Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 4, pp. 871-896, Warsaw 2009 MODEL AND ANALYSIS OF THE PROCESS OF UNIT-LOAD STREAM SORTING BY A MANIPULATOR WITH TORSIONAL DISKS Tomasz Piątkowski University of Technology and Life Sciences, Department of Postal Technology, Bydgoszcz, Poland e-mail: tomasz.piatkowski@utp.edu.pl In the paper, the author presents a proposal for the modelling of the process of sorting of a stream of unit loads realised by means of a ma- nipulator with torsional disks. In the developed model, one takes into account three zones of friction influencingmotion of the load in thema- nipulatorworking space: the first one associatedwith interactions on the manipulator active carrying surface, and two zones located on the belt conveyor right in front of, and behind the manipulator. Frictional pro- perties of the object are represented by a nonlinear friction coefficient defined based on a cubic b-spline curve. On the basis of numerical expe- riments performed on the sortingmodel, one determined the influence of fundamental structural and operational parameters of the manipulator on precision and reliability of the process of unit-load stream sorting. The obtained data can be used as guidelines for designing new solutions of sortingmanipulators andmayprovidehints necessary for optimization of already-existing devices. Key words: unit load, dry friction, handling process, belt conveyor, sorting 1. Introduction In transportation centres, where concentration of transported goods is high, one needs to automatically handle the stream of unit loads (i.e. objects having the form of cuboidal parcels). For this purpose, one uses highly efficient no- grip type manipulators, i.e. manipulators constructed on the basis of a belt conveyor or a link-belt conveyor, which have no gripping devices, and act on the objects through a push, or through a sequence of pushes or strikes (Akella et al., 2000; Mason, 1999; Piątkowski, 2004). 872 T. Piątkowski One of typical manipulation actions is, among other things, the process of sorting. It consists in dividing the stream of loads (according to the criteria recognised by the scanning device in the transport system) into several new transportationways. However, forcing a new direction ofmotion requires that a force impulse of adequate magnitude is exerted on the load at the right moment.One of theways for realising this task is application of devices, whose working elements are active carrying surfaces of the conveyor on which the transported loads lie. A practical implementation of this concept are stream manipulators with trays (built on the basis of link-belt conveyors – Tilt-tray Sorter1, CrossBelt Sorter2 and stationarymanipulators (incorporated into the belt conveyor) in the form of a system of driven disks or rolls which allow for controlling (programming) the direction of the friction field that exerts force on the load (ProSort SC13, Autosort 54, ProSort SRT5, Single Powered Pivot Diverter6). In the available literature, Böhringer et al. (2000), Laowattana and Sa- tadaechakul (1999), Luntz et al. (1997), disk manipulators are considered as devices realising the process of positioning and rotating the objects on convey- ors. The research works concerned with this problem deal with the analysis of control systems of microactuators (independently driven disks of two degre- es of freedom) equipped with an elaborated system of sensors controlling the current position of load and taking it to a precisely defined destination posi- tion. In the subject literature, however, one can not find any description of application of this kind of devices in a highly efficient process of load sorting, 1Tilt-tray SorterDDS, accessed 2007-10-27,Commercial folder published byMan- nesmannDematic AG, Offenbach, Germany, www.dematic.com 2Cross Belt Sorter, accessed 2008-07-24, Commercial folder published by Wally Transport Systems Inc, Longboat Key, FL, USA, www.wallysorter.com/pdfs/ally Cross Belt Sorter Overview.pdf 3ProSort SC1, accessed 2008-07-28, Commercial folder published by Hytrol Co- nveyor Company Inc., Jonesboro, AR, USA, www.hytrol.com/mediacenter/catalog sheets/ca prosortsc.pdf 4Autosort 5 – Pop-Up Wheel Sorter, accessed 2008-07-24, Commercial folder pu- blished by Automation Inc., Oak Lawn, IL, USA, www.automotionconveyors.com/media/printmedia/products/sc5 popupwheel.pdf 5ProSort SRT, accessed 2008-12-10, Commercial folder published by Hytrol Co- nveyor Company Inc., Jonesboro, AR, USA, www.hytrol.com/mediacenter/catalog sheets/ca prosortsrt.pdf 6Single PoweredPivotDiverter, accessed 2008-12-10,Commercial folder published by Hytrol Conveyor Company Inc., Jonesboro, AR, USA, www.mckessockconveyors.com/PDF/Hytrol/overheadpush singleppd.pdf Model and analysis of the process of unit-load stream... 