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Load-independent control of a hydraulic excavator Eugeniusz Budny*, Miroslaw Chlosta, Witold Gutkowski Institute of Mechanized Construction and Rock Mining, ul. Racjonalizacji 6/8, 02-673 Warsaw, Poland Accepted 23 August 2002 Abstract The primary focus of this study is to investigate the control of excavation processes by applying load-independent hydraulic valves. This approach allows avoiding closed loop control system with sensors and transducers mounted on the excavator attachment. There are, then, no sensor cells mounted on the machine attachment. The considered system is composed of two subsystems: a microcomputer and a hydraulic unit (a pump and load-independent valves). In the microcomputer unit, the bucket velocity vector is related to the oil flow into three cylinders through the application of inverse kinematics. Then, flows are transferred into the electric signals actuating the load-independent valves. Their motion is presented by applying transfer function. The performance of the system is verified by assuming an abrupt change of the oil flow into cylinders. The last part of the paper is devoted to the obtained experimental results. The first result deals with vertical drilling. The second result deals with an excavation along a horizontal trajectory. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Excavator; Hydraulic systems; Control; Trajectory execution 1. Introduction Due to encouraging results of recent research, there are increasing possibilities for enhancement of a large spectrum human efforts in excavation pro- cesses. This may occur mainly through control of repetitive work tasks, such as trenching and drilling, requiring constant attention of machine operators during the performance of each task. Particular attention, in research, is paid to excavation along prescribed trajectories subjected to varying soil envi- ronment. Fundamentals dealing with controlled excavation processes are discussed by Vaha and Skibniewski 1, Hemami 2, and Hiller and Schnider 3. An inter- esting approach to piling processes by a direct angular sensing method is proposed by Keskinen et al. 8. Budny and Gutkowski 4,6 proposed a system, applying kinematically induced motion of an excavator bucket. In this approach, influence of a small variation of hydraulic oil flow into cylinders, applying sensitivity analysis, is discussed by Gut- kowski and Chlgosta 5. Huang et al. 7 presented an impedance control study for a robotic excavator. They applied two neural networks: first, as a feed- forward controller and the second as a feedback target impedance. Another impedance system, apply- ing a hybrid position/force control, is proposed by Ha et al. 9. The first generation of robots was conceived as open loop positioning devices. This implied that all parts had to be manufactured with a very high 0926-5805/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S0926-5805(02)00088-2 * Corresponding author. E-mail address: .pl (E. Budny). URL: .pl. Automation in Construction 12 (2003) 245254 and costly accuracy. Next, positioning robots, with sensors, reduced this accuracy requirement consider- ably. Here were several approaches, mentioned in above references, to extend the industrial robots capabilities to robotic excavator. Systems of force cells, longitudinal and angular sensors have been applied. However, two main differences between requirements for manufacturing robots and robotic excavators should be noted. The first difference is that manufacturing robots are working in almost perfect conditions, free of vibrations, protected against shocks, humidity, and other possible damag- ing conditions. The second difference is the require- ments for high accuracy of manufacturing robots, often within microns. On the contrary, robotic exca- vators are working in very difficult construction site conditions, and required accuracy of the executed trajectories, comparing with industrial robots, is limited, say within centimetres. With difficult con- ditions of excavations works, all sensors attached to the boom, arm, and bucket have to be very well protected. Bearing in mind the above differences, it would be of interest to investigate the possibilities of controlling excavation trajectory by a hydraulic module com- posed of a pump and load-independent valves. In other words, to investigate a system free of sensor cells mounted at the excavator attachment, combined with a feedback controller, included in the hydraulic unit of the machine. The main objective of the present paper is to extend the discussion, initiated by the authors 10, on the possibilities of applying load- independent valves installed inside of operator cabin only. Under this assumption, the system is free of sensors located on the excavator attachment. After discussing mathematical model of the system, pre- liminary experimental results are presented at the end of the paper. 