1988_IEEE Trans.on Robotics and Automation.vol.4 Issue 2 Pp.361-367

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO . 3, JUNE 1988 36 1Dynamic Analysis of a Three-Degrees-of-FreedomIn-Parallel Actuated Manipulator

KOK-MENG LEE AN D DHARMAN K. SHAHAbstract-Despite the voluminous publications on robot dynamics and

control, the literature to date is based solely on the serial link manipula-tors. Little attention has been given to the alternative manipulator designbased on the concept of in-parallel actuated mechanism, which ischaracterized by its excellent rigidity, high strength-to-moving-weightratio, and relatively simple inverse kinematics. This communicationpresents the dynamic analysis o f a three-degrees-of-freedom in-parallelactuated manipulator. Th e equations of motio n have been formulated injoint space using Lagrangian approach. The analysis provides thesolution to predict the forces required to actuate the links so that themanipulator follows a predetermined trajectory. A dynamic simulationprogram which has been developed illustrates the influence of the linkdynamics on the actuating force required. The dynamic analysis providesa basis for future theoretical research to develop the control sch eme, f orexperimental research to estimate the inertia parameters, and for designoptimization of the prototype manipulator.

I. INTRODUCTIONRecently, some effort has been directed towards alternativemanipulator designs based on the concepts of closed kinematic chainmechanism to improve manipulator rigidity and strength-to-weightratio. The closed kinematic chain mechanism, in general, hasrelatively simple inverse kinematics as compared to the conventionalopen kinematic chain mechanism. The closed kinematic chainmanipulator has potential applications where the demand on work-space and maneuverability is low but the dynamic loading is severeand high speed and precision motion are of primary concern.Typical examples of in-parallel mechanism are the camera tripodand the six-degrees-of-freedom Stewart platform. T he Stewartplatform was originally designed as an aircraft simulator [ 1, later asa robot wrist [2], and as a tendon actuated in-parallel manipulator [ 3],[4]. Various applications of the Stewart platform were investigatedfor use in mechanized assembly [5] and as a compliance device [6].The kinematics and practical design considerations of the Stewartplatform for use as a manipulator were considered in [7], [8]. Asystematic review on possible alternati ve in-parallel mechanisms andother combinations in which part of the manipulator is serial and part

parallel were addressed in [9], [lo]. Recently, some efforts weredirected towards the dynamic analysis of the Stewart platform usingthe Newton-Euler method [l l ] and screw theory [12]. A moregeneral analysis for the six-degree-of-freedom (DOF) m ultiloopparallel manipulators was discussed in [131and [141.Apart from the Stewart platform manipulator, Landsberger and hisco-workers at MIT [4] have constructed a 3-DOF in-parallel tendon-actuated positioner as a first step to Stewart platform implementation.Since the tendon i s essentially massless and the spine of the 3-DOFpositioner is hydraulically driven, the positioner dynamics areprimarily due to the hydraulic servo and the unknown payload.Although no analytical dynamic simulation result was presented,experimental simulation was made to prove the concept feasibility.The 3-DOF positioner has the advantages of being lightweight,simple dynamics and control, and is well-suited for tasks (such aslifting objects or welding) where the demands on compression loadManuscript received November 3, 1986; revised October 21, 1987. Thiswork was supported by the Georgia Institute of Technology under the generalresearch fund and by the Computer Integrated Manufacturing Systems (CIMS)Program.The authors are w ith The George Woodruff School of MechanicalEngineering, Georgia Institute of Technology, Atlanta, GA 30332.IEEE Log Number 8718915.

are of less concern as compared to that on tension load through thetendons.Small working space is commonly recognized as one of thedrawbacks of the Stewart platform manipulator as compared to aserial link manipulator. The authors have perform ed the kinematicanalysis of a 3-DOF tripod-like manipulator which has two orienta-tion freedoms and one translatory freedom based on the concept of in-parallel actuated mechanism [151, [161. In particular, the closed -formsolution of the inverse kinematics were presented and the influencesof physical constraints on the range of motion were discussed in [151.Also, various potential applications where the 3-DOF in-parallelactuated manipulator may be used as part of the 6-DOF manipulatorsystem to enlarge the working space were highlighted in [15] an d

