外文翻译--应用整体滑动面重型机器人挖掘机的延时控制 英文版

Control Engineering Practice 10 (2002) 697711Control of a heavy-duty robotic excavator using time delay controlwith integral sliding surfaceSung-Uk Lee*, Pyung Hun ChangDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Science Town, 373-1 Koosung-dong, Yusung-ku,Taejon 305-701, South KoreaReceived 22 March 2001； accepted 14 December 2001AbstractThe control of a robotic excavator is difcult from the standpoint of the following problems: parameter variations in mechanicalstructures, various nonlinearities in hydraulic actuators and disturbance due to the contact with the ground. In addition, the morethe size of robotic excavators increase, the more the length and mass of excavators links； the more the parameters of a heavy-dutyexcavator vary. A time-delay control with switching action (TDCSA) using an integral sliding surface isproposed in thispaper forthe control of a 21-ton robotic excavator. Through analysis and experiments, we show that using an integral sliding surface for theswitching action of TDCSA is better than using a PD-type sliding surface. The proposed controller is applied to straight-linemotions of a 21-ton robotic excavator with a speed level at which skillful operators work. Experiments, which were designed forsurfaces with various inclinations and over broad ranges of joint motions, show that the proposed controller exhibits goodperformance. r 2002 Elsevier Science Ltd. All rights reserved.Keywords: Time-delay control； Robust control； Switching action； Robotic excavator； Trajectory control1. IntroductionA hydraulic excavator isa multi-functional construc-tion machine. Workers in the construction industry useit for tasks such as excavating, dumping, nishing,lifting work, etc. However, operatorswho controlhydraulic excavatorsmust be trained for many yearsto do such work quickly and skillfully. A hydraulicexcavator hasthree links: boom, arm and bucket； andthe operator hastwo arms. Thus, it isnot easy forbeginnersto execute elaborate work that manipulatesthree links at the same time. Moreover, because theoperatorshave to run work in variousdangerousanddirty environments, the number of skillful operators isever decreasing. For that reason, studying the automa-tion of hydraulic excavators is necessary for improvingproductivity, efciency, and safety.The automation of hydraulic excavatorshasbeenstudied by several researchers (Singh, 1997). Among theseveral tasks to be automated, Bradley and Seward(1998) developed the Lancaster University computerizedintelligent excavator (LUCIE) and used it to automatethe digging work. Stentz, Bares, Singh, and Rowe (1998)developed a complete system for loading trucks fullyautonomously on a 25-ton robotic excavator. Changand Lee (2002) automated straight-line motions on a 13-ton robotic excavator under working speed conditions.Here, the straight-line motion represents the importanttask of scraping or attening the ground and serves as afundamental element used as a basis for developingmore complicated tasks. As illustrated in Fig. 1, the end-effector of the manipulator needsto be controlled totrack a linear path on the task surface. An operatorshould manipulate three links simultaneously to executeit. Though an operator isskillful, performing thestraight-line motions for a long time results in thefatigue of an operator and decreases productivity.The control of robotic excavator isdifcult from thestandpoint of the following problems: parameter varia-tionsin mechanical structures, variousnonlinearitiesinhydraulic actuators, and disturbance due to the contactwith the ground. In mechanical structures, the inertial*Corresponding author. Tel.: +82-42-869-3266； fax: +82-42-869-5226.E-mail address: s sulee123cais.kaist.ac.kr (S.