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Direct calculation of minimum set of inertial parameters of seri....pdf
368
Communications
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION. VOL 6. kO 3 . JUNE 1YYO
Direct Calculation of Minimum Set of Inertial
Parameters of Serial Robots
Abstract-The determination of the minimum set of inertial param-
eters of robots contributes to the reduction of the computational cost
of the dynamic models and simplifies the identification of the inertial
parameters. These parameters can be obtained from the classical inertial
parameters by eliminating those that have no effect on the dynamic
model and by regrouping some others.
This paper presents a direct method of determining the minimum
set of inertial parameters of serial robots. The given method permits
determination of most of the regrouped parameters by means of closed-
form relations.
I. I\-rRODL CTIOY
In order to increase the dynamic performance of robots, many
control schemes based on the dynamic model have been presented
[1]-[4]. Two problems have to he solved for real time implementation
of these control laws:
1) The numerical values of the inertial parameters must be known.
The solution of this problem has been investigated by the use of iden-
tification procedures based on a dynamic model linear in the inertial
parameters [5]-[9]. The classification and the determination of the
identifiable parameters (the minimum inertial parameters) increases
the robustness of the identification process [8].
2) The computational cost of the dynamic model must be reduced.
The solution of this problem has been carried out by the use of the
customized Newton-Euler method [lo], [ 1 11. To increase the number
of parameters that are equal to zero and, consequently, to reduce the
number of operations of the dynamic model, the use of the minimum
set of inertial parameters in a customized Newton-Euler method has
been proposed [ 121, [ 131.
The determination of the minimum set of inertial parameters has
been investigated in [ 121-[15]:
Khosla [14] has used a symbolic Newton-Euler algorithm. Some
results concerning some parameters in the case of robots whose suc-
cessive axes are either parallel or perpendicular are given. The detec-
tion of most of the combined parameters requires a long examination
of the symbolic Newton-Euler model on a case-by-case basis.
Khalil et al. [13], [15] have determined the minimum set of inertial
parameters by the examination of the symbolic expressions of the
inertia matrix and the gravity forces of a Lagrangian dynamic model.
Although these expressions are generated automatically by computer
[16], the detection of the combined inertial parameters may take
several hours.
The aim of this paper is to present a general and direct method to
determine the minimum set of inertial parameters of serial robots.
Most of these parameters (or in most cases, all of them) will be
determined by the use of the closed-form solution.
Coincidently with our work [17], similar results concerning the
special case of rotational robots whose successive axes are perpen-
dicular or parallel have been given by Mayeda et al. [ 181.
Manuscript received April 20, 1988; revised June 9, 1989.
The authors are with the Laboratoire d'Automatique de Nantes URA
C.N.R.S., E.N.S.M., Nantes, France
IEEE Log Number 8933674.
11. DESCRIPTION
OF THt~ ROBOT
The system to be considered is an open-loop mechanism of n +
1 links, where link 0 is the base, whereas link n is the terminal
link. The description of the system will be carried out by the use
of the modified Denavit-Hartenberg notation [ 191, [20]. Thanks to
this description, the closed-form solution of regrouping the inertial
fixed with respect to link j . The z , axis is along the axis of joint j ,
parameters has been obtained. The coordinate frame j is assigned
and the x, axis is along the common perpendicular of E , and z , ,
The frame j is defined with respect to frame j - 1 as a function of
the parameters (a,, d,, O,, r,). The type of joint j will be defined
by a:, where a, = 0 for j rotational, a, = 1 for j translational, and
f.7, = ( 1 -a,).
111. THL MINIMUM
INERTML PARAMETERS
The minimum inertial parameters are defined as the minimum set
of constant inertial parameters that do not conrain the zero element
and are sufficient to calculate the dynamic model of the robot. They
can be obtained from the classical inertial parameters by eliminating
those that have no effect on the dynamic model and by regrouping
some other parameters.
