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推土机的工作装置结构及液压系统设计,推土机,工作,装置,结构,液压,系统,设计
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.Automation in Construction 9 2000 421435rlocaterautconImpedance control of a hydraulically actuated robotic excavatorQ.P. Ha), Q.H. Nguyen, D.C. Rye, H.F. Durrant-WhyteAustralian Center for Field Robotics, The Uniersity of Sydney, J07, Sydney, 2006 NSW, AustraliaAbstractIn robotic excavation, hybrid positionrforce control has been proposed for bucket digging trajectory following. In hybridpositionrforce control, the control mode is required to switch between position- and force-control depending on whether thebucket is in free space or in contact with the soil during the process. Alternatively, impedance control can be applied suchthat one control mode is employed in both free and constrained motion. This paper presents a robust sliding controller thatimplements impedance control for a backhoe excavator. The control law consists of three components: an equivalent control,a switching control and a tuning control. Given an excavation task in world space, inverse kinematic and dynamic modelsare used to convert the task into a desired digging trajectory in joint space. The proposed controller is applied to providegood tracking performance with attenuated vibration at bucketsoil contact points. From the control signals and the jointangles of the excavator, the piston position and ram force of each hydraulic cylinder for the axis control of the boom, arm,and bucket can be determined. The problem is then how to find the control voltage applied to each servovalve to achieveforce and position tracking of each electrohydraulic system for the axis motion of the boom, arm, and bucket. With anobserver-based compensation for disturbance force including hydraulic friction, tracking of the piston ram force and positionis guaranteed using robust sliding control. High performance and strong robustness can be obtained as demonstrated bysimulation and experiments performed on a hydraulically actuated robotic excavator. The results obtained suggest that theproposed control technique can provide robust performance when employed in autonomous excavation with soil contactconsiderations. q2000 Elsevier Science B.V. All rights reserved.Keywords: Robotic excavator; Hybrid positionrforce control; Sliding controller1. IntroductionThe usual task of a backhoe excavator is to freeand remove material from its original location and totransfer it to another location by lowering the bucket,digging by dragging the bucket through the soil, then)Corresponding author. Department of Mechanics and Mecha-tronic Engineering, The University of Sydney, J07, 2006 NSW,Australia. Tel.: q 61-2-9351-3098; fax: q 61-2-9351-7474.E-mail address: quang.ha.au Q.P. Ha .lifting, slewing and dumping the bucket. In movingtowards automatic excavation, there is a need for thedevelopment of a controller that is robust to uncer-w xtainties associated with these operations1 . Forcontrol purposes, kinematic and dynamic models ofexcavators that assume the hydraulic actuators act asinfinitely powerful force sources are presented inwxRefs.24 . Position control with a conventionalproportional and derivative controller is used in Refs.wx4,5for simulation of the digging process withlimited soil interaction.Excavators are, however, subject to a wide varia-tion of soiltool interaction forces. When digging,0926-5805r00r$ - see front matter q2000 Elsevier Science B.V. All rights reserved.PII: S0926-5805 00 00056-X()Q.P. Ha et al.rAutomation in Construction 9 2000 421435422the bucket tip motion is effectively force-constrainedby the nonlinear constitutive equations of the envi-ronment, and by the hydraulic forces. Compliancecontrol approaches may therefore be considered tobe more suitable than position control for the shap-ing of excavator dynamics. Compliant motion con-trol can generally be classified into two broad classes:hybrid positionrforce control and interactive controlorimpedanceradmittancecontrol.Inhybridforcerposition control, the Cartesian space of theend-effector co-ordinates is decomposed into a posi-tion sub-space and a force sub-space. Separate posi-tion- and force-trajectory tracking objectives arespecified in each sub-space. Excessive force tran-sients may, however, occur at the instant of contactbetween the tool and the environment. Rather thantracking