DYNAMIC ANALYSIS AND DESIGN OF ROBOT MANIPULATORS U S I N G I N E R T I A ELLIPSOIDS

HARUH I KO ASADA Department o f Mechanical Engineering

Laboratory fo r Manufacturing and Productivity Massachusetts In s t i t u t e o f Technology

Cambridge, MA 02139 U.S.A.

ABSTRACT

An analysis of robot arm dynamics and a graphical method of

representing these dynamics suitable for computer aided design is

presented. The inert,ia ellipsoid, which is used for graphically

representing the mass properties of a single rigid body, is extended

to a generalized ellipsoid for a series of rigid bodies such as a

robot arm. By drawing the Generalized Inertia Ellipsoid (GIE) on

a computer display, one can visualize the mass properties and

dynamic behaviour of a robot manipulator. This method is

applied to aid the design of a mechanical arm; the dimensions of

the arm structure and its mass distribution are optimized on the

basis of the evaluation of the arm dynamics displayed on a

graphics terminal.

1, Introduction

Robot manipulators have complicated dynamic behaviour

including interactions between multiple joints, nonlinear effects such

as Coriolis and centrifugal forces, and varying inertia depending on

the arm configuration. Designing a robotic manipulator needs

efficient tools for modeling and analyzing such complicated

dynamics. There have been a number of papers reported which

deal with modeling and analysis of manipulator dynamics in either

Lagrangian formalisms [l - 41 or Newton-Eulers formalisms [5 - 81.

Using these dynamic models it is possible to simulate the response

of a highly nonlinear and coupled robot manipulator, on a

computer. Walker and Orin [9] have developed simulations for

manipulator control system design, in which control strategies and

algorit.hms are tested and evaluated using the dynamic responses

which have been computed. Orlandea and Berenyi [lo] applied a

simulation program for mechanism dynamics (ADAMS) to the

manipulat,or dynamics analysis. Thomas and Tesar Ill] have

analyzed the effect of joint t.orques on arm dynamics in order to

est.imate t.hP torque requirements for selecting actuators.

Recent progress has enabled us to generate the dynamic

equations of robot manipulators efficiently and has made a variety

of simulation techniques possible. However, designing a robot is

not a simple process because of the highly nonlinear and conpled

dynamics of the structure. We need a way of representing the

manipulator itruct,ure and its dynamics so that they can be easily

understood by the designer and so that the structural modifications

necessary to improve the dynamics of the arm are apparent. A

drawback of most current simulation techniques is that,since the

dynamic models contain so many variables and parameters which

are configuration dependent, it is difficult to represent the arm

dynamics in a way which is both comprehensive and yet easily

comprehended. Also, if the computation time for the simulation is

long, the designer cannot easily try out many designs and

modifications. For these reasons, current simulation techniques

may not be efficient tools for designing and evaluating robot

94 CH2008-1/84/0000/0094$01.0001984 IEEE

manipulators.

The goal of this paper is to fill the void between dynamic

modeling and design with a new analytical tool which provides a

representation of the dynamic characteristics of a manipulator.

This analytical tool can then be applied to the efficient design and

evaluation of a robot arm. The inertia ellipsoid [12], which is used

for representing the mass properties of a single rigid body, will be

applied and extended to the dynamic analysis of a manipulator

which consists of a series of rigid bodies. This method, combined

with computer graphics, provides a comprehensive and easily

comprehended representation of the dynamic behaviour of a robot

manipulator.

2. Kinetic Energy [2,3,11]

The inertia ellipsoid is associated with the kinetic energy stored

into a rigid body. In this section, we derive the expression for the

kinetic energy of a manipulator which consists of a series of rigid

bodies numbered 0 to n from the base to the tip (Figure 1).

Motion of a single rigid body is decomposed to a translation with

respect to its center of mass and a rotation about this center. Let

us denote the velocity of translation by vi and the angular

velocity of rotation by wi. Then the kinetic energy tha t the t th

body has is given by 1 1

Ti = - Miv;v. + - w.~I.w. 2 I 2 I

(1)

i Wi=CFb.i. J J

j=1

i

v i=cbj;j j=1

In case of articulated joints, vector a. is a unit vector pointing

in the direction of the j-th joint axis, and b. is the vector product

between a position vector r{ and the unit vector a., as shown in

Figure 1. Substituting eq. (2) into eq. (l), the total kinetic energy

I

st.ored into the series of rigid bodies is then given by

1 . . T = - etHe

2

Fig. 1 Manipulator

where M and I are the mass and inertia tensor respectively,

vi and wi are 3x1 vectors and represents the transpose of a

vector or a matrix. Each body in a series of rigid bodies is

constrained in motion due to the linkage. Motion of body i , for

where e and H are an n-dimensional vector and a nxn

symmetric matrix given by

6 = col( el , ..., on ) . . example, is rrlated to the movement of preceding joints 1 through