-U. Lee).0967-0661/02/$-see front matter r 2002 Elsevier Science Ltd. All rights reserved.PII: S 0967-0661(02)00027-8force and gravitational force varieslargely with jointmotions. Hydraulic actuators, massively coupled andcomplexly connected, have variousnonlinear compo-nents. For such reasons, various difculties exist incontrolling a robotic excavator.To solve these problems, several research works havebeen performed, which may be categorized aseithersimulation studies or experimental studies. In terms ofsimulation studies, for instance, Chiba and Takeda(1982) applied an optimal control scheme to the controlof the manipulator of an excavator. Morita and Sakawa(1986) used PID control with feedforward control basedon inverse dynamics. Medanic, Yuan, and Medanic(1997) proposed a polar controller-based variablestructure control. Song and Koivo (1995) used afeedforward multiplayer neural network and a PIDcontroller over a wide range of parameter variations. Asfor experimental studies, Bradley and Seward (1998)used a high-level controller that was based on rulesobtained by observation of skilled operators, and a PIDlow-level motion controller that moved the end-effectorin response to a demand from the high-level controller.Lee (1993) used P control together with a fuzzy controltechnique that used response error and its derivative onthe phase plane. Sepehri, Lawrence, Sassani, andFrenette (1994) analyzed the phenomenon of couplingin the hydraulic actuator, and proposed a feedforwardscheme that compensates coupling and load variation byusing a simple valve model and measured pressure.Yokota, Sasao, and Ichiryu (1996) used disturbanceobserver and PI control, and applied it to a miniexcavator. Chang and Lee (2002) used time-delaycontrol (TDC) and compensators based on the dy-namicsof the excavator and applied it to straight-linemotionsof a 13-ton excavator with a bucket speed of0:5m=s； a speed level at which skillful operators work.However, almost all the research works above tend tobe limited to experimentson a mini excavator underrelatively lower speed conditions. Among the experi-mental research works above, only that of Chang andLee (2002), which wasperformed on the control of aheavy-duty 13-ton robotic excavator, wasperformedunder working speed conditions. The more the size ofexcavators increase, the more the length and mass ofexcavatorslinksincrease, and the more the parametersof a heavy-duty excavator vary. Therefore, the controlof a heavy-duty excavator becomesmore difcult thanthe control of a mini excavator. The control of a heavy-duty excavator (a 21-ton robotic excavator (Fig. 2) usedin thispaper) requiresa robust controller.In thispaper, we apply time-delay control withswitching action (TDCSA) using an integral slidingsurface (ISS) to the control of a 21-ton robotic excavatorand validate the proposed control algorithm throughexperimentson a straight-line motion tracking control.In addition, we show the advantage of the TDCSA usingan ISS. TDCSA, which wasproposed by Chang andPark (1998), consists of a TDC and a switching action.The switching action based on sliding mode control(SMC) compensates for the error of the time-delayestimation (TDE) and makes the TDC more robust.Chang and Park (1998) used a PD-type sliding surface(PDSS) for the switching action and applied the TDCSAusing a PDSS to a pneumatic system for compensatingthe stick-slip, but we use an ISS for the switching actionto improve the control performance in thispaper(Slotine & Li, 1991； Utkin & Shi, 1996). Asa similarcontroller, a new integral variable structure regulationcontroller, designed using an integral sliding surface anddisturbance observer, was proposed by Lee and Youn(1999). Lee and Youn, however, have demonstrated theusefulness of their proposed algorithm by simulationsabout regulation controlsof a two-link manipulator. Incontrast, we will perform experiments on the control ofa robotic excavator.Fig. 1. Illustration of straight-line motion.Fig. 2. Appearance of Robex210LC-3 excavator.S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711698The rest of this paper is organized as follows. Section2 will briey analyze the characteristics of the roboticexcavator system, Section 3 presents the design ofcontroller. In Section 4, the effectiveness of the proposedcontroller will be veried through experimentson a 21-ton robotic excavator. Finally, conclusions will bedrawn in Section 5.2. Overview of robotic excavator systemThis section describes briey the characteristics of therobotic excavator. More detailsabout the model of arobotic excavator can be found in Chang and Lee(2002). In thispaper, the swing motor together with thetraveling motor is not considered for straight-linemotion； only the boom, arm and bucket are considered.The mathematical model that isneeded for designing acontroller isdescribed in Appendix A. Since a roboticexcavator consists