desired position and force trajectories, inter-active control seeks to regulate the relationship be-tween the end-effector position and the interactionforce. It is known that impedance control provides aunified approach to both unconstrained and con-w xstrained motion 6 . If hybrid positionrforce controlis adopted, the control mode should be switchedbetween position control and force control accordingto whether the excavator bucket is in free space or incontact with the soil during an excavation task.Impedance control is believed to be better suitedto excavation tasks in the sense that it can be appliedcontinuously to both free and constrained motionsw x1 . An impedance controller has recently been re-w xported for an excavator arm 7 . This paper proposesa robust sliding mode control technique to imple-ment impedance control for an excavator using gen-eralised excavator dynamics. The bucket tip is con-trolled to track a desired digging trajectory in thepresence of environment and system parameter un-certainties. In impedance control of hydraulic exca-vators, the piston position and the ram force of eachhydraulic cylinder for the axis control of the boom,arm, and bucket can be determined. The problem isthen how to find the control voltage applied to theservovalves to track these desired commands. Takinginto account friction and nonlinearities, a discontinu-ous observer is developed for estimating both pistondisplacement velocity and disturbance including loadforce and friction. With an observer-based compen-sation for the disturbance force, robust tracking thepiston ram force and position is guaranteed usingrobust sliding mode controllers for electrohydraulicsystems. The validity of the proposed method isverified through simulation and filed tests performedon the Komatsu PC-05 mini-excavator. The remain-der of this paper is organised as follows. Section 2 isdevoted to the derivation of the excavator dynamicmodel. The problem formulation and the develop-ment of impedance control for excavator dynamicsare presented in Section 3. The control of electrohy-draulic systems is addressed in Section 4. The hard-ware organisation for the robotic excavator is de-scribed in Section 5 together with computer simula-tions and experimental results. Finally, conclusionsare provided in Section 6.2. Excavator dynamicsThe equations of motion for a generic excavatorcan be derived by applying the EulerLagrangeequations to a Lagrangian energy function, or bywriting the NewtonEuler equations successively foreach link of the machine. In the latter approach, thedynamics of each link are described by equationsthat recurse through the link index. The driving jointtorques of the boom, arm, and bucket are generatedby the forces of the hydraulic ram actuators. Thetranslational and rotational motions of these links aredescribed by the dynamic model of the excavatorsystem. Dynamic models of excavators are presentedw xw xin Ref. 2 with refinements in Ref. 4 . Firstly, a?4Cartesian co-ordinate frame O x y zfixed to the0000excavator body is chosen. Other Cartesian co-ordinateframes are systematically assigned by applying thewxDenavit and Hartenberg procedure as in Refs. 24 .?4?4?4The framesO x y z ,O x y z ,O x y z ,111122223333?4andO x y zare respectively attached to the4444boom, arm, bucket, and bucket tip as seen in Fig. 1.Note that the movements of the excavator mecha-nism during digging usually occur in the verticalplane. It is therefore assumed that no boom-swingmotion occurs during excavation, so that the boom.swing angleuis therefore held constantus011during digging. The model equations can be writtenfor each link of the excavator by considering the linkas a rigid free body. By combining Newton andEuler equations for all links, the dynamical modelfor the excavator can be expressed concisely in a()Q.P. Ha et al.rAutomation in Construction 9 2000 421435423Fig. 1. Excavator joint variables.well-known form of manipulator equations of motionw x4 : Du uqCu,u uqBuqGu.sAuFyTF ,F,1. .LtnwxTwhereusuuuis the vector of measured234shaft angles:ufor the boom joint,ufor the arm23joint, andufor the bucket joint; Trepresents the4Lload torques as functions of the tangential and nor-mal components, F and F , of the soil reaction forcetnat the bucket, and F are the ram forces of thehydraulic actuators that produce the torques actingon the joint shafts. The tangential component, F,twhich