This communication focuses on the dynamic analysis of a 3-DOFin-parallel manipulator using Lagrangian approach. In particular, thecommunication presents the formulation of the dynamic equations injoint space and the solutions which determine the forces/torquesrequired to follow a prescribed trajectory. An example of tracing ahelical path is chosen to illustrate the dy nam ic simulation and to showthat the Cartesian position of the moving platform may be controlledat a sacrifice of orientation freedoms. In applications such asunattended precision machining and fixturing, where both highdynamic compression and tension loading are required, the tendon-driven in-parallel actuated manipulator is less rigid than optimum.Hence, the influences of link dynamics are highlighted by means ofdynamic simulation results.

~ 7 1 .

11. KINEMATICSA schematics of the 3-DOF in-parallel actuated manipulator isshown in Fig. 1. The manipulator consists of a base platform, thre eextensible links, and a moving platform which houses the drivingmechanism of the gripper. The moving platform is connected to thelinks by means of ball joints which ar e equally spaced at 120" and at aradius r from the center of the moving platform. T he other ends of thelinks are connected to the base platform through equally spaced pinjoints at a radius R from the center of the base platform. By varyingthe link lengths, the moving platform can be manipulated with respectto the base platform.As shown in Fig. 1 , a base coordinte frame which is designated asX Y Z frame is fixed at the center of the base platform w ith its Z-axispointing vertically upward and the X-axis pointing towards the pinjoint 1 , P 1 . Similarly, a coordinate frame xy z is assigned to thecenter of the upper platform, with the z-axis normal to the platformand the x-axis pointing towards the ball joint 1, b 1. The coordinateframe xyz with respect to the base coordinate frame X Y Z can bedescribed by the homogeneous transformation [TI

where ( x c ,y , , z , ) ~ escribes the position of the origin of the xyzframe and the orientation vectors ( n l , n 2 , n3)', (01, 0 2 , 0 3 ) ~ , an d( a l ,a 2 , a 3 ) are the directional cosines of the axes x , y , an d z withrespect to the base frame X Y Z .In the dynamic analysis of the manipulator, both the inverse andforward kinematics are necessary. As the derivation of the kinematicequations has been discussed in [15], only the result of kinematicanalysis are presented here.A . Inverse Kinematics

In terms of Z-Y-Z Euler angles, it has been determined in [15]that the two orientatio n freedoms of the moving platform with respectto the base platform are the precession angle a and the nutation angle0882-4967/88/0600-0361$01.OO 0 988 IEEE

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO . 3 , JUNE 1988 36 3Local rigidity, however, does not imply uniqueness; multiplesolutions of O 1 , 2 , and O3 corresp onding to a given s et of link lengthsare possible. Th e further mathematical con straint which is necessaryto ensure uniqueness is

oo< e , < 1800.In other words , the Cartesia n position z,of the moving platform mustbe positive or the moving platform must always be on one side of thebase platform. This criterion is a physical constraint on thehardw are,not just mathematical artifacts . The physical constraints areimposed by the range of pin joints and the ball joints, which werediscussed in [15].Fo r a given set of link lengths, the corresponding angles 0; can becomputed numerically from (8)-( lo), which are implicit relationshipsbetween L; and e;, where i = 1 , 2 , 3 .Cartesian Position ofMoving Platform: Since the ball joints areplaced at the vertices of an equilateral triangle, the Cartesian positionor the origin of the xyz rame can be determined as

To determine the orientation of the moving platform, the directionalcosines, which ar e denoted as the components of the vectors n, , anda in the homogeneous transformation [ T I , are expressed in terms of0;. By equating (12) and ( 1 5 ) , he componen ts of the normal vector ncan be determined as