of a manipulator and actuators, thecharacteristic of these two parts will be described briey.2.1. ManipulatorIn the dynamic equation (Eq. (A.1), the inertialforcesand gravitational forcesaswell asthe centrifugaland Coriolisforcesvary nonlinearly with the change ofanglesof the links, and have coupling elementsbetweenlinks. Among these terms, the centrifugal and Coriolisforceshave a smaller effect on the control performance,since the velocity of each link is not that great. Incomparison, the inertial forces and gravitational forcesvary largely, since the total weight of boom, arm andbucket used in this research is 2:67 ton and the range ofjoint anglesare broad. The size and variation in each ofthe inertial and the gravitational forcesin a straight-linemotion with incline of 01 are shown in Fig. 3. We canobserve that the inertial forces and gravitational forcesvary largely.2.2. Hydraulic actuatorsThe hydraulic actuator of the robotic excavator usedin thispaper hasat least three kindsof nonlinearitiesasfollows: valve characteristics, dead zone and time lag.2.2.1. Valve characteristicsHydraulic valvesare devicesthat transfer the owfrom the pump to cylinder. From the general valve owequation Q cdADPp； the ow that istransferredfrom the pump to cylinder isdetermined by owcoefcient, the area of the valve and pressure difference.The area of a spool valve has a nonlinear shape asshown in Fig. 4. Therefore, valves have nonlinearcharacteristics according to the nonlinear area of thevalve andDPp:2.2.2. Dead zoneThe geometry of the spool valve used in a Ro-bex210LC-3 excavator is an overlapped shape as shownin Fig. 4 and causes the dead zone nonlinearity. Theoverlapped region isdesigned for the convenience of anoperator. Therefore, when the spool is displaced in theoverlapped region, the valve becomes closed: this causes0 2 4 6 8_3000_2000_10000100020003000timesecforceNewton(a) Inertial forceboomarm0 2 4 6 8_505101520x 104timesecforceNewton(b) Gravitational forceboomarmFig. 3. Inertial forcesand gravitational forcesof boom and arm.S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711 699the dead zone nonlinearity. The overlapped region isabout 30 percent of the whole spool displacement.2.2.3. Time lagA phenomenon similar to a dead zone occurs becauseof the time taken for the pump output pressure to reachthe pressure level that is sufcient to move the link. Notethat thisphenomenon issomewhat different from thepure time delay often found in transmission lines. Fig. 5illustrates this phenomenon with the experimentalresults. In the presence of maximum control input, theboom doesnot move until the time is0 :13 s； when thepump pressure begins to exceed the pressure of theboom cylinder plus the offset pressure, as shown inFig. 5. Thisphenomenon occursonly when the boomlink beginsto move and it doesnot exist any more oncethe pump output pressure reaches the pressure levelsufcient to move the link. Moreover, this phenomenoncan be compensated by the compensator which will beproposed in Section 3.2.3. Controller designA robotic excavator hasthe following nonlinearities:variationsin the inertial and gravitational forcesin themanipulator； and nonlinear valve characteristics, deadzone and time lag in the hydraulic actuator. Toovercome these aforementioned nonlinearities, Changand Lee (2002) suggested the TDC and compensatorsand used these to control a 13-ton robotic excavator, butwe need a more robust controller to control a 21-tonrobotic excavator effectively. The greatest differencebetween the 21-ton robotic excavator used in this paperand the 13-ton robotic excavator used in Chang and Lee(2002) exists in the length and mass of excavators links.The linksof the former are one and half timesthe lengthand weight of those of the latter. Therefore, theparameter variationsof a 21-ton excavator are moreserious than those of a 13-ton excavator. A more robustcontroller than TDC isrequired to control the straight-line motion of a 21-ton robotic excavator.For controlling a 21-ton robotic excavator, we haveconsidered TDCSA, which is more robust than TDC.Spool displacementVavleAreaoverlap regionFig. 4. Rough shapes for areas of spool valve.0 1050100150200(a) Pressures of pump and head side of boom cylindertimesecpressurebarpumphead side of boom cylinder0 1949698100102104106(b) Boom responsetimesecangledegFig. 