AbstractIn the process of automating an earthmoving machine, wehave developed a model of soil-tool interaction that predictsresistive forces experienced at the tool during digging. Thepredicted forces can be used to model the closed loop behav-ior of a controller that servoes the joints of the excavator soas to fill the bucket. In this paper, we extend the state of theart in two ways. First, we present a reformulated version ofthe classical Fundamental Equation of Earthmoving oftenused to model soil-tool interaction. The new model includesconsideration of previously unaccounted phenomena in theinteraction of an excavator bucket as it moves through soil.Secondly, given that soil properties can vary even within awork site, we present an on-line method to estimate soilparameters from measured force data. Finally, we show howthe predicted resistive force is used to estimate bucket trajec-tories.

1 IntroductionAutomation of earthmoving offers improved efficiency, con-sistency in work quality and improved safety. This is partic-ularly the case in mining sites where mass excavation isperformed. Even reducing the execution time of a singlecycle of an operation by a few seconds can translate into alarge savings over the entire job. We have developed anautonomous excavator that is able to sense its environment,make plans, and execute trajectories to dig and to load trucksat speeds comparable to a human operator [14].

Automation of earthmoving with an excavator, presentsunique challenges. The dynamics of the mechanism and link-ages are complex, there is uncertainty in the shape of the ter-

rain and soil parameters, and, the interaction forces betweenthe excavator and the environment are very large. Planningearthmoving operations requires the ability to model theeffect of actions when the actions are carried out open-loop,and, the result of initial conditions when a closed loop con-troller is involved. A key problem in the modeling of diggingtrajectories is the estimation of resistive forces that are notonly a function of the shape of the terrain, soil and toolparameters, but, also the controller that servos the jointsusing force feedback, to fill the bucket.In this paper we present a model of the interaction betweenthe tool (the excavator bucket) and the terrain. This modelreformulates a classical formula that has been proposed tomodel flat blades moving through flat ground, as in the caseof a bulldozer blade or a tilling device used in flat terrain. Ourmodel accounts for some phenomena that are particular toexcavation. Since such models are not analytically invertible,we have developed a numerical scheme that uses measureddata from excavation experiments to extract parameters ofthe model. These parameters are then used to predict resistiveforces and bucket trajectories for future candidate actions[13]. In addition, since the soil parameters can vary signifi-cantlythe difficulty in digging often varies with the stratabeing excavated it is necessary to determine the soil-parameters on-line. We present results that shows the utilityof a method that uses data from a small number of digs topredict forces for candidate digs in the future.

2 Related WorkThere has been some research on the operation of earthmov-ing machinery [16][1] that explicitly addresses the issue ofestimating forces necessary to overcome the shear strength ofsoil. Unfortunately, this work is mostly stated in empiricalterms for specific types of machines and it is not clear how toextrapolate the methodology for arbitrary mechanisms. Aconsiderable amount of research has explored finite elementanalysis of the soil plasticity, for example [2]. While thiswork performs a painstaking analysis of soil displacement, itis not suited for our purposes since it requires a very largenumber of iterations of numerical integration for each prob-lem. Faster, but more approximate methods have been devel-oped for the purposes of simulation [7] but the emphasis ison realistic looking motion of soil, not on accuracy of theforce model.There has been some work in the field of agricultural engi-neering that has been directed at producing estimates of cut-ting resistance for tilling implements [10],[3], [4] using wellunderstood physical principles. This work is important inunderstanding the mechanics of simple motions of a blade

Fig. 1 The testbed excavator is based on a commercial grade 25ton machine. It is equipped with onboard computing, joint sensingand range sensing that allows mapping of surrounding terrain.

Modeling and Identification of Soil-tool Interaction in Automated ExcavationO. Luengo (oluengo@disam.upm.es)

DISAM, Universidad Politecnica de Madrid (Spain) S. Singh (ssingh@ri.cmu.edu), H. Cannon (hcannon@ri.cmu.edu)

Robotics Institute, Carnegie Mellon University, Pittsburgh, PA (USA)

