moves on horizontally aligned linearbearings, either freely, driven or braked, to simulatevarious scenarios that might arise from walking onthe level or up and down slopes.

(iii) The loading of the leg / soil interface is adjusted di-rectly using the leg actuators and controlled by mea-suring the leg load using a sprung telescopic leg andlinear potentiometer sensing.

(iv) The leg / foot assembly is equipped with multi-axisforce sensors to directly measure forces arising.

(v) An infra red camera based motion capture system isused to capture and analyse sinkage, and the kine-matics of the motion at the soil interface.

(vi) The design of the testbed has been developed ini-tially using CAD tools and then simulated using a

Figure 5. Force / Slip relation

virtual reality visualisation of the testbed incorporat-ing the Open Dynamics Engine rigid body dynamicslibrary [21], to verify that the testbed will operate asexpected.

3. LPTPT MODEL PREDICTIONS

LPTPT enables the Forces / Slip relation to be plotted,giving insights into the dynamics of legged vehicle mo-bility.

The plot in Fig. 5 demonstrates how total force availableat the soil interface increases strongly with slip ratio. Un-like a wheeled rover, available traction will increase withslip provided sinkage is not so large as to overwhelm therover.

The mass and geometry of the rover will have a signifi-cant effect on both the force available at the soil interface,and the force necessary to produce adequate forward trac-tion, particularly where slope climbing is required.

To illustrate the effect of mass variation, total force perleg available at the soil interface as a function of both slipratio and mass of the rover is shown in the plot in Fig 6.The effect of mass changes on the dimensions is basedon the same illustrative 20kg hexapod described in Sec-tion 2.2 with critical dimensions scaled proportional tothe cube root of the ratio of actual mass to the mass ofthe example rover; thus assuming, for comparability pur-poses, a constant density for the rover construction mate-rials.

These data could, for example, be used to compare theeffect of changes in rover mass for a particular applica-tion, factoring in the effect of mass change on force re-quired from the soil interface for slope climbing and toovercome other resistances.

Understanding the range of predicted performance acrossa class of terrain materials could be extremely useful, asin many cases information on the material’s precise in situphysical properties will not be available.

Figure 7. Trafficability Prediction

Fig. 7 illustrates that a minimum trafficability perfor-mance can be derived by analysing a broad populationof a particular soil type. Thus even if the precise physi-cal properties of the soil to be travered are not known, theupper and lower bounds of the total force available can becomputed for all tested examples of that soil type. Usingthis information, the lower bound position is assessed inthe light of the demands on the vehicle such as climbingrequirements. This process may need to iterate to eval-uate the effect of vehicle geometry changes (varied legpositioning) and gait modification.

One of the objectives of LPTPT development has beento explicitly derive the relation between force, slip, andsinkage, so that these relationships can be directly utilisedby the vehicle controller.

This could be deployed by implementing an LPTPT de-rived model of terrain interaction in the vehicle controller.The rover would employ on - line estimation of slip andsinkage and compare predicted (by the model) slip andsinkage with that actually observed. This comparisoncould then be used to refine the model parameters on anon - line basis, indirectly deriving the characteristics ofthe actual terrain, and enabling greater confidence in un-derstanding vehicle - terrain interaction to be achievedand better vehicle performance realised.

3.1. Wheeled Rover Drawbar Pull Comparison

Drawbar Pull (DP) for Sojourner, the 11.5kg vehicle usedin the Mars Pathfinder mission, was estimated by Elleryat 6.88N [14]. In contrast, LPTPT computes DP of a11.5kg hexapod rover, walking with a wave gait in soiltypical of the Mars Viking Lander 2 site soil [22], as31.5N, a very substantial increase.

This comparison gives an indication of how signifi-cant are the resistances encountered by a wheeled rover;whilst in this example the force available for thrust at thesoil interface in both cases is of a similar amount, theresistances to motion of a legged rover typically do notact to impede forward motion as it can simply pick up its

Figure 6. Effect of Mass changes

Figure 8. University of Surrey nanorover SAFER-1 onsandy terrain

legs and step across intervening obstacles. Additionallythe principal forces given a high (≈ 90o) stepping anglereinforce rather than degrade DP, as described in Section2.2.