5. Illustration of the nonlinearity due to time lag.S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711700Then, instead of a PDSS used by Chang and Park(1998), we use an ISS in this paper for improving thecontrol performance (Slotine & Li, 1991； Utkin & Shi,1996).Among the aforementioned nonlinearities, however,the dead zone and the time lag cause the large trackingerrors. Hence, proper compensation is demanded. Forthat reason, we design the two controllers: rst theTDCSA using an ISS as the baseline control and second,compensators to overcome the dead zone and thetime lag.3.1. Design of the TDCSA3.1.1. TDCSA using an ISSIn order to apply the controller to a roboticexcavator, Eq. (A.5) isrearranged into the following:%M.lt%Htut: 1Note that%M isa constant matrix representing theknown angle of MK； whereas%Ht consists of termsrepresenting uncertainties and time-varying factors,which are expressed as%HtHtMKtC0%M.lt: 2Now we dene the desired dynamics of the closed-loopsystem with the following error dynamic:.etKvetKpet0； 3where etldtC0lt denotesthe position error vectorwith ldt denoting the vector of desired piston displace-ments, Kvthe derivative gain matrix, and Kptheproportional gain matrix. The TDC law that meetstherequirement isobtained asutdct%M.ldtKvetKpetC138 #Ht； 4where#Ht denotesan estimate of%Ht:The estimated#Ht can be obtained by using bothEq. (1) and the fact that%Ht isusually a continuousfunction. More specically, when L issmall enough,then#HtE%Ht C0 Lut C0 LC0%M.lt C0 L: 5Combining Eq. (5) with Eq. (4), the TDC law isobtained asfollows:utdct%M.ldtKvetKpetC138 utdct C0 LC0%M.lt C0 L: 6More details about the stability condition and the designof TDC can be found in Youcef-Toumi and Ito (1990)and Hsia and Gao (1990).L should be sufciently small for TDC to meet thedesired error dynamics of Eq. (3). The valve used for L；however, is set to be that of the sampling time, whenTDC isimplemented in a real-time controller. Thevariation of system nonlinearities and disturbances,occurred during the time delay L； caused TDE errorasfollows:%HtC0#Ht%HtC0%Ht C0 LDHt: 7More specically, the friction dynamics cause large TDEerror. Because of the TDE error, TDC does not have thedesired error dynamics of Eq. (3), but the followingerror dynamics:.etKvetKpet%MC01DHt； 8where the right term %MC01DHt denotesthe effect ofthe TDE error.The TDCSA is proposed by adding the switchingaction based on the sliding mode control to TDC, asfollows:utdcsat%M.ldtKvetKpetC138 utdcsat C0 LC0%M.lt C0 LKwsgns； 9where s represents the sliding surface and Kwisaswitching gain matrix. The TDCSA has the followingerror dynamic:.etKvetKpet%MC01DHtC0%MC01Kwsgns: 10In Eq. (10), we see that the switching action can reducethe TDE error.In order to match the desired error dynamics (Eq. (4)with the sliding surface (s), we use the integral slidingsurface as follows:stetKvetKpZt0et dtC0 e0C0Kve0； 11where the sliding surface (s) hasthe initial value of zeroand itsderivative isequal to desired error dynamics(Eq. (3). The necessity and advantage of using anintegral sliding surface will be shown in Section .1.2. Stability analysis of TDCSA using an integralsliding surfaceFor the stability analysis of the overall system, we usethe second method of Lyapunov. If the Lyapunovfunction isselected as V 12sTs； itstime derivative isasfollows:V sTs sT.e Kve KpeC138sT.ldC0%MC01u %MC01%H Kve KpeC138sTf.ldC0%MC01%M.ldKve Kpe#H KwsgnsC138%MC01%H Kve KpegsTC0%MC01#H %MC01%H C0%MC01KwsgnsC138sT%MC01DH C0%MC01KwsgnsC138: 12Therefore, the following condition isneeded so that thetime derivative of the Lyapunov function should benegative denite:Kwii jDHij for i 1；y；3: 13S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711 701In other words, the magnitude of the switching gainKw must be larger than that of the term due to the TDestimation error.3.1.3. Saturation functionTDCSA uses a switching action for compensating theTDE error, but the switching action in TDCSA causes achattering problem. Therefore, we use a saturationfunction to reduce the chattering problem (Slotine & Li,1991). A saturation function can be used as follows:satst；fstf if jstjof；sgnst otherwise；8 jDHij； the minimum tracking guarantee isjessijo%MC01iDHitKpi%MC01iKwili=fi: 22For a constant right-hand side of Eq. (21), however, thesteady-state