In Proceedings, IEEE/RSJ International Conference on Intelligent RoboticSystems, October 13-17, 1998 Victoria, B.C., Canada

moving through soil. We have found that this analysisaccounts for first order effects and allows for order of magni-tude predictions. Singh used insights from these physicalmodels (such as the force is quadratically related to depth)to develop a memory-based scheme to predict resistive forcesfor an excavator bucket operating in dry sand [12]. He savedmeasured resistive forces along with a topological descrip-tion of the terrain excavated in a look-up table that was inter-polated to predict resistive forces for candidate actions. Incomparison, the reported research depends on a strongermodel of the interaction between the soil and tool. The inputsto the model are bucket trajectories, a topological map of theterrain excavated, and pressures measured in the hydrauliccylinders of the machine. The output is a compact set of soilparameters that are used to instantiate the model in the future.The advantage of such a scheme is that a small set of data canbe used to extract meaningful parameters. Extraction ofparameters can be slow (although we present a scheme thatcan be used on-line) but estimation once the parameters areobtained is very fast. The disadvantage of such a method isthat it does not deal well with respect to systematic unmod-eled phenomena.One of the main distinctions of the work reported here and in[12] is the use of the shape of the terrain in the model. Wehave used a topological map of the terrain to be excavated asshown in Fig. 2 using range sensors onboard the excavator.Such accurate measurement of the terrain has not been pos-sible until recent advances have made non-contact rangesensing reliable and inexpensive.

In our work, the forces obtained are used as an input to theidentification process, but can be applied to a buried obstacledetection algorithm as in [5] that compares real forces andpredicted forces to detect major changes that imply somekind of buried object in the bucket path.

3 Soil-Tool ModelIn this section we present the classical soil-tool model calledthe Fundamental Earthmoving Equation along with areformulated version that accounts for other phenomena.

3.1 The Fundamental Earthmoving Equation.Our work is based on the well known Fundamental Earth-moving Equation (FEE), described by Reece[10] as:

Where F is the resistive force experienced at a blade, is thesoil density, g is the gravity, d is the tool depth below the soil,c is the soil cohesion, q is the surcharge pressure verticallyacting on the soil surface, w is the tool width, and N, Nc, andNq are factors which depend not only on the soil frictionalstrength, but also on the tool geometry and soil-tool strengthproperties [8].If we assume a static equilibrium and that the shape of thefailure surface can be approximated by a plane (of unitwidth), as shown in Fig. 3.

The components of the resistive force can be written as:

Solving the equations for F:

After some manipulation, the force equation can be writtenas Reeces equation. Assuming the surcharge (the amount ofmaterial that has been displaced in the past) is uniformly dis-tributed we obtain that:

Fig. 2 A topological map of the terrain that is being excavated isused in the identification and estimation process. This map is builtusing range sensors located onboard the excavator.

section of the terrain used inestimation and prediction

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Fig. 3 Static equilibrium analysis using an approximation of thefailure surface. W is the weight of the moving soil wedge, Lt is thelength of the tool and Lf is the length of the failure surface, Q is thesurcharge pressure, is the angle of soil-soil friction, c is thecohesion of soil, ca is the adhesion between the soil and blade and is the friction between the metal and the blade, R is the force resistingmovement of the wedge and F is the total resistive force. From[8].

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Lt cos R +( )sin cL f cos+ 0=Fz F +( )cos= caLr sin cL f sin R +( )cos W Q 0=+ ++ +

FW Q cd 1 +( )cotcot+[ ] c

ad 1 +( )cotcot[ ]+ + +

+( )cos +( ) +( )cotsin+------------------------------------------------------------------------------------------------------------------------------------------------------=

Ncot cot+

2 +( )cos +( ) +( )cotsin+[ ]--------------------------------------------------------------------------------------------N

c1 +( )cotcot+

+( )cos +( ) +( )cotsin+------------------------------------------------------------------------------------Nq

cot cot+ +( )cos +( ) +( )cotsin+------------------------------------------------------------------------------------

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=

=

This is a two dimensional model. The justification a twodimensional model for our application is that the sidewalls ofthe bucket do not allow shearing in direction transverse tobucket motion. We also assume that the inertial forces arenegligible. This assumption is acceptable because the accel-erations involved during digging are typically low.3.2 The reformulated soil-tool modelThe total force acting on the bucket has been decomposedinto three main forces. They are the shear or cutting force(Fs), the gravity force (Fg), and the remolding force (Fr). Theshear force is the force required to shear the soil away fromitself. This force is encompassed within a modified Reeceequation. For the case of earthmoving in flat ground the FEEis just the sum of Fs and Fg. In our case we have modified thecomputation of Fs by adding a term for the remolding force.As can be seen from Fig. 3, the FEE assumes that the soilprofile is horizontal. Since this assumption is not alwaysvalid (in fact only rarely the case in our application), a mod-ification was made in which the terrain profile angle isincluded within the rake angle as shown in Fig. 4.

