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Compliance effects in a parallel jaw gripperA.J.G. Nuttall, A.J. Klein Breteler*Faculty of Design, Construction and Production, Department of Transportation Technology, University of TechnologyDelft, Mekelweg 2, 2628 CD Delft, The NetherlandsReceived 17 April 2002; received in revised form 24 April 2003; accepted 30 June 2003AbstractThis paper discusses mechanical compliance effects in a gripper with parallel jaws. In it a case study of adedicated gripper design is presented to analyse two different design elements influencing the compliantbehaviour: the flexibility introduced by preloaded springs and the resistance caused by friction.The gripper manipulates semi-automatic twistlocks used for securing seagoing cargo containers. Thecompliance effects are effective to reduce misalignment and overload of the gripper.? 2003 Elsevier Ltd. All rights reserved.Keywords: Mechanical compliance; Twistlock manipulator; Preloaded springs; Friction force1. IntroductionRobots grasp and manipulate objects with the aid of a gripper. Usually the object is presentedat a predefined pickup location where the robot can grasp and move it to another predefinedlocation.Difficulties arise when the pickup location and destination are part of a heavy rigid body thatcan move due to external disturbances. If the robot cannot adapt to this movement during pickupor release, the full force of the movement will be transferred into the robot?s components, whichcan result in damage. Therefore the robot should be flexible or compliant where the environmentis stiff3.Compliance can be introduced to the robot by using a compliant end-effector or gripper. Thiscan be done in different ways. In literature a variety of subjects on gripper compliance canbe found 1,3,4, these are mainly focussed on control theories for universal grippers and fine*Corresponding author. Tel.: +31-15-278-3130; fax: +31-15-278-1397.E-mail address: a.j.kleinbretelerwbmt.tudelft.nl (A.J. Klein Breteler).0094-114X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0094-114X(03)00100-9Mechanism and Machine Theory 38 (2003) 15091522/locate/mechmtmanipulation. In 5,6 such a compliance is investigated using stiffness models and in 7 a remotecompliance centre is introduced. These investigations integrate compliance into the control sys-tem, with the aid of special sensors and actuators making reliable force and position controlpossible. With this form of electronic control the universal gripper can manage many differenttasks and objects.This is in contrast to the special-purpose end-effector that is to be designed for a specific taskand object. By making use of simple sensors and actuators combined with a mechanical form ofcompliance an effective, reliable and robust gripper can result, which will also be able to adapt (allbe it in a limited manner) to a moving pickup point. The adaptation of this form of compliance forgripper configurations has proven hard to find in literature.This paper gives an insight into the effects of mechanical compliance in a gripper. A case studyof a gripper design will aid as example to discuss two different modes of mechanical compliance.This example case consists of a parallel jaw gripper configuration intended for the manipulationof semi-automatic twistlocks.2. Background to the twistlock manipulatorA manipulator was required to automatically connect and remove semi-automatic twistlocks toand from a container?s bottom corner castings. In Fig. 1 a semi-automatic twistlock is shown onthe left. This type of twistlock is a lashing device that is used to secure sea-going cargo containersto the deck of a ship. It consists of a body, an upper and lower rotating cone and a handle formanual operation of the cone positions.The upper cone can be inserted into the bottom corner casting depicted on the right side of Fig.1 by unlocking it through rotating the lower cone. The top collar fixes into the hole of the cornercasting, because it matches the shape of the hole. When the cones are rotated back to their originalposition, the twistlock is secured to the bottom corner casting. The handle is intended for manualoperation. If it is pulled the shaft rotates that connects the cones together.Fig. 1. A semi-automatic twistlock and a corner casting of a container.1510A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 15091522For automation of this securing procedure and the reverse operation a gripper had to be de-signed that can hold different types of twistlocks by their collars with sufficient grasping force 10.The jaws also have to open far enough, to prevent collisions with the cones while the manipulatoris positioning over the twistlock with open jaws.The container can move during the pickup or release operation due to external disturbances,because it will be hoisted up in the air by cables or resting on a rolling chassis with pneumatictyres. The wind is an example of a disturbance that can generate fluctuating forces on the side ofthe container, which can result in an oscillating movement. Due to the possible movement of thelarge container mass (30 ton) and the robust construction of the twistlock the gripper will have tobe compliant to prevent damage to itself or other components of the robot. Mechanical com-pliance will also help tackle the problem of the moving pickup point on the container and keep therequired