solution of Eq. (21) is eit-0 andRt0eit dt-0: Therefore, TDCSA using an ISS candrive the tracking errorsresulting from biasinuncertainties (such as constant and slowly varyingparametric errors) to zero.From Eqs. (20) and (21), the relationship in theLaplace domain between the TDE error and positionerror et isasfollows:where p isthe Laplace operator. Eq. (23) isthat of theTDCSA using a PDSS and Eq. (24) is that of theTDCSA using an ISS. The bode plot of Eqs. (23) and(24) is shown in Fig. 6. The TDCSA using an ISS has thesame high-frequency behavior as the TDCSA using aPDSS and TDC. In the low-frequency range, however,TDCSA using an ISS has a lower gain than the othercontroller. Thus, the TDCSA using an ISS reduceseffectively the position error, which is caused by TDEerror in the low-frequency range, and then the TDCSAusing an ISS is more robust than the other controlleragainst the disturbances and variation of parameterswhich occur in the low-frequency range.3.2. Design of compensatorsCompensators are designed to overcome the deadzone and the time lag.Since the size of the dead zone coming from theoverlapped area of the spool valve is constant, we addthe size of the dead zone to TDCSA input as follows:u utdcsa ucomp1； 25EipDHip%MC01ip2Kvi%MC01iKwi=fip Kpi%MC01iKwili=fi； 23EipDHip%MC01ipp3Kvi%MC01iKwi=fip2Kpi%MC01iKwiKvi=fip %MC01iKwiKpi=fi%MC01ipp %MC01iKwi=fip2 Kvip Kpi； 2410_210_110010110210310_610_510_410_310_210_1|Ei(s)|/|delHi(s)|FrequencyHzMagnitudeTDC TDCSA using a integral sliding surfaceTDCSA using a PD type sliding surface Fig. 6. Bode diagram of closed-loop error dynamics.S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711 703where u denotesthe overall control input and ucomp1the3 C2 1 vector whose elements are constants equivalent tothe dead zone of each link.To compensate for the time lag, Chang and Lee(2002) designed the compensator using the pressuredifference of the pump and cylinder. Thiscompensatorincreases the pump pressure to cylinder pressure quicklyand works until pump pressure begins to exceed thepressure of the cylinder plus the offset pressure, but itneeds pressure sensors. In this paper, as shown in Fig. 7,we add the constant value ucomp2 to the control lawuntil the pump pressure is increased to the pressure levelsufcient to move the link, and then decrease the valueslowly to zero once the link begins to move.The whole control input, which now consists of theTDCSA input and the compensation inputs, is obtainedasfollows:u utdcsa ucomp1ucomp2: 26Note that ucomp2isused for the control inputsof boomand arm.4. ExperimentTo evaluate TDCSA using an ISS in real circum-stance, we have experimented the method in a heavy-duty excavator carrying out realistic tasks. The task ofconcern is primarily a straight-line motion in free spaces；yet, we have applied the straight-line motion to scrapingthe ground with the bucket in contact with the ground.The excavator used is a Hyundai Robex210LC-3, whichhasthe following specications: it weights21 ton ； thetotal length of the manipulator is10 :06 m and the totalweight of boom, arm and bucket is2 :67 ton: In theexperiments, the average speed of the bucket is set to be0:5m=s 4 m in 8 s), the speed level at which only anexpert operator can perform a task.4.1. Experimental setupFig. 8 shows the overall structure of the excavatorcontrol system. The trajectories of boom, arm andbucket for a task are calculated on a DSP controller.The angles of each links are measured with a resolverand then converted by an A/D converter for generatinga control input. The control input for each link iscalculated on a DSP controller according to the controllaw, by using the trajectories generated and the anglesmeasured. The control inputs obtained are convertedthrough a D/A converter and fed to operate the electri-proportional pressure reducing (EPPR) valve, whichtransforms the electrical signal into the hydraulicpressure signal. The pilot pressure from the EPPR valvemovesthe spool of each main valve, thusmaking eachlink move. Here, the sampling frequency of a DSPcontroller isselected as100 Hz :4.2. Trajectory generationTo automate the straight-line motion, the trajectoriesof the end-effector and the joint angle are needed.Once the incline of the task surface is given, the end-effector path isdetermined, and then