1 - L I cos e,-x,nl= P

(18)where the ball-joint coordinates with respect to the base frame are L , si n O1-2,Pn3=x b , = 1 -L1 co s 81Similarly, equating (13) and (16), the orientation vector o isYbl=o o1=n 2

J 3 - 6 ~ ~OS e2-3y ,Zb l= L1 in 0, (12) 0 2 =1 J 3 P

(19)As the unit vectors n, 0 , and a form an orthogonal set, thecomponents of the approach vector a can be determined as

2Lz si n O2+LI in 01-3Z,0 3 =J? P

z b 2 =L2 sin O2 (13 ) al = n 2 0 3 - 2113a2= -n lo3+o ln3a3= nl o2-n2o 1 (20)

Hence, for a given set of link length, (8)-(10) are computednumerically for the angles 0; . The Cartesian position is thencomputed from (1 1) and the orientation is obtained by computing thedirectional cosines of the axes x, , and z with respect to the baseframe X Y Z from (18)-(20).

&yb3= -- ( 1 -L3 COS 0, )2(14)b 3 =L3 sin &.

111. FORMULATIONF DYNAMICQUATIONSOrientation of Moving Platform: With the Car tesian position ofthe ball joints defined in (12)-(14), the orientation of the movingplatform can be determined by noting that the Cartesian position of The equations of motion which describe the actuating forcesthe ball ioints can also be exmessed as required to cause motion are derived using Lagrangian approach. Inthe following dynamic analysis, the dynamics of the gripper are notincluded and the moving platform is assumed to be a circular plate.The mass of the moving platform is therefore assumed to have thecenter of gravity at the origin of the xyz rame. Since the actuatingforces acting on the links Fl, F 2 , and F3 , are to be found, thenormalized link lengths L I , L2, and L3 , are chosen to be the1 5 )

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364 IEEE JOURNAL OF ROBOTICS AN D AUTOMATION, VOL. 4, NO. 3, JUNE 1988independent generalized coordinates and e l , e2 , and e3 are thedependent generalized coordinates. The three constraint equations by directly differentiating (1 1) with respect to time such that

(25)relating L; an d 0; are given in (8)-(10). ax dL . axc&v = c L L + c--e aLj dt ; = I aei dthe kinetic co-energy of the mechanism can be written as I = 11 12 2T = - M(Xf+jf i f ) + -I , c o : + I ~ ~ w F + Z ~ W : ) where X,(X, , Y,, Z,) an d Vc(X,, Yc,Z,) are position and velocityvectors, respectively. The angular velocity, o ( w x , w u , w z ) , in termsof the generalized coordinates can be determined by noting that the

velocity of the ball joint may be written asvbi= Vc+ ox rj

where r; s the line vector directed from the ith ball joint to the centerof the moving platform. From the geometry, r; ca n be written as

l 32 (21)- m d:Of (26)= I

whereMm mass of the link,d,w,, a y ,w,

mass of the moving platform,distance between the pin joint and the centroid ofeach link,angular velocity of the body axes of the movingplatform with respect to a moving frame parallelto the XYZ frame,moment of inertia of the moving platform about

Due to the symmetry of the circular platform, Zxy, Zyz , an d Is ar eidentically equal to zero. The moments of inertia are

rl = p i

r 2 =(-; + $ j ) pI,, Z I ,

x , y , and z, respectively. r 3 = ( - ; i - $ j ) p . (27)The velocity of the ball joints with respect to the base frame is

I , = Iyy=- 1,, = -Mr2. vb i= i ;4 / ; 8i4,r x Li4/i (28)2 4 where tPli an d 4ziare the unit vectors along the ith link length andalong the axis of the ith pin joint, respectively. In terms of the unitvectors of the base frame , the vectors areThe potential energy of the mechanism is given by3P=Mgz,+ mg di si n Oi . (22) 4/;= cos OIZ+sin OIKi = 1

