NEWTON - EULER FORMULATION OF

MANIPULATOR DYNAMICS FOR COMPUTER

CONTROLl

J. Y. S. Luh, M. w. Walker2 and R. P. C. Paul

School of Electrical Engineering Purdue University, West Lafayette,
Indiana 47907, U.S.A.

Abstract. This paper presents a new approach to obtaining the equations by

the method of Newton-Euler formulation which is independent of the type of
manipulator-configuration. The method involves the successive transformation
of velocities and accelerations from the base of the manipulator out to the
gripper, link by link, using the relationships of moving coordinate systems.
Forces are then transformed back from the gripper to the base to obtain the
joint torques. Using this formulation, the amount of computation increases
linearly as the number of joints increases while the conventional Lagrangian
formulation causes the increase proportional to the quatic of the number of
joints. With a program written in floating point language, it is possible
to achieve an average execution time of 4.5 milli-seconds on a POP 11/45 com-
puter for a Stanford manipulator.

Keywords. Computational methods; computer control; industrial robots;

manipulators; nonlinear systems; robots; robot dyanmics.

INTRODUCTION "computed torque" technique (Markiewicz,

1963; Bejczy, 1974), resolved-acceleration
Industrial robots are mechanical manipula- control (Luh, 1979), etc. In most cases,
tors which become increasingly important in the control scheme involves the computation
automation. The manipulator usually has six of the appropriate input generalized forces
joints, seven links and a gripper. Let q{t) ! by equation (I) using the measured values
be a six-dimensional vector representing-the of q and q, and the computed values of Cj.
actual displacements of the six joint vari- The-execution time for the computation Poses
ables. The equation of motion for the ma- one of the main problems because: (1) for
nipulator can be written as: on-l ine control, the computation must be per-
formed on-line, and (2) for achieving the
J(q)q + Hq + f(q.
- - - -- I
q.,
j
q; i,j=I,2, ••• ,6)
- convergence of the control algorl thm, the
+~(9)+~=! (1) computation must be repeated very frequently
(Luh, 1979), preferably at the sampling rate
where ~(9) = 6x6 inertia matrix,
of no less than 60 Hz since the bandwidths
~ = 6x6 diagonal viscous friction of most of the mechanical manipulators are
matrix, around 10 Hz. Consequently the formulation
f(q. q., q) = 6xl vector defining of the equation of motion partially deter-
- I j - mines the feasibility of the control scheme.
cori 01 is and centri fuga 1 te rms,
The equations of motion for a manipulator
g(q) = 6xl vector defining the
- - gravi ty terms, with six joints and seven links has been de-
rived by Uicker (1965), Paul (1972), and
cr = 6xl static friction vector. Lewis (1974) usIng Lagrangian generalized
, = 6xl vector of input generalized coordinates. For manipulators having n
forces. joints, the equation for joint i is:
n j
This equation represents a nonlinear system '·1 E {E Tr[U· k J.{U· i )"] qk}
j=i k .. 1
which is difficult to control. The known -j -j-j
control schemes include, for example, re- n j j
solved rate control (Whitney, 1969, 1972), + E {E E Tr[~·km~·(~·i)"] ci m qk}
"inverse problem" technique (Paul, 1972), j .. 1 k=l m=l j j j
n
E m.
j
g"
-
U ••
-jl
r.
-j
(2)
lSupported by NSF Grants ENG 76-18567 and j=i
APR 77-14533. where Tr .. trace operator,
2Now with Aerospace Corporation, Los Angeles, ( ) .. = transpose of ( ),
California.