2. Statement of the problem The paper deals with a controlled, stable motion of an excavator bucket along a prescribed path. The problem is based on previous authors theoretical investigations 4 of quasi-static, kinematically in- duced excavation processes for assumed trajectories. In this study, the following assumptions are made. . The excavator attachment is a planar mechanism, composed of a boom, an arm, and a bucket. Three, independently driven, hydraulic cylinders operate the system. They are assuring a unique representation of the three degrees of the planar bucket motion, two displacements and a rotation. . The excavation process, in the experiments per- formed,isassumedtobeslowenoughtoconsideritasa quasi-static one. Inertia terms in motion equations of attachment can be then neglected. Only spool of the servomechanism is assumed to move with accelera- tions, which cannot be neglected. . The force (pressure) disturbances are assumed to have sinusoidal form. The acceptable parameters of the sinusoid are defined from stability conditions of the system. . The soil is assumed homogeneous. Some small inclusions in the form of stones are acceptable. . The proposed control system of excavation is operator-assisted. It means that in a case of a larger obstacle, the operator has to intervene. . If successful, the proposed control setup could apply to standard excavators with the aim of enhance- ment of a large spectrum of human efforts in repetitive processes such as trenching and drilling. . The experiment is considered as a system com- posed of three subsystems, namely: microcomputer with PLC; hydraulic arrangement (a pump, valves, cylinders); and the mechanism with three degrees of freedom of the bucket. Next, the subsystems are considered as sets of components. In the first sub- system, the following components are recognised: personal computer with appropriate software, trans- forming introduced equations and inequalities of motion and trajectory planers into electric signal. The latter is send to a PLC unit, which in turn causes an electrical actuation of solenoid valves. Pressures from the solenoid valves are causing changes in spool positions, assuring assumed flow of the hydraulic oil into cylinders. The spool position, in turn, is con- verted through a transducer to an electric feedback signal sent to the solenoid valves. Opened spools are letting the hydraulic oil to flow into the third sub- system, namely cylinders of the excavator mechanism. Finally, the last subsystem is composed of three components: the hydraulic cylinders, the boom, the arm, and the bucket. With the motion of the excavator, arms and the bucket itself, the pressures in cylinders E. Budny et al. / Automation in Construction 12 (2003) 245254246 are changing. Information about these changes is sent to the second, hydraulic subsystem, where the feed- back signal corrects position of spools assuring the oil flow according to the designed trajectory. In the paper, transfer functions of all system components are investigated from the point of view of stability. The functions are defined theoretically, or numerically from diagrams presented in catalogues of hydraulic equipment. Joining all transfer function of particular component, the transfer function of the whole system is discussed from the point of view of performance under abrupt unit signal. Several experiments were performed, showing that it is possible to assure stable, assumed motion of the bucket. Among experiments, one was devoted to drill- ing. In other words, the kinematically induced trajec- tory was a straight, vertical line. Experimentally obtained line is presented in Refs. 6 and 10. It is interesting to note that the variation of experimental line does not exceed 10 cm. 3. Three subsystems of the experimental setup The discussed system is divided in three sub- systems, namely: microcomputer, hydraulic valves, and excavator arms with a bucket. Below, they are discussed separately and then a joint control prob- lem is defined. 