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A New Approach to the Dynamic Models of Robot M anipulatorswith Closed Kinematic Chains

Shih-Liang WangAssistant Professor

Department of Mechanical EngineeringNorth Carolina A&T State University

Greensboro, N.C. 27411

Absrracr-An &ective way to formulate the dynamic model of arobot with a closed kinematic chain is introduced in this paper.Joint variables of the linkages passive joints are eliminated byloop closure equations, and a dynamic model similar to that ofa serial link robot can be obtained by the work and energymethod. The new approach applies to planar or spatial linkages,and the systematic derivation of the dynamic model can be

executed by a symbolic mathematic software package.

1. Introduction

Closed kinematic chains, or linkages, are used in many robotmanipulators to replace flexible transmission elements likechains and belts. Asada [1,2] developed a 3 dof direct driverobot with a planar five bar parallel linkage. Kazerooni [3]employed a parallel planar four bar linkage for a direct driverobot with the elbow motor as the counterweight of the arm.ORNLs CESARm manipulator [4] used two concatenateparallel four bar linkages to locate the elbow joints actuatorand to accommodate the counterweight of the arm.

The stiff members of the linkages eliminate the problems ofbacklash and compliance associated with gears, chains, andbelts. As a result, the positioning precision of the robot isgreatly improved. Additionally, the dynamic modeling oflinkages is easier than that of gears, chains, or belts as thelatter have the nonlinear characteristics of backlash and hys-terias.

Luh and Zheng [5] developed an algorithm for robots withclosed kinematic chains to formulate equations of motion.The linkage is first cut to a open chain system with a treestructure, and then the equation of motion is formulated foreach joint, active or not. Constraint equations are then ob-tained by connecting the hypothetically cut joints.

Manuscript received October 3 1, 1990. This work was supported in partby U.S. Army Research Office, DAAL03-89-G-0109, and Martin MariettaEnergy Systems, 19X-SE790.

This approach requires many simultaneous equations to besolved. Moreover, a dynamic model, similar to that of aserial link robot, is not available. The dynamic model shouldinclude the inertial tensor, the Coriolis and centrifugal coeffi-cient, and the gravitational coefficient.

An alternative method for dynamic modeling of robots withclosed kinematic chains is presented here to answer the&?problems. Constraint equations are solved first to express therates of passive joints as functions of those of active joints.The rate of passive joint s can hen be eliminated in the equa-tions of motion. Hence a dynamic model in terms of activejoints can be obtained. The dynamic model obtained has aform similar to that of a serial link manipulator. Since th elatter is well documented, it will be presented here first.

2. The Dynamic Model of a Serial Link Manipulator

The equations of motion of a serial link manipulator can beexpressed as [ 6 ] :

Fi = D, 8, +n n

Dij gi +j = l , i # j i j , k = 1

Dijk bj b k +Di (1)

where

Fi is the joint forceltorqueb the joint rate8 the joint accelerationDG he effective inertia of Joint iD, the coupled inertia between Joints i and jDijk the Coriolis and centrifugal terms at Joint iDi the gravitational force at Joint i

The dynamic coefficients, D,, D,, Dijk, and Di, can be de-rived from link properties and equations of motion [ 6 ] .Acalternative way, the work and energy method, is introducedhere which will also be used to derive dynamic coefficientsof closed kinematic chains later.

7803-O078/91/06O0- 18 10$01.OO 01991 IEEE

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The inertial tensor, composed of D, and D,, will be derivedfrom the lunetic energy, and the gravitational term Di will beobtained from the principle of virtual work. The Coriolis andcentrifugal terms, Dijk,will not be discussed here.

The kinetic energy of a serial link robot with n joints can beexpressed as:

n n

T = $5 WT& wi + 1h m, v vi (2)i = 1 i = 1

whereT is the kinetic energy5 the inertial tensor of Link im, he mass of Link iwi the angular velocity of Link ivi the centroidal velocity of Link i

wi and vi are functions of joint rates and angles, and they canbe expressed as:

i- 1

p = l

i-1

p = 1

wi = e,k,

vi = 1 6, k, x rp,i (3 )

where k, is the unit vector in the direction of Joint p, rp,i sthe distance from Joint p to the centroid of Link i, and is afunction of inboard joint angles. All vectors are defined inthe world coordinate frame.