A . The Classical Inertial Parameters
The inertial parameters of link j are given by the vector X J , which
is denoted as
X J = IXX,XY~X~,.YY,.YZ,.ZZ,.~X,.~Y,~.~Z,.~,I~(I)
where ( X X j , . . ,ZZ,) are the elements of the inertia matrix ' J , ,
which defines the inertia of link j around the origin of frame j ,
( m X j , m y , , mZ,) are the elements of JmS,, which defines the first
moments of linkj, and m j is the mass of link j .
Using this set of parameters and denoting E as the kinetic energy
and U as the potential energy of the robot, we deduce that
where m is equal to 10n, XI is an inertial parameter, and W', is
constant.
B. Parameters Having No Eflect on the Dynamic Model
Using the Lagrange equation, one can derive that if
DE, - 0
(4)
the corresponding inertial parameter X I has no effect on the dy-
namic model. Thus, X , is not an element of the minimum inertial
parameters.
and d U , = 0
'The inertial parameters satisfying these conditions belong to the
links near the base side. They can be determined easily by hand
or automatically by computer [16]. Supposing that r l is the first
rotational joint and r2 the first succeeding rotational joint not parallel
to r I , one can derive most of the parameters having no effect by the
following rules:
1) If joint j
is translational and j < rI, then X X , , X Y , ,
X Z , , Y Y , , Y Z , , ZZ,, mX,, m y , , mZ, have no effect on the dy-
1042-296X/90/06OO-0368$01 .OO 0 1990 IEEE
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 6 , NO. 3, JUNE 1990
369
namic model due to the fact that the angular velocity of link j is
equal to zero.
2 ) If the axis of joint r l is parallel to the direction of gravity and
the axes of joints j < r l are parallel to the axes of joint r l , or if
r l = 1 , then m X r I , m y r I have no effect on the dynamic model.
Moreover, in the case of r l = 1, m I has no effect on the dynamic
model.
3) For a linkj such that r l 5 j < r2 and the axis of link; is paral-
lel to that of r l , the parameters X X , , X Y , , X Z j , Y Y , , Y Z , , mZ,
have no effect on
the dynamic model. The elimination of
m X , , mY J , m j must be studied on a case-by-case basis. The pa-
rameters to be eliminated if the axis of joint j is perpendicular to
that of r l can be deduced easily.
4) For the links j 2 r2, all the parameters of the inertia tensor
have an effect on the dynamic model. Therefore, if the parameters
mXk, mYk, mZk appear in the model, then all the parameters of
links j > k will appear in the dynamic model.
C . Conditions for Regrouping the Inertial Parameters
An inertial parameter X I can be regrouped to some other param-
eters X;l , . . . ,Xir if
DE; = ~ ; I D E , I + . .. + a,,DE,, = x f f , k D E , k
r
(Sa)
and
dU; = f
f ; ~ dU;i + . . . + f f j r dUjr - m;k dU;k
k = l
-2
k = l
(5b)
with (Y;k = constant; therefore, we get the same value of E and
dU/dq (and consequently, the dynamic model) if we use the pa-
rameters x;, x;~,
. . . ,Xir or if we put X i equal to zero and the
parameters X;k equal to XR;k where
XRjk = Xjk + (YjkXj.
(6)
Relation (5) is similar to that given in [13]. Its use by calculating
DE; and dU; is time consuming and error prone, especially for
robots having n 2 3. In the following section, we present a method
that overcomes this difficulty.
IV. CLOSED-FORM SOLUTION OF REGROUPING
and j = l , . . . , n . Denoting DE'
aE/dX! and DUJ the vector of the components dU/dXj
Let X j be the ith inertial parameter of link j , with i = 1 , . . . , l o ,
the vector of the components
THE INERTIAL PARAMETERS
(7)
The main idea is to calculate DEJ and DUJ as a function of DE'-'
and D U J - l , respectively.