873 in which the role of sensors is reduced to bi-state detection of the presence of load in the working space of the manipulator. The general knowledge of application of the sorting manipulators with torsional disks is based only on the data contained in information booklets publishedby the companies specializing inproductionanddeliveryof complete distribution systems (ProSort SC1, Autosort 5, ProSort SRT, Single Powered PivotDiverter). The contents of these booklets donot allowone to get through to the information characterising the course of the realised process of sorting, neither can one learn about imperfections of the process and their causes. There are no data that would facilitate introduction of adequate changes, e.g. constructional improvements or changes in operational parameter settings, which could remove the imperfections appearing in the course of the sorting process. In order to objectively assess the basic utility features of the concept of a manipulator with torsional disks, one undertook an attempt of developing a dynamic model of the load stream distribution process. The data obtained from numerical experiments carried out on the proposed model can be used for formulating necessary assumptions and guidelines, applicable when one designs a disk manipulator satisfying concrete exploitation requirements. 2. Working conditions of the manipulator The sorting manipulators with torsional disks are allocated along the belt conveyor andbuilt into its structure (Fig.1a).The role of executing elements is played by a system of power-driven disks, which constitute the active carrying surface for the transported loads. For the purpose of planned investigations, we assumed that the disks have two degrees of freedom: rolling motion about the horizontal axis x1, and rotary motion about the vertical axis x2 made when a change of the load transport direction is forced (Fig.1b – according to Luntz et al. (1997)). In the neutral position, the disks take the angle of α = 0◦, which makes it possible to send the loads to more distant reception lines. The working position of the disks corresponds to the inclination angle of the reception lines with respect to the main stream. We assume that this angle is held in the range of α=30◦-90◦. The idea of active carrying surface with power-driven disks rotating about theaxes x1 and x2, havingagreat rangeof theworking angle α is complicated from the constructional point of view. For the sake of simplification of the disk 874 T. Piątkowski Fig. 1. Illustrative diagram of a manipulator with torsional disks: (a) working space of the manipulator, (b) degrees of freedom of the manipulator executing element (rotational motions about axes x1 and x2); 1 – unit load, 2 – main conveyor, 3 – executing element of the manipulator (system of power-driven disks), 4 – chute, v – transportation velocity of the main conveyor, α – working angle of disk setting, s – width of the main conveyor drive, one applies two variants of solutions, dependingon the assumedworking angle α of the disks. In the case of devices which carry away the loads from the main conveyor at an angle of α = 30◦, one uses pairs of coupled torsional disks (Fig.2a). Each pair is set in rotational motion about the axis x1 bymeans of a driving strand of circular cross-section, taking the drive from the main conveyor belt (ProSort SC1) or from an electricmotor (Single Powered PivotDiverter). The torsional motion of individual pairs of disks about the axis x2 is effected by a lever system connected to a pneumatic cylinder. These devices are adapted for distributing loads to both sides, relative to the main conveyor axis. When the carry-away angle in the sorting device takes a high value, i.e. α=90◦ (Fig.2b –ProSort SRT), or α=45◦ (Autosort 5), one usually applies another solution in which all the disks have the same constant working angle (α= const) and are driven by a common drive and make rotational motion about the axis x1. The ability of sorting the loads is achieved through action that consists in either preventing the manipulated object and the disk from being in contact or causing that they enter into suchacontact. This is achieved byexposingorhidingthediskswith respect to the surfaceof themain conveyor through translational motion in the vertical direction x2; themotion is made by the set of disks or by some segments of the carrying surfaces (the segments placed in the spaces between the disks responsible for the flowof loads tomore distant chutes – (3) Fig.2b). The systems of disks having the working angle Model and analysis of the process of unit-load stream... 875 Fig. 2. Example solutions of power drive transfer onto disks of the manipulator with respect to the working angle of disks α: (a) when α=30◦ – ProSort SC1, Single PoweredPivotDiverter, (b) when α=90◦ – ProSort SRT, or α=45◦ – Autosort 5; x1 and x2 – degrees of freedom of the manipulator executing element, 1 – surface of main conveyor, 2 – carrying surface of the disk system, 3 – carrying surfaces responsible for the flow of loads (alongmain conveyor) to more distant chutes, 4 – electric motor, v – transportation velocity of the main conveyor of α=90◦ are adapted to carrying-away themanipulated loads to both sides, while those having the angle of α=45◦ – only to one side. In the assessment of the course of load manipulation process, two basic criteria have essential meaning: protection of the load from damage and relia- bility of the operations the manipulator is expected to realise. As far as the first criterion is concerned, one can take the assumption that the manipula- ted loads should not be endangered to any mechanical damage – the course of interaction forces exerted on the manipulated loads must then have a gen- tle character (they result only from frictional contact between the carrying surfaces of the object and the manipulator). To assess the process according to the second criterion, one must acquire data describing the influence of parameters of the sorting process onmotion of themanipulated object. Thesedata have been obtained bymeans of numerical experiments on the mathematical model described in a further part of the paper. One of the most essential components of this model is description of dry friction phenomenon. 