a new parallelogram linkage for gravity compensation using torsional springs

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Nica,W UM.y 1992A New Parallelogram Linkage Configuration for GravityCompensation Using Torsional SpringsAjay Gopalswamy, Pramod Gupta and M.VidyasagarCentre for Artificial Intelligence and Robotics (CA IR)Ra j B havan Circle, Bangalore 560 001 INDIA

AbstractA11 articulated robots sufle r fr om th e adverse eflectsof gravity loading, namely increased actuator size anddegraded performan ce. Parallelogm m linkage manip-ulators are uniquely suited for counterbalancing tech-niques due to the decoupled nature of gravity terms intheir dynam ic equations. In this paper, we present anew parallelogram linkage configuration where the twoactuated degrees-of-freedom in the vertical plane aregravity compensated using torsional springs. We showthat the robot design is thus improved in two ways:i) the peak torques required to be output by the QC-tuators are reduced, ihereby allowing smaller motorsto be selected and ii) by ensuring that the equilibriumposition for all robot joi nts assume safe configura-tion on power-off i.e. position where the robot linksdo not collide with one another or with other objects,the need for fail-safe brakes is eliminated. A threedegree-of-freedom direct drive robot using the proposednew configuration is under development t CAIR todemo nstmte the feasibility of these concepts.

1 IntroductionClosed loop linkages have been widely adopted inindustrial robots to enable the drive motors to be 10cated near the robot base. The most popular of the

closed loop linkages is the parallelogram mechanismbecause its kinematics and dynamics are easy to ana-lyze. The line diagram of the three degree-of-freedomd o f ) parallelogram linkage manipulator (character-ized by the MIT Direct Drive Arm I1 11) is shown inare presented below to provide insight into the relativeeffects of the inertial, centrifugal, Coriolis and gravita-tional torques (the manipulator dynamics is derivedbased on the assumptions that - the robot links arerigid, friction at the joints is negligible, gyroscopic ef-fects due to motor rotations are negligible, there areno link offsets, and that the wrist-gripper-workpieceentity is a point mass).

Figure 1. The dynamic equations o f tI, manipulator

Figure 1: Line Diagram of Parallelogram Linkage Ma-nipulator

3)where dij are the elements of the inertia matrix (Cirefers to cosqi, Si to sinqi and Ci j to cos(qi - j ).di i =dl2 =d22 =dm =

d13 =and 4is are given by

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Examining the manipulator dynamic equations for71 and Q one can conclude that gravitational torques41 and 4 2 comprise a significant portion of the torquewhich must be output by the motors. Elimination orreduction of the gravitational torques through coun-terbalancing can lead to a more efficient robot designdue to the following reasons:Smaller motors can be used with counterbalanceddesigns, leading to lighter robots.

Counterbalancing can lead to statically balancedrobots, eliminating or reducing the size of brakesrequired. This is an important aspect of coun-terbalancing for two reasons. First, reduction inbrake size leads t o substantial weight saving, es-pecially in direct drive robots, since in the absenceof any friction in the joints, the entire gravity loadhas t be supported by the brakes themselves.Secondly, fail-safe brakes are 'normally closed'in operation, implying that brakes are functionalwhen power is off. During normal robot opera-tion, power supplied to the brakes acts to disen-age them. Thus large amounts of power must beiissipated by the brakes leading to overheating atthe joints and degrading performance.Due to static balancing, torque required by themotors to maintain a commanded position atstandstill. is reduced. Subsequently, heat dissi-pation in the motors is reduced.In the event that the robot is perfectly coun-terbalanced, gravity terms in the dynamic equa-tions of the manipulator can be ignored, reducingthe computational burden for implementation ofmodel-based control systems.