is parallel to the digging direction, representsthe resistance to the digging of ground by excavatorsbucket teeth. This resistance is considered as the sumof soils resistance to cutting, the friction betweenthe bucket and the ground, and the resistance tomovement of the prism of soil and soil movement inthe bucket. The tangential component can be calcu-w xlated according to Ref. 8 asF sk bh,2 .t1wy2xwhere kis the specific digging force N m, and1h and b are respectively the thickness and width ofw xthe cut slice of soil m . The normal component, F ,nis calculated asF sCF ,3 .nt.wherecs 0.10.45 is a factor depending on thedigging angle, digging conditions, and the wear andw xtear of the cutting edge 8 . The determination of the .matrices of inertia, Du, of Coriolis and centripetal . .effects, Cu,u u, of gravity forces, Gu, and of .functions of the moment arms, Au, is comprehen-wxsively described in Refs. 24 . All entries in thesew xmatrices are given in Ref. 4 . The 3=1-matrix of .viscous friction Buis treated in this paper as asource of uncertainty. .In the digging plane, the Jacobian Judefined asxsJu u,4. .w xwcan be obtained from Ref.3 , where xs xz44xTurepresents the Cartesian co-ordinates and orien-O4.tation of the bucket tipOwith respect to4?4 .O x y z . Assuming that the Jacobian matrix J e0000 .is non-singular, Eq. 1 in joint space can be rewrit-ten in Cartesian space as:H x xqCx,x xqBx qGx. xxxyTsJAFyF ,5 .ewhereHsJyTDJy1,C sJyTCyDJy1J Jy1,.xG sJyTG,B sJyTB,6 .xxand F sJITTdenotes the generalised forces ofeL.interaction between the end-effector bucket tip and.the environment soil . They consist of digging forcesacting on the bucket with force entries for the co-.ordinatesx ,zand a torque entry around y .444The determination of the forward and inverse .y1 .kinematic relationships, xsLuandusLxw x .is detailed in Ref. 3 . As Eq. 5 has the form of thegeneralised robotic manipulator dynamics where x isa vector of the co-ordinates of the contact point ofthe manipulator with the environment, below and inthe next section, we will consider in general xgRnand ugRn. We assumeA1:HsHqAH,C sC qDC ,xxxG sG qDG ,AsAqDA,F sF qDF ,xxxeee7 . where matrices H, C , G , and A are known, F is toxxebe measured by force sensors such as load pins, andDH, DC , DG , DA, and DF are uncertainties.xxeDenoting friction and uncertainties byDf x,x,x sDHxqDC xqDG qB x. xxxyTqDF yJDAF,8 .e()Q.P. Ha et al.rAutomation in Construction 9 2000 421435424 .Eq. 5 can be rewritten asH x xqCx,x xqGx. xxsuyF yDf x,x,x ,9. . ewhereyTusJAF10.is the control input. .Remark 1. As Duis a 3=3-symmetric positive-definite matrix satisfying the skew symmetric prop-w x.erty 9 , for the nominal dynamics Df x,x,x s0 w .xof the excavator, H x -2Cx,xis also a skew-xsymmetric matrix, i.e.TxH x y2Cx,xxs0,;x.11.x3. Excavator dynamics impedance control3.1. Problem formulationOne of the excavating task elements is penetrationof the soil by an excavator bucket to follow apre-planned digging trajectory. During digging, threemain tangential resistance forces arise: the resistanceto soil cutting, the frictional force acting on thebucket surface in contact with the soil, and theresistance to movement of the prism of soil ahead ofand in the bucket. The magnitude of the diggingresistance forces depends on many factors such asthe digging angle, volume of the soil prism, volumeof material ripped into the bucket, and the specificresistance to cutting. These factors are generallyvariable and unavailable. Moreover, due to soil plas-ticity, spatial variation in soil properties, and poten-tial severe inhomogeneity of material under excava-tion, it is impossible to exactly define the forceneeded for certain digging conditions.The objective of impedance control is to establisha desired dynamical relationship between the end-ef-.fectorbucket tipposition and the contact force.This dynamical relationship is referred to as the .target impedance. Let xt be the desired trajectoryrof the end-effector. Typically, the target impedanceis chosen as a linear second-order system to mimicmass-spring-damper dynamics:Zs e s M s2qB sqKe .tPtttPsM e qB e qK e se ,12.tPtPtPFwhere s is the derivative operator and the constantpositive-definite n=n-matrices M , B and Karetttrespectively the matrices of inertia, damping andstiffness. The position error, e , and the force error,Pe , are defined asFe sx yx,e sF y yF,13.PrFre .where F t sM x qB x qK xis the forcertrtrtrset-point.The control problem is to asymptotically drive the.system state to implement the target impedance 12even in the presence of uncertainty. If the positionerror eapproaches zero, the force error ealsoPFapproaches zero and vice versa, according to a speci-fied dynamical relationship defined by the numerical.values of the matrices M , B and K in Eq. 12 . Intttsome contact tasks, the force set-point, F , will berspecified to be constant rather than time-varying.During free-space motion where there is no contactwith the environment, F syF s0, so etends toreP .2zero since Z s sM s qB sqKis stable. Thettttchoice of the matrices M , B and K will determinetttthe shape of the desired transient response of thesystem. When the end-effector contacts the environ-ment, the interaction is characterised by the target.impedance 12 , which results in a compromise be-tween the position error and the force error. If theend-effector position tracks the desired trajectory.xxthen the contact force follows the forcer.set-point yF F .er3.2. Controller deelopmentConsider a manipulator dynamical model of the . .form 5 with uncertainty satisfying condition 7 . Itwxis well known 10 that robustness is the most distin-guished feature of variable structure control withsliding mode. In this section, a robust sliding modecontroller will be developed for the manipulator . .dynamics5 . As Eq.5represents a 2n-dimen-()Q.P. Ha et al.rAutomation in Construction 9 2000 421435425sional system with an n-dimensionalcontrol input, asliding surface in the state space will be a manifoldwxwof dimension 2nynsn 10 . Let us define ss s1 . . .xTx , sx ,., sx, the sliding functions, as2nwxfollows 11 :ssye yMy1B e yMy1Ke dtHPttPttPqMy1e dtsxyx ,14.HtFswherex sx qMy1B e qMy1Ke dtyMy1e dt.HHsrttPttPtF15.The existence of a sliding mode, ss0, requires thatssye yMy1B e yMy1K e qMy1e s0.PttPttPtF16.It can be seen that once the system state is in the.sliding mode associated with Eq. 14 , condition 16.guarantees that the target impedance 12 is reached. .Thus, in the sliding mode s x s0, is1,2,.,n ,i .the force error tends to zero. The magnitudes s xi.is1,2,.,nrepresent then the deviations of thesystem state from the sliding surface. We assumefurther that:.A2: Each entry of the uncertainty Df x,x,xis bounded:D fx,x,xFb,. ii; x,x,xis1,2,.,n .17. Let us now define the control inputusuyQsgn s ,18 .whereusHx qC x qG qF ,19.sxsxeTQs Q sgn s,.,Q sgn s,.11nnQ )bis1,2,.,n .20.ii.Remark 2. The control law18consists of ponents. The component19 , calculated withthe nominal system dynamic model, is called theequialent control. The other component, with the.discontinuous gain given in Eq.20 , is called theswitching control.( )Theorem 1. Consider the system of Eq. 5 associ-()ated with sliding functions14and the target()impedance 12 . If the assumptions A1 and A2 are()satisfied, and the control law 18 is employed, then()the impedance error 14 asymptotically conerges to zero 9 .Implementation implies that sufficiently largeswitching gains, Q , are available. Large values ofiQ will, however, tend to excite chattering. To accel-ierate the reaching phase and to reduce chattering, the.control law 18 is added by a tuning component:usuyQsgn s yKs,21 .whereKsdiag Ksis1,2,.,n .22.iiEmploying the fuzzy tuning technique proposed inwx.Ref. 12 , the expressions for Ks )0 are choseniias:K sK1yexp y s rd.ii maxiiis1,2,.,n ,23.where Kanddare some positive constants.i maxi( )Theorem 2. Consider the system of Eq. 5 associ-()ated with the sliding functions 14 and the target()impedance 12 . If the assumptions A1 and A2 are()satisfied, and the control law 21 is employed then()the impedance error 14 asymptotically conerges to zero 9 .Remark 3. The switching component is for ensuringrobust stability only. It can be omitted in practice.Note that from the geometry of the excavator,there exists a trigonometric mapping between eachjoint angleuand the corresponding linear displace-iment y of each hydraulic cylinder piston, is2,3,4iwx.13 . Using this relationship and Eq. 10 , the cylin-wxTder positions ys yyyand the ram forces F234of the hydraulic actuators can be determined. Theyare considered as the control references to the exca-vator hydraulic systems. The following section isdevoted to the control of electrohydraulic systems inorder to track these desired commands.