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LINEAR, NONLINEAR AND CLASSICAL CONTROL OF A1/5TH SCALE AUTOMATED EXCAVATORE. Sidiropoulou, E. M. Shaban, C. J. Taylor, W. Tych, A. ChotaiEngineering Department, Lancaster University, Lancaster, UK, c.taylorlancaster.ac.ukEnvironmental Science Department, Lancaster University, Lancaster, UKKeywords: Identification; model-based control; proportional-integral-plus control; state dependent parameter model.AbstractThis paper investigatesvariouscontrol systems fora laboratoryrobot arm, representing a scale model of an autonomous exca-vator. The robot arm has been developed at Lancaster Univer-sity for research and teaching in mechatronics. The paper con-siders the application of both classical and modernapproaches,including: Proportional-Integral (PI) control tuned by conven-tional Ziegler-Nichols rules; linear Proportional-Integral-Plus(PIP) control, which can be interpreted as one logical exten-sion of the conventional PI approach; and a novel nonlinearPIP design based on a quasi-linear model structure, in whichthe parameters vary as a function of the state variables. Thepaper considers the pragmatic balance required in this context,between design and implementational complexity and the po-tential for improved closed-loop performance.1IntroductionConstruction is of prime economic significance to many indus-trial sectors. Intense competition, shortfall of skilled labourand technological advances are the forces behind rapid changein the construction industry and one motivation for automa-tion 1. Examples of excavationbased operationsinclude gen-eral earthmoving,digging and sheet-piling. On a smaller scale,trenching and footing formation require precisely controlledexcavation. Fullorpartialautomationcanprovidebenefitssuchas reduced dependence on operator skill and a lower operatorwork load, both of which are likely to contribute to improve-ments in consistency and quality.However, a persistent stumbling block for developers is theachievement of adequate fast movement under automatic con-trol. Here, a key research problem is to obtain a computercontrolled response time that improves on that of a skilled hu-man operator. This presents the designer with a difficult chal-lenge, which researchers are addressing using a wide range ofapproaches; see e.g. 2, 3, 4.This paper considers a laboratory robot arm, a 1/5th scale rep-resentation of the more widely known Lancaster UniversityComputerised Intelligent Excavator (LUCIE), which has beendeveloped to dig trenches on a construction site 4, 5. De-spite its smaller size and light weight, the 1/5th model has sim-ilar kinematic and dynamic properties to LUCIE and so pro-vides a valuable test bed for the development of new controlstrategies. In this regard, the paper considers both classicaland modern approaches, including: Proportional-Integral (PI)control tuned by conventional Ziegler-Nichols rules; linearProportional-Integral-Plus (PIP) control, which can be inter-preted as one logical extension of the conventional PI ap-proach 6, 7; and a novel nonlinear PIP design based on aquasi-linear model structure in which the parameters vary asa function of the state variables 8.Further to this research, the paper briefly considers utilisationof the robot arm as a tool for learning and teaching in mecha-tronics at Lancaster University. In fact, the laboratory demon-strator provides numerous learning opportunities and individ-ual research projects, for both undergraduate and postgraduatestudents. Development of the complete trench digging systemrequiresa thoroughknowledgeofa widerangeoftechnologies,including sensors, actuators, computing hardware, electronics,hydraulics, mechanics and intelligent control.Here, control system design requires a hierarchical approachwith high-level rules for determining the appropriate end-effector trajectory, so as to dig a trench of specified dimen-sions. In practice, the controller should also include modulesfor safety and for handling obstructions in the soil 4. Finally,the high-level algorithm is coupled with appropriate low-levelcontrol of each joint, which is the focus of the present paper.2HardwareThe robot arm has a similar arrangement to LUCIE 4, 5, ex-ceptthatthislaboratory1/5thscale modelisattachedtoawork-bench and the bucket digs in a sandpit. As illustrated in Fig. 1,thearmconsistsoffourjoints,includingtheboom,dipper,slewand bucket angles. Three of these are actuated by hydrauliccylinders, with just the slew joint based on a hydraulic rotaryactuator with a reduction gearbox: see 9 for details.The velocity of the joints is