3. Dry-sliding friction The coefficient of dry friction depends on the kind of surfaces of bodies enga- ged in frictional contact, quality (state of roughness) of the surfaces, ambient temperature and humidity, and the rubbing speed. Additionally, for a given 876 T. Piątkowski pair of bodies rubbing one against another in specific ambient conditions, the most important factor influencing the friction coefficient value is the speed of rubbing (Hensen, 2002). Fig. 3. Examplary courses of the dry friction coefficient in function of the rubbing speed: (a) constant two-parameter – Tarnowski and Bartkiewicz (1998), (b) constant single-parameter – Tarnowski and Bartkiewicz (1998), (c) linear – Krivoplas (1980), (d) polynomial – Awrejcewicz (1984), (e) two Stribeck curves – Awrejcewicz and Olejnik (2002), (f) b-spline curve – Piątkowski and Sempruch (2006b); parameter values (α, β, b1, b2, b3, c, µo, µmin, vmin) are taken from the cited literature where A=                µmin+(µo−µmin)exp(−b1|vo|) if |vo|vmin µo otherwise In contemporary literature, one canfindmanydiverse characteristics of the friction coefficient proposed for description of dry friction. Examplary courses of these characteristics (drawn in function of the rubbing speed vo =0-1m/s – the interval pertaining to a sorting process ofmoderate intensity, Piątkowski and Sempruch (2006a,b, 2008a,b)) are presented in Fig.3. Inmost cases, they have an idealised linear character (Krivoplas, 1980; Tarnowski and Bartkie- wicz, 1998 – Fig.3a-c), or sometimes a nonlinear course, and are usually valid in a narrow range of the rubbing speed (Awrejcewicz, 1984; Awrejcewicz and Model and analysis of the process of unit-load stream... 877 Olejnik, 2002 – Fig.3d,e). The applicability of the presented descriptions of friction (to the considered application) can be preliminarily assessed on the basis of analysis of friction coefficient variability shown in graphs. The graphs in Fig.3a-c present a logical and real range of variability of the friction coeffi- cient. On the other hand, the courses of characteristics in Fig.3d,e show that the applicability of these characteristics is limited to cases when only low rub- bing speeds may appear – for the speed approaching vo =1m/s, the friction coefficient would reach an unbelievably high value. The characteristic of friction coefficient should be valid in a wide range of the rubbing speed. The transport of the load stream via lines of automa- ted sorting may result in rubbing of the objects against working surfaces of the manipulator with the speed exceeding vo = 0-2.5m/s (Piątkowski and Sempruch, 2008a,b). The results of experimental investigations on friction in this range of velocity and a proposal for a new characteristic of the friction coefficient (for an object made of cardboard sliding on a belt made of rubber, Fig.3f) are presentedbyPiątkowski andSempruch (2006b).The characteristic is defined by a cubic b-spline curve with six control points. From among the hereinafter presented characteristics of the friction coef- ficient, we select the characteristic of Fig.3f as the one which most faithfully represents the physical nature of friction in the process of manipulation of unit loads. The analysis of results of model simulation of the sorting process, taking into account different ways of description of the friction coefficient, is presented in a further part of this study (in Section 6). Particular difficulties in determining the friction force one can encounter in the range of rubbing speeds close to zero, where the phenomenon of stick-slip appears (Tarnowski andBartkiewicz, 1998). This phenomenon is accompanied by a rapid change of frictional resistance, and it has an important meaning i.e. in robotics, when one deals with precise positioning of robot’s members (Korendo and Uhl, 1998). For the needs of the present study, we used the following expression to describeanalytical relationships taking into accountnonlinearity of the friction force resulting from static and kinetic friction F =        Fkin sgn(vo) if |vo|>vmin { Pext if Fstat > |Pext| Fstat sgn(Pext) otherwise } otherwise (3.1) where 878 T. Piątkowski Pext – external force exerted on the object, tangent to resistance surface Fkin – kinetic friction force Fstat – limiting static friction force. The quantity vmin (in expression (3.1)) makes up the threshold speed of a small magnitude (assumed in the paper, vmin = 10 −6m/s), below which the rubbing speed is considered zero (Kikuuwe et al., 2005). The use of the threshold speedallows one to overcome difficultieswhich occur duringnumeric integration of equations of motion containing a non-continuous frictionmodel (when vo =0). Fig. 4. Results of dry friction simulation: (a) diagram of the mechanical system, (b) trajectorymotion of the object and spring end B, (c) course of the friction force, external force and rubbing speed, (d) detail A of Fig.4c in magnification; mp =1kg, Fkin =1N, Fstat =1.5N, k=2N/m, ẋs =0.1m/s, vmin =0.001m/s, Pext = k(xs−xp), tI – duration of stage F(3) in the first cycle of stick-slip friction Activity of respective rows of formula (3.1) is illustrated inFig.4b,c,d. The graphs are assigned on the basis of simulation of the object motion in stick- slip friction conditions (Fig.4a). The following data was accepted during the simulation: mp =1kg, Fkin = 1N, Fstat = 1.5N, k =2N/m, ẋs =0.1m/s, Model and analysis of the process of unit-load stream... 