()Q.P. Ha et al.rAutomation in Construction 9 2000 4214354264. Electrohydraulic systems control4.1. Hydraulic modelling.The control 10 requires the ram force generatedat each cylinder of the excavator arms follow adesired function of time when executing diggingtasks in impedance control. Nonlinear effects occur-ring during the toolsoil interaction, and in thehydraulic system itself, complicate the control strat-egy requirements. It is known that gravitational andfriction between the piston and cylinder should becompensated for to achieve high performance ofheavy-duty hydraulic machines, such as excavatorswx13 . Furthermore, oil viscosity, oil flow through thehydraulic servovalve, and variable loading, will causehydraulic control systems to suffer from highly non-linear time-variant dynamics, load sensitivity, andwxparameter uncertainty 14 . Thus, these factors haveto be taken into account in servo hydraulic modellingand control. The hydraulic actuators incorporated inthe blade, boom swing, boom, arm, and bucketattachments of the excavator are axial hydrauliccylinders. The flow of hydraulic oil to the cylinder isregulated by a direct drive servovalve with an elec-trically controlled closed loop that controls spoolposition. This system could be generally describedby a six-order differential equation. For simplicity,the following linear expression can be used withlittle loss of accuracy for frequencies up to 200 Hz:x sK u ,24.vvvwhere xis the spool valve displacement and uisvvthe valve input voltage. Thus, a nonlinear state modelwxcan be obtained as in Ref. 15 , based on the relation-ship between the valve displacement xand thevfluid flow to the head side Q and from the rod side1Q , the continuity of flow in the cylinder, and the2force balance equation for the piston:ys1wxsA P yA P yF1122LMbewP sKus xP yP.(1vvS1A yqV1hx4qs yxPyCP yPyA .(v1ip121ybew?P sKus xP.(2vv2ALyy qV.2hx4qs yxP yPqCP yPyA ,.(vS2ip12225.w xwhere y is the piston displacementm , is itsvelocity, Pand Pare respectively the fluid pres-12wxsures at the head and rod sides of the cylinder Pa ,w xuis the control input V , Fis a load disturbancevLw xon cylinder including external forces and friction N ,wxPis the supply pressure Pa , K is a fixed constant,SwxM is the equivalent moving mass kg , Cis theipinternal leakage coefficient, A and Aare respec-12w2xtively the area of the piston head and rod m , Vishthe hose volume between the servovalve and thew3x.cylinderm , and s xis a switching functionvdepending on the extension or retraction of the pis-wxton 15 :1x G0vs xs.26.v0x -0vBy assigning the ram forceFsA P yA P27.1122.as a state variable, an equivalent form of Eq. 25can be obtained as followsys1wxsFyFLMFsa qa FqaP qb u12131411vP sa qaFqaP qb u28.122232412vwhereA2bA2b1e2ea syq,12/A yqVALyy qV.1h2hA rA112a syqbC ,13eip/A yqVALyy qV.1h2ha saA yA,.141321A2b1rA1e2asy,asybC ,2223eipA yqVA yqV1h1h()Q.P. Ha et al.rAutomation in Construction 9 2000 421435427asaA yA,.242321AbK1eb ss xP yP qs yxP.(1vS1v1A yqV1hAbK2eqALyy qV.2h= s xP qs yxP yP,.(v2vS2AbK1eb ss xP yP qs yxP.(2vS1v1A yqV1h.In Eq.28 , information of the states can beobtained from the piston position and pressure trans-ducers. Our experiments indicated, however, thatdifferential pressure readings would be inadequatewhen used to represent the external force F exertedon the hydraulic piston. Taking static friction intoaccount, load pins inserted at the cylinder rod eyesare used for force measurements in our mini-excava-tor for a more accurate representation of the force F.The piston velocity, , and load disturbance FwillLbe estimated using a discontinuous observer as de-scribed in the next section. Numerical values ofparameters for the excavator arm-axis hydraulic con-wxtrol system can be found in Ref. 16 .4.2. Discontinuous obserer for disturbance estima-tionIn tracking control of electrohydraulic manipula-tors, there is a need to compensate for the influenceof external disturbance such as the acting forcecoming from outside the hydraulic cylinder, gravity,and friction. Hydraulic servos are affected by avariety of frictional forces including linear and non-linear viscous frictions and Coulomb friction. Anwxadaptive nonlinear observer 17 is proposed to com-pensate for Coulomb friction in tracking control of amanipulator actuated by a linear hydraulic cylinder.Simultaneous estimation of velocity and Coulombwxfriction 15 is obtained via the observer where