Reference [2] cites various techniques to countergravity torques: (i) Mass counterbalancing (ii) Springcounterbalancing and (iii) Controlled force counter-balancing. Mass and spring counterbalancing are byfar the most popular means of gravity compensationin manipulators, being passive in action. In contrast,force counterbalanced systems make use of actuatedmechanisms to compensate gravity torques and areusually restricted to massive robots which handle largepayloads. Spring counterbalancing has the advantageover mass counterbancing in that a different payloadwill simply result in a changed manipulator equilib-rium position. With mass counterbalancing, the ma-nipulator links will collapse when a different payloadfrom the designed value is used; direct-drive robotswill collapse rapidly, while geared robots will collapseyore slowly due to friction present in the transmis-sion. Another disadvantage of mass counterbalanc-ing is that it requires either a heavy counterweightor a large moment arm. The former results in in-creased robot weight whereas the latter causes the us-able robot workspace to be reduced.2 Theory of Gravity Compensation us-ing Torsional SpringsOriginally, the idea of using spring counterbalanc-ing for a five-bar linkage manipulator was suggested

in [3]. The unique property of the parallelogram link-age manipulator which makes counterbalancing conve-nient is that the gravity term for joint 1 (41 is inde-terbalancing technique is to affix a torsional sprinbetween the frame of motor 1 and link 1, as depict2in Figure 2, such that the spring is in its undeformedstate when q1 is 90 (i.e. link 1 s vertical). At this p e

pendent of joint 2 position and vice versa. I,e coun-

Figure 2: Torsional Spring Affixed to Link 1sition (where q1 is go ), the gravity torque experiencedby joint 1 is zero. Any deviation from this positionresults in a torque that tends to pull the joint awayfrom its 90 position, causing the spring to deform bycoiling or uncoiling and exert a restoring torsional m ement. The spring can be designed in such a way thatthis torsional moment cancels out the gravity torque,resulting in a statically balanced design about joint 1However, since the spring characteristic is linear, andthe gravitational torque is non-linear, being a functionof the cosine of the angle q1, the two cannot exactlycancel each other out. By restricting the range of q1such that the difference between the slope of the ap-plied torque versus spring deflection (assumed to belinear) and the gravitational torque is substantiallyreduced, the peak torque required by the motor canbe reduced. Other techniques have been proposed inthe literature [4,5,6], where linear springs are used toobtain exact gravity compensation over the completerange of motion of the joint. However, all three meth-ods make use of additional components like gears, beltdrives or linkage mechanisms, increasing both systemcost and complexity. Torsional spring counterbalanc-ing seems to be superior to the above mentioned linearspring counterbalancing methods because of its sim-plicity. The modified gravitational term for joint 1with the torsional spring affixed to the manipulatoris:41 = (m11c1+m31c3+m4h

(6)?rmpll)gCl h(q1 - ,where tl is the torsional spring constant. In order todetermine k l , the following procedure is adopted:

1. The motion of joint 1 is restricted to the rangewhere the profile of the cosine function of q1 ap-

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2.

proaches a straight line. The range of motion,however, must not be so restrictive as to reducethe useful workspace of the manipulator.kl is expressedas7lg mll,l+m31,3+m411+mpll>so that Equation (6) may be rewritten as41 = mllcl m3lc3 m411

(7)17+mph g[C1 + 7 l b 1 - 3where, 71 is a non-dimensional constant whosevalue lies between 0.6 and 1.0 for practical mo-tion ranges of q 1 . The torsional spring is designedfor the maximum rated payload of the manipula-tor, since the peak torque to be exerted by joint 1occurs when payload is maximum. Then, by min-imizing the expression C1 + yl q1- t betweenthe specified joint limits the gravitational torqueis minimized.

While joint 1 is counterbalanced using the abovetechnique, a valid question is: Can join t 2 also begravity compensated using a similar technique o r oth-erwise? Rewriting 4 2 with the torsional spring term,and employing the same technique for expressing kzas before in Equation 7) for AI we get4 2 = g[m2lc2 m312 - m41c4

- mpQ4- 12)1[C2 +YZ P2 - 31 (8)It is clear that for m41,4 +mp 14- 2) > m2lC2+m3l2,the procedure outlined above would yield a negativespring constant. With existing manufacturing tech-nology, it is not possible to realize a negative springconstant torsional spring. In [3], joint 2 is mass coun-terbalanced by assuming tha t the parallelogram link-age manipulator is used in applications where varia-tions in payload are negligible and by satisfying thefollowing equation (obtained by substituting mp = 0in Equation 5) and equating 952 to zero):m21c2 -k m312 = m41c4However, there is no reason to exclude parallelogramlinkage manipulators from pick-and-place operationswhich form an important class of robot applications.We have come across at least one citation in the lit-erature 71 where a parallelogram linkage manipulatorAs robot links become lighter through use of advancedmaterials e .g . carbon fibre composites) and thrdughimproved optimisation methods e .g . finite elementanalysis), the load that a robot is capable of han-dling will comprise a significant percentage of its ownweight. Hence, the mass counterbalancing techniquecan be applied on1 for a particular payload (maxi-mum rated payload$, leaving the joint unbalanced forother payloads.3