165
166 J. Y. S. Luh, M. W. Walker and R. P. C. Paul

Ti • input generalized force for joint i, the manipulator-configuration. Essentially

the process involves the successive trans-
m .. mass of link j,
"i formation of velocities and acceleratlons
r . • a vector describing the center of from the base of the manipulator to the hand,
-j
mass of link j with respect to j-th link by link, using the relationships of
coordinate system, moving coordinate systems (Symon, 1971).
2 Forces are then transformed back from the
[0,0,9.8 m/sec ,0] is a gravitational
hand to the base to obtain the joint torques.
acceleration vector at a sea level
Because of the nature of the formulation,
base,
computations are much simpler which makes it
J. inertia matrix for link j, possible to achieve a short computing time.
-j

"fTo-_~-I) 9.k (!~-I)' for j ~ k, JOINTS, LINKS AND COORDINATES

Although the discussion in this paper is on a

otherwise,
general manipulator with n joints, Figure 1
is used as a reference for clarity. The ma-
(!~-I) gk(T~:) 9-m(!~-I)' nipulator in the fIgure has seven links and
six joints in addition to the gripper. Link
for j ~ m~ k , o is the stationary pillar which is bolted
on the platform, while link 6 is the band.
( m-I) (k-I ) (j) For j-l ,2, ••• ,6, joint j is the joint connects
~jkm !o gm !m-I gk !k-I '
link j -I to link j.
for j ~ k ~ m , (4)
The coordinate systems are assigned according
Q otherwise to Denavit and Hartenberg convention (1955) .
There are six orthonormal coordinate systems
(x., y., z.) such that z. = x. X y. where X
-I -I -I -j -I -I
-I 0
denotes the cross-product. For i=0,1, ••• ,5,

9- k
[! 0
0
0
0
0
0 !J
if joint k is
rotat i ona I ,
system (x., y . , z.) associates with the
joint i+l; and
- I -I -I
~i

if joint i+1 is rotational, or,

is the axis of rotational
~i is the
0 0 center line of link i+1 and is in the di-
0 0 if joint k is
[1 0
0
0
0 !J trans lati onal,
rect ion of motion if joint i+1 is transla-
tional. System (x,-0
y,
_0
z) is the fixed
-0
base coordinate in which z is the axis of
4x4 matrix which transforms any vector rotation of joint 1. Therg is no link 7.
Thus, instead of defining and attaching sys-
expressed in k-th coordinate system to
the same vector expressed in j-th co- tem (~6' t6' !6) to link 6, the system (!:!, ~,
ordinate system. ~) is defined with ~6 replaced by the unit
qk = generalized coordinate (i.e., joint normal vector!:!'!:6 by the unit slide vector
displacement) •
~, and ~6 by the unit approach vector!. The
The computational complexity emerges from system (n, s, a) specified the orientation of
the evaluation of the trace operator Tr, the hand: - -
which involves the matrix multiplication U'
-j k
For j"'I,2,4,5, the origin of (x., y . , z.) Is
and -U'k'
j m
By the definition of these -j -j -j
at the intersection of -j-
z . I and -j
z . , and -j
x.
matrices shown above, each trace operator re-
quires one matrix multiplication in general. is perpendicular to -z.j - I' The origin of (x-0 ,
It follows that the first term in equation
y , z ) and the axis x are arbitrarily as-
(2) involves n(n+1) (2n+I)/6 matrix multi- -0 -0 -0
plications while the second terms n2 (n+I)2/4 signed. The origin of (~3' t3' ~3) does not
multiplications. Consequently the computa-
follow the above rule since ~3 coincides with
tional complexity is of the order 0(n 4 )
where n is the number of joints. ~2' But for convenience, it is assigned to

Based on equation (2), a FORTRAN program was be the same origin of (~4' Y4' !4)'
written for POP 11/45 computer which computes
the Input torques and forces for all the With the above definition, the system (~I'
joints. The program has an average execution y., z.) for i"'I,2, ••• ,5, is fixed on link I
-I -I
time of 7.9 sec. for a Stanford manipulator
and hence moves with 1 ink i, a I though ~i
as shown in Figure I. I t is too long to be
useful for on-l ine operations. represents the motion of link i+l. But the
system (x
-0
, y_0 , -0
z ) is fixed on Link 0, the
This paper presents a Newton-Euler formula-
stationary pi liar.
tion of equations of motion for mechanical
manipulators. The method is independent of
 Newton-Euler formation of manipula tor dynami c s 167

,
\ '
\ \ ' '\
\ \ X " - V3
"\ \ \ -3','
~-3
_Z
I \
\ \
\ \
.. \ ~4
X'
-4\~
\ 1
Y4 lf"
\

~5~'~5 I·
\ :(5

Fig. 1. A f;lanipulator with Si x Joints and Seven Li nk s .