the velocitytrajectory of the end-effector isdetermined by consider-ing the average speed of the end-effector. Here, thisvelocity prole isfor the direction tangential to the tasksurface, with the velocity in its normal direction beingkept at 0 m=s: The displacement trajectory of theend-effector isobtained by integrating the velocitytrajectory.To control each link, the displacement trajectory ofthe end-effector istransformed into the joint angletrajectory of each link. Moreover, during a straight-linemotion, the constraint exists that the attack angle ofbucket (the angle of the bottom surface of bucket)is kept constant with respect to the task surface.The joint anglesof boom and arm are calculated byDACTrajectoryGenerationControllerADCEPPRvalveExcavatorResolverDSPControllerAnglesFig. 8. Overall structure of control system.TimesecCompensatorInputCompensator for time lag(Ucomp2)Compensator for dead-zone(Ucomp1)Fig. 7. Compensator for dead zone and time lag.S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711704using excavator kinematics from Fig. 9, as Eqs. (27)and (28):yboom sinC01x2a y2a L21C0 L222L1x2a y2ap C0 tanC01yaxa； 27yarm cosC01x2a y2aC0 L21C0 L222L1L2； 28where xaand yaisthe position of the bucket joint. Fromthe aforementioned constraint, the angle of the bucket isdetermined asfollows:ybucket yinitC0yboomC0yarm； 29where yinitisthe sum of the boom angle, arm angle, andbucket angle in initial posture. From the above, we candetermine the joint trajectory of each link.We use the velocity trajectory obtained that meets thespecied average speed of 0:5m=sasshown in Fig. 10.Then, the joint trajectories for three task surfaces withrespective inclines of C0301； 01 and 301 isshown inFig. 11. Thistrajectory ismade to realize the practicethat straight-line motions are usually carried out from astretched posture to a folded one.4.3. Experimental resultsIn implementing the control law in Eq. (15), wedetermined each gain for utdcsaasfollows:%M %M1000%M2000%M326643775；Kv2x1w10002x2w20002x3w326643775；Kpw21000 w22000w2326643775； 30where xiand widenote the damping coefcient and thefrequency for desired error dynamics of each link. Inaddition,%Miis selected in the stable range throughexperiments. The switching gain Kwi； satisfying thecondition of Kwii jDHij； isselected by tuning. Theboundary layer of the saturation function fi isFig. 9. Excavator kinematics.0 1 2 3 4 5 6 7 8_0.7_0.6_0.5_0.4_0.3_0.2_0.10End Effector velocity trajector ytimesecvelocitym/sFig. 10. End-effectorsvelocity trajectory for horizontal direction.S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711 705properly chosen by tuning, without the chattering. Then,the magnitude of ucomp2isto be tuned from a small valueuntil satisfactory performance is achieved.4.3.1. Noncontact conditionIn free spaces, we experimented with three controllers:TDC, TDCSA using a PDSS and TDCSA using an ISS；for task surfaces with inclines of C0301； 01 and 301；respectively. Here, among the gains of the TDCSA, thedesired error dynamics are the identical with those ofTDC. The switching gain of the TDCSA using an ISS isidentical with that of the TDCSA using a PDSS.Fig. 12 shows the experimental results for TDC. Asshown in Fig. 12, the vertical distance error of the end-effector iswithin 73 cm for the task surface with anincline of C0301: For the task surface with an incline of301； however, the vertical distance error is within76cm: According to the task surface and jointpositions, the tracking error of the boom varies greatly.Namely, the control performance for the TDC variesgreatly with the task surface and joint positions.The experimental results for the TDCSA using aPDSS and for the TDCSA using an ISS are shown inFigs. 