879 vmin = 0.001m/s, initial conditions – xp(t=0) = 0, ẋp(t=0) = 0, xs(t=0) = 0. Symbols F(1),F(2) and F(3) presented in Fig.4 relate to the successive rows of equation (3.1). The first row F(1) represents kinetic friction, the second F(2) and third F(3) – static friction. The row marked as F(2) is responsible for the body maintenance at rest (when the external force Pext is smaller than the static friction force Fstat), and row F(3) – for the initiation of the body transition from the state of static friction to kinetic friction (when the external force Pext becomes larger than the static friction force Fstat). The value of threshold speed is chosen on the basis of experience and the explorer’s intuition. In Fig.5, the influence of the threshold speed vmin on duration of friction static stage F(3) is shown. A strict relationship between the speed threshold value vmin and duration of stage F(3) occurs during the first body transition from the state of static friction to the kinetic one (curve marked by tI). The next cycles of the stick-slip friction (curves tII and tIII) do not show such a dependence. The obtained effect is caused by the fact that the first stick-slip cycle occurs when the initial speed of the object ẋp = 0, and the next – when |ẋp| ¬ vmin. Fig. 5. Graph of duration of static friction stage F(3) in function of the threshold velocity vmin (when Fstat ¬ |Pext|, according to equation (3.1)); tI, tII, tIII – duration of static friction stage F(3) of successive stick-slip cycles 4. Model of load motion The motion of a load in the working space of the manipulator is evoked by the carrying surfaces which can be divided into three zones (Fig.6): b – that includes the manipulator-load interaction area, and a and c that are located on the main conveyor just in front of the manipulator and behind it. 880 T. Piątkowski Depending ondimensions andposition of the load andwidth of the zone b, the load can be in contact with one, two or with all zones simultaneously. Fig. 6. Scheme of forces acting on the object during sorting by themanipulator with torsional disks; A×B – load dimensions, φ – angle of load rotation with respect to gravity center Cs, dS – infinitesimal surface, voa, vob, voc – resulting rubbing speed of the infinitesimal friction surface dS in zones a, b and c, ξa, ξb, ξc, ηa, ηb, ηc – components of the rubbing speed, dFa, dFb, dFc – infinitesimal friction force in the zones a, b and c (friction force has the direction of the rubbing speed and has an opposite sense in relation to it), dFξa, dFξb, dFξc, dFηa, dFηb, dFηc – components of the friction force, v – conveyor velocity of transportation In the physical model of load sorting, one assumes a rectangular reference coordinate system Oxoyo connectedwith themanipulator frame,whose origin coincides with the border of the zone b, and the direction of the axis xo is consistent with the axis of the main conveyor (Fig.6). Additionally: Model and analysis of the process of unit-load stream... 881 • the friction zones a, b and c of the conveyor lie in one plane, • the power-drivendisks of themanipulator are free of axis-direction error, i.e. they do not exhibit any lateral whip, • the disks and dead zones located between them have dimensions much smaller than those of the manipulated objects, • the load is treated as a rigid body with uniformly distributedmass, • the influence of random disturbances is neglected, • the frictionphenomenon is describedaccording toCoulomb’smodel (and its modifications, Kikuuwe et al. (2005)), • one takes into account the existence of static and kinetic friction which can appear in the following configurations: – lack of static friction in all zones simultaneously, – kinetic friction in the zones a and c, and static friction in the zone b, – kinetic friction in the zone b, and static friction in the zones a and c, • one assumes identical frictional properties in the zones a and c, and the same conveyor transportation speed v in these zones. The planar motion of the load on the conveyor and the manipulator disk surfaces is described in the rectangular system of coordinates Oxoyo by the following system of equations (according to Fig.6) mpẍo =Fξac+Fξb mpÿo =−Fηac−Fηb (4.1) Ipφ̈=−Tac−Tb where: F(j)ac,F(j)b are components of the load friction force in the zones a, c and b, j= ξ,η F(j)ac =                Facmax jac voac if voac >vmin        −Fbmax jb vob if Facmax >Fbmax −Facmax jb vob otherwise        otherwise 882 T. Piątkowski F(j)b =                Fbmax jb vob if vob >vmin        −Facmax jac voac if Fbmax >Facmax −Fbmax jac voac otherwise        otherwise (4.2) Facmax =Famax+Fcmax F(i)max is the limit friction force acting between the load and carrying surface of the manipulator in the zones i= a,b,c F(i)max = mpg S ∫ Si µiwi dS (4.3) Tac, Tb –moments of load friction forces in the friction zones a, c and b Tac =          Tacmax if |φ̇|>φmin { −Tbmax if Tacmax > |Tbmax| −Tacmax sgn(Tbmax) otherwise } otherwise Tb =          Tbmax if |φ̇|>φmin { −Tacmax if Tbmax > |Tacmax| −Tbmax sgn(Tacmax) otherwise } otherwise (4.4) Tacmax =Tamax+Tcmax T(i)max –moment of the limit friction force of the load in the zones i= a,b,c T(i)max =            mpg S ∫ Si µi wi voi [ηi(xs−xo)+ ξi(ys−yo)] dS if vio >vmin mpg S µi ∫ Si wir dS otherwise (4.5) wi – function describing geometry of the load surface; wi = 1 in cases when the load is in contact with the conveyor in the zone i, otherwise – wi =0 S=AB – carrying surface of the load r – distance between the infinitesimal surface dS and gravity centre of the load Cs r= √ (xs−xo)2+(ys−yo)2 (4.6) Model and analysis of the process of unit-load stream... 