control system simple.The collar was chosen as contact surface for the gripper, because it is the common element indifferent twistlock designs. It has to fit into the standardised hole of the corner casting, so theshape and size will be roughly the same. Although the width of the hole is only allowed a toleranceof 1.5 mm the collar widths found in practice can vary between 57 and 62 mm. This 5 mm range incollar sizes had to be taken into account for a reliable operation of the gripper.In Fig. 2a the forces applied by the jaws during a grasp are presented. The frictional forcesgenerated on the collar sides will have to be sufficiently large to compensate the static and dy-namic forces created on the twistlock. The total force (Ftot) that has to be compensated in a di-rection parallel to the collar surface is 200 N. This was calculated by determining the dynamicforces caused by movement of the manipulator and the static gravitational force. With a fric-tional coefficient (l) of 0.125 the grasping force exerted by each jaw (Fjaw) can be calculated asfollows:Ftot 2 ? lFjaw) FjawFtot2l 800 NThis is the minimal force that has to be guaranteed during the manipulation of a twistlock, for allcollar sizes.Fig. 2b shows the open and closed position of the jaws. It shows how far the jaws have to open,during the positioning of the open gripper. There has to be enough clearance between the coneFig. 2. Grasping forces on twistlock and the required jaw travel.A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 150915221511and jaw to prevent a collision, because the cone diameter is larger then the collar width. To get aclearance of 15 mm the displacement of a jaw has to be 40 mm.3. Finding a suitable gripper configurationAn existing gripper that is capable of producing a rather large clamping force and large dis-placement is illustrated in Fig. 3a 2. It consists of two parallel jaws, actuated by a double actingpneumatic cylinder. Attached to the cylinder?s piston rod is a dual rack gear, which drives twopartial sectors of pinion gears. Two pairs of the symmetrical arranged parallel closing linkages aremounted directly on the partial sectors of the pinions and provide the clamping force.This design only features compliant behaviour with respect to the width of the grasped object.If the grasped object is larger then the distance between the closed jaws, they will come in contactwith the object before they are fully closed. Therefore the piston will not travel to its end positionduring this closing operation. This makes it possible to grasp different sized objects. It willhowever be more difficult to sense the closed position of the jaws. A special sensing method likeforce detection will be required to measure the closed position.The gripper configuration of Fig. 3a can be given additional compliance as shown in Fig. 3b.Preloaded springs have been added to the jaws, to get compliant behaviour in the horizontaldirection. Preloading the springs gives two advantages. First of all the stroke required to build upsufficient grasping force can be short. If the preload is set to the minimal required grasping force,after contact with the object the springs hardly need any travel for a secure grip. Secondly theminimal required grasping force can be guaranteed with the aid of a proximity sensor that candetect the end position of the pneumatic cylinder. If the cylinder reaches the end of the closingstroke with an object between the jaws, the springs will have been pressed in and the graspingforce would at least have to be equal to the set preload.Fig. 3. Parallel gripper configurations with compliance.1512A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 15091522This gripper design with springs in the jaws was not used for the twistlock manipulator, becausethe springs take up too much space. Special measures would have to be taken to keep the jawconstruction sufficiently compact.An alternative configuration with spring elements can be seen in Fig. 3c. The preloaded springsare not directly connected to the jaws, but they have been placed between the actuator and thelever of the jaw parallelograms. The mechanism amplifies the force of the cylinder, when the jawsare closing, if the springs would have been ordinary bars. The generated grasping force is largestwhen the jaws are nearly closed. The more they are opened the smaller the possible force, but thelarger their displacement versus the cylinder displacement.In this configuration another effect is introduced with respect to compliant behaviour. If ahorizontal force is applied, a resistance is generated by friction in the contact surfaces. Thehorizontal force has to be large enough to overcome this resistance and to move the jaws with theobject in between.The cause of this effect is illustrated in Fig. 3d. When the grasped object moves to the right theleft jaw swings up and the right jaw swings to a lower position while they both remain parallel toeach other. This causes the jaws to slide over the surface and generate frictional forces if thegrasped object does not change orientation. In the example case the object or twistlock will notchange orientation, because it is fixed to the container during the grasping manoeuvre. It can onlymove with the container in the horizontal direction.The design in Fig. 3c will be considered further for the twistlock manipulator and will beanalysed using the theory