A . Recursive Relations of DE' and DUJ
The recursive relations of DEJ and DUJ can be written as
DEJ = XiDEJ-' + q J j ( q , q )
and
or in the expanded form, we have
DUJ = P D u J - l
(9)
(10)
As dU/dXi-' = 0 for k = 1,. . . ,6, and taking into account that
(see the Appendix) p:,k = A;,, ( k = 7 , . . . , lo), (12) can be written
as
The expressions of hi and f' are given in the Appendix.
B . General Regrouping Relations of Inertial Parameters
Let us examine the regrouping of the inertial parameters of link j
on those of link j - 1 by the use of the recursive relations of Xi and
f J given in the Appendix. We consider the following two cases:
1) If joint j is rotational, we see that
a) the coefficients A{,
are constant for k = 1, . . , l o , whereas
f {o(q, q ) = 0. Thus, DE',, can be obtained as linear combinations of
DE;-', i.e., DE{, verifies the condition (5a). Taking into account
(13), (5b) is also verified. Thus, the tenth parameter m, can be
regrouped to the ,inertial parameters of link j - 1. Using (6) and the
expressions of A{o, given in the Appendix, we get
X X R , - I = X X J - , +r?m,
X Y R J P I = X Y j - , + d j r j S a j m j
X Z R j - , = X Z , - , - d,r,Ca,m,
YYRj-, = Y Y , - I + (d: + r:Ca:)m,
Y Z R J - , = Y Z J P l + r ~ C a , S a j m ,
ZZR,-I = Z Z j - I + (d: + r:Sa:)m,
mXR,-I = mX,-l + d j m j
mYRj-I = m Y j - , - r,Sajmj
mZRjPl = mZ,-, +r,Ca,m,
mRj-, = m j P l + m,.
b) The coefficients A;,
(14)
are constant for k = 1,. . . ,lo, whereas
fi(q, q ) = 0. Similar to the foregoing case, the parameter mZ, can
be regrouped to the inertial parameters of link j - 1 .
c) The sum of DE: and DE:, corresponding to the parameters
X X j and YY j , respectively, can be expressed through constant co-
efficients as a function of DE$-', k = l , . . . , 10. Thus, X X , or
Y Y j can be regrouped to the parameters of link j - 1 and Y Y , or
X X j , We will always choose to regroup Y Y , .
As a conclusion,
the parameters
( Y Y , , m Z j , m j ) can be eliminated, whereas the combined parame-
ters of links j and j - 1 are given as
is rotational,
if joint j
X X R j = X X j - YYj
X X R j - , = X X , - I + Y Y j + 2rjmZ, + r:mj
X Y R j - , = X Y j - , + d j S a j m Z j + d j r j S a J m j
X Z R j - I = X Z j - , - d,CajmZj - d,r,Ca,m,
YYRj-I = Y Y j - , + Ca:YYj + 2rjCa:mZ, + (d: + r:Caj)m,
Y Z R j P I = Y Z , - , + C a j S a j Y Y j + 2rjCa,SajmZ,
+r:Ca,Sa,m,
ZZRj-I = Z Z j - , +Sa:YYj + 2rjSa;mZj + (d: + r:Sa?)m,
m X R j - , = m X j P l + d j m ,
m Y R j P l = m y , - , - Sa,mZ, - rjSa,m,
mZRj-, = mZ,-, + C a j m Z j +r,Ca,mj
mRj-, = m j - l +m,.