The Lagrangian equations of motion become

whereC = T - P

I;, i = l , 2 , 3e;, i = 4 , 5 , 6i =

1 d3412= 2 os e2Z-- cos OzJ+sin B2K21 J34[3=jCoS e 3 Z + - c o s 0 3 J + s i n 0 3 K2

and4 z 1 = J4 J3 l J

2 22 -

(30)d3 14,3=3- I - - J.{Fi, i = l , 2 , 3 2n.z t (c-3, i = 4 , 5 , 6 By equating (26) and (28) for the ith ball joint and noting that the unitvector (i, j , k ) may be transformed to (I, J , K) through thehomogeneous transformation [ T I , the angular velocities o ( w x , mu ,w,) can be derived by equating the appropriate vector components asand where F; s the actuating force along the ith link and T j - 3 , = 4,5 , 6 are frictional torques of the ith link in the 0; direction. In thefollowing discussion, the frictional torques are assumed to be zero.The constraint equations are 1co s e 2 - - L2 in e2 BI-X,)

fk(L1, L ~ ,,, el , e2, e 3 ) = o (24))where k = 1 , 2 , 3. Note that fk(L; , ;, i = 1 , 2, 3) are the constraint +a2( $-i2 co s e,+- L 2 sin ez 4 P

+ a 3 ( i 2 inequations (8)-( lo) , respectively. 2

co s e2 . t j 2 - Z c ) - - w yIv . DETERMINATIONF THE VELOCITY COMPONENTS 2JAs the link lengths and the angles Li an d Oi have been chosen as 1U,= -- a l ( - i l os e l +L l si n el . OI-%,)-azPeneralized coordinates, the Cartesian velocity and the angularvelocity must be expressed as a function of L ; and 0; and theirderivatives with resp&t to time. The Ca rtesian position is a functionof L ; and 0; as shown in (1 1). The Cartesian velocity can be obtained +a3(tl sin el+L~ co s el . dl- , ) ] (32)



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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO . 3, JUNE 1988 365

TABLE ITRAJECTORY OF THE SIMULATION

))

1P 2,= -2 [ o , (i i2os e 2 - - L 2 si n 0 2 4 2 - 2 c

(33) @max*'k7\. &

+ 0 3 ( i 2 si n e2+L~ co s e2 . 4, - Z,)].L~ co s e 2 + - L2 si n e2e2- Yc+ 0 2 (-y 2

v. DETERMINATIONF ACTUATINGORCESIn many real-time applications of on-line control of the manipula-tor, the Car tesian positionlorientation and the respective velocity andacceleration of the moving platform a re known or predetermined. It isof interest to determin e the forces required to actuate the links so thatThe kinetic coenergy T, he potential energy P, and the constraintequationsfk, are functions of Li , i, nd their t ime derivatives. From(23), the force required to actuate the ith link is

1I II I

the manipulator follows a predetermined trajectory. 0 t a tb T* t

1 Ua= -- 1 4 d t+z2(34)

where i = 1, . . . 6 and the three unknown Lag rangian multipliers Xjcan be solved from the three simultaneous equations, i.e., (23) with i= 4, 5 , 6 , using Cramer's rule. T he actuating force along the ith linkcan be computed by directly differentiating T, P, an d k with respectto li and its derivatives.

VI . A N EXAMPLEF DYNAMICIMULATIONAn example of tracing a helical path is simulated to illustrate theabove dynamic analysis. Although it is more practical to assume themanipulator has two orientation freedoms in addition to a thirdfreedom in the z direction, the example illustrates that the Cartesiancoordinates of the center point of the moving platform may becontrolled at the sacrifice of orientation freedoms. T he example issimulated with the following assumptions: 1) The pin and ball jointsare assumed to be frictionless. 2) The position variation of the centerof gravity of each link is negligible, i.e., di is constant.The helical path to be traced has a radius r* and a pitch h . Th ehelical path with respect to the base frame can be described by thefollowing equation:

1 .a i = - - 42.. 1 ..a = - - 42

p = c o s - 1 [z ($) 1 1hr*c=zi+-

Z c = Ox = r * co s I$y=r* sin I $z=z i+- thT*

TABLE I1PARAMETERS FOR SIMULATION

where T* is the time required to travel one pitch, t is the timevariable, and zi is any particular starting z . As the center point of themoving platform is to follow the helical path, (3) an d (4) re equatedto the x and y components of (35) yielding