3.1. Microcomputer as a subsystem We start with defining a model of the end- effector (bucket, drill, hammer) motion. The end- effector, in its plane motion, has three degrees of freedom aj(j=1,2,3) (Fig. 1). They are rotations of the boom, of the arm, and of the effector. Denoting by x1p, x2pposition of the end-effector tip, and by x3its rotation, the kinematics of the considered mechanism is represented by vector relation: x1p x2p x3p 2 6 6 6 6 4 3 7 7 7 7 5 c1c2c30 s1s2s30 000a3 2 6 6 6 6 4 3 7 7 7 7 5 ? l1 l2 l3 2 6 6 6 6 4 3 7 7 7 7 5 ;1 where cjand sjdenote cos ajand sin aj, respec- tively. In further considerations, the sub index p is omitted as the position of only one point is con- sidered. Velocity of the point P, v=v1, v2, v3T=x 1, x 2, x 3Tis obtained by taking time derivative of Eq. (1), and by reducing 3?4 matrix to a 3?3 matrix: x v A a a a Aw;2 where A ?l1s1?l2s2l3s3 l1c1l2c2l3c3 001 2 6 6 6 6 4 3 7 7 7 7 5 :3 Taking inverse of A matrix equal to: A?1 l2c2l1c10 ?l2s2?l1s10 l2l3f23l1l3f13l1l2f12 2 6 6 6 6 4 3 7 7 7 7 5 ? 1 l1l2c1s2? s1c2 4 with fij=sicj?cisj, we find the inverse kinematics, relating angular velocities of mechanism elements to the tip displacement vector w A?1v:5 Angular velocities xj, in turn, are dependent on the elongation velocities h i of hydraulic cylinders. This dependence has to be determined from geometrical relations between cylinder lengths, constant param- eters of attachment, and aj. We start with the first cylinder. From Fig. 2 we find coordinates of two cylinders hinges, A1and B1. They are: x1A1 a0;x2A1 b0;x1B1 b1c1 a1s1; x2B1 b1s1? a1c1: Taking h2 1 x1B1? x1A1 2 x2B1? x2A22; E. Budny et al. / Automation in Construction 12 (2003) 245254247 after transformation we obtain h2 1 p01 q01c1 r01s1;6 where p01 a2 0 a 2 1 b 2 0 b 2 1; q01 2a1b0? a0b1; r01 ?2a0a1 b0b1: Taking time derivative of Eq. (6) we find: h1 ?q01s1 r01c1 2h1 ? x1 G111 2h1 ? x1:7 Repeating the same consideration for the second cylinder length (Fig. 3) we obtain h2 2 p02 q02f12 r02g128 where p02 a2 2 a 2 3 b 2 2 b 2 3; q02 ?2a2a3 b2b3; r02 2a2b3? b2a3; Fig. 1. The mini-excavator considered. E. Budny et al. / Automation in Construction 12 (2003) 245254248 and fij cicj? sisj;gij sicj cisj:9 Taking again time derivatives of Eqs. (8) and (9), we arrive at h2 ?q02g12 r02f12 2h2 ? x1 x2 G212 2h2 ? x1 x2:10 An expression representing the length h3of the third cylinder is more complex, and requires intro- duction of an auxiliary variable a4(Fig. 4). With a new variable, there is a need to introduce an addi- tional relation. In this case, the relation joins varia- bles a2, a3, and a4, through the condition that distance between B3and D3is constant and equal to b7. After some lengthy transformation, these relations take the following form: h2 3 p03 q03f24 r03g24;11 b2 7 p04 q04f23 r04g23 q05f24 r05g24;12 where p03 a2 4 a 2 5 a 2 7 b 2 4 b 2 5 b4b5; q03 ?2a7a4? a5; r03 2a7b4? b5; p04 a2 5 a 2 6 a 2 7 b 2 5 b 2 6; q04 2b5b6? a5a6? a6a7; r04 ?2a5b6 a6b5; q05 2a5a7; r05 ?2a6a7: Fig. 4. The length h3of the third cylinder. Fig. 2. The length h1of the first cylinder. Fig. 3. The length h2of the second cylinder. E. Budny et al. / Automation in Construction 12 (2003) 245254249 Taking time derivative of Eq. (11) and recalling that aj=xj, the velocity h 3can be presented as: h3 ?q03g24 r03f24 2h3 ? x2 x4 G324 2h3 ? x2 x413 The mentioned condition for b7in the form of Eq. (12) allows to find a4, and eliminates it from the other equations. Taking now time derivative of Eq. (12), we can express x4in terms of x2and x3 x4 ? G423 G524 1 ? ? x2? G423 G524 ? x3;14 where G423 ?q04g23 r04f23; G524 ?q05g24 r05f24: Combining, now, together Eqs. (7), (10), (13), and (14) in a vector notation, we can write: h H ? w15 with H12 H13 H23 H31 0; H11 G111 2h1 ; H12 H22 G212 2h2 ; H32 H33 ? G324G423 2h3G524 : The flow of the hydraulic fluid into jth cylinder, denoted by qj, is equal to h jSj, where Sjis the cross- section area of the cylinder. With above notations, we can write the final relation between assumed velocity vector of the end-effector and flow vector q as q S ? H ? A?1? v16 where S is diagonal matrix with components Sj (j=1, 2, 3). The flow (Eq. (16) is a