In addition, the volume of material swept by the bucket, Vs,is assumed to result in surcharge and the material shown inthe shaded region above. Assuming that the surcharge is uni-formly distributed above the shaded region, the FEE can berewritten as follows:

and the factors can be written as:

At any moment, the swept volume or the amount of soildisplaced into the bucket, is assumed to account for the entire

gravitational force acting on the bucket. Therefore:

Note that the gravitational force has been subtracted from thecutting force equation so that it is not accounted for twice.The gravitational force is represented separately so that it canbe applied when the cutting force equation is not relevant,such as when the bucket comes up out of the ground.The remolding force is the force required to remold the soilin the bucket. As the bucket begins to fill up, additional forceis needed to form the soil within the bucket, and then to com-press the soil. We write this force as follows:

Fig. 5 gives an indication of the relative magnitude of theseforce components for four separate digs. Note that the shearforce is clearly the dominant force.

Note that the cutting force is referenced relative to a coordi-nate frame parallel to the ground. In the rest of this paper, wewill examine the forces after they have been transformed to acoordinate frame relative to the bucket. This new coordinateframe is shown in Figure 6. Note that the origin of the coor-dinate frame is at the bucket tip, and the x axis is parallel tothe blade. This choice for the coordinate system is more intu-itive in terms of representing the forces that affect the bucketbehavior.

4 Mechanism modelSince the method of reliably measuring forces during opera-tion uses pressure sensors that are mounted inside thehydraulic cylinders, we need a method that converts cylinderpressures into forces experienced at the bucket. (9) represents

Fig. 4 The reformulated model of soil-tool interaction

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Fig. 5 The three force components in the soil model. is theshear or cutting force, is the gravitational force, and is theremolding force.The gravity force (dashed line), is limited because itis based only on amount of soil swept into the bucket.

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the excavator dynamics in joint space:

Where M is the inertia matrix, V is a matrix containing thecentripetal and Coriolis terms, G is a vector containing thegravity terms and f is a vector representing the end-effectorcontact forces, is a vector used to denote the torque at eachjoint and J is the jacobian matrix of the manipulator. Underthe assumption that acceleration and velocity terms are neg-ligible during digging, (9) can be reduced to:

The forces at the tip of the bucket are obtained from theabove expression. The masses of the links of the excavatorare known, as well as the jacobian matrix, so J and G areobtained through these data and the joint positions. isobtained through the pressure measurements and some datafrom the cylinders. The calculation of the force and thetorque that each cylinder exerts on each joint are obtainedthrough a simple fluid mechanics analysis.

Based on Fig. 7, the resultant force applied on the fulcrum ofthe excavator link can be written [15] as:

where Pi are the measured pressures in the cylinder cham-bers, Ai the areas on both sides of the piston mr the mass ofthe rod, y its displacement, and F the exerted force. Once Fis obtained from pressure and displacement measures, theeffective force is computed based as shown in Fig. 8 showsan schematic drawing of the boom and stick links.Distances a and b are known and fixed, angle is also known(from resolver measurement plus a constant angle), so dis-tance L is obtained as follows:The rod acceleration is obtained from the above equation,and with the value of L, the sine of angle is calculated:

Finally, the torque is obtained:

In this section we have developed a framework with a newsoil model and a method to obtain the forces at the tip of thebucket. The former is necessary to understand the soil reac-tion forces, and the latter is very necessary to understand howthe measurements are related to the soil-tool interactionforces.

5 Estimation MethodsThis section explains how the above model and the measure-ments are used to extract missing parameters. The parame-ters identified are: , , and c. As can be seen in the abovesection, the equation of the soil model presents a highly non-linear behavior. This makes it impossible for the equation tobe solved analytically for the characteristic parameters. Moreover the form of the equation does not allow estimation usingmethods like linear least-squares. Below we consider threemethods that can be used to extract the soil parameters.5.1 Exhaustive SearchA simple solution is to test a range of each of the four param-eters and use the set that minimizes the difference betweenthe predicted and measured forces. The cost of such a methodis that it is necessarily discrete valued and in general it is dif-ficult to determine the range of parameters to be searched.This method is also computationally taxing. We have usedthis method to get a baseline in terms of performance.5.2 Efficient gradient descentIf the function is well behaved it is possible to use gradientdescent to minimize the errors. One option is the use of con-jugate gradient methods, but they require the calculation of

Fig. 6 Coordinate system used to relate resistive forces in thefollowing graphs.

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Fig. 7 Simplified cylinder model.