MOVING COORDINATES
YI+I
As Indicated above, there is one coordinate
system attahced to each link of the manIpu-
lator. Thus the coordinates move together
with links. Consider the following three
coordinate systems as shown in Figure 2:
the base coordinates (x-0 , y-0 , -0
z ) attached
to link 0 with origin 0, the coordinates
(x-I. , Y.,
-I
z.)
-I
attached to link i with origin
0*, and the coordinates (~i+I' Yi+I' ~I+I)
attached to link 1+1 with origin O~. Thus
O~ is located by the position vector r re-
lati·ve to origin 0, or the vector s relative Fig. 2 . Relationships Among Three Coordinate
to origin 0*. As shown in Figure 2, these
vectors are related by: Systems.

- -
r • s + e
- (5) cross-product.
Is given by
Thus the I i near acce I e ra t i on

Let ~e and ~e be, respectively, the linear

and angular velocity of coordinate system v
-r = d*2~/dt2
+ W x ~ + 2!!:le x (d *~/dt)
-e
(xI'
-
y.,
-I
z.) with reference to the base co-
-I
+ w x (w x ~) + -e
-e -e
v ( 7)
ordinates (x , y , z). Then the linear
-0 -0 -0 in which the third and fourth tenns on the
velocl ty of r wl th reference to base co- right side of the equal sign are, respective-
ordinates Is- ly, the corlolis and centrifugal accelera-
tions.
v • d*s/dt + w x s + v (6)
-r - -e - -e Let wand w
-r -s be the angular velocity of co-
where d*/dt denotes the time derivative with
respect to the movl ng coordl nates (~I' ri' ordinate system (~I+I ' YI+I' ~i+l) with
reference to, respectively, the systems (~o'
~I) (Symon, 1971), and x denotes the
168 J. Y. S. Luh, M. W. Walke r and R. P. C. Paul

z ) and (x. , z.) • Then where pt+1 is the position of the origin of
Yo' -0 -I :il' - I
( 8) coordinates (~i+I' Yi+I' ~i+I)' which is
W
-r
= W
-e + W
-s
attached to link i+I, relative to the origin
and
W = W-e
. + W (9)
of (x.,
-I
y-I. , -z.).
I
If link i+1 is transla-
-r -s tional in coordinates (x., y., z . ), it trav-
-I - I -I
But = d"'w Idt + (10) ~i
W
-s -s -e x W
W
-s els in the direction with a joint

hence (9) becomes velocity qi+I' If is is rotational in co-

. + d '" -s Idt + ( 11) ordinates (x., 't., z.), it has an angular
-e x W
W = W W W -I I -I
-r -e -s
velocity ~s' Thus
Equations (6), (7), (8) and (11) are basic
relations for velocities and accelerations '" f~Sx~i+l*' if link i +1 is rotational,
among links and the base of the manipulator.
d ~/dt =l ~i qi+I' if link i+1 is translation-
KINEMATICS OF THE LINKS
ai,
( 18)
Using the coordinate systems defined pre-
viously, coordinates (x., y., z.) and (x'+ I ' (d'~w-s Idt)x~'+"'
I
I+w-s x( w
-s xO'+I
~I
'~)'
- I _I - I - I
if link i+1 is rotational,
:ii+I ' ~i+l) are attached to links i and i+l,
respectively. Thus
if link i + lis
translational . ( 19)

( 12) Combine (8), (12), (17) and (18), and sub-

stitute them into (6) to yield
where Yi and ~i are, respectively, the linear

{
and angular velocities of link i with (!.l i+IXEi+~+Yi' if link i+1 is
reference to the base coordinates (~o' :io' rotat i ona I ,
z). Since, by the definition given pre- Yi+1 =. '"
-0 ~i qi+I+~i+lxei+I+Yi' if I ink i+1
viously, an angular motion of link i+1 is
is translational, (20)
about ~i axis, then

z. if link i+1 is rotational, in which ~i+1 = ~ i for the translational link

-I
W =
-s { 0
i+l. By using (8), (10), (12), (17), (18),
if link 1+1 is translational, (19) and the identities (axb) x c = b(a·c) -
a(b.c) and ax(bxc)=b(a·c)=c(a.b): wh~r~ ;'."
( 13) denotes the-dot product~ equation (7) can be
.
where qi+1 is the velocity of joint i+I, or written as:
in other words, the magnitude of angular
velocity of link i+1 with respect to (x-I. ,
y ., z . ). Th us
-I -I