i . Let us denote the displacement of joint i by Oi and its time-

derivative by B i then the translational velocity and angular velocity

of body i are given by the following linear combinahions of Bjs

n

H = C(A;I~A~ + M~B;B~ ) i= 1

Ai = [ al ,..., ai$ ,..., 0 1

Bi = I b, ,...., b,O ,..., 0 ]

95

The kinetic energy of a series of rigid bodies can also be

represented in any generalized coordinates, ql, ...,qn , that have one-

to-one correspondence to a set of joint displacements, 01, ..., On,

within a specified region in the joint Coordinate space. Let R be

the inverse of the Jacobian matrix associat.ed with the

transformation from 8 to q=col( ql, ...,q, ) then the kinetic energy

represented in generalized coordinates is given by

1 . rn T = - q"Gq

2 (4)

where the matrix G is given by

G= RtHR

We call the symmetric matrix the generalized inertia tensor of a

series of rigid bodies.

Fig. 2 Generalized inertia ellipsoid for two-degree-of-freedom arm

3. Generalized Inertia Ellipsold

The ~11.: has principal axes along which the inertia tensor is

ciiagonal. The principal axes of the GIE are aligned with the

eigenvectors of the matrix G, and the length of each principal axis

is the reciprocal of the square root of the corresponding eigenvalue.

Figure 2 shows an example, in which the GIE of a two-degree-of-

frrcdom rnanipulat,or is illust,ratrd i n space. In most cases, we are

interest(4 in the motion of an end effector mounted at the tip of

the arm. Therefore we investigate the manipulntor dynamics with

respett. lo the tip motion being referred to a Cartesian coordinate

system fixed in space. First, we obtain the generalized inertia

tensor in terms of the Cartesian coordinates, Computing bhe

eigenvalues and eigenvectors of the tensor, we draw a G E locat,ing

its center at the location of the arm tip in space, as shown in

Figure 2. The doband-dash lines in the figure shows the reachable

region of the arm tip. Unlike Ihe inertia ellipsoid of a single rigid

body, the GIE varies its configuration depending on the location in

the generalized coordinate system. We draw the GIE a t each

point in the reachable region so that the global characteristics of

the arm inertia can be represented in the whole.

In the following sections, a graphical analysis of the manipulator

dynamics using the GIE will be shown. The geometrical

configuration of the GIE depicts the characterist.ic features of the

manipulator dynamics.

96

1 1 1

2

axis. For motions with the same kinetic energy the velocity is T = - w t vector' Wvector = - nt I n w2scalar = - I W2scalar 2 2 minimum if it is in the direction of the minor axis. On the other

where I is the matrix of inertia tensor and the scalar quantity I

is called the moment of inertia about the axis of vector n. The

above equation implies that the moment of inertia can be defined

by the expression: I= 2T/w2scalar. We extend this expression to

a scries of rigid bodies. To this end, it is necessary that there

e.sists a scalar quantity that measures the speed of multi-degree-of-

freedom motion, as we needed the scalar angular velocity wsaelar

for the moment of inertia of a single rigid body. In this section,

we assumc the following velocity norm can be defined in the

gcmcralizcd coordinat,es.'

In the case of Figure 2, the velocity norm stands for the speed

of the arm tip with reference to the base coordinates. In

accordance with t,he expression; I= 2T/wZscalar the moment of

inrrtis can be generalized to a series of rigid bodies by the

fo!lowing esprpssion,

The generalized moment of inertia h varies depending on the

direction O f motion as well as the location in the generalized

coordinates. Our questions are; which direction gives the largest

inertia and how much it is. This problem is a kind of eigenvalue

problems. The solution is that the maximum (minimum) of h is

the maximum (minimum) eigenvalue of matrix G and that the

direction in which h is maximum (minimum) is aligned with the

direction of t.he corresponding eigenvector. Since the largest

eigenvalue of the inertia tensor corresponds to the minor axis Of

the GIE, the generalized moment of inertia is maximum along that

hand, the generalized moment of inertia in the direction of the

major axis is the smallest; therefore, the speed is fastest in that

direct,ion. If the lengths of the principal axes are the same,

namely the GIE is a pure sphere, bhe resultant inertia is isotropic.