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Y X

M q( ) q V q q,( )q G JT f+ + + =

f J T G( )=

P1,A1P2,A2 ymr

F P1 A1 P2 A2 mr y=

Fig. 8 Schematic drawing of boom-stick links

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b

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a2 b2 2ab cos+ L2=

sin 1 b2 L

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=

Fa 1 b2 L

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the gradient of the function, an expensive process. We haveused a variation of gradient descent called Powells method[9] that uses the concept of conjugate directions to do gradi-ent descent starting from the basis vectors of the searchspace.5.3 Stochastic searchThe method of simulated annealing [6] is specially suited forproblems of large scale where a desired global extremum ishidden among many local extrema. It consists of a pseudo-random search for the minimum (or maximum) of an objec-tive function J (in this case the error between the measure-ments and the output of the model). Each trial starts at aninitial random point within the searched parameters interval.At each step a random movement is made. If it improves theerror, the new point is chosen. If it doesnt, it is still chosenbased on a decaying function. This way, initially, many stepsare taken even though they do not improve the objectivefunction. As the search proceeds, the algorithm is less likelyto choose steps that increase the error.5.4 A combined methodWe want our method to be fast (both in identification andestimation). Ideally, we would like the method to be used on-line so that it can react to changing soil characteristics. Wehave found that a combination of the simulated annealing andPowells method works well. Each trial of simulated anneal-ing is followed by a gradient descent from the point in thefunction space that had the lowest error for the trial. UsingPowells method in conjunction with simulated annealingprovides both accuracy and speed. The former because itcovers the search space with integer discrete steps, and thelatter because it picks up isolated points in a random way.

6 ResultsIn this section we present a sensitivity analysis given a set ofdata collected on our testbed. We find that some parametersare more sensitive than others. We show how the predictedforces compare with the measured forces for both the FEEand the reformulated version.6.1 SensitivityWe conducted a sensitivity analysis to determine which ofthe parameters were most sensitive. We used as the startingpoint, the set that produced minimum error from an exhaus-tive search. Then the value of each parameter (one each time)is varied from the starting point, and the error between thepredicted force and the real one is calculated. The results for, , are shown in Fig. 9. This shows that is the most sen-sitive among the three angles. In contrast, c, is not very sen-sitive a 50% change in c from the best value, only changesthe error by 25%.6.2 Results with the FEESeveral tests were carried out with this model, and the esti-mation process of the soil parameters was made throughexhaustive search and simulated annealing. These times were

obtained with data from 8 digs (about 800 measurements).The root mean square error between predicted and true forceis about 30000 N. The parameters found are =37.31=51.947 =44.85 degrees and cohesion=30000 Pa.6.3 The reformulated modelUsing the reformulated model with the same data, the rootmean square error is approximately 13000 N. Using thismethod, slightly different soil parameters were found(=29.15 =51.5 =36.18 degrees and cohesion=29234.9Pa.) There seems to be generally greater agreement betweenthe measured and predicted forces as shown in Fig. 10.6.4 Estimation MethodsWe used the same data set as above to compare exhaustivesearch and the method from 5.4 (referred to as just simulatedannealing below). Fig. 11 and Fig. 12 show the convergenceof exhaustive search and simulated annealing to the best fitparameters when the modified FEE is used. Convergence ofresults in exhaustive search is more uniform than in simu-lated annealing. This is essentially because simulated anneal-ing is a stochastic procedure. The main difference is thatsimulated annealing produces results in minutes that areequivalent to the results produced after hours of exhaustivesearch. In our trials simulated annealing runs for 4 minutes asopposed to 20 hours for exhaustive search.6.5 Correlating Over Extended OperationsIn this section we show how predicted and measured forcescompare over a long run. We took data from 20 digs and firstused this data to extract soil parameters. Even though thecharacteristics of the soil changes as the excavationprogresses, we get a reasonable prediction for resistive forcesas shown in Fig. 13. The error is clearly visible in the sectionswhen a new strata is encountered (around 45 secs).Next we used a sliding window, in which the parameters usedto calculate the force for a given dig is extracted from the datacorresponding to the previous three digs. Using this method,the comparison between predicted and measured forces is

Fig. 9 Sensitivity analysis of , , , and c. These graphs showhow the error increases when one of the parameters is perturbedfrom nominal values.

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shown in Fig.15. The error in this case is noticably reducedfor the last 3 digs.6.6 Predicting Bucket TrajectoriesOne of the main objectives of our work has been to improvethe prediction of digging trajectories in order to select opti-mal locations for digging [13]. Our autonomous excavator isequipped with a closed-loop controller that uses resistiveforces (as measured by cylinder pressures) to fill the bucket.

The predicted trajectory of the bucket tip is dependent on themodeled soil reaction forces. We have shown that by refor-mulating the soil model and by correlating the model param-eters, that the predicted trajectory can be much improved asillustrated by an example in Figure 16.