(9)

is used t or machine loading and unloading operations.

Gravity Compensation for Joint 2 -A New ConfigurationWe propose a new configuration for the parallelo-gram linkage manipulator to enable gravity compen-sation for both joints 1 and 2. The line diagram of

the proposed new configuration is shown in Figure 3.With reference to Figure 1, this configuration is ob-

Figure 3: The New Parallelogram Linkage Configura-tiontained by moving the motors from the common axisof links 1 and 2 to the common axis of links 3 and 2.The st ructure of the dynamic equations of the newconfiguration is exactly the same as in Equations 1)-3); however, some elements of the inertia matrix andthe gravity term, 4 2 are changed and are presentedbelow (all symbols refer to Figure 3):

dl2 =d22 =d a =

and

Since 4 2 is now positive for all possible values of robotlink parameters, it is possible to design a torsionalspring using the method outlined in Section 2 forjoint 2 also. Peak torques for motor 2 is consequentlyreduced and the need for a fail-safe brake is eliminated.An immediate observation can be made about the newconfiguration by observing the dlz term of the iner-tia matrix t is not possible to decouple the inertiamatriz using the technique originally suggested intechnique applicable only for negligible payload vari-aiions) is to design the robot links such that dl2 .re-duces to zero. However, consequent to our contention(in Section 2) that a parallelogram linkage manipula-tor need not be restricted to applications where thepayload variations are negligible, by examining theexpression for d12 in Section 1, it can be concludedthat it is not possible to decouple the inertia mat riz of

for the conventional parallelogram configuration. T

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the con ventio nal pam llelogm m configuration itself forpick-and-place op emiion s. Other issues, as identifiedbelow, need t be resolved before the new configura-tion can be practically realized.0 It must be verified that the greater moment armof link 4 as seen by joint 2 in the new configuration

does not cause an increase in inertia, centrifugaland Coriolis torques, offsetting the advantages ofcompensating the gravity torque on joint 2.The theory of torsional spring counterbalancingassumes that the range of motion of joints 1and 2is restricted between a range where the cosinefunction approaches linearity. However, if therobot links are coplanar, and not offset (as n theMIT direct Drive Arm II), q1 and q2 must satisfythe following identity to avoid collisions betweenthe links:

where 8 is the smallest included angle at thevertices of the parallelogram linkage before thelinks collide. The range of motion of joint 2 willhave to be different from the range of motion ofjoint 1; the cosine of 42 will therefore deviatesignificantly from a straight line. Joint motionsfor the robot must be designed to satisfy oppos-in requirements i) maximum workspace and(iif constrained joint motions for spring counter-balancing.One of the advantages being mooted of the newconfiguration is that the need for fail-safe brakesis eliminated. The springs must be designed insuch a way that on power-off, the manipulatorassumes a 'safe' configuration i.e. the robot linksdo not collide with objects in its environment aswell as with one another.

To examine these issues, a parallelogram linkagerobot incorporating the propoeed ideas is being devel-oped at CAIR.4 Design of CAIR Direct Drive RobotA computer generated schematic of the CAIR robotis illustrated in Fi ure 4. The robot joints are to be ac-tuated by direct frive motors in anticipation of usingmodel-based schemes for the control of the manipu-lator; absence of friction and backlash in direct-driverobots results in better correlation between the mod-elled and actual robot dynamics. It should however benoted that the theory developed above isequally validfor geared robots. Joints 1 and 2 are to be actuatedby 60 N-m motors, while joint 3 is to be actuated by a100 N-m motor. All three motors are of the shaflless,ouiside-rotor, hollow-stator construction. The maxi-mum rated payload of the CAIR robot is 7.5 kgs.The link parameters of the CAIR robot are listedin the following table:

Figure 4: Schematic of the CAIR Direct Drive Robot

4.1 Equilibrium PositionAs mentioned in Section 3, the equilibrium posi-tions of joints 1 and 2 must satisfy Equation (11) toprevent the links from colliding with one another (forthe CAIR robot, 8 = 30 ). If the undeformed posi-tions of the torsional springs of joints 1and 2 are 90,the equilibrium position of both the joints are identi-cal at 90 over a range of payloads. This leads to aphysically invalid manipulator configuration. In orderto prevent such an eventuality, the unstrained positionof the joint 2 spring is k e d at 60 . This particularvalue of the undeformed spring position for joint 2 en-sures that Equation (11)is satisfied without diminish-ing the advantages of gravitational torque reduction.Figure 5 shows variation of the equilibrium positionfor joint 1 for different payloads. The stable equilib-

Figure 5: Equilibrium Positions of Joint 1for VaryingPayloadsrium positions are the points at which the 41 versusq1 graph croeses the x axis with a negative slope. As



the payload is increased from 0 kgs the equilibriumposition remains unchanged at 90, till a certain 'crit-ical' payload value is reached. At the critical payload,the 41 versus q graph exhibits an inflection, whenthe tangent to the curve at q1 = 90 coincides withthe abscissa of the graph, i e

Beyond the critical payload (analytically determinedto be 4.72 kgs by solving Equation (12)), the equi-librium position trifvrcates into two stable and oneunstable equilibrium position.Joint 2 exhibits a single equilibrium point whichvaries continuously as the payload increases from 0 to7.5 kgs (Figure 6). The equilibrium position(s) of

E

I.IO 4 -IO 10 2 1 U Hqz

Figure 6: Equilibrium Positions of Joint 2 for VaryingPayloadsjoint 1 is plotted against that ofjoint 2 in Figure 7 forpayloads in the range 0 - 7.5 kgs. The graph trifur-

Figure 7: Equilibrium Positions of Joint 1and Joint 2for Varying Payloadcates at the critical payload value of joint 1 indicatingexistence of one unstable and two stable equilibriumpositions for joint 1when the payload at the manipula-tor tip exceeds the critical payload value. The equilib-rium position that the manipulator actually assumes

depends on the initial position of joint 1. However,both equilibrium positions satisfy Equation (11).4.2 WorkspaceThe workspaces of the new and conventional par-allelogram configurations is plotted in Figures 8 (a)and (b) respectively to compare their reach and acces-sible regions. The workspace of the new configuration

Figure 8: Workspace of New and Conventional Paral-lelogram Manipulator Configurationsis obtained by plotting the position of the manipula-tor tip for q1 and 42 varying between their joint limits(15" 2 q 5 150" and -30" 5 42 5 105' respectively)and ensuring that Equation (11) is satisfied. Since thebase motor (joint 3) simply rotates the vertical planecontaining the parallelogram structure about a verti-cal axis, the effect of 43 on the workspace is ignored.The workspace of the conventional parallelogram con-figuration is plotted by constraining joint 1 (which isspring counterbalanced as in the new configuration)to move between 15" and 150, and allowing joint 2 tomove freely without any counterbalancing constraint?.For the conventional configuration, Equation 611) ismodified as follows to ensure that the links o notcollide with one another.

From the workspace plots, it can be clearly seen thatwith the new configuration the maximum reach is in-creased by almost 20%. However, the conventionalmanipulator is able to better access regions closer tothe robot base.4.3 Joint TorquesIn the new parallelogram configuration, link 4 piv-ots at the far joint for joint 2 motions, whereas inthe conventional configuration, link 4 is pivoted at thenear joint. Consequently, the inertia seen by motor 2is numerically greater in the new configuration. In or-der to verify if the new configuration actually results inlesser joint torques because of both joints in the ver-tical plane being counterbalanced, the joint torquesare enumerated by simulating the dynamic behaviorof the two configurations for the sa me pick-and-placetask. The pick-and-place task is specified in worldspace with data points being specified with respect