controlled by means of the appliedvoltage signal. Therefore, the whole rig has been supportedby multiple I/O asynchronousreal-time control systems, whichallow for multitasking processes via modularisation of codewritten in Turbo C+R?. The computer hardware is an AMD-K6/PR2-166 MHz personal computer with 96 MB RAM.The joint angles are measured directly by mounting rotary po-tentiometers concentric with each joint pivot. The output sig-nal from each potentiometer is transmitted with an earth lineto minimise signal distortion due to ambient electrical noise.Figure 1: Schematic diagram of the laboratory excavator showing the four controlled joints.These signals are routed to high linearity instrumentation am-plifiers within the card rack for conditioningbefore forwardingto the A/D converter. Here, the rangeof the input signal just af-ter conditioning does not exceed 5 volts. This A/D converteris a high performance 16 channel multiplexed successive ap-proximation convertercapable of 12 bit conversion in less than25 micro seconds. At present only eight available channels arebeing used. In the future, therefore, there would be no prob-lem for incorporating additional sensors into the system; e.g. acamera for detecting obstacles, to be used as part of the higherlevel control system, or force sensors.Valve calibration is essential to provide the arm joints withmeaningful input values. This calibration is based on normal-izing the input voltage of each joint into input demands, whichrange from -1000 for the highest possible downward velocityto +1000for the highest possible upwardvelocity of each joint.Here, an input demand of zero corresponds to no movement.Note that, without such valve calibration, the arm will gradu-ally slack down because of the payload carried by each joint.In open-loop mode, the arm is manually driven to dig thetrench, with the operator using two analogue joysticks, eachwith two-degrees of freedom. The first joystick is used to drivethe boom and slew joints while the other is used to move thedipper and bucket joints. In this manner, a skillful operatormoves the four joints simultaneously to perform the task. Bycontrast, the objective here is to design a computer controlledsystem to automatically dig without human intervention.3KinematicsThe objective of the kinematic equations is to allow for con-trol of both the position and orientation of the bucket in 3-dimensional space.In this case, the tool-tip can be pro-grammed to follow the planned trajectory, whilst the bucketangleis separatelyadjustedtocollectorreleasesand. Inthisre-gard, Fig. 1 shows the laboratoryexcavator and its dimensions,i.e. i(joint angles) and li(link lengths), where i = 1,2,3,4for the boom, dipper, bucket and slew respectively.Kinematic analysis of any manipulator usually requires devel-opment of the homogeneous transformation matrix mappingthe tool configuration of the arm. This is used to find the po-sition, orientation, velocity and acceleration of the bucket withrespect to the reference coordinate system, given the joint vari-able vectors 10. Such analysis is typically based on the well-known Denavit-Hartenberg convention, which is mainly usedfor robot manipulators consisting of an open chain, in whicheach joint has one-degree of freedom, as is the case here 9.3.1Inverse kinematicsGiven X,Y,Z from the trajectory planning routine, i.e. theposition of the end effector using a coordinate system origi-nating at the workbench, together with the orientation of thebucket = 1+2+3, the following inversekinematic algo-rithm is derived by Shaban 9. Here Ciand Sidenotes cos(i)and sin(i) respectively, whilst C123= cos(1+ 2+ 3).X=X l4C4C4 l3C123(1)Y=Y l3S123(2)1=arctan?(l1+ l2C2)Y l2S2X(l1+ l2C2)X l2S2Y?(3)2=arccos?X2+Y2 l21 l222l1l2?(4)3= 1 2(5)4=arctan?ZX?(6)3.2Trajectory planningExcavation of a trench requires both continuous path (CP)motion during the digging operation and a more primitivepoint-to-point (PTP) motion when the bucket is moved outof the trench for discharging. In particular, each digging cyclecan be divided into four distinct stages, as follows: positioningthe bucket to penetrate the soil (PTP); the digging process ina horizontal straight line along the specified void length (CP);Figure 2: Trajectory planning for the laboratory excavator.picking up the collected sand from the void to the dischargeside (PTP); discharging the sand (CP).For the present example, the CP trajectory can be traversed ata constant speed. Suppose v0and vfdenote, respectively, theinitial and final position vector for the end-effectorand that themovement is required to be carried