calculated flow, which in our model is needed to move the end effector according to its assumed motion. In a real system, this amount of oil has to be supplied to real cylinders through valves. The latter must be then actuated by an electrical signal vector u. The relation of qj=qj(uj) between this signal and oil flow is given by valve characteristic, which in general has the form presented in Fig. 5. The positive values of qj are related to the elongation of the cylinder. The negative ones are related to its shortening. The curve representing graphically qj(uj) can be assumed to be represented by the following function: qj a1u ? b a3u ? b3 a5u ? b5;17 with constraints d imposed on maximum openings of the valve. Coefficients a1, a2, and a3can be deter- mined by fitting the function (17) at three points of the characteristic curve. In order to find electrical signal uj in terms of qj, we have to take the inverse of Eq. (17). In general, this can be achieved only through a numerical solution method. 3.2. Hydraulic valve subsystem (HVS) The calculated in microcomputer, reference elec- trical signal is now converted into real electrical signal, actuating the valve. In the problem discussed here, this is a load-independent, proportional valve PVG 32 by DanfossR. The discussed subsystem is presented in Fig. 6. Below, all of its parts and their transfer functions are discussed. Fig. 5. The oil flow q leaving the valve, as a function of uj. E. Budny et al. / Automation in Construction 12 (2003) 245254250 The difference between reference signal ujand ud, and a signal coming from the feedback, is actuating the controller. The controller in turn, is adjusting the pump pressure ppto a pressure pcneeded for an adequate position of the spool. This adjustment is done by four solenoid valves. Denoting by capital letters the Laplace transforms, we find: Ucs Ujs ? Uds Ujs ? Huds ? Ds; 18 where D(s) is Laplace transform of spool displace- ment d; Hudis a transfer function between the spool displacement and feedback signal ud. The latter is obtained by a transducer, with constant multiplier, giving: Huds Uds Ds Kd:19 The relation between UCentering the controller and pc leaving it is also constant: Gpus Pcs Ucs Kc:20 The pressure acting on the spool is causing its motion, defined by an equation for one degree of freedom, with a spring constant ks, spool mass m, damping coefficient c, and cross-section area on which the pressure is acting As: md cd ksd pcAs:21 The transfer function between spool displacement d and pressure pcis then as follows s2m sc ks ? Ds Pcs ? As:22 Considering now Eqs. (18)(21), we obtain the rela- tion between the transformed output of spool displace- ment and transformed reference input of electrical signal: Djs As? Kc AsKcKd ks sc s2m ? Ujs;23 or considering feedback electrical signal Ujd, we have Ujds AsKcKd s2m sc AsKcKd ks ;24 With a constant nominator and denominator, in the form of a second order polynomial, we can verify the performance of our control setup by assuming elec- trical signal equal to a unit step function ujt ujut25 which implies an abrupt change in the cylinder length. Considering now Eqs. (24) and (25), carrying a partial fraction expansion, and taking inverse Laplace transforms, we find the error e(t) as a function of time: et e?fxntcosxdt fxn xd sinxdt ? ujdt; 26 where 2fxn c m ; x2 n AsKcKd ks m ; xd 1 ? f21=2xn; f 1 weak damping: Fig. 6. Hydraulic valve subsystem. E. Budny et al. / Automation in Construction 12 (2003) 245254251 The relation (26) shows that the error asymptoti- cally tends to zero with the increase of time. 4. Experimental realization 4.1. The mini-excavator used for experiments The mini-excavator K-111 is used for experiments. It was assumed to minimize part replacements, in a serial machine, needed to perform the considered control. The main components in the hydraulic system to be replaced were valves. Moreover, the hydraulic cylinders are supplied with additional valves assuring required pressure. This ensures that unpredicted motion of the attachment is not taking place. The hydraulic load-independent valves used in this experi- ment were supplied by DanfossR. The transfer of information from a microcomputer to load-independ- ent hydraulic valves is conducted by a Controlled Area Network (CAN). After modification of the hydraulic system, the excavator can be controlled in two diffe