ve-locity is estimated independently using a first orderlow-pass filtered differentiator and static friction istaken into consideration using an approximationmodel. Variable structure control theory has broughta new insight to nonlinear observers in terms of highwxinsensitivity to uncertainties 18 . It is from this meritthat a discontinuous observer is developed to providerobust estimate of the piston velocity and disturbanceforce. Assuming that the disturbance force is slowlytime-varying, the state model for can be written withy as the output,swyF ,F s0,ys,29.LLwhere w is the acting force measured by a forcewxsensor. Note that in Ref. 17 , w denotes the forcedue to all sources other than friction. In our applica-.tion, the ram force 27 , measured from load pins, isused for estimating external disturbance F .Lwx.The Utkin observer19for Eq.29can bedesigned to have the schematic diagram shown in.Fig. 2, wheressmsgn ewithm)0, e syyyyy .is the output error, and the circumflexdenotes theestimates. The convergence rate of the proposedobserver depends on a proper choice of the observergains l , l , or, in other words, on selecting eigen-12values of the following observer characteristic equa-wxtion 20 :s2yl sql s0.30.12In order to reduce chattering associated with thesliding motion of the output error, the signum func-.tionssmsgn eis replaced by a sigmoidal oneyFig. 2. Observer schematic diagram.()Q.P. Ha et al.rAutomation in Construction 9 2000 421435428resulted from a fuzzy reasoning technique presentedwxin Ref. 21 :ssmtanh e rg31.yewheregis some positive constant.e4.3. Hydraulic controller designVarious advanced control methods addressing theelectrohydraulic servo control problem have beenreported in literature. Among these methods, variablestructure control with a sliding mode has been pro-wxmoted by many authors 22,23 . The reason is mainlybecause of its robustness to uncertainties. This sec-tion describes the design of a chattering-attenuatedsliding mode controller that provides robust trackingof the desired ram force F and piston position yrrrequired to implement impedance control of the ex-cavator bucket tip in the presence of soil uncertain-ties. Let us define the control errorse syyy ,e sy ,e sFyF ,1r2r3re sP yP,32.411rwhere and Pare respectively the desired pistonr1rvelocity and head-side fluid pressure. Assuming thatestimates of the piston velocity and disturbance forceobtained from the observer are adequately accurate,.from Eqs.28and32 , the following nonlinearwdynamics can be derived for the error vector es e1xTeee:234esA x,xqB uqf ,33.rwherey ser2FyFe yFL3Ly srMMA x,xs,.ra qa FqaP yF1213141r?0a qaFqaP yP22232411r00Bs,34.b1? 0b2and f is an uncertain source taking into accountparameter variations and modelling mismatch.A3: It is assumed that f is unknown but boundedby a known positive functionr:sw f Fr.35.swLet us now define the following switching func-tionSsCese qc e qc e ,36.32211wx.where Cs cc1 0 and cis1,2 are positive12iconstants to be specified according to the desired.dynamics of the closed-loop system. From Eq. 34 ,it can found that the sliding mode Ss0 is associatedwith the eigenvalues of the following characteristicequation:12s qc sqc s0.37.21MRemark 4. With the choice of the sliding function.36 , the undesirable influence of the pressure errorecan be excluded from the servovalve control4voltage u .vA necessary condition for the state trajectory tostay on the sliding modes Ss0 is Ss0. The fol-lowing control lawu suququ38.eqswftwxis proposed 24 . It consists of three components: anequivalent control uto assign desired dynamics toeqthe closed-loop system, a switching control utoswguarantee a sliding mode, and a fuzzy control utoftenhance fast tracking and to attenuate chattering. Theequivalent control can be obtained from the nominal .system parameters, denoted with bars, as followswx12,24 :y1usy CBCA.eqc2y1wsybc qaqqa.111213/M=c2xFqaP yryF,39.141LM()Q.P. Ha et al.rAutomation in Construction 9 2000 421435429Fig. 3. Teleoperated robotic excavation.where rsc x qc x qF and the coefficients c1r2rri.is1,2 are chosen according to the desired roots of.the characteristic equationEq.37 . In order toforce the system trajectory back to the sliding modein the presence of uncertainties and to smooth outthe control action, the switching and fuzzy controlwx24 are proposed:usyrsgnw,40.swswu syutanhwrg,41.ftfmtwherewsSCBsSb , and uandgare some1fmtpositive constants.Fig. 4. Electrohydraulic servovalves.Remark 5. The law39can be considered as acombination of state feedback and feedforward con-Fig. 5. Digging responses.