out in T seconds. In thiscase, the uniform straight-line trajectory for the tool-tip is,v = (1 St)v0+ Stvf0 t T(7)Here, Stis a differentiable speed distribution function, whereS0= 0 and ST= 1. Typically, the speed profileStfirst rampsup at a constant acceleration, before proceeding at a constantspeed and finally ramping down to zero at a constant deceler-ation. In the case of uniform straight-line motion, the speedprofile will take the formSt= 1/T. By integrating, the speeddistribution function will be St= t/T.For this particular application, the kinematic constraints of thelaboratoryexcavatorallow for digging a trench with length anddepth not exceeding 600 mm and 150 mm, respectively. Fig. 2shows one complete digging cycle, illustrating the proposedpath for the bucket. Note that each digging path is followed bypicking up the soil to the point (270,150,0) with an orien-tation of 180 degrees using PTP motion. This step is followedby another PTP motion to position the bucket inside the dis-charging area at coordinate (100,100,400). The last stepin the digging cycle is the discharging process which finishesat (600,100,700)with an orientation of -30 degrees.4Teaching and learningOne of the most important features of engineering education isthe combination of theoretical knowledge and practical expe-rience. Laboratory experiments, therefore, play an importantrole in supporting student learning. However, there are severalfactors that often prevent students from having access to suchlearning-by-doinginteractionwith roboticsystems. Thesein-clude their high cost, fragility and the necessary provision ofskilled technical support. Nonetheless, the utilization of robotspotentially offers an excellent basis for teaching in a number ofdifferent engineering disciplines, including mechanical, elec-trical, control and computer engineering; e.g. 11, 12, 13, 14.Robotsprovideafascinatingtoolforthedemonstrationofbasicengineering problems and they also facilitate the developmentof skills in creativity, teamwork, engineering design, systemsintegration and problem solving.In this regard, the 1/5th scale representation of LUCIE pro-vides for the support of research and teaching in mechatronicsat Lancaster University. It is a test bed for various approachesto signal processingandreal-time control;and providesnumer-ous learning opportunities and individual projects for both un-dergraduate and postgraduate research students. For example,since only a few minutes are needed to collect experimentaldata in open-loop mode, the robot arm provides a good labo-ratory example for demonstrating contrasting mechanistic anddata-based approaches to system identification.With regards to control system design, various classical andmodern approaches are feasible.However, the present au-thors believe that PIP control offers an insightful introductionto modern control theory for students. Here, non-minimal statespace (NMSS) models are formulated so that full state vari-able feedback control can be implemented directly from themeasured input and output signals of the controlled process,without resort to the design and implementation of a determin-istic state reconstructor or a stochastic Kalman Filter 6, 7.Indeed, a MEng / MSc module in Intelligent Control taught inthe Department covers all these areas, utilising the robot armas a design example.5Control methodologyThe benchmark PID controller for each joint is based onthe well known Ziegler-Nichols methodology. The system isplaced under proportional control and taken to the limit of sta-bility by increasing the gain until permanent oscillations areachieved. The ultimate gain obtained in this manner is subse-quently used to determine the control gains. An alternative ap-proach using a Nichols chart to obtain specified gain and phasemargins is described by 15.Linear PIP control is a model-based approach with a similarstructure to PID control, with additionaldynamic feedbackandinput compensators introduced when the process has secondorder or higher dynamics, or pure time delays greater than onesample interval. In contrast to classical methods, however, PIPdesign exploits the power of State Variable Feedback (SVF)methods, where the vagaries of manual tuning are replaced bypole assignment or Linear Quadratic (LQ) design 6, 7.Finally, a number of recent publications describe an approachfor nonlinear PIP control based on the identification of the fol-lowing state dependent parameter (SDP) model 8,yk= wTkpk(8)where,wTk=?yk1yknuk1ukm?pk=?p1,kp2,k?Tp1,k=?a1kank?p2,k=?b1kbmk?Here ykand ukare the output and input variables respectively,while aik(i = 1,2,.,n) and bjk(j = 1,.,m)are state dependent parameters. The latter are assumed to befunctions of a non-minimalstate vector Tk. For SDP-PIP con-trol system design, it is usually sufficient to limit the model (8)to the case that Tk= wTk. The NMSS representation of (8) is,xk+1=Fkxk+ gkuk+ dyd,k(9)yk=hxkwhere the non-minimal state vector is defined,xk=?ykykn+1uk1ukm+1zk?Tand zk= zk1+ yd,k yk is the integral-of-error betweenthe command input yd,kand the output yk. Inherent type 1servomechanism performance is introduced by means of thisintegral-of-error state. For brevity, Fk,gk,d,h are omittedhere but are defined by e.g. 9, 16.The state variable feedback control algorithm uk= lkxkissubsequently defined by,lk=?f0,k.fn1,kg1,k.gm1,kkI,k?where lkis the control gain vector obtained at each samplinginstant by either pole assignment or optimisation of a LinearQuadratic (LQ) cost function. With regard to the latter ap-proach, the present research uses a frozen-parameter systemdefined as a sample member of the family of NMSS modelsFk,gk,d,h to define the P matrix 9, with the discrete-timealgebraic Riccatti equation only used to update lkat each sam-plinginstant. Finally,notethatwhiletheNMSS/PIP linearcon-trollability conditions are developed by 6, derivation of thecomplete controllability and stability results for the nonlinearSDP system is the subject on-going research by the authors.6Control designFor linear PIP design, open-loop experiments are first con-ducted for a range of applied voltages and initial conditions, allbased on a sampling rate of 0.11seconds. In this case, the Sim-plified Refined Instrumental Variable (SRIV) algorithm 17,suggests that a first order linear model with samples timedelay, i.e. yk= a1yk1+ buk, provides an approximaterepresentation of each joint. Here ykis the joint angle and ukis a scaled voltage in the range 1000, while a1,b are timeinvariant parameters. Note that the arm essentially acts as anintegrator, since the normalised voltage has been calibrated sothat there is no movement when uk= 0. In fact, a1= 11000800600400200020040060080010000.0050.010.0150.020.0250.030.0350.04ParameterScaled voltageFigure 3: Variation of bagainst input demand for the boom.is fixed a priori, so that only the numerator parameter bisestimated in practice for linear PIP design.With = 1, the dipper and bucket joints appear relativelystraightforward to control using linear PIP methods. In thiscase, the algorithm reduces to a PI structure 6, hence the im-plementationresults are similar to the PI algorithm tuned usingclassical frequencymethods. As would be expected, the differ-ence between the classical and PIP methods for these joints isqualitative. Such differences relate only to the relative ease oftuning the algorithm to meet the stated control objectives.By contrast, with = 2, the slew and boom joints are bettercontrolledusing PIP methodssince (as shownin numerousear-lier publications) the latter automatically handles the increasedtime delay 9. Of course, an alternative solution to this prob-lem would be to introduce a Smith Predictor into the PI controlstructure. The authors are presently investigating the relativerobustness of such an approach in comparison to PIP methods.However, further analysis of the open-loop data reveals limi-tations in the linear model above. In particular, the value ofbchanges by a factor of 10 or more, depending on the ap-plied voltage used, as illustrated in Fig. 3 for the case of theboom. Here, numerous experiments are conducted for a rangeof applied voltages and, in each case, SRIV methods used toestimate linear models. Fig. 3 illustrates these estimates of bplotted against the magnitude of the step input (the solid tracerepresents a straightforward polynomial fit).In fact, SDP analysis suggests that a more appropriate modelfor the boom takes the form of equation (8) with,wTk=?yk1uk1uk2?pk=?a1k0b2k?T(10)where,a1k=0.238 106u2k2 1b2k=5.8459 106uk2+ 0.0189880901001101201301401502002040608090100110120130140150100050005001000Figure 4: Top: linear PIP (thin trace), nonlinear SDP-PIP(thick) and command input (dashed) for the boom angle, plot-ted against sample number. Bottom: equivalent control inputs.The associated SDP-PIP control algorithm takes the form,uk= ?f0,kg1,kkI,k?ykuk1zk?T(11)where the gains f0,k, g1,kand kI,kare updated at each sam-pling instant in the manner of a scheduled controller. Full de-tails of this approach and the equivalent SDP-PIP algorithmsfor the dipper, bucket and slew joints are given by Shaban 9.7ImplementationTypical implementation results for the boom arm are illus-trated in Fig. 4, where