883 dS – infinitesimal surface ξa, ξb, ξc, ηa, ηb, ηc – components of the rubbing speed of the infinitesimal friction surface of load dS in the zones a, b, and c ξa = ξc = v− ẋo+ φ̇r sinβ ξb = vcosα− ẋo+ φ̇rsinβ ηa = ηc = ẏo+ φ̇rcosβ ηb =−vsinα+ ẏo+ φ̇rcosβ (4.7) vo(i) – resulting rubbing speed of the infinitecsimal friction surface dS in the zones i= a,b,c vo(i) = √ ξ2i +η 2 i (4.8) β – inclination angle of radius r β=arctan ys−yo xs−xo (4.9) xo, yo,φ – coordinates of the gravity centre Cs and rotation angle of the load, respectively xs, ys – coordinates of the infinitesimal surface dS mp, Ip – mass andmass moment of inertia of the load, respectively µi – coefficient of friction between the load and conveyor in the zone i. 5. Numerical experiments In the performed numerical analyses, we assumed the characteristic of the friction coefficient described with a single parameter – according to Fig.3b. The analysis of discrepancies resulting from the application of the classic, single-parameter characteristic of the friction coefficient in the sorting process modelwith respect to that applying a curvilinear characteristicmodelledwith the cubic b-spline curve (Fig.3f), is presented in Section 6. Figure 7 depicts the trajectory of motion of the load gravity centre deter- mined as the load scraped by the system of disks set at the angle of α=90◦. One neglects here the influence of the zone c on the course of the sorting process assuming an infinitesimally great width of the zone b. Such an ap- proach allows us to determine the distance at which the manipulated object forces its way into the working space of the manipulator without considering the disturbances resulting from the influence of the zone c on the work of the manipulator. This influence will be considered in a further part of this study. In Fig.7, the continuous line (reference 1) denotes the trajectory of load motion obtained by simulation of the sorting process carried out with the use 884 T. Piątkowski Fig. 7. Trajectory of the load gravity centre during sorting by the manipulator with frictional disks; 1 – moment of friction forces taken into account, 2 – moment of friction forces neglected; A×B=0.4×0.2m, v=1m/s, α=90◦, b→∞, µa =µb =0.6 of equations (4.1), while the broken line (reference 2) pertains to the case, when the influence of moment of friction forces is neglected. The presence of themoment of friction forces, which are produced by reactions of themanipu- lator carrying zones, causes that the load is rotated by the angle φk, and the distance at which the load forces its way into the manipulator working space is shortened. Neglecting the moment of friction forces in analysis of the load motion significantly speedsupnumerical experiments.The estimates ofminimalwidth of the zone b (necessary for the sorting process to be carried out correctly), which have been determined based on the simplified model, will be taken to further considerations with some excess in order to increase the probability that the object is scrapped correctly. The transportation speed assumed during the investigations equals v = (0.5-1.5)m/s, and the dimensions of the load projection on the conveyor sur- face are A×B = (0.2-1.2)× (0.2-0.8) m. The assumed conveyor speed is equal to that used in roller conveyors (Sempruch and Piątkowski, 2002). The minimal dimension of the object was assumed due to the necessity of taking into account the dead zones in the carrying space located between the disk axes. Figure 8presents the results of simulation investigations on the load sorting process aimed at assessing the hazards resulting from difficulties of satisfying the condition of reliable scrapping of the load onto a proper chute. This condi- Model and analysis of the process of unit-load stream... 885 tion could be violated because of inadequate translocation of the load towards the chute, in the direction transverse to the conveyor axis. Fig. 8. Distance covered by the gravity centre of the object (during sorting) in the direction of conveyor axis xo determined in function of: (a) dimensions of the object A×B, (b) transportation speed of the load stream v and dimension of the load A, (c) inclination angle of torsional disks α and dimension of the load A, (d) friction coefficients µa, µb; s=0.7m – width of the main conveyor, φ=0 ◦, b→∞ It was assumed in the analyses that for the load to be scrapped onto a chute its centre of gravity must cover a distance equal to the width of the conveyor s=0.7m. Themain reason why the load fails to reach the assumed destination place is the insufficient width of the manipulator-load interaction space (zone b) relative to the assumed parameters of the sorting process (load dimensions,working angle of the systemof disks, transportation speed, frictio- nal properties of the belt conveyor and its operating elements). The necessary 886 T. Piątkowski width of the interaction zone is a function of the distance at