()Q.P. Ha et al.rAutomation in Construction 9 2000 421435430trol. Experience shows that the observer dynamics.30 should be chosen three to five times as faster as.the system dynamics 37 .5. Simulation and field test results5.1. The robotic mini-excaatorFig. 3 shows the robotic excavator used for exper-iments. The machine, originally a 1.5-tonne KomatsuPC05-7 mini-excavator, has been retrofitted with.electrohydraulic servovalvesFig. 4 , spool valveposition and pressure transducers, 12-bit joint angleencoders, and digital axis controllers. Additional an-cillary equipment needed to support the servovalvesis also fitted: an accumulator with unloading valve,solenoid check valves and an air radiator. The ma-chine has eight hydraulic axes: two rubber tracks,cab slew, arm swing, boom, arm, bucket, and a smallblade for back filling.Closed-loop control of all axes is achieved byfour proprietary programmable digital controllers thatare commanded and coordinated by an industrial.IBM-compatible personal computerPC . The PCcommunicates with the digital controllers through a. .control area networkCAN bus, and issues trackvelocity commands and axis position set-points. Atthis time, system motion commands are inputted viaa joystick that is interfaced to the PC.5.2. Simulation resultsThe matrices of inertia, damping and stiffness of.the target impedance 12 are chosen respectively asM s100I , B s5000I , K s10000I , where It3t3t33is the 3=3 identity matrix. The controller parame-.ters for Eq. 21 are Qs0, Ks1000,ds20i maxi.is1,2,3 . The soil model used in the simulation issandy loam with characteristic parameters givenwxin Ref. 25 . For the multi-axis control of the excava-.tor arm, the desired eigenvalues of Eq.37are?4chosen at y10, y10 , which gives a settling timeof 0.4 s and no overshoot. The controller parametersfor electrohydraulic systems are: l sy70, l s122500,ms5,gs0.01, c s100, c s20,rs3,e12swus3, andgs0.1.fmtThe proposed impedance controller is applied to aposition tracking process when the bucket is in freespace and in force-constrained motion. A desiredFig. 6. Responses to a planned digging trajectory.()Q.P. Ha et al.rAutomation in Construction 9 2000 421435431Fig. 7. Rotary cutting tracking responses.trajectory is first specified in Cartesian space. Thew xinverse kinematic equations of the excavator 3 areused to calculate the desired joint trajectories. Con-sider first the soil reaction during the digging processwhen the bucket contacts the soil and begins topenetrate the surface. Fig. 5 shows the load force andbucket ram force; the arm joint angle; and the xztrajectory. As changes to the bucketsoil contactconditions may result in tracking oscillations, it seemsprudent to choose large values for the dampingmatrix Bin the target impedance. Good trackingtperformance is exhibited during both free motionand constrained motion. Oscillations and errors inthe joint angles at the instant of contact with the soilare significantly suppressed. Now let us simulate thetask of collecting a bucket of sandy loam bycontrolling the excavator in such a way that thebucket follows a given trajectory in a predefinedposture. The trajectory template used in the simula-tion can be considered as a penetratedragcurlmotion. Fig. 6 depicts the responses of the bucketram force and load force, the arm position, and theplanned trajectory and bucket trajectory. As pre-w xsented in Ref.2 , significant vibrations in jointangular positions, velocities and accelerations occurat the points where the bucket begins to penetrate thesurface or when it is removed from the surface whenemploying a pure position controller. With the use ofan impedance controller, it is seen that these oscilla-tions can be substantially reduced.In impedance control of the robotic excavator, ourobjectives focus on the execution of common exca-vation tasks such as digging building footings, orloading haul trucks from an open-cut mine bench.For our experimental mini-excavator, the diggingforce for a cut depth of about 0.2 m into sandywxloam can be estimated 25 to be about 3.4 kN.Fig. 8. Hardware organisation.