Substituting (3) to (2) and collecting terms, we can get:

n n

T = 95 D, e,'+ 55 1 D, e,bj (4)i = l j = l , i# jwhere

n

Dii = c I ,& + nf, rp,Trp,i)p = i

n

D, = 1 ($T I, kj + mp rpA%jirpd) here j > i ( 5 )P = j

The gravitational coefficients can be obtained by the principleof virtual work.

n

Di de i + 1 j dzj = 0, for i = l,..,nj = l

where f j is the weight of Link j, and 3 s the z component ofLink j's centroid coordinates. Rearrange this equation, weget:

3. The Dynamic Model of a Robot Manipulator withClosed Kinematic Chains

Similar to ( l ) , he dynamic model of a robot with a closedkinematic chain can be expressed as:

n n

1 D,' e , +i = D: 8, + DijC bj 4, + D:j = l , i # j i j , k = 1

where the superscript c denotes closed kinematic chains.

By cutting a closed h e m a t i c chain into two serial llnkchains, the dynamic coefficients of the closed kinematicalchains, D,' and D;, can be expressed in terms of coeffici-ents of open loop chains, D, and D,, as:

n n n n

D,'=

1[1 D, (dO,/deJ (ae,/d4)]r = l s = 1 t = l u = l

n n

D = - 1 [ D, (aejaejlr = l s = l

The relations of dO,/dO, and dO,/dt9, can be derived from loopclosure equations.

A planar closed kinematic chain with two degrees of free-dom, i.e. a five bar linkage, is discussed here for illustration.The linkage can be viewed as two connected serial links asshown in Figure l (b) . Velocities wi and vi can be expressedas functions of joint rates as:

w1 = d , , w2 = eI2, w4 = e5, wj = e4 5

v, = rlbl,

v4 = r46,,

v2 = (dI2bl2 + r;b,? + 2d,r,~od,b,b,J'~

v3 = (d,28,2 + r2b4; + 2d4r3~~s8405045)'h. .

Then the kinetic energy can be expressed as:

T = 1hl,6,2 + 1hm,r,26,2 + Lh12b,,2 %m2(dI2bl2r?b,; + 2d,r,cos02~,b,J + Ih13e4? + 'hm,(d,28,2+ r,2b4,' + 2d4r3cos04b,b4,) + 1h14b52 1hm4r42b52

Collecting terms with respect to joint rates, we can rewritethis equation as:

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These two equations can be used to solve for 8, and 8, interms of 8, and e the joint rates of the two active joints.

8, = [[d,~in(8,$~,)/d,~in(8,$,J]- 13 8 1 +[d,~in8,/d+in(e.&,J] 8 5

m < y 4

9 1 95~

/ // / 8, = [ [d ,~in(8 , ,$~) /d ,~in(8~~~,J] - l ]~~d 5 [d4sin8,/d3sin(8,~,J] 8,

(a>The relation of a8i/a8j can then be obtained from these twoequations. Substituting these relations to (7), we can get theinertial coefficients Dl lC , D5;, and D,,.

The gravitational coefficients can be obtained by the principleof virtual work and be expressed as:

D,= = -f,r,cosO, - f,d,cos8, -f,r,~0~8,(a8,/a8,)f , r,~~~~, (ae , /ae , )

Figure 1. A Five Bar Linkage and Its Serial Link ModelD, = -f,r,cos8,

-f,d,cos8, -f,r,~0~8,(a8,/a8,)

-f,r,~~ss,(ae,/ae,)

D,; = D, + D,,(a8,/~38~)~ 2DM(a8,/a8,) + D,(a8,/a8,)2DIJC = DI2(a8,/a8,) + D22(a82/a8,)(a82/a8,) + DM(a8,/a8,)

+ ~,,(ae,/ae,)(ae,/ae,) (7) D, = D, + D2(a8,/a8,) + D3(a8,/a8,)

Alternatively, DI C nd D can be expressed in terms of Disas:

where D, is the inertial coefficient of the two serial linkmodels and can be expressed as:

D,, = I, + mlr12 +I, + m,(d12+r,2+2d,r2cos8JD, = I, + m2r2D,, = I, + mz(r,2 + d,r,cos8,)

D = D, + D3(a8,/a8,) + D2(a8,/a8,)

whereD, = -f,r,cos8, - f,d,cos8,, D, = -f,r,cos8,

D, = -f,r,cos8, - f4d4cos8,, D, = -f,r3cos8,

D, = I, + qr : +I 3 + m,(d,2+r,2+2d4r3~~~84)D,, = I, + m3r:

