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Then, flows aretransferred into the electric signals actuating the load-independent valves. Their motion is presented by applying transferfunction. The performance of the system is verified by assuming an abrupt change of the oil flow into cylinders. The last part ofthe paper is devoted to the obtained experimental results. The first result deals with vertical drilling. The second result dealswith an excavation along a horizontal trajectory.D 2002 Elsevier Science B.V. All rights reserved.Keywords: Excavator; Hydraulic systems; Control; Trajectory execution1. IntroductionDue to encouraging results of recent research,there are increasing possibilities for enhancement ofa large spectrum human efforts in excavation pro-cesses. This may occur mainly through control ofrepetitive work tasks, such as trenching and drilling,requiring constant attention of machine operatorsduring the performance of each task. Particularattention, in research, is paid to excavation alongprescribed trajectories subjected to varying soil envi-ronment.Fundamentals dealing with controlled excavationprocesses are discussed by Vaha and Skibniewski 1,Hemami 2, and Hiller and Schnider 3. An inter-esting approach to piling processes by a directangular sensing method is proposed by Keskinen etal. 8. Budny and Gutkowski 4,6 proposed asystem, applying kinematically induced motion ofan excavator bucket. In this approach, influence ofa small variation of hydraulic oil flow into cylinders,applying sensitivity analysis, is discussed by Gut-kowski and Chlgosta 5. Huang et al. 7 presentedan impedance control study for a robotic excavator.They applied two neural networks: first, as a feed-forward controller and the second as a feedbacktarget impedance. Another impedance system, apply-ing a hybrid position/force control, is proposed by Haet al. 9.The first generation of robots was conceived asopen loop positioning devices. This implied thatall parts had to be manufactured with a very high0926-5805/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.PII: S0926-5805(02)00088-2* Corresponding author.E-mail address: mchimbigs.org.pl (E. Budny).URL: http:/ in Construction 12 (2003) 245254and costly accuracy. Next, positioning robots, withsensors, reduced this accuracy requirement consider-ably. Here were several approaches, mentioned inabove references, to extend the industrial robotscapabilities to robotic excavator. Systems of forcecells, longitudinal and angular sensors have beenapplied. However, two main differences betweenrequirements for manufacturing robots and roboticexcavators should be noted. The first difference isthat manufacturing robots are working in almostperfect conditions, free of vibrations, protectedagainst shocks, humidity, and other possible damag-ing conditions. The second difference is the require-ments for high accuracy of manufacturing robots,often within microns. On the contrary, robotic exca-vators are working in very difficult construction siteconditions, and required accuracy of the executedtrajectories, comparing with industrial robots, islimited, say within centimetres. With difficult con-ditions of excavations works, all sensors attached tothe boom, arm, and bucket have to be very wellprotected.Bearing in mind the above differences, it would beof interest to investigate the possibilities of controllingexcavation trajectory by a hydraulic module com-posed of a pump and load-independent valves. Inother words, to investigate a system free of sensorcells mounted at the excavator attachment, combinedwith a feedback controller, included in the hydraulicunit of the machine. The main objective of the presentpaper is to extend the discussion, initiated by theauthors 10, on the possibilities of applying load-independent valves installed inside of operator cabinonly. Under this assumption, the system is free ofsensors located on the excavator attachment. Afterdiscussing mathematical model of the system, pre-liminary experimental results are presented at the endof the paper.2. Statement of the problemThe paper deals with a controlled, stable motionof an excavator bucket along a prescribed path. Theproblem is based on previous authors theoreticalinvestigations 4 of quasi-static, kinematically in-duced excavation processes for assumed trajectories.In this study, the following assumptions are made.The excavator attachment is a planar mechanism,composed of a boom, an arm, and a bucket. Three,independently driven, hydraulic cylinders operate thesystem. They are assuring a unique representation ofthe three degrees of the planar bucket motion, twodisplacements and a rotation.The excavation process, in the experiments per-formed,isassumedtobeslowenoughtoconsideritasaquasi-static one. Inertia terms