/w Idt ={~i ci i + l , if I ink i+1 is rotational

... *
~i qi+l~i+lx~I+I+2~i+lx ~i qi+1
( )
-s + ~i+lx(~i+lx~i+7)+Yi '
Q , i f link i+1 is translational. if link i+1 is translational.
( 14) (21)
Combine (8), (11) , (12) , (13) and (14) to Again, in (21) for the translational link
yield i+I, note that w'+
-I 1
= w .• Equation (15),
-I

{wo.,
-I -I qi+I' If I ink i+1 Is rotational
(16), (20) and (21) give the recursive re-
lations of velocities and accelerations be-
~i+l= tween links i and i+l.
w if I ink i+1 is trans la-
i
tional, EQUATIONS OF MOTION FOR MANIPULATORS
( 15)
D~Alembert's Principle will be applied to
w.+z.
-I - I
q'+I+w.x(z.
I -I -I
q'+I)'
I each link of the amnipulator. Let
W = if link i+1 is rotational, m total mass of link i, a scalar,
-i+1 i
{ W., if link i+1 is translational, '"r.
-I = position of the center of mass of
-I link i with reference to base co-
(16)
ordi nates,
The linear velocity and acceleration can be
obtained from (6) and (7) in which ~-I. .. dr./dt,
-I
linear velocity of the
.. p* ( 17)
l! _1+1 center of mass of link i, with
 New t on- Eul er fo rma ti o n of ma n i pul ator dynam i cs 169

reference to base coordi nates, Figure 3. Let 0' be situated at its center
of mass. Then, by the convention shown in
F.
-I
= total external vector force
* A A
exerted on link i, Figure 2, I?i-l' I?i' I?i' !"i' and ~i are de-
N. total external vector moment ex- fined and indicated in Figure 3. Thus to
-I
e rted on 1 ink i, compute ~. and
-I
Q.
-I
from (6) and (7), replace
inertia matrix of link i about its
J.
-I
center of mass in (~o' lo' ~o)'
S
-
by s.
-I
~..
and -r by -I Now -r
v and -r in (6), v
(7), and Figure 2 are the 1 inear velocity
viscous friction coefficient
associated with translational and acceleration of r with reference to
base coordinates, and their counter parts in
motion, Figure 3 are the linear velocity and accel-
viscous friction coefficient eration of 1. (or the linear velocity and
associated with rotational motion. accelerationlof the center of mass of link
i) with reference to base coordinates . Thus
Then replace -r
v in (6) by and -r in (7) by -I v. v v..
-I
F.
-I
dem oI ~.)/dt
I
+ g .I q.I ~./
-I
I I~- .I
I
I Also, based on (10), replace -e
(! v.,I - e
by - w by

mi ~i + gi qi Y/ II ~ill (22) w.,

-I -e
w
by w
-I
., and note that d*s./dt
-1-
=0 =
d,~2s./dt2
-I
since both coordinates (x.,
-I
't.,
I

N.
-I
d(J. W.)/dt + h .I
- I -I
<i .I ~ ·I-l z.) and s.
-I -I
are fixed on l i nk i. Consequent-
l y (6) and (7) become
J . w.
-I -I
+ w.
-I
x (J.
-I
~ .I) + h ·1 q ·1 _z ·l _l(23)
A A

where w. and w. can be computed from (15) v.