which the lo- ad forces its way in the direction of the axis xo. The function is determined without taking into account the influence of the friction zone c. From the analysis of Fig.8a it follows that the distance xo depends on the length of the manipulated object (dimension A – in the case of load position before scraping parallel to the manipulator axis, φ= 0), and is not sensitive to its width B. The longer the load, the greater the required width of the zone b. The load transportation speed v has the influence on the course of the sorting process similar to that of load dimension A. The greater the speed value v, the greater the distance xo (Fig.8b). Theworking angle α of the torsional disk systemsetting also decides about the required length of the space in which the load is scrapped onto a chute. The influence of this angle on the sorting process is illustrated in Fig.8c (on the assumption that the new direction of load transportation is reached after the load covers the distance of the conveyor width, s=0.7m). The lower the disk setting angle α, the greater the requiredwidth of the zone b that realises transfer of the load onto a chute. The relationship between the load friction coefficients in the contact zones a and b and the achievable distance xo is illustrated in Fig.8d. From the analysis of Fig.8d, it follows that the friction coefficient µb (in the zone b) has greater influence on the course of the sorting process than that of the coefficient µa (in the zone a). The greater the value of friction coefficient µb, the shorter the distance xo reached by the load.The friction coefficient µa has an opposite effect – the lower the value of this coefficient, the more effective translocation of the load onto the chute. The graphs in Fig.9 illustrate relationships between the transportation speed of the loads, their lengths and capacity of the sorting process. In these analyses, it was assumed that the load takes a position parallel to the axis of the main conveyor (φ = 0), the working angles of the manipulator disks equal α = 90◦ (Fig.9a), α = 45◦ (Fig.9b), or α = 30◦ (Fig.9c), and that the load must cover a distance of s = 0.7m in order to be scrapped onto the chute. According to what is shown in Fig.9a, for the transportation speed v = 1.5m/s and load length equal to A = 0.7m, one can achieve technical capacity of the sorting process of approximately Wt =3600pcs/h. When the disk setting angle is set to a lower value, α = 45◦(Fig.9b), the sorting capacity slightly deteriorates (in comparison to that presented in Fig.9a – especially in the case of longer objects). When the working angle of the disks equals α=30◦ (Fig.9c), the drop of sorting capacity looks similar: Model and analysis of the process of unit-load stream... 887 Fig. 9. Influence of the transportation velocity of load stream v and length of load A on capacity of the sorting process for: (a) α=90◦, (b) α=45◦, (c) α=30◦; data: µa =µb =0.6,B=0.2m, b→∞ it is more significant in the case of short loads, and less significant for longer loads. The change of the working angle α does not have any radical effect on the time of scrapping the load onto the chute (especially in the case of long loads), despite the fact that the load scrapped by the system of disks of the working angle α = 45◦ (and α = 30◦) must travel much longer way to the chute than that for the system in which α = 90◦. The obtained effect can be explained by the conditions that exist during the load sorting process – a substantial part of time of load translocation onto the chute is the time of transient motion (stage E2 – Fig.10). The load scrapped by the system of disks of the working angle α=45◦ (and α=30◦) remains for amuch shorter period in the state of transient motion (stage E2 – Fig.10b,c) than that in the case of α=90◦ (Fig.10a). Additionally, the component of rubbing speed in the direction of the axis yo (ηb), shown in Fig.10b,c (setting angles of the disk system α = 45◦, α = 30◦), is much greater than the component in the direction of the axis xo (ξb). In the same proportion, the component of the friction force in the direction of the axis yo (Fηb, Fig.6) is much greater than 888 T. Piątkowski the component of that force in the direction of the axis xo (Fξb), which results in a greater acceleration of the load motion towards the chute. If the working angle of the disk system equals α=90◦, the components of the load rubbing speed in the directions of the axes xo and yo are the same (ξb = ηb, Fig.10a), and consequently, the components of the load friction forces (as well as the accelerations) in both directions have the same values. Fig. 10. Graphs of components of the load rubbing speed (A×B=0.7×0.2m) for the working angle of the torsional disk system: (a) α=90◦, (b) α=45◦, (c) α=30◦; stages of loadmotion: E1 – steadymotion I (sliding in zone b, no sliding in zone a), E2 – transient motion (load sliding in zones b and a), E3 – steadymotion II (no load sliding in zone b); v=1.5m/s In the determination of the required size of the manipulator-object inte- raction space, one aims at assuming possibly small width of the zone b, which follows from the condition ofminimizing construction costs of the load sorting system. Selecting smaller and smaller widths of the zone b (and, at the same time, widening the influence of the zone c) can be continued, however, up to the moment when (during translocation of the loads onto the chute) the manipulator is able to effectively cause that rubbing of the load against the system of disks disappears (i.e. to obtain vab =0). The graph of minimal width of the zone b versus transportation velocity of the conveyor v and length of the load A is presented in Fig.11 (determi- Model and analysis of the process of unit-load stream... 