in motion equations ofattachment can be then neglected. Only spool of theservomechanism is assumed to move with accelera-tions, which cannot be neglected.The force (pressure) disturbances are assumed tohave sinusoidal form. The acceptable parameters of thesinusoid are defined from stability conditions of thesystem.The soil is assumed homogeneous. Some smallinclusions in the form of stones are acceptable.The proposed control system of excavation isoperator-assisted. It means that in a case of a largerobstacle, the operator has to intervene.If successful, the proposed control setup couldapply to standard excavators with the aim of enhance-ment of a large spectrum of human efforts in repetitiveprocesses such as trenching and drilling.The experiment is considered as a system com-posed of three subsystems, namely: microcomputerwith PLC; hydraulic arrangement (a pump, valves,cylinders); and the mechanism with three degrees offreedom of the bucket. Next, the subsystems areconsidered as sets of components. In the first sub-system, the following components are recognised:personal computer with appropriate software, trans-forming introduced equations and inequalities ofmotion and trajectory planers into electric signal.The latter is send to a PLC unit, which in turn causesan electrical actuation of solenoid valves. Pressuresfrom the solenoid valves are causing changes in spoolpositions, assuring assumed flow of the hydraulic oilinto cylinders. The spool position, in turn, is con-verted through a transducer to an electric feedbacksignal sent to the solenoid valves. Opened spools areletting the hydraulic oil to flow into the third sub-system, namely cylinders of the excavator mechanism.Finally, the last subsystem is composed of threecomponents: the hydraulic cylinders, the boom, thearm, and the bucket. With the motion of the excavator,arms and the bucket itself, the pressures in cylindersE. Budny et al. / Automation in Construction 12 (2003) 245254246are changing. Information about these changes is sentto the second, hydraulic subsystem, where the feed-back signal corrects position of spools assuring the oilflow according to the designed trajectory.In the paper, transfer functions of all systemcomponents are investigated from the point of viewof stability. The functions are defined theoretically, ornumerically from diagrams presented in catalogues ofhydraulic equipment. Joining all transfer function ofparticular component, the transfer function of thewhole system is discussed from the point of view ofperformance under abrupt unit signal.Several experiments were performed, showing thatit is possible to assure stable, assumed motion of thebucket. Among experiments, one was devoted to drill-ing. In other words, the kinematically induced trajec-tory was a straight, vertical line. Experimentallyobtained line is presented in Refs. 6 and 10. It isinteresting to note that the variation of experimentalline does not exceed 10 cm.3. Three subsystems of the experimental setupThe discussed system is divided in three sub-systems, namely: microcomputer, hydraulic valves,and excavator arms with a bucket. Below, they arediscussed separately and then a joint control prob-lem is defined.3.1. Microcomputer as a subsystemWe start with defining a model of the end-effector (bucket, drill, hammer) motion. The end-effector, in its plane motion, has three degrees offreedom aj(j=1,2,3) (Fig. 1). They are rotations ofthe boom, of the arm, and of the effector.Denoting by x1p, x2pposition of the end-effectortip, and by x3its rotation, the kinematics of theconsidered mechanism is represented by vectorrelation:x1px2px3p266664377775c1c2c30s1s2s30000a3266664377775?l1l2l3266664377775;1where cjand sjdenote cos ajand sin aj, respec-tively. In further considerations, the sub index p isomitted as the position of only one point is con-sidered.Velocity of the point P, v=v1, v2, v3T=x 1, x 2,x 3Tis obtained by taking time derivative of Eq.(1), and by reducing 3?4 matrix to a 3?3 matrix: x v A a a a Aw;2whereA ?l1s1?l2s2l3s3l1c1l2c2l3c3001266664377775:3Taking inverse of A matrix equal to:A?1l2c2l1c10?l2s2?l1s10l2l3f23l1l3f13l1l2f12266664377775?1l1l2c1s2? s1c24with fij=sicj?cisj, we find the inverse kinematics,relating angular velocities of mechanism elementsto the tip displacement vectorw A?1v:5Angular velocities xj, in turn, are dependent on theelongation velocities hiof hydraulic cylinders. Thisdependence has to be determined from geometricalrelations between cylinder lengths, constant param-eters of attachment, and aj.We start with the first cylinder. From Fig. 2 we findcoordinates of two cylinders hinges, A1and B1.They are:x1A1 a0;x2A1 b0;x1B1 b1c1 a1s1;x2B1 b1s1? a1c1:Takingh21 x1B1? x1A12 x2B1? x2A22;E. Budny et al. / Automation in Construction 12 (2003) 