(15)
370
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION. VOL. 6, NO. 3, JUNE 1990
2 ) If joint j is translational, we see that A:,,
are constant, whereas
f:(q,q) = O for i = 1 , . . . , 6 and k = 1 , . . . , 1 0 . Thus, the param-
eters X X , , X Y , , X Z , , Y Y , , Y Z j , Z Z j can be regrouped to the
inertial parameters of link j - 1 by the use of the following rela-
tions:
XXR,-1 = XXj-1 +C8:XXJ -2CS8jXYj +SB:YY,
XYRj-1 = XYj-1 + CS8jCajXXJ + (C8: - SO:)CajXY,
- c t l j S ~ ~ j X Z j - CS8jCajYYj + SBJS~,YZj
XZR,-l = XZj-1 +CS8jSaJXX, + (C8: -SB?)Sa,XY,
+CBjCajXZj - CSBjSajYY, - S8jC~ijYZj
YYR,-I = Y Y j - , +S8:CajXX, +2CSOjCaSXY,
- 2SBjCSajX2, + CB~ZCCY~YY,
- 2C8jCS~rjYZj + S a s Z Z j
YZR,-l = YZj-1 +S8:CSa,XX, + 2CS8,CSaJXY,
I
I
I
i
i
+SB,(Ca: -Sa:)XZ, +C8:CSa,YYJ
+ C8j(Ca: - Sa:)YZ, - CSa,ZZ,
ZZRj-, = ZZj-I +SBfSa:XX, + 2CS8,Sa;XYj
+2S8jCSa,XZJ + C B ~ S a ~ Y Y , + 2COjCSajYZj + CajZZ
(16)
where CS( .) is equal to cos ( . ) sin ( ).
Remarks: 1) We note that (16) corresponds to the following
= J - I J , - ,
J-lJRj-l
+'-'A I .JJ J JA I - 1
matrix equation:
(17)
where J-IA , is the (3 x 3) matrix defining the orientation of frame
j with respect to frame j - I .
This relation can be obtained directly by noting that the angular
velocity of link j is equal to that of link j - 1 if joint j is translational.
2) From the general regrouped results and taking into account
the eliminated parameters of link 1, we deduce that the number of
minimum inertial parameters is equal or less than
7n, +4n, - 3 - 51
(18)
where n, is the number of rotational joints = Eaj and n, is the
number of translational joints = Coj .
C. Particular Regrouping of the Inertial Parameters
As we have seen in Section 111-B, some of the inertial parameters
of the first links may have no effect on the dynamic model; there-
fore, particular regrouping, other than that denoted in Section IV-B,
may take place. The detection of these particular cases have to be
studied on a case-by-case basis using (5). We find that the particular
regrouping will concern only the parameters m X , mY, and mZ of
the translational links lying between the joints rl and r2 defined in
Section 111-B. For the general case, it is not easy to calculate this
regrouping by a closed-form solution. If we consider that the joints
between r l and r2 are either parallel or perpendicular such that the
angular velocity of the links referred to the link frame will have only
one component different than zero, then the particular cases will take
place if the axis of the joint is parallel to r l ; in this case, the param-
eter mZj has no effect on the model, whereas the parameters m X j
and mY will be regrouped to the parameters of link j - I . Two
cases are considered.
and
1) If Sa, = 0, then mZ,-I has no effect on the dynamic model,
ZZRj-, = Z Z j - , + 2d,CBjmXj - 2d,SBjmYJ
m X R j P I = m X J P I + C8,mX, - SBjmYj
mYRj-, = m Y j - , +SBjCajmXi +C8,CajmYj.
(19)
Fig. 1. PUMA robot.
TABLE I
GEOMETRIC PARAMETERS
OF THE PUMA ROBOT
3
4
0
0
0
-90
D 3
D4
e3
84
6 I
o
I
o
I e 6
I
I
I
1 - 9 0
I
R3
R4
o
2) If Ca, = 0, then m y ,
has no effect on the dynamic model,
and
YYR,-l = YY,-I + 2dJC8,mX, - 2dJS8,mY,
mXR, = M X , -, + C8, mXJ - S8, m y ,
mZR,-, = MZ,-I +Sa,SB,mX, +Sa,CB,mY,.
(20)
V. PRACTICAL UTILIZATION
OF THE REGROUPING RELATIONS
c
The proposed algorithm of the detection of the set of minimum
parameters will be carried out as follows:
1) Find the parameters having no effect on the dynamic model, as
is given in Section 111-B.
2 ) Apply (15) and (16) from link n to link 1.
3) Apply (5) to the parameters of translational links lying between
r l and r2 to detect the existence of supplementary regrouping. If
these joints are parallel or perpendicular, we can use (19) and (20).