889 ned based on numerical optimization). The analysis of this graph shows that application of the disk setting angle α = 90◦ (Fig.11a) leads to the expec- ted shortening of the required width of the manipulator working space b – as compared tomanipulators inwhich α=45◦ (Fig.11b) and α=30◦ (Fig.11c). A particularly high difference in the selected zone width (showing the di- sadvantage of angles α=45◦ and α=30◦) appears in the case of low trans- portation velocity of the conveyor v and loads of small length A. Fig. 11. Minimal width of the manipulator working space b versus transportation velocity v and load length A for: (a) α=90◦, (b) α=45◦, (c) α=30◦; data: φ=0◦, µa =µb =µc =0.6 6. Evaluation of the influence of friction coefficient characteristics on the course of unit-load manipulation process The appropriateness of utilizing the classical constant-function characteristics of load coefficients (shown in Fig.3a,b) in the model of the sorting process is decided on the basis of analysis of numerical investigation results in which 890 T. Piątkowski we take into account the characteristics of Fig.3a,b, and the ”reference” one, of Fig.3f. In these analyses, we assume that the friction coefficient functions shown in Fig.3f, Fig.3a and Fig.3b will be identified as: model 1, model 2 and model 3, respectively. The parameters µ0 and µG that appear in these functions have the values of 0.65 and 0.46, respectively. The temptation for using the classic friction coefficients follows from the simplicity of mathematical expressions. Frictional properties of bodies repre- sentedby themodel ofFig.3a need only twoparameters to bedetermined, and those of Fig.3b – just one parameter. The characteristic of friction coefficient shown in Fig.3f requires as much as 12 parameters be determined – 6 control points on the b-spline curve. The characteristic of Fig.3f was determined on the basis of experimental investigation results, which consisted in positioning the object by means of a system of two, inversely-driven belts of the conveyor (Fig.12 – Piątkowski and Sempruch (2006b)). In these investigations, the object was set in damped oscillatorymotion (with respect to the equilibriumposition that occurs at the boundaryof influences of the two frictional sections of the conveyor –Fig.13a). The simulation of this process with the use of classical friction coefficient models always leads to the same effect – to non-damped oscillatory motion, Fig.13b.This result proves imperfection ofmodels 2 and3 in their application. The discrepancy between the results obtained in such away leads to radically different conclusions. The lack of damping in the oscillatory motion would indicate uselessness of such a method of object positioning and failure of the whole idea. Fig. 12. Scheme of the stand for object positioning: 1 – unit load under test, 2 – conveyor belt, 3 – tension roll, 4 – powered roll, 5 – bed; v – linear velocity of belt The courses of friction forces appearing between the object and the car- rying surface of the sorting manipulator with torsional disks are shown in Fig.14. One took the following assumptions: α=45◦, v=1.5m/s,mp =5kg, A×B=0.4×0.26m, s=0.7m, b→∞ (denotations as in Fig.1 and Fig.6). The results represented in Fig.14a pertain to the simulation that utilizes mo- del 1, and those in Fig.14b – to that based on models 2 and 3. The basic Model and analysis of the process of unit-load stream... 891 Fig. 13. Trajectory of motion of the load positioned by a system of two inverse fields of friction forces: (a) according to model 1, (b) according to models 2 and 3; v=0.37m/s,A×B=0.4×0.3m Fig. 14. Friction force F exerted on the object during sorting by the manipulator with torsional disks: (a) for model 1, (b) for models 2 and 3; stages of loadmotion, according to Fig.10: E1 – steadymotion I, E2 – transient motion, E3 – steady motion II; remaining data: α=45◦, v=1.5m/s,mp =5kg,A×B=0.4×0.26m, s=0.7m, b→∞ difference between the presented graphs concerns the course of the friction force Fb just before the end of its activity. In the case ofmodel 1 (Fig.14a), in the final interval of stage E2 (denotation as inFig.10) one observes a rapid in- crease of this force.The influenceof theobserveddifferenceon thebasic sorting process parameters (time of load-scrapping cycle tc and the required length of the manipulator working space L) is illustrated in Fig.15. The discrepan- cies between the sorting process parameters determined based onmodels 1, 2 and 3 are insignificant. The data shown in Fig.16 (derived based on Fig.15) justify this conclusion. These data represent relative working parameters of the manipulator, i.e. the results calculated on the basis of models 2 and 3 referred to those obtained by usingmodel 1. The relative discrepancy between 892 T. Piątkowski the results based on classic models 2 and 3 and those obtained with model 1 does not exceed 1%. Fig. 15.Working parameters of the manipulator with torsional disks: (a) time of load-scrapping cycle, (b) length of the manipulator working space Fig. 16. Relative working