The effect of gait variations can also be analysed. Forthis analysis, to show the applicability to different roverclasses, a much smaller vehicle has been chosen; the sub-1kg nanorover SAFER-1 shown in Fig. 8. Using theLPTPT tool, the predicted performance of SAFER-1 iscomputed under Mars gravity assuming a 90◦ steppingangle on soils with the characteristics of those seen at theViking 1 and 2 (VL1 and VL2) Mars lander sites [22].

The potential impact of gait changes was evaluated byassessing two gait scenarios; the wave gait, a highly sta-

Figure 9. predicted nanorover soil /leg interaction, wavegait, VL1 soil

ble gait under which only one leg is lifted at any one time,and the tripod gait, where three legs are lifted at any time,and which represents the maximum number of legs liftedconsistent with a statically stable gait. Although highlystable, the disadvantage of the wave gait is that it is quitea bit slower than the tripod gait. The plot in Figure 9shows the individual and combined forces on the leg fora range of sinkage amounts. The active force Fa is negli-gible and has been omitted for the sake of clarity.

Predicted static sinkage for this configuration using theBernstein-Bekker formula is 6.5mm, and in practice sink-age will somewhat increase once leg slip occurs due toslip sinkage. At what might be considered a typical sink-age level of 10mm, the draught force Fd as the leg pushesthrough the soil is the dominant force providing forwardthrust, followed by soil friction and the shear force at thesoil interface which are approximately equal.

VL1 VL2Force Type Wave Tripod Wave Tripod

(N) (N) (N) (N)Ho 0.31 0.41 0.44 0.67Fd 0.81 0.81 0.76 0.76Fa -0.03 -0.03 -0.01 -0.01Ff 0.32 0.32 0.22 0.22

Soil Force on Leg 1.41 1.51 1.41 1.64Vehicle DP 7.05 4.53 7.05 4.92

Table 1. SAFER-1 Forces Summary, VL1 & VL2 soils,10mm sinkage

Table 1 summarises outputs from LPTPT showing theforce components and calculating the DP available forthis vehicle on both VL1 and VL2 soils, using wave andtripod gaits, at an estimated sinkage of 10mm.

The results demonstrate that whilst varying the gait hassome effect on DP per leg, this is modest given the lim-ited contribution the soil interface shear force (the ele-ment primarily affected by the gait pattern) has on thetotal DP.

Total DP is therefore maximised by using the slowerwave gait option as more legs are in ground contact atany time. Where high thrust is needed, such as for slopeclimbing, a wave gait scheme is likely to be required.It can also be seen that total DP of SAFER-1 (7.05N,mass 850g) compares favourably with that of a muchlarger wheeled vehicle such as Sojourner (6.88N, mass11.5 kg) [14], despite the low mass.

4. CONCLUSIONS AND FUTURE WORK

LPTPT is presented as a comprehensive tool to predict,analyse and quantify the forces available at the leg / soilinterface of a walking rover. It enables performance pre-dictions to be made of a wide variety of vehicles operat-ing on many soil types, and suggests that a legged rover,in addition to demonstrating superior agility over rough,rocky terrain, can also be an effective vehicle to traversesoft sands and other types of deformable materials.

Incorporation of Slip / Sinkage analysis enables the dy-namics of the Force / Slip relationship to be modelled,and shows that slip and associated sinkage, rather thanbeing a disadvantage, can positively aid legged vehicletraction.

LPTPT can reduce the risk that incorrect “stop / go” de-cisions are made in challenging terrain scenarios, by in-creasing confidence that a traverse is feasible despite in-complete information on terrain physical characteristics.

It is planned to develop the model further to incorporateLPTPT in the vehicle controller, working with an on -line estimator of slip and sinkage parameters to reducethe uncertainty arising from unknown terrain parametersand enable improved performance to be achieved.

ACKNOWLEDGEMENT

The authors would like to thank the European SpaceAgency, the German Research Center for Artificial Intel-ligence, the German Aerospace Center and the Universityof Surrey for supporting the presented work.

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