The difference between the lengths of the major and minor axes

stands for the anisot,ropy of the resultant inertia. The shape of

the GIE, in Figure 2, is long and narrow in the peripheries of the

rearhablc region, and thick and round in the center. Therefore,

the rcsulhnt inertia is more isotropic in the center than at the

peripheries. At point S in the figure, the arm cannot move in the

radial dircction. The point S is known as a singular point. The

GlE becomes thinner and converges to a line, when the tip of the

arm approaches the singular point. The width of the GIE

represents the degree of singularity.

5. Nonlinear Forces

The motion of a mechanical arm is highly nonlinear including

Coriolis and centrifugal forces. In t.his section, these nonlinear

forccs are analyzed geometrically using the generalized inertia

ellipsoid. We fix a point A in space. If the principal axes of the

N E , a t A, are used as the coordinate axes to describe the motion

of the arm in the vicinity of the point, then the inertia tensor is

diagonal with referenee to .these axes. Let D A be the diagonal

matrix whose diagonal elements are the eigenvalues of G , X, ,...,

X,,, and let ;=cot ( b l , ...,bn ) be the generalized velocity referred to the principal axes, the kinetic energy is then given by

T = 1

2

97

The second term represents the inertia forces which have no

interaction along the principal axes. The third term stands for the

forces caused by the change in the inertia tensor, which are given

by

The diagonal components of DB are the eigenvalues 01 the

tensor GB, which correspond to the lengths of the principal axes

and determine the shape of the GIE. The orthonormal matrix C

represents the rotation of the Gffi and determines its orientation.

Therefore, the change of GIE configuration is classified into the

change in shape and the change in orientation, as shown in Figure

3. The former is described by the change of lengths along the

principal axes, and the latter is described by rotation angles about

the principal axes. If the length of the principal axis slightly

changes from , to 4 +Uj as shown in Figure s a , the diagonal matrix changes as follows,

On the other hand, the othonormal matrix C stands for the

rotation of the GtE from A to B. Let Wrj be a small angle of

rotation in the plane that includes the i-th and j-th principal axes,

p. and p., measured from the pi to the pj, the orthonormal matrix

that represents the rotation is given by,

... i ...... 1 -6O,, -6& WI2 1 C = . . . . . . . . . . . . . . 1

Fig. 3 Change of GIE configuration

The force F, consists of forces proportional to the products of

velocities, hence it is nonlinear.

Substit.ut,ing eqs.(l2) and (13) into (11) and neglecting the

higher-order small quantities, the change of the inertia tensor,

6D,=G, - D,, is then given by

Since the varying inertia tensor yields the nonlinear forces, as

mentioned above, the nonlinearity can be analyzed by investigating

the geometrical change in the GIE configuration. As shown in

Figure 3, the GIE changes its configuration, while the arm moves

from A to B. Let GB be the inertia tensor of the GIE a t point B,

referred to the principal axes at point A, then GB is standardized

to a diagonal matrix D, by an orthonormal matrix C.

GB = CD,Ct (11)

6D, =

Now we discuss the case where the arm moves along the Cth

principal axis. The velocity vector in this case is

; = c d ( O , .... O,;,,O, ... 0). This motion causes the following nonlinear

force. Substituting eq.(14) to eq.(lO),

98

F N

where a+../api is the curvature of a curve along which the Cth

principal axis changes i t s direction, and aX,/ap, represents the

change of the GIE in terms of the length of the bth axis. From

the Cth component in eq.(15) it follows that, when the arm moves

in the direction of the Gth principal axis along which Che length of

the axis becomes longer, a nonlinear force - b:/2.aXi/api acts on

the arm in the same direction as the arm motion. Also from the

j t h component it follows that a nonlinear force ;:/2 Vhi/apj acts

in the direction of t h e j t h axis as the length of the Cth axis

varies along t h e j t h axis and that, if the GIE rotates in p.-p.

plane, a nonlinear force - (xi-hj)a+ij/api.;: acts along the bth axis.

The nonlinear forces due to the change of the GIE orientation are

proportional to the difference of the eigenvalues. Therefore, if the

GIE is isotropic, those nonlinear forces do not appear. In the case

of Figure 3, the nonlinear forces shown in the figure are developed.

The length of the major axis, in Figure S a , is increasing for

manipulator motions along this axis. That means -Vh,/ap, > 0.