-I
w.
-I
x s.
-I
+ v.
-I
(24)
-I -I
and (16). For convenience, the gravity is
A
v.
-I
w. x
-I
A
s.
-I
+ w.
-I
X (w.
-I
x 5.)
-I
+ v.
-I
(25)
included in v.,
-I
the linear acceleration of
The total external force -F.I and moments N.
-I
the center of mass of 1 ink i, and wi 11 be
are those exerted on 1 ink i by gravi ty and
shown at the end of this section. Now Yi 1 inks i-I and i+ 1. Wi th reference to
and v.
-I
can be derived from (6) and (7) as Fi gure 3, let

follows. Consider link i alone as shown in Xi = force exerted on 1 ink i by 1 ink i-I,

o* .. origin of
(~i' ~ i ' ~ i)

Link 1+1

!!i .. I?i

o - Origin of
(~i -1' 't i -1' ~ i-I )

Link 1-1 ~o

Fig. 3. Forces and t-1oments Exerted on and from Lin k i.

170 J . Y. S. Luh, M. W. Walker and R. P. C. Paul

n.
-I
moment exerted on link i by link Yo = Q. However, to Include the gravity in
i-I. the equations of motion, v is defined as
-0
Then

F.
-I

N.
-I [) 'I-[)'I+I + {p .1- I-~·)xf.
-I -I - (.e.-~
(26)

I -I. )xf·+1
-I
v
-0
~.806" !.t..",.c'] (31)

,.. 'I<
for a level base at sea level. Re fe r ring to
[)i-[)i+1 + (Pi-I-r) x ~i-.ei x !i+1 equations (21) and (25) reveals that its ef-
fect wi II be added to each link acceleration
(27) and hence to each link force F. th rough
equation (22). -I
•
Since (r.-pA *
= (o.+s.),
A

- I .I - I) I: I -I
then from (26)
COMPUTATIONAL SCHEME
and (27) one obtains the recursive rela-
tions: The equations of motion derived in the pre-
ceding section are referenced to the base
f.
-I
(28) coordinate systems. Unfortunately the
inertia matrix ~i is dependent on the ori-
n.
-I
~'I+I + 2'1* x -I * ,..s . ) xF. + N.
f ' + 1 + {2·+
I -I -I -I entation of link i, which is changing. Thus
the computation is quite complicated. A
more efficient technique for computing the
For manipulators with n joints and n+1 links, joint input forces and torques is to have
!n+1 and [)n+1 are respectively the forces each link I s dynami cs re fe renced to its own
link coordinates as follows.
and moments exerted by the hand (i.e. link
n) upon an external object. Equations (28) i
Let ~i-I be a 3x3 matrix which transforms any
and (29) are similar to those described by
Paul (1972) for computing the joint forces vector with reference to (x.,
-I 't., z.) co-
I -I
or torques requi red to exert a given force ordinate system to a new coordinate system
or moment on an external object with the whose coordinates are parallel to (~i-I'
hand. The difference is the inclusion of
the terms involving -I
F. and N. in (28) and 't.1- I' -1-
z. I)' but its origin is the same as
-I
(29), which represent the additional terms that of (~i' 't ·I , z.), i.e., a pure rotation.
needed to move link i. Thus -I
Ai = Al 2 i
By convention, if joint i {and hence link i} -0 -0 ~I ... ~i-I
is rotational, then its motion is about the
~i-I axis. Therefore, the input torque 'i Since each coordinate system is orthonormal,
then
at that joint is the projection of [)i onto
i -I i" i-I
~i-I' If, however, joint i {and hence link (~i-I) = (~i-I) = (~i )
i} is translational, then its motion is a- where ( )" = transpose of ( ).
along the z . I axis. Then the input force
"1 at that-JOint is the projection of -I
f. on- Instead of computing ~i+I' ~i+I' etc., com-
to ~i-I' Thus pute ~~+I ~i+I' ~~+I ~i+I' etc. Hence the
[) i • ~ i -I' i f link is rotational complete set of equations of motion becomes:
,.
I
(30) (IS) :
is translational, i (o 0
~i+1 ~i ~i + ~i ~i qi+I)'
where ' i is the input torque for link i (if if link i+1 is rotational,
it is rotational) or the input force for o
link i (if it is translationa1), and "." ~i+1 ~i+1 =
i
A'+ (o
A. ~I)' if link i+1 is
denotes the dot product.
-I I -I translational,
In summary, the complete set of equations of
motion for the manipulator with n joints and (16) :
n+1 links consists of equations (IS), (16),
(20) through (25), and (28) through (30),
for i=O,I, ... ,n. Note that wand cO are
i
A'+
0
-I I [A.
-I -I
w.
A~ -I
+ -I z. qi+1
-0 -0
the angular velocity and acceleration of the + (~~ ~i)x{~~ -I
z. qi+I)),
base, while Yo and Yo are the linear velocity if link 1+1 is rotat i ona I ,
o •
and acceleration of the base. ~i+1 ~i+1
A~+I{A~
-I
w ),
-1- l
if link i+1 is
If the base is bolted on the platform and translational.
link 0 is stationary, then ~o • ~o ~ Q, and (35)
 Newton-Euler formation of manipulator dynamics 171