given below.4. Modeling the gripper, FEM approachWhen the frictional compliancy is in effect, the jaws will slip relatively to the twistlock surface.The friction forces under slip will be considered proportional to the contact force. This is areasonable assumption for the conceptual design phase of the gripper.The theory needed concerns just the equilibrium of static forces, as for instance can be de-scribed with the principle of virtual work. The spring forces and the friction in the grippermechanism are considered as internal forces. Their virtual work must be equal to the virtual workof the compliant force, which is considered as the driving force.To perform the actual calculations, a general computer program for kinematic and dynamicanalysis can be used in which this theory has been embedded. The portion of the theory used toperform the analysis calculations, is described below briefly. The theory is also known as finiteelement approach 8,9.From FEM the two maps displacements on deformations, and applied forces on internal forcesare known as dual maps, indicating that both relations can be described with the same matrix.Here it means that the contact forces of the jaws (internal forces) will be calculated with the samematrix as used for kinematic motion analysis.A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 150915221513A model of the gripper mechanism can be built with truss elements each having constant length.In the FEM-concept a constant length is considered to have deformation zero. For kinematicsit concerns just a mathematical variation of the length, for force analysis a normal force exists asan internal force.The length itself is a continuous function of global co-ordinates (position vector x) of the el-ement. For a truss element, defined by the end-points P and Q and numbered k, the continuityequation can be written asxk jxPyPxQyQjTkxk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyQ? yP2 xQ? xP2q1Moving with deformation zero can be expressed in kinematics with a continuity equation of thefirst order, like here for constant lengthokoxk?T?oxkok? 12Written out with the help of (1)?cosbk?sinbkcosbksinbk?oxkok? 13where b is the angle of the element that can be obtained from the given xkvector.Comparable continuity equations can be constructed to prescribe a fixed angle of the trusselement or a fixed angle w between two truss elements.Those partial derivatives concerning fixed nodes are known (zero); those concerning movingnodes are the unknowns in a linear system of first order continuity equations. In a correctmechanism model there are as many equations (prescribed deformations, vector ep) as unknownpartial derivatives (the total amount of co-ordinates of moving nodes, vector xc). Following theFEM-approach the mechanism input is also to be modelled as a prescribed deformation. In thegripper mechanism this concerns the elongation of the pneumatic cylinder.To calculate all unknown partial derivatives, co-ordinates with respect to deformations, thematrix called Dc(the known coefficients of the first order continuity equations) can be invertedoxcoep? Dc?14and this determines implicitly the kinematic transfer function of first order as one column (ormore columnsin case of a multiple DOF mechanism) of the inverse of matrix Dc.Applied forces (vector fc) can be exerted at the co-ordinates. Their amount of virtual work willbe consumed by the internal forces (vector rp), which should be regarded as multipliers for theprescribed deformations. This equilibrium condition yieldsrp DcT?1? fc5known in the FEM for stress analysis of statically determined structures. Eqs. (4) and (5) showclearly the dual use of the maps: the matrix Dccan be used both for position analysis and for force1514A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 15091522analysis. The deformation modelled for input has a corresponding r, which is then the drivingforce. As with a pneumatic cylinder, this force should be interpreted either as tensile or com-pressive force.Position analysis of the mechanism needs a numerical procedure with prediction and correctionof the co-ordinate values. Starting at a given position (all co-ordinate values given), the input canbe incremented (given a finite deformation), which can iteratively be reduced to zero to find theneighbouring position. The Newton/Rapson method is suited because the required partial de-rivatives are available.A spring element, in the form of a coil spring, can be modelled using the continuity equation forthe length of a truss element. Now the internal normal force rkis to be given as a function of thelength, which means a spring characteristic must be known. Length of this spring element shouldnot be prescribed, but can be calculated in the known mechanism position. This spring force canbe converted to applied forces at the connection points, using (3)fkokoxk? rk6The theory given above is available in a computer program 11, which has been used for theinvestigations.5. Force amplification on the driving cylinderThe driving concept assumes a fixed stroke of the pneumatic cylinder. If the end position can bedetected with a simple on/offswitch the required contact forces between jaws and twistlock can beguaranteed by (preloaded) springs.The jaws need a relatively wide opening (see Fig. 2b) and a high force at the end of the stroke tohold the twistlock. This combination tends to