()Q.P. Ha et al.rAutomation in Construction 9 2000 421435432Fig. 9. Experimental free motion responses.When rotary cutting with the excavator arm, the armcylinder piston has to track square motion sequencesbetween 0.07 and 0.17 m. The cylinder position andthe servovalve control voltage responses of the armhydraulic system are shown in Fig. 7. It can be seenthat robust tracking and a significant reduction ofchattering are obtained with the proposed controlscheme.It should be noted that here, the force exerted bythe excavator on the soil is controlled not by regulat-ing the position error nor the force error but through.the imposed relationshipEq.12between them.Fig. 10. Experimental tracking responses when digging sandyloam.The simulations illustrate that the proposed controlschemes can perform adequately in an autonomousexcavating task. . .Fig. 11. Loading a truck with impedance control: a bucket, b .arm, and c boom joint angles.()Q.P. Ha et al.rAutomation in Construction 9 2000 421435433Fig. 12. Bucket ram force during a truck loading task.5.3. Experimental resultsExperiments have been performed on the Ko-matsu PC05-7 mini-excavator shown in Fig. 3. Thehardware organisation of electrohydraulic systemsfor axis control of to implement impedance controlfor excavator dynamics is presented in Fig. 8. Dataacquisition and control algorithms are written inCqq and executed under the Windows NT operat-ing system. The sampling time is chosen to be 0.010w xs . The CAN bus communication rate is set at 250wxkBitrs .The arm position, valve voltage, and load pres-sure responses in free space with the use of theproposed sliding mode controller are depicted in Fig.9. Very accurate tracking can be observed. Todemonstrate robust tracking in constrained motions,Fig. 10 shows the arm position and valve controlvoltage responses when digging sandy loam ac-.cording to a prescribed target impedance 12 of thebucket tip.Let us now consider the excavation task of load-.ing a truck. With the target impedance12 , thereference and actual responses the boom, arm, andbucket joint positions are respectively shown in Fig.11. The ram force exerting at the bucket joint duringthe excavation task is represented in Fig. 12 for oneloading pass, i.e. digging and unloading the soil. Theresults obtained verified the validity and feasibilityof our proposed robust control schemes for au-tonomous operations of the robotic excavator takinginto account toolsoil interactions.6. ConclusionIn moving towards autonomous excavation in themining, earth moving and construction industries, wehave proposed a robust sliding mode controller forimpedance control of excavators dealing with uncer-tainties in its dynamical model, friction and bucketsoil interactions. The controller design consists of the()Q.P. Ha et al.rAutomation in Construction 9 2000 421435434choice of a target impedance, and the determinationof an equivalent control, a switching control, and atuning control. The control outputs and the jointangles are then converted to the commands, namelythe desired ram force and piston position, to the axiscontrol of electrohydraulic servo systems of the ex-cavator. Sliding mode control incorporating a fuzzytuning approach has been successfully implementedfor the control of the ram force and the cylinderposition the excavator hydraulic actuators. A 1.5-tonne robotic mini-excavator is simulated and exper-imentally tested to verify the validity of the proposedcontrol schemes. Given a desired trajectory, excavat-ing tasks such as digging and loading exhibit goodperformance. Vibrations in joint angular positions,velocities and accelerations due to the contact be-tween the bucket and the soil can be significantlyreduced using the proposed controllers. High perfor-mance and strong robustness of the excavator elec-trohydraulic servo systems are achieved in simula-tion and field tests. The results obtained demonstratethe feasibility and efficacy of the proposed techniquefor the control of excavator dynamics and its hy-draulic actuators in execution of robotic excavationtasks with soil contact considerations.AcknowledgementsThe support of the Australian Research Council,of NS Komatsu, and of the Co-operative ResearchCentre for Mining Technology and Equipment isgratefully acknowledged.Referencesw x1 A.T. Le, Q.H. Nguyen, Q.P. Ha, D.C. Rye, H.F. Durrant-Whyte, M. Stevens, V. Boget, Towards autonomous excava-tion, in: Proceedings of the International Conference on Fieldand
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