1-5号规格自动挖掘机的线性、非线性和经典的控制器@中英文翻译@外文翻译

LINEAR, NONLINEAR AND CLASSICAL CONTROL OF A1/5TH SCALE AUTOMATED EXCAVATORE. Sidiropoulou, E. M. Shaban, C. J. Taylor, W. Tych, A. Chotai Engineering Department, Lancaster University, Lancaster, UK, c.taylorlancaster.ac.uk Environmental Science Department, Lancaster University, Lancaster, UKKeywords: Identification; model-based control; proportional-integral-plus control; state dependent parameter model.AbstractThis paper investigates various control systems for a 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 modern approaches,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.1 IntroductionConstruction 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 excavation based operations include gen-eral earthmoving, digging and sheet-piling. On a smaller scale,trenching and footing formation require precisely controlledexcavation. Full or partial automation can provide benefits suchas 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 systemrequires a thorough knowledge of a wide range of technologies,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.2 HardwareThe robot arm has a similar arrangement to LUCIE 4, 5, ex-cept that this laboratory 1/5th scale model is attached to a work-bench and the bucket digs in a sandpit. As illustrated in Fig. 1,the arm consists of four joints, including the boom, 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 asynchronous real-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 conditioning before forwardingto the A/D converter. Here, the range of 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 converter capable 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 +1000 for the highest possible upward velocity 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.3 KinematicsThe 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 bucketangle is separately adjusted to collect or release sand. In this re-gard, Fig. 1 shows the laboratory excavator 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 9.3.1 Inverse 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 inverse kinematic algo-rithm is derived by Shaban 9. Here Ci and Si denotes cos(i)and sin(i) respectively, whilst C123 = cos(1 + 2 + 3).X = X l4C4C4 l3C123 (1)Y = Y l3S123 (2)1 = arctanbracketleftbigg(l1 + l2C2) Y l2S2 X(l1 + l2C2) X l2S2 Ybracketrightbigg(3)2 = arccosbracketleftbigg X2 + Y 2 l21 l222l1l2bracketrightbigg(4)3 = 1 2 (5)4 = arctanbracketleftbiggZXbracketrightbigg(6)3.2 Trajectory 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 v0 and vf denote, respectively, theinitial and final position vector for the end-effector and that themovement is required to be carried out in T seconds. In thiscase, the uniform straight-line trajectory for the tool-tip is,v = (1St)v0 + Stvf 0 t T (7)Here, St is a differentiable speed distribution function, whereS0 = 0 and ST = 1. Typically, the speed profile St first 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 form St = 1/T. By integrating, the speeddistribution function will be St = t/T.For this particular application, the kinematic constraints of thelaboratory excavator allow 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.4 Teaching 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-doing interaction with robotic systems. These in-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.Robots provide a fascinating tool for the demonstration of basicengineering 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 processing and real-time control; and provides numer-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.5 Control 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 additional dynamic feedback andinput 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 = wTk pk (8)where,wTk = bracketleftbig yk1 ykn uk1 ukm bracketrightbigpk = bracketleftbig p1,k p2,k bracketrightbigTp1,k = bracketleftbig a1k ank bracketrightbigp2,k = bracketleftbig b1k bm k bracketrightbigHere yk and uk are the output and input variables respectively,while ai k(i = 1,2,.,n) and bj k(j = 1,.,m)are state dependent parameters. The latter are assumed to befunctions of a non-minimal state 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 = bracketleftbigyk ykn+1 uk1 ukm+1 zkbracketrightbigTand zk = zk1 + yd,k yk is the integral-of-error betweenthe command input yd,k and 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 = lkxk issubsequently defined by,lk = bracketleftbig f0,k . fn1,k g1,k . gm1,k kI,k bracketrightbigwhere lk is 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 lk at each sam-pling instant. Finally, note that while the NMSS/PIP linear con-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.6 Control 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 0.11 seconds. 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 yk is 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 = 11000 800 600 400 200 0 200 400 600 800 10000.0050.010.0150.020.0250.030.0350.04ParameterScaled voltageFigure 3: Variation of b against input demand for the boom.is fixed a priori, so that only the numerator parameter b isestimated 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-plementation results are similar to the PI algorithm tuned usingclassical frequency methods. 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 bettercontrolled using PIP methods since (as shown in numerous ear-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 ofb changes by a factor of 10 or more, depending on the ap-plied voltage used, as illustrate