Example: In the following, we get the minimum inertial param-
eters of the six rotational joints of the well-known 560 PUMA robot
(Fig. 1). The geometric parameters of the robot are given in Table
I.
Step a): Using rules 2) and 3) given in Section 111-B, the following
parameters have no effect on the dynamic model:
X X I , X Y , , X Z I , Y Y , , Y Z 1 , m X 1 , m y 1 , m Z , , and m l .
Step b): Use (15) for links 6, . . . , 1:
link 6:
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION. VOL. 6. NO. 3. JUNE 1990
37 I
The minimum parameters of link 6 are X X R 6 , X Y 6 , X Z 6 , Y Z 6 .
ZZb, m x b , myb.
link 5:
XXR5 = X X s +YYt, - Y Y s
XXR4 L X X 4 + YYs
ZZRj ZZ4 + YYS
mYRJ = mY4 - mZs
mR4 = m4 t mRS = m4 + ms + m 6 .
The minimum parameters of link 5 are X X R S , X Y 5 , X Z s , Y Z s ,
ZZR5, m X 5 , mYRs.
link 4:
XXR4 = X X J +YY5 - YYI
X X R , = X X 3 + Y Y 4 + 2R4mZJ + R4'(mJ -t mc + m6)
XYR3 = X Y 3 - D4mZJ - D4R4(m4 + mc t m,)
Y Y R ? =-YY-\ + D4'(m4 + m5 + m6)
ZZR3 = ZZ3 + Y Y J + 2R4mZ4 + (04' + R4')(m4 + m5 t m6)
m X R s = mX3 + D4(m4 + m5 + m6)
mYRi = my3 + m Z j + R4(m4 + ms + mb)
mR3 = m3 + m4 + m5 + mb.
The minimum parameters of link 4 are X X R j , X Y 4 , X Z 4 , Y Z 4 .
ZZR4, m X 4 , mYR4.
link 3:
X X R S = X X 3 + Y Y 4 t 2R4mZ4 + (R42 - 0 4 ' )
.(m4 + m 5 + m 6 ) - Y Y 7
XXR2 = X X 2 + Y Y 3 + 2R3mZ3 + R32(m3 + m4 + m5 + mh)
-E M 5 + mg)
+ D 4 2 ( ~ 4
XZR2 = XZ2 - DSmZ, - D3R3(m3 + m4 + ms + mb)
YYR2 = yy2 + y y 3 + D42(m4 + m5 + 1716) + 2R3mZ3
+(D32 + R3')(m3 + m4 + ms + mg)
ZZR2 = ZZ2 + D3'(m3 t m, t mc + mb)
m X R 2 = m X 2 + D3(m3 + m4 -t ms + m 6 )
mZR2 = m Z z + mZ3 + R3(m3 + m4 + ms + m b )
mR2 = m2 + m3 + m4 + m5 + mb.
The minimum parameters of link 3 are X X R 3 , X Y R 3 , X Z 3 , Y Z 3 ,
ZZR3, m X R 3 , mYR3.
link 2:
X X R 2 = X X 2 - Y Y 2 - D32(m3 + m4 + m5 + m e )
Z Z R , = Z Z I + Y Y 2 t Y Y R 3 + 2R3mZ3
The regrouping relations on
+(R3* + D3')(m3 + m4 + m5 + m6).
the parameters X X , , X Y , ,
x z i , y y ~ , Y Z I , m X , , m y , , m Z , , m l are not written because
these Parameters have no effect on the dynamic model. The minimum
parametersoflink2areXXR2, X Y 2 , X Z R 2 , Y Z 2 , Z Z R 2 , m X R 2 ,
m y 2 .
link 1:
The minimum parameter of link 1 is Z Z R , .
Since mZ2 and mz do not appear on Z Z R , , they have no effect
on the dynamic model.
Step cJ: This step does not have to be calculated since the robot
has only rotational joints. The final results arc summarized as fol-
lows:
1) The parameters having no effect on the dynamic model are
X X , , X Y , , X Z I , Y Y , , Y Z , , m X l , m y , , m Z I , m , , m Z z , m z .