13 and 14, respectively. The control performanceof TDCSA isbetter than that of TDC. The controlperformance of TDCSA doesnot vary greatly with thetask surface and joint positions. The whole verticaldistance error of the end-effector is within 74 cm forTDCSA using an ISS, whereas the vertical distanceerror isover 74 cm for TDCSA using a PDSS.Fig. 15 shows the experimental results of the abovethree controllers for the task surface with an incline of01； and Fig. 16 shows the power spectral density of thetracking errorsin Fig. 15. In Fig. 15, the difference dueto TDCSA using an ISS is not that signicant； yet thereexists a noticeable trend. As shown in Fig. 16, TDCSAusing an ISS results in the DC components of the powerspectral densities considerably smaller than those of theother two controllers. This trend is a direct outcome ofthe offset of the tracking error of TDCSA using an ISSsmaller than those of the other two controllers. A closeinspection of the tracking errors in Fig. 15 (a) revealsthat the tracking error due to TDCSA using an ISStendsto uctuate around zero, whereasthose due to theother controllerstend to drift from zero. These resultssuggest that TDCSA using an ISS can decrease theoffset of the tracking error effectively, which still occursfor TDC and TDCSA using a PDSS.4.3.2. Contact conditionThe above experimentsdo not involve any contactwith the ground； but the straight-line motion in freespaces. Therefore digging or excavating work is not ourimmediate concern. Nevertheless, we have applied thestraight-line motion to leveling work, scraping theground with the bucket in contact with the ground.When leveling work isbeing done, friction due to thecontact behaves as a disturbance to position control ofTDCSA using an ISS. Since TDCSA using an ISS has0 2 4 6 8405060708090100(a) Boom angledegreetimesec0 2 4 6 8406080100120140(b) Arm angledegreetimesec0 2 4 6 8020406080100120(c) Bucket angletimesecdegreeFig. 11. Angle trajectories for task surface (solid line stands for incline of 01； dashed line for incline of 301； and dotted line for incline of C0301).S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711706shown effectiveness in rejecting the disturbances, it doesnot require additional force control scheme.We have applied TDCSA using an ISS to levelingworksfor the ground surfaceswith inclinesof C0301； 01and 301； respectively. The soil used at the experimentsis classied into gravel (GP group)poorly gradedgravel and graded-sand mixturesby the United SoilClassication, a generally used classication in civil0 2 4 6 8_6_4_20246time(sec)errorcm(a) Bucket vertical error0 2 4 6 8_1_0.500.51time(sec)errordeg.(b) Boom angle error0 2 4 6 8_2_1012time(sec)errordeg.(c) Arm angle error0 2 4 6 8_3_2_1012time(sec)errordeg.(d) Bucket angle errorFig. 13. Experimental results of TDCSA using a PDSS for task surface (solid line stands for incline of 01； dashed line for incline of 301； and dottedline for incline of C0301).0 2 4 6 8_6_4_20246time(sec)errorcm(a) Bucket vertical error0 2 4 6 8_1_0.500.51time(sec)errordeg.(b) Boom angle error0 2 4 6 8_2_1012time(sec)errordeg.(c) Arm angle error0 2 4 6 8_3_2_1012time(sec)errordeg.(d) Bucket angle errorFig. 12. Experimental results of TDC for task surface (solid line stands for incline of 01； dashed line for incline of 301； and dotted line for inclineof C0301).S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711 707engineering (Bowles, 1982). Fig. 17 (a) shows theexperimental results that the bucket vertical error ismostly within 74 cm for three task surfaces, which issimilar to the results for non-contact condition. On thecontacting condition, the ground will prevent the bucketvertical error from having the large error of (-) directionby supporting the end-effector of the excavator, and thefriction force between the end-effector and the ground0 2 4 6 8_6_4_20246time(sec)errorcm(a) Bucket vertical error0 2 4 6 8_1_0.500.51time(sec)errordeg.(b) Boom angle error0 2 4 6 8_2_1012time(sec)errordeg.(c) Arm angle error0 2 4 6 8_3_21012time(sec)errordeg.(d) Bucket angle errorFig. 14. Experimental results of TDCSA using an ISS for task surface (solid line stands for incline of 01； dashed line for incline of 301； and dotted linefor incline of C0301).0 2 4 6 8_3_2_1012345time(sec)errorcm(a) Bucket vertical error0 2 4 6 8_0.4_0.6time(sec)errordeg.(b) Boom angle error0 2 4 6 8_1.5_1_0.500.51time(sec)errordeg.(c) Arm angle error0 2 4 6 8_2_1.5_1_0.500.51time(sec)errordeg.(d) Bucket angle errorFig. 15. Experimental results for task surface with incline of 01 (solid line stands for TDCSA using an ISS； dashed line for TDCSA using a PDSS；and dotted line for TDC).S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711708deterioratesthe control performance of the arm link；however, it doesnot affect largely the error of the bucketvertical error due to the posture of the excavator. Theseresults verify that the position control approach basedon the TDCSA iseffective in handling the contactcondition.From the experimental results of applying TDCSAusing an ISS to a 21-ton robotic excavator, the vertical00.8 1050010001500FreqHzPower spectral density(a) Bucket vertical error00.8 102468FreqHzPower spectral density(b) Boom error00.8 10102030405060FreqHzPower spectral density(c) Arm error00.8 1010203040FreqHzPower spectral density(d) Bucket errorFig. 