Therefore, a positive force -(aX1/ap1);)12/2 acts along the major

axis, while the tip moves along the axis. On the other hand, the

GIE in Figure 3-b rotates in the counterclockwise direction in the

pl-p2 plane, where the length of the p, axis is longer than the p2

axis. Therefore, a positive force -(x,-h2)(a+12/apl)~,2 acts along

the p2 axis, while the tip moves along the major axis.

I J

6. Design of a Mechanleal Arm

The GIE configuration represented in generalized coordinates

Fig. 4 Dimensions and mass distribution of two d . 0 . f . arm

depicts the global characteristics of manipulator dynamics as a

whole. From the graphical representation, one can understand the

inertial effect and nonlinearities of mechanical arms, which are

characteristic features of the arm. In this section, this technique is

applied to the design of a mechanical arm. The inertial effects

and nonlinearities depicted by the GIE configuration provide useful

da t a for designing the structure of the arm. We discuss the arm

shown in Figure 4, where the dimensions and mass distributions

are described by the following parameters,

11,12 = l i n k l e n g t h s of bodies 1 and 2

g1,g2 = d i s t ances be tween t he cen te r s of mass and t h

11,12 = moments of i n e r t i a

ml, 3' mass

Figure 2 shows the GIE configuration when the arm has the

same length for the upper and the lower arms with the parameters

listed in the figure. From this configuration, it follows that the

arm dynamics have the following problems

(1) The large difference in axial lengths between the major and

the minor axes shows that the generalized moment of inertia at

the tip of the arm varies significantly depending on the direction

of motion.

(2) Since the changes of the GIE configuration, both in shape

and in orientation, is significantly large, the arm dynamics involve

large nonlinear forccs.

By modifying the link lengths and the distribution of mass, we

try to improve the arm dynamics so that the generalized moment

of inert,ia is uniform in any direction over a wide range of

reachable region and the nonlinear forces are reduced as well. If

the GIE is a pure circle at any point in the reachable region, the

arm is isotropic and has no nonlinearity. This uniform and

isotropic configuration reduces the complexity in controlling the

arm. Therefore the improvement of control performance can be

expected. It is also desirable for painting robots because of the

following reason. Namely, most of the painting robots are

controlled in teach-in-playback mode, where a human operator

shows an exemplary motion by moving the spray gun mounted at

the tip of the arm, and then the robot repeats the motion. In

order to acquire good data from the operators motion, the arm

must not impede the operators motion. If the arm has

significantly different characteristics depending on the moving

direction as well as large nonlinearities, the arm does not respond

to the operators motion accurately. Therefore, uniform and

isotropic characteristics are desirable. From Figure 4, the

generalized inertia tensor with respect to the joint coordinates is

given by

H1=Il+m1gl2ll2 H,=&+?g; H3=m,I,g2cos( B,-B,

where

The Jacobian matrix associated with t.he transformation from the

joint coordinates to the base coordinates is given by

The generalized inertia tensor with respect to the tip of the arm

is derived from substituting the above equations into eq.(4). In

order that the generalized inertia tensor is isotropic in the base

coordinates, the matrix G=RtHR should have the same

eigenvalue. Namely, the components of the matrix must satisfy

the following equations,

G,=G, , G3=0

where

G= :: :: 1

e1-e2 = + 90 deg

The first equation shows that the isotropic dynamics are realized

when the arm configuration is such that the bending angle of the

forearm is 90 degrees from the lower arm. In the second equation,

HI represents the resultant moment of inertia about the first joint

a.xis, when the mass of the forearm is lumped at the tip of the

lower arm, and H, is the resultant moment of inertia of the

forearm about the second joint axis. Therefore the second

condition requires that the ratio of the resultant inertias about

both axes must I,- equal to the squared ratio of the link lengths.

Fig. 5 Modified GIE configuration 1

(isotropic on center line)

Figure 5 shows a modified design, where the GIE is isotropic on

the broken line on which the bending angle of the second joint is

90 degrees. The configuration of the GIE, however, changes

rapidly from the pure circles to long and narrow ellipses when the

arm moves from the center to the peripheries. The rapid change

of the GIE configuration leads to large nonlinear forces and

unbalanced inertia characteristics. The dynamic characteristics of

the arm can be improved by enlarging the region of isotropic GIE

t.o include more of the reachable region. As the angular difference

of the second joint from 90 degrees becomes larger, the GIE

becomes slender. Therefore, we limit the bending angle to a

smaller range where the GIE is almost isotropic. At the same

time, we extend the link length of body 2 so that i t covers a

larger region with the small bending angle. Figure 6 shows the

modified configuration, where the GIE is almost isotropic and

uniform over the wide range of the reachable region. The

improvement is noticeable.