2) The parameters to be eliminated by regrouping are
YY2, YY3, mZ3, m3, YYq, m Z 4 , m4,
Y Y s , mZs, ms, y y b , m Z 6 , mb.
VI. APPLICATION
TO T H E I D E N T I F I C A T I ~ N A N D CONTROL
The minimum set of inertial parameters are the only identifiable
parameters by the use of the dynamic model [SI, and they are only
needed for the calculation of the control law. They represent, in the
case of the given example, 36 parameters, i.e. 24 parameters less
than the classical parameters. This will greatly facilitate the identifi-
cation process. The regrouped relations may not be important in this
case; only the knowledge of the parameters to be eliminated either
by regrouping or because they have no effect is needed. The identi-
fication process would give the values of the identifiable parameters
directly.
In many control schemes, such as computed torque, we need
to calculate the dynamic model at the servo rate. The use of the
minimum set of inertial parameters in an algorithm of customized
Newton-Euler will contribute to reducing the cost of calculating the
dynamic model [13].
VII. CONCLUsroN
This paper presents a direct method of determining the minimum
set of inertial parameters of serial robots. The given method per-
mits determination of most of the regrouped parameters by means of
closed-form relation function of the geometric parameters of the
robot. The parameters that may need particular study concern the
first moments of the translation links between r I and r2. If the joints
between r l and r2 are parallel or perpendicular, we can obtain all of
the minimum inertial parameters directly without calculating the dy-
namic model or the energy. The method is integrated to our software
package of automatic symbolic modeling of robots SYMORO [ 161.
Generalization of this symbolic method to tree-structured robots and
to robots with parallelogram closed loops are given in [22] and [23],
respectively. Numerical approaches to determine the minimum iner-
tial parameters are given in [24] and [25).
A wENi)rX
c
RECLIRSIVE
RSLA'TIONS OF THE DERIVATIVES
OF THE ENERGY
In this Appendix, we present directly the results of the calculation
of the derivatives of the energy with respect to the inertial parameters
of link j in terms of the derivatives of the energy w.r.t. the inertial
parameters of link j
A . Recursive Relation of DE'
We have the matrix relation
- 1. The details can be found in [21].
DEJ = XJDEJ-l + . s f
where Xi is a ( I O x IO) matrix, andf'
I . Expressions of XI: The matrix XI is given as
(AI)
is a (10 x 1) column matrix.
where
Om,n is the ( m x n) zero matrix:
' - ' A , is the ( 3 x 3) matrix defining the orientation of frame i with
respect to frame i - 1;
372
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 6. NO. 3. JUNE 1990
‘ - ’ P , is the (3 x 1) vector defining the position of the origin of
frame i with respect to frame i - 1;
and J-’PJ are functions of the geometric
DJ, DS, D m ,
parameters d j , r , , OJ , aJ . They are given as follows (the indexj has
been omitted to simplify the writing):
We initialize the recursive relation by
DUO = [ O 1 , 6
*T
OIT
(‘412)
where g is the acceleration of gravity.
csecff
CSOSff
ssocca
SSOCSCY
SSOSSa
(CCO - SSO)SCU ~CSOCCCY
ceca
-CSOSa
4OCa
0
2CSOCSa
2CSOSSa
-2SOCSff SO(CCff - S S a ) 2SOCSa
CCOSSa
CCOCCa
CCOCSa
ssa) 2cecsa
-2cocsff ce(ccff
cca 1
-csa
ssa
~
(-43)
*
~~~~
~~~
~~
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P. Khosla, ‘‘Real-time control and identification of direct-drive manip-
ulators,” Ph.D. thesis, Carnegie Mellon University, Pittsburgh, 1986.
W. Khalil, M. Gautier, and J. F. Kleinfinger, “Automatic generation
of identification models of robots,” Int. J . Robotics Automat., vol.