16. Power spectral density of tracking errors for task surface with incline of 01 (solid line stands for TDCSA using an ISS； dashed line forTDCSA using a PDSS； and dotted line for TDC).0 2 4 6 8_6_4_20246time(sec)errorcm(a) Bucket vertical error0 2 4 6 8_1_0.500.51time(sec)errordeg.(b) Boom angle error0 2 4 6 8_2_1012time(sec)errordeg.(c) Arm angle error0 2 4 6 8_3_2_1012time(sec)errordeg.(d) Bucket angle errorFig. 17. Experimental results for TDCSA using an ISS for task surface in contact the end-effector with the ground (solid line stands for incline of 01；dashed line for incline of 301； and dotted line for incline of C0301).S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711 709distance error of the end-effector is mostly within74 cm for the task surfaces with inclines of 0；C0301and 301； respectively. Considering that the accuracyachieved by an expert operator isusually within an errorof 75cm at 0:5m=s speed level, these results verifygood tracking performance of our proposed control law.5. ConclusionA TDCSA using an ISS was proposed in this paperfor the control of a 21-ton robotic excavator. Fromanalysis, we observed that the proposed control is morerobust against the disturbances and the parametervariations, which occur in the low-frequency range,than that of TDC and TDCSA using a PDSS. Throughexperiments, we observed that TDCSA using an ISS iseffective enough to control a 21-ton robotic excavator；and that our proposed control achieves better trackingperformances than an expert operator does. Consideringthat the experimentshave been made over a broadmotion range, under realistic working speed conditions,and on task surfaces with various inclines, we canconrm the validity of our proposed control algorithm.Appendix A. Dynamic modeling of the excavator systemWe will obtain the brief model of the roboticexcavator to design the controller. More details aboutthe model of a robotic excavator can be found in Changand Lee (2002).The dynamics of the manipulator consisting of boom,arm and bucket can be mathematically modeled asfollows:F Mll.l Vll；lGllFrll；l； A:1where F denotesthe 3 C2 1 vector of forcesacting on thepistons in the cylinders, and l isthe 3 C2 1 vector, eachelement of which represents the piston displacementrelative to the cylinder responsible for each link. Mll isthe 3 C2 3 inertial matrix, Vll；l isthe 3 C2 1 vector ofcentrifugal and Coriolisterms, Gll isthe 3 C2 1 vectorof gravity terms, and Frll；l isthe 3 C2 1 vectorconsisting of various friction terms.Fig. 18 shows the hydraulic actuator circuit of aHyundai Robex210LC-3. In Fig. 18, the ow frompump 1 isdivided and supplied to the boom cylinderand the bucket cylinder, via main valves； whereas pump2 supplies ow to the arm cylinder. Fig. 19 shows one setof a main valve and cylinder pertaining to a specic link.Now, consider the divide above, ignoring the com-pressibility of oil at junctions (a)(c) in Fig. 19. Then,from the continuity equation of ow at junctions(a)(c)and a linearized valve ow equation (Chang & Lee,2002), one can obtain linearized relationships of at eachlink asfollows:Kpa ai0 Kpa ci0 Kpb biKpb ciKpc aiKpc biKpc ci26643775PaiPbiPci26643775AailiC0 Ku aiuiC0 Qrm aiAbiliC0 Ku biuiC0 Qrm biQsec iC0 Ku ciuiC0 Kpc diPdiC0 Qrm ai26643775； A:2Fig. 18. Overall structure of the hydraulic circuit for boom, arm andbucket.Fig. 19. Detailed structure of the hydraulic circuit pertaining to a link(boom, arm and bucket).S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697711710where i 1；2；3 denote subscripts representing boom,arm and bucket, respectively； Aaiand Abithe areasoftwo sides of the piston in the cylinder； Qsec ithe owfrom the pump mainly for ith cylinder； uithe displace-ment of each spool, which is the input to the system；Kp:and Ku :denotesvalve ow gain and valve ow-pressure coefcient； Pai； Pbi； Pci； and Pdidenotepressures at junctions (a)(d), respectively. In addition,Qrm ai； Qr