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 mch imbigs org pl E Budny URL http www imbigs org pl Automation in Construction 12 2003 245 254 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 245 254246 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 v3 T x 1 x 2 x 3 Tis 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 l1l2 c1s2 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 x2A2 2 E Budny et al Automation in Construction 12 2003 245 254247 after transformation we obtain h2 1 p01 q01c1 r01s1 6 where p01 a2 0 a 2 1 b 2 0 b 2 1 q01 2 a1b0 a0b1 r01 2 a0a1 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 r02g12 8 where p02 a2 2 a 2 3 b 2 2 b 2 3 q02 2 a2a3 b2b3 r02 2 a2b3 b2a3 Fig 1 The mini excavator considered E Budny et al Automation in Construction 12 2003 245 254248 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 2a7 a4 a5 r03 2a7 b4 b5 p04 a2 5 a 2 6 a 2 7 b 2 5 b 2 6 q04 2 b5b6 a5a6 a6a7 r04 2 a5b6 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 245 254249 Taking time derivative of Eq 11 and recalling that a j xj the velocity h 3can be presented as h3 q03g24 r03f24 2h3 x2 x4 G324 2h3 x2 x4 13 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 w 15 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 v 16 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 a1 u b a3 u b 3 a5 u b 5 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 245 254250 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 Uc s Uj s Ud s Uj s Hud s D s 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 Hud s Ud s D s Kd 19 The relation between UCentering the controller and pc leaving it is also constant Gpu s Pc s Uc s 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 m d c d ksd pcAs 21 The transfer function between spool displacement d and pressure pcis then as follows s2m sc ks D s Pc s 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 Dj s As Kc AsKcKd ks sc s2m Uj s 23 or considering feedback electrical signal Ujd we have Ujd s 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 uj t uju t 25 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 e t e fxntcos xdt fxn xd sin xdt ujd t 26 where 2fxn c m x2 n AsKcKd ks m xd 1 f2 1 2xn f 1 weak damping Fig 6 Hydraulic valve subsystem E Budny et al Automation in Construction 12 2003 245 254251 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 different ways The first method consists in using joysticks mounted in the operator cab This way using a joystick the operator can move the mechanism in an arbitrary position and with desirable velocity The second method consists of programming the bucket motion in a microcomputer The information from it is then transformed in elongation rates of cylinders and in the flow of the oil moving them The latter is converted in an electrical signal send through CAN to load independent valves The organization of the electrical system for load independent valves is shown in Fig 7 The control algorithm is written in Borland Pas cal and executed under MS Window 98 operating system The CAN communication rate is assumed to be 250 kbit s with sampling time between 0 5 and 2 0 s Fig 7 Hardware of the control subsystem Fig 8 Experimental results of vertical drilling E Budny et al Automation in Construction 12 2003 245 254252 4 2 Experimental results To examine the
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