1, pp. 2-6, 1986.
W. Khalil, J. F. Kleinfinger, and C. Chevallereau, “SYMORO:
system pour ia modelisation des robots,” Notice d’utilisation, Note
interne LAN, Nantes, 1987.
M. Gautier and W. Khalil, “A direct determination of minimum iner-
tial parameters of robots,” in Proc. IEEE Conf. Robotics Automat.
(Philadelphia), 1988, pp. 1682- 1687.
H. Mayeda, K. Yoshida, and K. Ousuka, “Base parameters of manipu-
lator dynamics,” in Proc. IEEE Conf. Robotics Automat. (Philadel-
phia), 1988, pp. 1367-1373.
W. Khalil and J. F. Kleinfinger, “A new geometric notation for open
and closed loop robots,” in Proc. IEEE Conf. Robotics Automat.
(San Francisco), 1986, pp. 1174-1 180.
E. Dombre and W. Khalil, “Modelisation et commande des robots,”
Edition Hermes, Paris, 1988.
DJ =
-2CSO
0
SSO
0
0
-
(CCO
SSO)Ca
~
- a s f f
-cseca
SOSa
0
Dm
= [ r 2 drSa
I
- d r C a
d ? + r ’ C C a
r 2 C S a
/ - l p T - - [ d
- r s a
r c a ]
2) Expression of f J : Two cases are considered:
a) For j rotational
case
sase
J A ~ - ~ =
CMO
-sa
S ~ C O
cff
[
C O
-se
0
1
b) for j translational
f J = [ O 0 0 0 0 0
-wz,j
W I , J 0
( V 3 , J - ; q j ) I T
(‘49)
where 0, is the velocity of the joint variable J , whereas wk, J , and
V k , j denotes the kth component of J w j and ’ V , , respectively, and
where J V , is the velocity of the origin of the link j fixed frame,
referred to frame j , and Jo, is the angular velocity of link j , referred
to frame j .
3) Znitialization of the recursive relations of DE’: Since link 0
is fixed, then we initialize the calculation by
. . . 0 I T .
(A101
D E o = [ O 0
B . Recursive Relations of DUJ
We have the relation
DU’ = 03.6 JA,-l
( A l l )
[
0 6 , 6
0 6 , I
0 6 , 3
J-bY ‘y‘]
DuJ-‘’
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL 6 . NO. 3. JUNE 1990
373
1211
[22]
[23]
1241
[25]
M. Gautier, “Contribution a la modelisation et a I‘identification des
robots,” these d’etat, Nantes, 1990.
W. Khalil, F. Bennis, and M. Gautier. “Calculation of the minimum
inertial parameters of tree structure robots,” in Proc. 3rd ICAR Conf.
(Columbus), 1989.
F. Bennis and W. Khalil, “Minimum inertial parameters of robots
with parallelogram closed loops,” in Proc. IEEE Conf. Robotics Au-
tomat. (Cincinnati), 1990.
H. Sheu and M. W. Walker, “Basis sets for manipulator inertial pa-
rameters,” in Proc. IEEE Conf. Roborics Automat. (Scottsdale),
1989, pp. 1517-1522.
M. Gautier, “Numerical calculation of the base inertial parameters,”
in Proc. IEEE Conf. Roborics Automat. (Cincinnati), 1990.
Robustness Analysis of Nonlinear Decoupling for
Elastic-Joint Robots
WOLFGANG M. GRIMM. UEMBER. IEEE
Absmct- In this communication, a robustness analysis of feedback
linearization or nonlinear decoupling is proposed for a class of elastic-
joint robot manipulator control loops. it is shown that by splitting the
modeling-error nonlinearities of incomplete feedback linearization or
nonlinear decoupling by a suitable transformation, a robustness analy-
sis becomes feasible under weak assumptions without the necessity of
estimating bounds on derivatives of robot nonlinearities.