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to the ori ins of the 2-y-z coordinate frames in Fig-ures 1 an8 3. The operation involves starting fromrest at ( ~ ~ 0 . 0, y=0.25 m, z=1.0 m} with the mm-imum rated payload,.passing through (0 m,1.25 m,Om} apd ending the pick-and-place task by Coming torest at (0 m,0.25 mi-1.0 } all goal points are ac-cesible to both configurations). The joint anglea cor-respohding to these tip positions are (41 = 80.19',and {80.19,-29.280,180.00} or the new configura-tion and = 60.63', 92 = 147.45', q3 = 0.07 ,{123.56 ,2 6.44,90.00} and {60.63,147.450,180.0for the conventional configuration respectively, deter-mined by solving the respective kinematic equations.Cubic trajectories are specified between the variousgoal points according to standard trajectory planningschemes ( 1 The task is specified to be completed in2.5 secon for which the mmcimum tip velocity andtip acceleration are 2.1 m / s and 0.65 g respectively.Figures 9 and 10 show the joint torquea for the newand conventional parallelogram manipulators respec-tively for the pick-and-place task. It isseen that in thenew configuration, joints 1 and 2 exhibit a reductionin torques; joint 3 orque is almost identical for bothconfigurations for this particular motion trajec ory.

42 = -29.28', 43 = O.Oo , {143.13 ,53.13 ,90.0'}

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Figure 9: Joint Torques for the New ParallelogramManipulator Configuration

5 DiscussionA new parallelogram manipulator configuration hasbeen presented in the paper which enables both jointsactuated in the vertical plane to be gravity compen-sated using torsional springs. Simulations of the CAIRrobot indicate that the new Configuration has manyadvantages over the conventional parallelogram con-figuration including - reduced joint torques, increasedreach and elimination of fail-safe brakes by ensuring'equilibrium pasitions for joints 1 and 2 uch that therobot links do not interfere with one another.One aspect of the CAIR robot design is its modu-lariiy. Referring to Figure 4, ink 4 s composed of ahollow square section module and a detachable hollowcircular section module. The detachable module canbe interchanged with replacement modules of differentlengths, crass sections or materials. A consequence of

O J 1 1J 2i I

O J 1 1J 2

Figure 10: Joint Torques fdr Conventional Parallelegram Manipulator Configurationthis modularity is that the manipulator can be mademconfigumble. By affixing the detachable cylindricalportion of link 4 o the rear of the rismatic mod-ration is obtained An automatic implementation ofthe reconfiguration process can also be coqceptualized.The CAIR robot is undergoing fabrication h d current,work is focusing on its real-time control using model-baaed and adaptive approaches.Referencesl] Asada, H., nd Youcef-Toumi, K., Diwct DriveRobots: Theory and Pmc i ice, MIT Press, 1987.[2]Rivin, E., Mechanical Design of Robots, McGraw-Hill, 1987.[3]Huissoon, J. P., and Wang, D., On the Design ofa Direct Drive 5-Bar-Linkage Manipulator, Robot-ica, 1991, o be published.[4]Petrov, B.A., M an i pu l dors , Mashinostroenie Pub-lishing House, Leningrad, cited in [2].[5]Herve, J.M., Design of Spring Mechanisms for Bal-ancing the Wei ht of Robots, Proceedings of theSixth CISM-IF#OMM Symposium on Theory andPractice Robots and Manipulators, RoManSy6,Cracow, Poland, 1986, edited by Morecki, A.,Bianchi, G., and Kedzior, K., pp 564 - 567.[6]Mahalingam,S., and Sharan,A.M., The OptimalBalancing of the Robotic Manipulators, IEEE In-ternational Conference on Robotics and Automa-tion, 1986, p 828 - 835.[ Meyer, J.D.,-An Overview of Fabrication and Pro-cessing Applications, in Handbook of IndustrialRobotics; ediied by Shimon Y. Nof, John Wileyand Sons, 1985.[8]Craig, J .J., Introduction to Roboiics: Mechanicsand Conirol, Addison-Wesley, 2nd Edition, 1989.

ule of link 4, he conventional parallef gram configu-

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a new parallelogram linkage for gravity compensation using torsional springs

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