245254247after transformation we obtainh21 p01 q01c1 r01s1;6wherep01 a20 a21 b20 b21;q01 2a1b0? a0b1;r01 ?2a0a1 b0b1:Taking time derivative of Eq. (6) we find:h1?q01s1 r01c12h1? x1G1112h1? x1:7Repeating the same consideration for the secondcylinder length (Fig. 3) we obtainh22 p02 q02f12 r02g128wherep02 a22 a23 b22 b23;q02 ?2a2a3 b2b3;r02 2a2b3? b2a3;Fig. 1. The mini-excavator considered.E. Budny et al. / Automation in Construction 12 (2003) 245254248andfij cicj? sisj;gij sicj cisj:9Taking again time derivatives of Eqs. (8) and (9), wearrive ath2?q02g12 r02f122h2? x1 x2G2122h2? x1 x2:10An expression representing the length h3of thethird cylinder is more complex, and requires intro-duction of an auxiliary variable a4(Fig. 4). With anew variable, there is a need to introduce an addi-tional relation. In this case, the relation joins varia-bles a2, a3, and a4, through the condition thatdistance between B3and D3is constant and equalto b7. After some lengthy transformation, theserelations take the following form:h23 p03 q03f24 r03g24;11b27 p04 q04f23 r04g23 q05f24 r05g24;12wherep03 a24 a25 a27 b24 b25 b4b5;q03 ?2a7a4? a5;r03 2a7b4? b5;p04 a25 a26 a27 b25 b26;q04 2b5b6? a5a6? a6a7;r04 ?2a5b6 a6b5;q05 2a5a7;r05 ?2a6a7:Fig. 4. The length h3of the third cylinder.Fig. 2. The length h1of the first cylinder.Fig. 3. The length h2of the second cylinder.E. Budny et al. / Automation in Construction 12 (2003) 245254249Taking time derivative of Eq. (11) and recallingthat aj=xj, the velocity h3can be presented as:h3?q03g24 r03f242h3? x2 x4G3242h3? x2 x413The mentioned condition for b7in the form of Eq. (12)allows to find a4, and eliminates it from the otherequations. Taking now time derivative of Eq. (12), wecan express x4in terms of x2and x3x4 ?G423G524 1? x2?G423G524? x3;14whereG423 ?q04g23 r04f23;G524 ?q05g24 r05f24:Combining, now, together Eqs. (7), (10), (13), and(14) in a vector notation, we can write:h H ? w15withH12 H13 H23 H31 0;H11G1112h1;H12 H22G2122h2;H32 H33 ?G324G4232h3G524:The flow of the hydraulic fluid into jth cylinder,denoted by qj, is equal to hjSj, where Sjis the cross-section area of the cylinder. With above notations,we can write the final relation between assumedvelocity vector of the end-effector and flow vectorq asq S ? H ? A?1? v16where S is diagonal matrix with components Sj(j=1, 2, 3). The flow (Eq. (16) is a calculatedflow, which in our model is needed to move the endeffector according to its assumed motion. In a realsystem, this amount of oil has to be supplied to realcylinders through valves. The latter must be thenactuated by an electrical signal vector u. The relationof qj=qj(uj) between this signal and oil flow is givenby valve characteristic, which in general has theform presented in Fig. 5. The positive values of qjare related to the elongation of the cylinder. Thenegative ones are related to its shortening.The curve representing graphically qj(uj) can beassumed to be represented by the following function:qj a1u ? b a3u ? b3 a5u ? b5;17with constraints d imposed on maximum openings ofthe valve. Coefficients a1, a2, and a3can be deter-mined by fitting the function (17) at three points of thecharacteristic curve. In order to find electrical signal ujin terms of qj, we have to take the inverse of Eq. (17).In general, this can be achieved only through anumerical solution method.3.2. Hydraulic valve subsystem (HVS)The calculated in microcomputer, reference elec-trical signal is now converted into real electricalsignal, actuating the valve. In the problem discussedhere, this is a load-independent, proportional valvePVG 32 by DanfossR. The discussed subsystem ispresented in Fig. 6. Below, all of its parts and theirtransfer functions are discussed.Fig. 5. The oil flow q leaving the valve, as a function of uj.E. Budny et al. / Automation in Construction 12 (2003) 245254250The difference between reference signal ujand ud,and a signal coming from the feedback, is actuatingthe controller. The controller in turn, is adjusting thepump pressure ppto a pressure pcneeded for anadequate position of the spool. This adjustment isdone by four solenoid valves. Denoting by capitalletters the Laplace transforms, we find:Ucs Ujs ? Uds Ujs ? Huds ? Ds;18where D(s) is Laplace transform of spool displace-ment d; Hudis a transfer function between the spooldisplacement and feedback signal ud.The latter is obtained by a transducer, with constantmultiplier, giving:Huds UdsDs Kd:19The relation between UCentering the controller and pcleaving it is also constant:Gpus PcsUcs Kc:20The pressure acting on the spool is causing its motion,defined by an equation for one degree of freedom,with a spring constant ks, spool mass m, dampingcoefficient c, and cross-section area on which thepressure is acting As:md cd ksd pcAs:21The transfer function