Equations (34) and (35) can be simplified own coordinates (~I' ri' ~I)"
as
(24) :
A~+I(A~
-I -I w.
-I + -0
z q'+I)'
I
o'" (0 ) (0"') 0'"
A YI"
-i ~I ~i x ~i ~I + ~i ~i (43)
if link i+1 is rotational,
°
~i+1 ~i+1 .. ° '"
where ~I ~ i is the center of mass of 1 i nk i
A~+I(A~ w.), if link i+1 is
referred to its own coordinates (x., 'l" z.),
-I -I -I translational, -I I -I

( 36) (25) :
I [0.
~i+1 ~i ~i + ~o ql+1
.. 0:'
~I Yi"
(0,)(0"')
~i ~i x ~I ~i + (~i ~i)
°
+ (~~ ~i) x (~o ql+l»)' x [(A~ w.)x(A~
-I -I
s.)] + -A~I -V.I
-I -I
( 44)
if link i+1 is rotational,
° .
~i+1 ~i+1 ..
(28) :

i (0')
~i+1 ~i ~i ' if link 1+1 is A~
-I -I
A~+I (A~
f. = -I -1+1 _f.l + l ) + A~ F
-I -i
(45)
trans lational.
(37)
where
A~ -I
-I
n. = -A~+I[A~
I -1+1
(Ao *)
!:li+l + -i+1 ei
8°i J\~i )
(38) o )] + (Ao. *
x (A-i+1 -i+l
f
- I ei +
(20) :
x (A~ A~I -N.
F.) + - (46)
(~~+I~i+I)X(~~+le~+I)+~:+I(~~~I)'
-I -I I

(30) :
if link i+1 is rotational,

{
(A~ !:li)·(~:-l ~o), if link i is
-I rotational,

Ti = (A~ f.).(A~-1 z ), if link i is

-I -I -I -0 translational.(47)

(21) : Note that for parameters between two

adjacent links defined as in Figure 4, one
obtai nes

['~'
A.I -cos Cl.
I
sin 9 .
I
Ai = sin 9.I cos Cl. cos 9.I
- i-I I
0 sin Cl.
I

° Yi+1 =
~i+l sin Cl. sin
I (.. 0.) (0'
~i+1 ~oql+l+~iYi + ~I~i+l
) -sin Cl.
I

I
cos 9.
I
8J (48)
cos
° * °
x(~i+lei+l)+2(~I+I~i+l)
Cl.
I
and
x(~:+I~oqi+l)+(~~+I~i+l)
° ° *
x[(~i+l~i+l)x(~i+lei+I»)'
if link 1+1 Is translational.
(40)
~~ e~ = [i s~~ i]
r. cos
I
Cl
Cl.
I

(22) :
Finally the computational algorithm can be
.. m. stated as follows:
I
A. Given constants are:

A~ N.
-I -I
. (A~ J. i 0'
-I -I ~o)(~i ~i)
1,1
n .. nurrber of joints (i .e. n+1 I inks),
.. W - 0 and -v 0.. -0'
~o
~~)(~~ ~I)]
-0 -
+ (A~ W.)x[(A~ J.
-I -I -I-I