both a large cylinder diameter and a large stroke.Force amplification, such that the high forces apply only when needed, can reduce the cylinderdiameter. This is advantageous for space occupation of the moving end-effector. Not just thediameter, but also the overall length decreases because the piston length, bearing and end cap areshorter. A second advantage is the decreased volume of the air supply.To investigate the force amplification, mainly intended to help to choose the driving cylinder, anumerical experiment has been performed. Having in mind the mechanism of Fig. 3c and springsat the jaws (like in Fig. 3b), the whole subsystem of the two springs and the twistlock can bereplaced by one spring (see Fig. 4). With d as the width of the twistlock the spring characteristiccould be chosen as follows: Length greater than d 22 mm (11 mm clearance at both sides): the applied forces are zero. From d 22 to d 20 mm the applied forces build up to 400 N (the preload). From d 20 to d mm the applied forces increase linearly to 800 N.The closing motion including force analysis according the FEM-theory has been performedusing the mechanism model in Fig. 4. Some trials have been made before the final dimensions ofthe gripper mechanism were chosen. The spring force at the jaws and the driving force of theA.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 150915221515cylinder have been depicted in Fig. 5 for the largest and the smallest width of the twistlock. Thegraphs have been marked with DRIVE_57 (driving force for collar width d 57 mm) andGRASP_57 (contact force for collar width d 57 mm) etc. Apparently a driving force of about500 N is sufficient to fully compress the spring being responsible for the grasping force of 800 N.Assuming that a standard air pressure of 6 bar is available, the information obtained from thisinvestigation is sufficient to choose a cylinder diameter.6. Construction of the gripperThe gripper includes a device to rotate the cones of the twistlock while it is being grasped. Therotation device needs to occupy some space between the gripper links. Therefore it is advantageousFig. 4. Mechanism model for motion and force analysis.Fig. 5. Driving force and grasping force.1516A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 15091522to apply two identical gripper mechanisms, at parallel planes aside the rotation device, which canbe connected by the jaws (see Fig. 6, the rotation device and the frame are left out this figure).Now it is natural to use driving cylinders for each of the two parallel mechanisms. These twocylinders together will produce the total driving force. This embodiment avoids space conflictswith other driving elements of the gripper. The overall length of these smaller cylinders can bedecreased again and will reduce space conflicts with the gripper?s surroundings.The experiment of the previous section has also provided the normal forces in the connectionbars numbered 1 and 2 in Fig. 4. These bars have been selected to act in reality as the preloadedsprings, with the intention to generate grasping forces comparable with those presented in Fig. 5.It was decided to choose the following spring characteristic for the two bars (l0is the preloadedbut free length): Length l0 1 mm: spring force zero (actually loss of contact). Length from l0 1 mm to l0: building up to 80% of the preload. Length from l0to l0? 2 mm: building up linearly to 100%. This part determines the actualspring constant. Shorter length will numerically be accepted (no matter the technical realization of the spring).7. Compliance investigationsThe situation where the twistlock is grasped will be investigated with respect to side forces, aswould be caused by the oscillating container. Wrenching is left out the investigations. Since theFig. 6. Embodiment of the gripper mechanism.A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 150915221517gripper contains two identical mechanisms in parallel planes, only one of these mechanisms needsto be considered further. The driving force and the forces at the twistlocks are thus less by half.The driving cylinder of the mechanism is completely out, and one end-point of each springmay be considered as a fixed point. The model of this mechanism is depicted in Fig. 7.7.1. Investigation 1: compliance without frictionThis situation occurs typically for flat objects, which can easily have friction coefficients higherthan b=d (the ratio of height and width of the grasping object, here the collar of the twistlock).The twistlock will tumble slightly and no slip occurs. The twistlock can be modelled as a simplebar, which is connected with pin-joints to the jaws (Fig. 7).The symmetry axis of the twistlock will be moved from left to right over a distance of 20 mm(from )10 to +10 mm). The force required for this movement, the driving force should be re-garded as resistance force. The stiffness is the reverse of the compliance. The result of the FEMbased analysis is presented in Fig. 8. It clearly shows that the resistance force grows suddenlywhen one of the jaws looses contact, and then proceeds with a lower stiffness because only one ofthe springs is active. This type of behaviour is typical for grippers with preloaded springs. It can beremarked further: The sudden jump is advantageous because it can consume extra energy during overload. In the case of twistlock removal the gripper is upside