between spool displacement dand pressure pcis then as followss2m sc ks ? Ds Pcs ? As:22Considering now Eqs. (18)(21), we obtain the rela-tion between the transformed output of spool displace-ment and transformed reference input of electricalsignal:Djs As? KcAsKcKd ks sc s2m? Ujs;23or considering feedback electrical signal Ujd, we haveUjds AsKcKds2m sc AsKcKd ks;24With a constant nominator and denominator, in theform of a second order polynomial, we can verify theperformance of our control setup by assuming elec-trical signal equal to a unit step functionujt ujut25which implies an abrupt change in the cylinder length.Considering now Eqs. (24) and (25), carrying apartial fraction expansion, and taking inverse Laplacetransforms, we find the error e(t) as a function of time:et e?fxntcosxdt fxnxdsinxdt?ujdt;26where2fxncm;x2nAsKcKd ksm;xd 1 ? f21=2xn;f 1 weak damping:Fig. 6. Hydraulic valve subsystem.E. Budny et al. / Automation in Construction 12 (2003) 245254251The relation (26) shows that the error asymptoti-cally tends to zero with the increase of time.4. Experimental realization4.1. The mini-excavator used for experimentsThe mini-excavator K-111 is used for experiments.It was assumed to minimize part replacements, in aserial machine, needed to perform the consideredcontrol. The main components in the hydraulic systemto be replaced were valves. Moreover, the hydrauliccylinders are supplied with additional valves assuringrequired pressure. This ensures that unpredictedmotion of the attachment is not taking place. Thehydraulic load-independent valves used in this experi-ment were supplied by DanfossR. The transfer ofinformation from a microcomputer to load-independ-ent hydraulic valves is conducted by a ControlledArea Network (CAN).After modification of the hydraulic system, theexcavator can be controlled in two different ways.The first method consists in using joysticks mountedin the operator cab. This way, using a joystick, theoperator can move the mechanism in an arbitraryposition, and with desirable velocity. The secondmethod consists of programming the bucket motionin a microcomputer. The information from it is thentransformed in elongation rates of cylinders, and inthe flow of the oil moving them. The latter isconverted in an electrical signal send through CANto load-independent valves. The organization of theelectrical system for load-independent valves isshown in Fig. 7.The control algorithm is written in Borland Pas-cal and executed under MS Window 98 operatingsystem. The CAN communication rate is assumed tobe 250 kbit/s with sampling time between 0.5 and2.0 s.Fig. 7. Hardware of the control subsystem.Fig. 8. Experimental results of vertical drilling.E. Budny et al. / Automation in Construction 12 (2003) 2452542524.2. Experimental resultsTo examine the performance of the proposed con-trol system, the above-mentioned mini excavator withthree hydraulic cylinders and three degrees of freedomis used. An electro-hydraulic, load-independent, pro-portional valve controls separately each cylinder.Experiments were done in order to control themotion of the bucket tip along straight lines, a verticalone, and a horizontal one. As mentioned in Statementof the problem, the motion was controlled both in freespace and soil box, filled with homogenous mildlyhumid sand.The experiments were performed with relativelysmall velocity of about 2 m/min. Trajectories obtainedfor drilling along a vertical line and the movement ofthe bucket tip along a horizontal line are presented inFigs. 8 and 9.5. ConclusionsA relatively simple control system for an excavatoris proposed. The system is free of sensor mounted onthe attachment of the machine. The whole controlhardware is within load-independent valves, located inthe operator cabin. After some additional study andimprovements, the system could be applied on mass-manufactured excavators, helping in works with repet-itive processes, like digging trenches or drilling.The experimental results are showing good qual-itative performance. Also, quantitative results, in thecase of horizontal line, are promising. Here, theobtained trajectory varies from straight line by nomore than 4% of the traveled distance. Less accurateresults are in the second case, namely for vertical line.Here, errors, for some experimental runs, are even upto 15%. This might be caused, among others, by thefact that elongation of one of the cylinders is changingsign during the motion. It means that for a moment, itsspool is closing oil supply to both cylinder ends. Atthe same time, when motion of the cylinder in ques-tion is blocked, the two other cylinders are moving.These observations have to be considered by improv-ing the present model. The improvement should con-sist in a joint control system for all three cylinde