~o ~.ao621 l,m/J ·'0 - m

+ h. ql ~I i-I (42)
I ~o
-
where ~~ ~i ~~ is the Inertia matrix of link
I about Its center of mass referred to Its B. Joint variables are ql' ql and Ci l for
172 J. Y. S. Luh, M. W. Walker and R. P. C. Paul

i=I,2, ••• ,n. puter for a Stanford manipulator, which is

not short enough to be useful. However,
C. Link-wise variables are i, Ei' ~i' the progrCl11 is rewritten in floating point
Ii' !:!i' Ti • assembly language which has a shorter
average execution time of 4.5 mill i-seconds.
Step O. Set i '" I. At present time a control aglorighm (Luh,
Step I. Compute A~ w. , A~
-I -I -I
w.,
-I
A~ v. and
-I-I
1979) has also been developed which requi res
an additional average execution time of 7
A~
-I
v.
-I
by (36), (37), (39) and (40). milli-seconds. Thus the total required com-
Step 2. Compute A~ ~. and A~
- I -I -I
v.
-I
by (43) puting time for each sampling cycle is
0.0115 second which allows a possible sampl-
and (44). ing frequency of 87 Hz. Consequently the
Step 3. Compute A~ F. and A~ N. by (41) requirement of minimum 60 Hz is met.
-I - I -I -I
and (42).
Step 4. If i=n, continue . Otherwise set REFERENCES
i=i+1 and return to Step I.
Step 5. Compute A~ f. and A~ n. by (45) and Bejczy, A.K. (1974). Robot Arm Dynamics and
-I -I -I - 1 Control, Technical Memorandum 33-669,
(46) with In+1 and !:!n+1 being the Jet Propulsion Laboratory .
requi red forces and moments, Denavit, J., and R.S. Hartenberg (1955). A
respectively, for the hand (i.e. Kinematic Notation for Lower-Pair
link n) to carry the load. Mechanisms Based on Matrices, Journal of
Applied Mechanics, Vol. 22, pp. 215-221.
Step 6. Compute Ti by (47) • Lewis, R.A. (1974). Autonomous Manipulation
Step 7. If i=O, stop. Otherwise set i=i-I on a Robot: SummaJ: of Manipulator Soft-
and return to Step 5. ware Functions, Tec nlcal Memorandum 33-
679, Jet Propulsion Laboratory.
Obviously the amount of computation in- Luh, J.Y.S., M.W . Walker (1979). Controller
creases linearly as the number of l i nks in- for a Mechanical Manipulator, Proceedings
creases. of 1st International Symposium on Mini-
and Mi crocomputers in Control, January 8-
9,1979, San Diego, California.
Markiewicz, B.R. (1973). Analysis of the
Computed Torque Drive Method and Com -
parison with Conventional Position Servo
for a Computer-Controlled Manipulator,
Technical Memorandum 33-601, Jet Propul-
sion Laboratory.
Paul, R.C. (1972). Modeling, Trajectory
Calculation and Servoing of a Computer
Controlled Arm, A. I. Memo 177, Stanford
Artificial Intelligence Laboratory,
Stanford University.
Symon, K.R. (197\). Mechanics, 3rd Edition,
Addison-Wesley, pp . 271-281 .
Uicker, J.J. (1965). On the Dynamic
Analysis of Spatial Linkages Using 4x4
Matrices, Ph.D. Thesis, Northwestern
Un i ve rs i ty.
Whitney, D.E. (1969). Resolved Motion Rate
Fig. 4 . Parameters Relating Two Adjacent Control of Manipulators and Human
Link s. Prostheses, IEEE Transactions on Man-
Machine Systems, Vol. MMS-IO, No. 2,
CONCLUSION pp. 47-53.
Whitney, D.E. (1972). The Mathematics of
A FORTRAN program was written which uses Coordinated Control of Prosthetic Arms
these equations to compute the forces and and Manipulators, Journal of Dyanmic
torques needed to drive the manipulator. Systems. Measurement. and Control,
This program has an average execution time Transactions of ASME, December 1972,
of 33.5 milli-seconds on a POP 11/45 com- pp. 303- 309 •