Fig. 6. Modified GIE configuration I1 (isotropic and uniform)

l1 = 0.4m, 12 = 0.6m, gl = 0, gl = 0.42m ml = 50Kg, r n ~ = 30Kg,Il = 1.552Kgm2,1~ = 9Kgm2

7. Conclusion

A geometrical representation of manipulator dynamics was

presented and applied to the design of a mechanical arm. The

inertia ellipsoid, which is used to represent the inertia tensor of a

single rigid body, was extended to a mechanical arm that consists

of a series of multiple bodies. Uqlike the inertia ellipsoid of a

single body, the generalized inertia ellipsoid (GIE) changes its

configuration depending on the location of the arm. In order that

the global characteristics of the changeable inertial effects can be

understood in the whole, a graphical representation to depict the

GIE configuration in generalized coordinates was presented. The

relat.ion between the features of the GIE configuration and the

characteristics of manipulator dynamics were analyzed in terms of

resultant inertia and nonlinear forces. It was found that the

resultant inertia, which is a measure of inertial effect in geralized

coordinates, is maximum in the direction of the minor axis of the

GIE. The nonlinear forces are caused by the change of the GIE

configurat,ion. The presented graphical analysis of the manipulator

dynamics was applied to the mechanical design of a robot arm.

The lengths of links and their mass distribution were modified so

that the arm has isotropic inertia characteristics as well as less

non1inearit.y. An example of designing a two-degree-of-freedom arm

verified the efficiency of the presented approach.

Acknowledgement

References

[I] I-Iollerbaeh, J., "A recursive Lagrangian Formulation of

Manipulator Dynamics and a Comparative Study of Dynamics

Formulation Complexity", IEEE Trans. of Syst,em, Man, and

Cybernetics, Vol. SMC-10, No. 11, pp. 730-736, 1980.

[Zj Uicker, J.J., "On the Dynamic Analysis of Spatial Linkages

Using 4x4 Matrices", Ph.D. dissertation, Northwestern University,

1965.

[3] Huston, R.L., a,nd Keily, F.A., "The development of Equations

of Motion of Single-Arm Robots", IEEE Trans. of System, Man,

and Cybernetics, Vol. SMC-12, No. 3, pp. 259-2G6, 1982.

[dl Mahil, S.S., "On the Application of Lagrange's Method to the

Description of Dynamic Systems", IEEE Trans. of System, Man,

and Cybcrnetics, Vol. SMC-12, No. 6, pp. 877-883, 1982.

161 J Juh , J.Y.S., Walker, M.W., and Paul, R.P.C., "On-Line

Cornyutational Scheme for Mechanical Manipulators", ASME

Journal of Dynamic Systems, Measurement, and Control, Vol. 102,

No. 2, pp. 69-76, 1980.

[:I Orin, D.E., hlcGhee, R.B., et. al., "Kinematic and Kinet.ic Analysis of Open-Chain Linkages LJtiliring Newton-Euler Methods",

hlatllernatical Biosciences, No. 43, pp. 107-130, 1979.

[i?] Pennock, G.R., and Yang, A.T., "Dynamic Analysis of a

Multi-Rigid-Body Open-Chsin System", ASME Journal of

Mechanisms: Transmissions, and Automation in Design, Vol, 105,

NO. 1, pp. 28-34, 1983.

191 Walker, h.I.W., and Orin, D.E., "Efficient Dynamic Computer

Simulat.ion of Robotic Mechanisms:, ASME Journal of Dynamic

Sgstcms, Measurement, and Control, Vol, 104, No. 3, pp. 205-211,

1982.

[lo] Orlandea, N., and Berenyi, T., "Dynamic Continuous Path

Synthesis of Industrial Robots using ADAMS Computer Program",

AShlE) Journal of hfechanical Design, Vol, 103, No. 3, pp. 602-607,

1981.

111 Thomas, hf., and Tesar, D., "dynamic Modeling of Serial

Manipulator Arms", ASME Journal of Dynamic Systems,

Measurement, and Control, Vol. 104, No. 3, pp. 218-228, 1982.

112) Coldstein, H., "Classical Mechanics", Addison-Wesley, 1950.

[I31 Noble, E., and Daniels, J.W., "Applied Linear Algebra",

Prentice- Hall, Englewood Cliffs, 1977.

(51 Stepanenko, Y., and Vukobratovic, S., "Dynamics of

Articulated Open-Kinematic-Chain Active Mechanisms",

h.lathematica1 Biosciences, No. 28, pp. 137-170, 1976.

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