down. The twistlock does not directly fallout of the gripper when one of the jaws looses contact. The force characteristic is not precisely symmetrical to the origin. The two springs are the sameindeed, but the link chains left and right are not. They therefore do not equally contribute tothe resistance force. If this matters, the l0of both springs could be given unequal values. A small width seems to have little compliance. There is however more clearance between twist-lock and corner casting of the container.Fig. 7. Mechanism model for compliance investigation.1518A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 15091522A rough conclusion is that this gripper arrangement can function properly even with a mis-alignment of 3 mm. Dynamic overload protection can be much larger, but depends on thetechnical properties of the spring, like the minimum compressive length.7.2. Investigation 2: compliance with frictionIt will be assumed that the twistlock slips along the parallel jaws. This happens not only forrectangular objects that have a high value for b=d, but also when the object has a fixed orien-tation. The latter situation applies certainly when the twistlock is still in the corner casting of thecontainer. The model of the mechanism may be changed like in Fig. 7, alternative twistlockmodel.The symmetry axis of the twistlock will be moved from left to right and then back to the left,with the same displacement as in the previous experiment. Because the software available for themotion and force analysis does not yet support friction forces, the friction needs to be added in apost-processing action to obtain the driving forces with friction. This can be done as follows.Suppose F0is the driving force without friction in the mechanism model of Fig. 7. The contactforces at the left-hand and the right-hand jaw appear in the FEM approach as internal forces(normal forces) called further rLand rR.With s as the slip and l as the friction coefficient, the slip forces rw, which are of course re-garded as internal forces, can be expressed asrwL sign_ s sL ? lL? jrLjrwR sign_ s sR ? lR? jrRj7Fig. 8. Compliance without friction.A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 150915221519They contribute to the driving force, acting along displacement direction x, in accordance with thebalance of virtual workDF rwLdsLdx rwRdsRdx8The friction influences not only the driving force, but also the internal forces, and among themthe normal forces at the jaws. The friction equations (7) can be substituted into equation system(5) for rp, but this system will then become non-linear. This could be solved into the programitself, but not in a post-processing phase.For this purpose of investigation the normal forces can be estimated with the result of theprevious mechanism position. The horizontal force equilibrium of the twistlock in position i issatisfied accurate enough byrLi r0Li DFi?1andrRi r0Ri? DFi?19where r0indicates the normal force without friction.The driving force F F0 DF has been calculated for a width d 60 mm of the twistlock, fora range of friction coefficients l. The result has been depicted in Fig. 9. A hysteresis loop, which istypical for the friction effect, is clearly present. This effect is advantageous because of reasons like it helps to suppress the container oscillations, although the effect is probably small; it can prevent the system of the gripper holding a twistlock to oscillate; and It provides some initial resistance force against side movements of the twistlock. In this partic-Fig. 9. Compliance with friction.1520A.J.G. Nuttall, A.J. Klein Breteler / Mechanism and Machine Theory 38 (2003) 15091522ular case the gripper must place the twistlock upside down in a storage position. Halfway thismanipulation task the symmetry axis will be horizontal. From the picture it can be concludedthat a twistlock with a mass of 8 kg will not slip due to gravity forces if the friction coefficientl 0:3 or more.It can be concluded that friction is not only required to hold an object between parallel jaws,but it can also provide extra resistance against side forces. This compliance effect is extra to theelastic compliance of built-in preloaded springs and depends on the friction coefficient and thefirst order kinematic slip. The latter is a property of the gripper configuration, which could bedesigned with respect to the mentioned effect.8. Experimental verificationAn experimental twistlock manipulator was built using the described gripper configuration(Fig. 6) and it was tested for different types of twistlocks. During the test the manipulator proveditself, being capable of a secure and reliable grip using a simple sensing and control system.Tests were also done to give an indication of the gripper compliance. For these tests the pre-cision was measured with which the gripper has to be positioned at the pickup or release point fora problem free manipulation. Before the manipulator was given the command to get or put away atwistlock the position of the corner casting was moved from the predefined position with a staticoffset. The maximum amount of static offset was measured that would still allow a problem freemanipulation. The result for two twistlocks with different collar widths (59.5 and 62 mm) is twistlock 1 (collar width59.5 mm), maximum offset: 7 mm; twistlock 2 (collar w
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