A special case of five bar linkage is Asadas five bar paral-lelogram linkage, as illustrated in Figure 2. In this mecha-nism,

Therefore,D,, = I, + m,(r: + d4r3cos8,) 8, - 8 5 = 8 4 - 8 2

To get the relations of a82la81, a84/a8,, a82la85, and a84la85,we can use the closure equation of the linkage. The positionclosure equation can first be expressed as:

ae,iae,= 1, a84la81 = 1,

ae,/ae, = -1, ae,/ae, = 1

d,cos8, + dZcos8,, = d, + d4cos8, + d,cos8,,Substituting these relations into (7) , we get:

d,sin8, + d,sin8,, = d5 + d,sin8, + d3sin8,,D,, = [I , + mlr12] + m,d12 + [I, + m3r]

Taking derivatives of these equation, we get the velocityclosure equation as:

D,; = [I, + m4r:] + m,d,2 + [I, + m2r2]

d,sin8,8, + d,sin8,,8,, = d4sin8585 + d,sin8,,b4, D,; = (-mzdlr2 + m3d,r3)cos8,

These equations agree with those in [2]. As shown in thatpaper, an invariant inertia tensor can be obtained by eliminat-ing D,; with the proper placement of link centroids.

d l ~ ~ ~ 8 1 8 1d 2 ~ ~ s 8 1 2 8 1 2d4cos8,8, + d 3 ~ ~ ~ 8 4 5 8 4 5

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( a>t- c t i v e j o i n t

Figure 2. Asada's Direct Drive Robot and Its Serial Link Model

4. Conclusion

An effective way to formulate the dynamic model of a robotwith a closed kinematic chain is introduced in this paper.Joint variables of the linkage's passive joints are eliminatedby loop closure equations, and a dynamic model similar tothat of a serial link robot can be obtained by the work andenergy method. The equivalent dynamic model of a robotwith closed kinematic chains can be used in conjunction withthe equivalent kinematic model of the robot.

The linkage investigated in this paper is a planar linkage withtwo degrees of freedom. For a planar four bar linkage,which has one degree of freedom, like the one in the CES-ARm manipulator, the procedures to the get dynamic modelis identical.

For spatial closed kinematic chains, the procedures will besimilar except that the loop closure equations are more com-plex. Nevertheless, the systematic derivation of the dynamicmodel can be assisted by a symbolic mathematic softwarepackage like Mathematica [7].

The equivalent kinematic model of the five bar linkage inAsada's 3 dof robot can be represented by a two link serial

model, as illustrated in Figure 2@). The equivalent dynamicmodel is formulated about the two active joint actuators atthe shoulder . The equivalent dynamic model is still related to

the equivalent kinematic model because the joint angles ofthese two models are the same due to the parallelogramconfigurations.

References

[ l] H. Asada, T. Kanade, and 1. Takeyama, "Control of aDirect-Drive Arm, " ASME J. Dynamic Systems, Mearure-ment, and Control, Vol. 105, Sept. 1983, pp. 136-142.

[2] H. A d a , and K. Youcef-Toumi, "Analysi s and Designof a Direct-Drive Arm with a Five-Bar-Link Parallel DriveMechanism," ASME J. of Dynamic Systems, Measurement,and Control, Vol. 106, pp. 225-230, Sept. 1984.

[3] H. Kazerooni, and S . Kim, "A New Architecture forDirect Drive Robots," Proceedings of IEEE Conference onRobotics and Automation, pp. 442-5, 1987.

[4] R.V. Dubey, J.A. Euler, and S.M. Babcock, "A n Effi-cient Gradient Projection Optimization Scheme for a SevenDegree-of-Freedom Redundant Robot with a SphericalWrist, " Proceedings of IEEE International Conference onRobotics and Automation, April 25-29, 1988.

[SI J.Y.S., Luh, and Y-F., Zheng, "Computation of InputGeneralized Forces for Robots with Closed Kinematic ChainMechanisms," IEEE Journal of Robotics and Automation,Vol. RA-1, No.2, pp. 95-103, June 1985.

[6] S . Ezaquirre, and R.P. Paul, "Computation of the Inertialand Gravitational Coefficients of the Dynamics Equations Fora Robot with a Load," Proceedings of IEEE Conference onRobotics and Automation, 1985.

[7] S . Wolfram, "Mathematica: A system for Doing Mathe-matics by Computer," Addison-Wesley, 1988.

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