7 ConclusionsThe main advantage of the new formulation is that accountsfor the terrain slope during the excavation. Without this con-sideration, forces are either under or over estimated. In addi-tion we have presented an online method to extract soilparameters from reasonably sized data sets. Instead of hoursof computation required by an exhaustive search method, theonline method produces comparable results in a few minutes.We expect that the new force model, and the ability to adjustthe model parameters to changing soil conditions, willimprove the ability to effectively plan automated earthmov-

Fig. 10 X and Y components of the force. Measured force isshown a solid line. The FEE model is shown with a dashed line, andthe improved model with a dotted line. Only 4 digs shown for clarity.

Fig. 11 Error and parameters of best 100 trials using exhaustivesearch. Since exhaustive search is orderly, the parametersconverge linearly. The left most trial is the best. Asterisks represent, circles and pluses .

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Fig. 13 Measured vs. modeled force (magnitude) using the entiredata set of 20 digs to obtain the model coefficients. In the uppergraph, the solid line shows the actual measured force while thedotted line shows the predicted force. The lower graph shows thepercentage error in the prediction.

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AcknowledgmentsO. Luengo wishes to thank Universidad Politecnica deMadrid for its support during his visit to CMU, to A. Barri-entos and E.A. Puente for their encouragement, and, to S.Singh, H. Cannon, and J. Bares for their advice and support.

References[1] Alekseeva, T. V. and Artemev, K. A. and Bromberg, A.

A. and Voitsekhovskii, R. I. and Ulyanov, N.A., Machinesfor Earthmoving Work, Theory and Calculations, A. A.Balkema, Rotterdam, 1992.[2] Cundall, P., Board, M. A Microcomputer Program forModeling Large Strain Plasticity Problems, NumericalMethods in Geomechanics, Balkema, Rotterdam1988.[3] Gill, W. R., VandenBerg, G. E., Soil Dynamics in Tillageand Traction, Agriculture Handbook No. 316, AgriculturalResearch Service, US Department of Agriculture, 1968.[4] Hettiaratchi, D.R.P., Theoretical Soil Mechanics andImplement Design, Soil and Tillage Research, Vol. 11, 1988,pp.325- 347, Elsevier Science Publishers.[5] Huang, X.D. and Bernold,L.E. Robotic rock handlingduring backhoe excavation. Automation and Robotics inConstruction X. Elsevier Science Publishers, 1993.[6] Kirpatrick, S. Journal of Statistical Physics, vol. 34 pp.975-986.[7] Li, X. and Moshell, J. M. Modeling Soil: RealtimeDynamic Models for Soil Slippage an Manipulation, In Pro-ceedings of SIGGRAPH93, 1993.[8] Mckeyes, E. Soil Cutting and Tillage. Elsevier, 1985.[9] Press, William H. and Teukolsky, Saul A. and Vetterling,William T. and Flannery, Brian P. Numerical Recipes in C:The Art of Scientific Computing. Second Edition. CambridgeUniversity Press,1992.[10]Reece, A.R The Fundamental Equation of Earthmov-ing Mechanics, Proceedings of Institution of MechanicalEngineers, 1964.[11]Singh, S. Synthesis of Tactical Plans for Robotic Exca-vation. Ph.D. Thesis, Robotics Institute, Carnegie MellonUniversity. 1995.[12]Singh, S. Learning to Predict Resistive Forces DuringRobotic Excavation, Proceedings of the International Con-ference on Robotics and Automation, May 1995, NagoyaJapan.[13]Singh, S. and Cannon, H. Multi-Resolution Planningfor Earthmoving, Proceedings of the International Confer-ence on Robotics and Automation, May 1998, Leuven, Bel-gium.[14]Stentz, T, Bares, J., Singh, S. and Rowe, P. A RoboticExcavator for Autonomous Truck Loading, In Proceedings,IROS, Oct 98, Victoria, Canada.[15]Watton, J. Fluid power systems: modeling, simulation,analog and microcomputer control. Prentice Hall Interna-tional, 1989.[16] Zelenin, A. N. and Balovnev, V. I. and Kerov, L. P.,Machines for moving the earth, 1992.

Fig. 14 Measured force versus modeled force using a slidingwindow of 3 digs to obtain the model coefficients. In the uppergraph, the solid line shows the actual measured force while thedashed line shows the predicted force. The lower graph shows thepercentage error in the prediction.

Fig. 15 Actual versus predicted path of the bucket tip duringdigging. The solid line is the actual path of the bucket tip. The starsindicate the predicted path of the bucket tip using forces from theFEE soil model. The pluses show the predicted path using theforces from the reformulated soil model.

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