(35) Manipulator Parameters R = 0.2286 m

(36) Helix Parametersa+I$=n?r, n = O , + I , + 2 .. .r*ros p = 2 -+ 1. (37)

Differentiating (36) with respect to time(38)1 .&= -- $2

the rate of chang e of I$ with respect to time is linearly proportional to(Y an d p remains constant with respect to time for a constant r* . Thetrajectory can be planned based on I $ noting that the total timerequired for one helical path must be equal to T*. The parameters ofthe trajectory are summarized in Table I and the parameters used fo rsimulation are listed in Table 11.

p = 0 .5D, = 0.5334 m*M = 0.18 kgm/ M = 0. 5d, = 0.1524111

r* = 0.266 mmh = 0.03048 mz, = 0.3048 mTrajectory Parameters

T* = 1 0 sto = 1 stb = 9 s+mx = 2/ 9 T

Dm is the distance between the center of the moving platform and theintersection between the axis of the ball-joint socket and the normal of themoving platform through the center [15].

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 3, JUNE 1988

1.3 f

0.9I t

1 2 3 4 5 6 7 8 9 10T i m (second)

Fig. 2 . Actuating lengths as a function of time.

m - 0.0 kg

0.55 I 1 I I I1 2 3 4 5 6 7 E 9 10

T i m e (Second)Fig. 3. Computed actuating forces for m = 0.0.

The corresponding actuating lengths are computed from theinverse kinematics as shown in Fig. 2 and the angles Oi are determinedfrom the following relationship:(39)z b isin O i=-L ,

where z b i is given in (15)-(17).The actuating forces for the specified trajectory are computed forthe following two cases: 1) The m ass of eah link is assumed to be verysmall compared with the mass of the moving platform. This isparticularly true for tendon actuated manipulator and is a goodassumption for hydr aulic actuated manipulator with high payload atthe gripper. 2) The m ass of the link is not negligible but m / M = 0.5.The objective is to determ ine the effect of the mass dynamics of the

links. The simulation outputs for m = 0.0 and m = 0.09 kg areshown in Figs. 3 and 4, respectively. The result has shown that theforces required have been increased by approximateiy 20 percent dueto the mass dynamics of the links.

VII. CONCLUSIONThe dynam ic analysis of a 3-DOF in-parallel actuated manipu lator,which is characterized by its excellent rigidity, high strength-to-moving-weight ratio, and relatively simple inverse kinematics, hasbeen formulated using Lagrangian approach. The inverse dynamicmodel, which predicts the forces required to actuate the links so thatthe manipulator follows a predetermined trajectory, are derived injoint space. The actuating link lengths of the manipulator are chosenas generalized coord inate s and the velocity c omponents have beenexpressed in terms of the generalized coordinates.



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IEEE JOURNAL OF ROBOTICS AND A UTOMATION, VOL. , NO. 3, JUNE 1988 3670.85

0.80

0.75

h10g 0708v

a)02 0.65

0.60

0.55 1 , I I I -1 2 3 4 5 6 7 8 9 10

T i m e (second)Fig. 4. Computing actuating force on m/ M = 0.5.

In addition, a dynamic simulation program h as been developed anda numerical example of tracing a helical path has been chosen todemonstrate the dynamic simulation. The result of the simulationillustrates the influence of the link dynamics on the actuating forcesrequired. T he dynamic model, which is essential for feedforwardcontrol of the manipulator, will serve as a basis for prototype design,control scheme development, prediction of inertia parameters. Futurework will include prototyp e design and real-time simulation comp utercontrol scheme development and performance evaluation in anindustrial environment.ACKNOWLEDGMENT

The authors would like to thank Dr. W . Book, CIMS director, forestablishing their contact with the CIMS progra m and fo r providing astimulating research environment.