parameters of the manipulator with torsional disks, tc(1)/tc(j) and L(1)/L(j), derived based on Fig.15 (where j=1,2,3) It follows from the performed investigations that the nonlinear model of the friction coefficient, proposed by Piątkowski and Sempruch (2006b), has meaningwhen one needs to representmotion of objects in the case of rubbing speeds close to zero, which remain in this range during a substantial part of realisation of the manipulation process. An example of the process in which such conditions exist is the positioning of loads by means of inverse fields of friction forces. In this process, motion of the load relative to the manipulator carrying surface is made cyclically and frequently at low rubbing speeds. The classic, constant-function friction coefficients are then completely ineffective in this application. A different situation exists in the case when we deal with the description of the sorting process. This process usually involves high rubbing speeds of lo- Model and analysis of the process of unit-load stream... 893 ads and the number of transitions between the states of static and kinematic friction isminimal. The result is that the discrepancies between the simulation results, appearing due to application of different friction coefficient models – type 1, 2 and 3, are insignificant. For this reason, the friction coefficient de- scribed by the one-parametermodel (Fig.3b) can be considered as an effective tool for representing the course of the load sorting process. 7. Conclusion The results of investigations on the sorting process model, presented in this work, have a theoretical and practical value as they give a possibility of influ- encing the design and exploitation of load-distributing systems with frictional disks. • The classic one-parameter functiondescribing the friction coefficient, ap- plied in thismodel, realistically represents the nature of the load sorting process. Capacity of this description can be attributed to high rubbing speeds at which the object moves with respect to the manipulator wor- king elements, and to the low number of transitions between the states of static and kinematic friction. • An increase in the working angle of the disk system in the range of α∈ 〈30◦,90◦〉 causes that: – the required width b of the manipulator working zone becomes shorter, – capacity of the sorting process increases; this increase is more pro- nounced in the case of short loads and less significant for longer loads, – acceleration ofmotion of themanipulated object towards the chute decreases, and the period of transient motion of the object in this direction becomes longer. • For the sake of capacity of the sorting process, one should assume possi- bly highvalues of the friction coefficient for thedisk system, andpossibly low values for the carrying surface of the main conveyor; the course of the sorting process is definitely more sensitive to changes of frictional properties of the disk system than to those of the carrying surface of the main conveyor. 894 T. Piątkowski • A decrease in length of the manipulator working space (friction zone b) can be obtained by dividing the sorting process into two stages: the pre- liminary and the final one. Themanipulator, which realises preliminary sorting, should be installed at the beginning of the unit load stream – before the manipulators that carry out the process of final sorting. The preliminary sorting will allow for shifting the load closer to one of the conveyor borders, according to the predicted direction in which the load will be carried away to a new transportation line. Acknowledgement This work has been financed from the funds for science as a Research Project in the years 2008-2010. References 1. Akella S., Huang W.H., Lynch K.M., Mason M.T., 2000, Parts feeding onaconveyorwith aone joint robot,Algorithmica, Springer-Verlag,26, 313-344 2. 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Sempruch J., Piątkowski T., 2002,Technical Facilities ofWorks Transport, WydawnictwoUczelniane ATRwBydgoszczy [in Polish] 19. Tarnowski W., Bartkiewicz S., 1998,Modelowanie matematyczne i symu- lacja komputerowa dynamicznych procesów ciągłych, WydawnictwoUczelniane Politechniki Koszalińskiej Model i analiza procesu sterowania strumienia ładunków jednostkowych manipulatorem z krążkami skrętnymi Streszczenie Wartykule przedstawionopropozycjęmodelowania przebiegu procesu sortowania strumienia ładunków jednostkowych realizowanego za pomocą manipulatora z krąż- kami skrętnymi.W opracowanymmodelu uwzględniono, iż na ruch ładunku w prze- strzeni pracymanipulatoramająwpływ trzy strefy tarcia: pierwsza strefa obejmująca 896 T. Piątkowski oddziaływanie aktywnej powierzchni nośnej manipulatora oraz dwie strefy znajdują- ce się na przenośniku taśmowym tuż przed i za manipulatorem. Właściwości cierne obiektu reprezentowanesąnieliniowymwspółczynnikiemtarciawykorzystująckrzywą cubic b-spline. Na podstawie przeprowadzonycheksperymentównumerycznychmode- lu sortowania określono wpływ podstawowych parametrów konstrukcyjnych i eksplo- atacyjnych manipulatora na precyzję i niezawodność przebiegu procesu sortowania potoku ładunków jednostkowych. Uzyskane dane mogą być wykorzystane jako wy- tycznepodczasprojektowanianowychrozwiązańmanipulatorówsortującychoraz jako wskazania niezbędne do optymalnej eksploatacji urządzeń już istniejących. Manuscript received January 16, 2009; accepted for print May 11, 2009