it is clear that the SDP-PIP algorithmis more robust than the fixed gain, linear PIP algorithm (orequivalentclassical PIcontroller)tolargestepsinthecommandlevel. Furthermore, the nonlinear approach yields a consider-ably smoother control input signal.Note that the linear and nonlinear controllers are designed toyield a similar speed of response in the theoretical case, i.e. thedifferences seen in Fig. 4 are due to the variation in b2(Fig. 3)which is only taken account of in the SDP-PIP case. It shouldpointed out that the response time for this example has beendeliberately increased to the practical limit of robust linear PIPdesign, in order to emphasis these differences.Fig. 5 shows controlof the dipperarm fora similar experiment.Although the differences between the linear and nonlinear ap-proachesare oftenrelativelysmall when each joint is examinedin isolation for movement in air, as in Fig. 5, such differencesare multiplied up when the bucket position is finally resolvedin the sandpit. In this regard, Table 1 compares the responsetime of the linear PIP and SDP-PIP approaches, representedby the number of seconds taken to complete three completetrenches, each consisting of 9 digging cycles. Here, the im-proved joint angle control allows for a faster SDP-PIP design,typically yielding a 10% improvement in the digging time.204060801001201401601802001201008060402040608010012014016018020020015010050050100Figure 5: Top: linear PIP (thin trace), nonlinear SDP-PIP(thick) and command input (dashed) for the dipper angle, plot-ted against sample number. Bottom: equivalent control inputs.Table 1: Time taken to complete one trench.TrenchLinear PIPSDP-PIP1338.46s369.01s2334.39s370.43s3336.13s372.86sFinally, Fig. 6 illustrates typical SDP-PIP implementation re-sults for one cycle of the bucket showing a 3D co-ordinate plotof the end-effector. This graph shows the bucket being firstlowered into and subsequentlybeing draggedthroughthe sand,followed by extraction, displacement and release.8ConclusionsThis paper has described the control of a laboratory robot arm,representing a 1/5th scale model of an autonomous excavator,developed for both research and teaching at Lancaster Univer-sity. In contrast to earlier research 15, the present paper con-siders the implementation of the complete control system forautomatically digging a trench in the sandpit.Classical and modern approaches have been evaluated for jointcontrol, including: Proportional-Integral (PI) control tunedby conventional Ziegler-Nichols rules; linear Proportional-Integral-Plus (PIP) control; and a novel, nonlinear PIP designbasedontheidentificationofStateDependentParameter(SDP)models. Here, linear PI algorithms tuned using either classicalor PIP methods initially appear sufficient for controlof the dip-per and bucket angles. By contrast, the slew and boom jointsare better controlled using the higher order PIP algorithm thatautomatically handles the pure time delays.However, inherent nonlinearities in all these joints prove prob-lematic when the feedback controllers are combined with kine-matic equations to control the position of the end effector, par-ticularly once the bucket in movedinto the sand. In fact, the in-creasedcomplexityofthenonlinearSDP-PIPapproachappearsjustified here, given the improved closed-loop performance. Inparticular, the time taken to complete an entire digging cycleis reduced by 10%. Finally, the experience gained usingthe laboratory excavatorhas recently been exploited for the de-velopment of a full scale vibro-lance system used for groundimprovement on a construction site; see e.g. 16.AcknowledgmentsThe authors are grateful for the support of the Engineering andPhysical Sciences Research Council (EPSRC). The statisticaltools used in this paper have been assembled as the CAPTAINtoolbox 18 within the MatlabTMsoftware environment,avail-able for download at: http:/www.es.lancs.ac.uk/cres/captain/References1 R. L. Tucker. Construction automation in the USA. InProceedings 16th Internation Symposium on Automationand Robotics in Construction, Madrid, Spain, 1999.2 Q. Ha, Q. Nguyen, D. Rye, and H. Durrant-Whyte.Impedance control of a hydraulic actuated robotic exca-vator. J. Automation in Construction, 9:421435,2000.3 E. Budny, M. Chlosta, and W. Gutkowski.Load-independent control of a hydraulic excavator. Journal ofAutomation in Construction, 12:245254,2003.4 J. Gu, C. J. Taylor, and D. W. Seward. The automation ofbucket position for the intelligent excavator LUCIE us-ing the proportional-integral-plus (PIP) control strategy.Journal of Computer-Aided Civil and Infrastructure E