REFERENCES[ I ] D. Stewart, A platform with six degrees of freedom, Proc. Inst.Mech. Eng., vol. 180, pt. 1, no. 15, pp. 371-386, 1965/1966.[21 W. M. Bennett, A mechanical wrist for a robot arm , B.S. thesis,MIT, Cambridge, MA, 1968.[3] K. H. Lim, Control of a tendon arm, MIT, Cambridge, MA, MITA. I. Memo 617, Feb. 1981.[4] S . E. Landsberger, Design and construction of a cable-controlled,parallel link manipulator, MIT, Cambridge, MA, S . M. thesis, Sept.1984.H. McCallion and P. D. Truon g, T he a nalysis of a six-degree-of-freedom work station for mechanized assembly, in Proc. 5th WorldCongr. for the Theory of Machines and Mechanisms (an ASME

publ.), pp. 61 1-616, 1979.H. McCallion, G . R . Johnson, and D. T. Phan, A compliant devicefor inserting a peg into a hole, The Industrial Robot, June 1979.D. C. H. Yang and T. W. Lee, Feasibility study of a platform type of

[5]

[6][7]

r91

11211131

r141

1151

1181

robotic manipulators from a kinematic viewpoint, J. Mech. Trans-miss. Automat. Des., vol. 106, pp. 191-198, June 1984.E. F . Fich ter, A Steward platform-based manipulator: General theoryand practical construction, Int. J. Robotics Res., vol. 5, no. 2,Summer 1986.K. H . Hunt, Structural kinematics of in-parallel-actua ted robotarms, Trans. ASME, J. Mechanisms, Transmiss. Automat. Des .,E. F. F ichter and E. D . McDo well, A novel design for a robot arm,Adv. Comput. Technol. (an ASME Fubl.) pp. 250-256, 1980 .W. Q. DoDo and D. C. H . Yang, In verse dynamics of a platform typeof manipulating structure, presented at the ASME Design EngineeringTechnical Conf., Columbus, OH , Oct. 5-8, 1986, publ. 86-DET-94.K. S ugimoto, Kinematic and dy namic analysis of pa rallel manipula-tors by means of motor algebra, presented at the ASME EngineeringTechnical Conf., C olumbus, OH, Oct. 5-8, 1986, publ. 86-DET-139.Z. Huang and H. B. W ang, Modeling formulation of six-DO F multi-loop parallel manipulation: Part 2 . Dynamic modeling and examp le,in Proc. 4Th IFTONM Symp. on Linkage and CAD DesignMethods, paper 21, vol. 2-1 (Bucharest, Romania, July 4-9, 1985).Z. Huang and H. B. Wang, Dynamic force analysis of six-DOFparallel multi-loop robot manipulators, presented at the ASMEEngineering Technical Conf ., Columbus, OH., Oct. 5-8, 1986, publ.K. M. Lee and D. Shah, Kinematic analysis of a three degrees offreedom in-parallel actuated manipulator, in Proc. I987 ZEEE Znt.C o nf . on Robotics and Automation (Raleigh, NC). Also, this issue,K. M. Lee, A. Chao, and D. K. Shah, A three degrees of freedom in-parallel actuated manipulator, in Proc. Znt. Conf. on AppliedControl and Idenfifcation (Los Angeles, CA, Dec. 10-12, 1986).G. Vachetsevanos, K. Devey, and K. M . Lee, On the development ofa novel intelligent roboric manipulator, in Proc. I986 ZEEE Znf.Conf. n System, Man-Machine and Cybernetics (Atlanta, GA, Oct.1986). Also, in IEEE Contr. Syst. Mag., vol. 7, no. 3, June 1987.J. J. Craig, Introduction to Robotics, Mechanics and Control.Reading, MA : Addison-Wesley, 1985.

vol. 105 , pp. 705-712, 1983.

86-DET-168.

pp. 354-360.

1988_IEEE Trans.on Robotics and Automation.vol.4 Issue 2 Pp.361-367

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Transcript of 1988_IEEE Trans.on Robotics and Automation.vol.4 Issue 2 Pp.361-367