I. INTRODUCTION
The robustness analysis for rigid robot manipulators, which is out-
lined in [1]-[5], is based on feedback linearization [6] or nonlinear
decoupling [7], where the rigid robot model is always feedback lin-
earizable. This may not be the case for elastic-joint robot models
in general [8]; however, under assumptions outlined in [9] or [IO],
the conditions for feedback linearization, or equivalently, for the ex-
istence of a completely observable nonlinear decoupling, can always
be fulfilled. Then, the globally linearized elastic-joint robot model
can be represented by a chain of four integrators. In [ 1 I], the robust-
ness of such nonlinear-decoupled robot control loops is discussed.
It turns out that the robust design method in [I]-[5] for rigid robot
models cannot be carried over to the elastic-joint case without ma-
jor difficulties when the system description of the globally linearized
system is used. Either bounds on derivatives of robot nonlinearities
have to be estimated, as is also suggested in [9], or a loss of per-
formance has to be tolerated when, for example, a compensation of
gravitational forces is abandoned [IO].
In this contribution, the results in [IO] are modified and developed
further. The idea is not to transform a given nth-order nonlinear
system into a chain of n integrators for the robustness analysis but
to compensate nonlinearities “in between” the integrators exactly
where they physically appear. This means that the summation point
of modeling error terms that contain derivatives is shifted beyond
integrators such that derivatives in these terms vanish.
Feedback linearization or nonlinear decoupling applied to rigid
robot models, which is equivalent to the well-known computed-
torque method, yield control laws that contain no derivatives of
robot nonlinearities. Thus, it is obviously not necessary to apply
such a nonlinear transformation to the case of rigid robots; however,
it may be advantageous for any system that has a control law derived
from feedback linearization or nonlinear decoupling that contains
derivatives of nonlinearities. As a result, the integrator chain will be
partitioned into groups, where nonlinear decoupling is accomplished
in stages. This transformation does not change the nature of the con-
trol law, but it is attractive for a robustness analysis, and it facilitates
the controller design.
It is shown in this communication that a robustness analysis of
feedback linearization, or equivalently, of nonlinear decoupling, be-
comes feasible under weak assumptions. The features of the trans-
formed globally linearized system are twofold: First, derivatives of
the control-loop nonlinearities due to model uncertainties vanish. Sec-
ond, measurements of the external-joint accelerations and jerks that
would be needed for control [9] are reconstructed by the proposed
transformation and are not used for control. However, robust ob-
servers as proposed in [12]-[I31 can still be used to overcome mea-
surement difficulties [9].
This communication is organized as follows: In Section 11, the con-
trol law of feedback linearization or nonlinear decoupling is stated for
a class of elastic-joint robot models. The globally linearized system
is then transformed in Section 111 for the robustness analysis, which
is sketched in Section IV. The robustness condition to be used is an
extension of those previously obtained for the “rigid case” in [4].
Finally, some remarks in Section V conclude this communication.
11. NONLINEAR DECOUPLINC
FOR A CLASS OF ELASTIC-JOINT ROBOT
MODELS
A class of robot manipulators that can be described by a set of
linear and decoupled and a set of nonlinear and coupled differential
equations is considered according to [9]-[ 1 11:
vector of applied torques at the motor shaft
diagonal matrix with gear ratios ng; < 1 (i =
1 , . ’ . ,n)
diagonal matrix with constant motor inertia
external joint position vector
= q M - N;’qa vector of angular displacements at
the motor shaft, where qM is the vector of motor shaft
positions
diagonal matrices of spring stiffness and damping co-
efficients, respectively, measured at the motor shaft
nonlinear and coupled robot inertia matrix without any
entries of motor mass
vector of centripetal, Coriolis, and frictional forces
vector of gravitational forces.
C and D
The model description of (1) and (2) can be shown to include the
rigid robot model as C 4 x or D 4 x. It represents a class of
elastic-joint robots under the assumptions stated in [9]-[IO]. A major
assumption is that the kinetic energy of the ith motor mass due to the
motion of all jth robot links, where j < i, can be neglected ( i , j =
1, . . , n). As is shown by Spong [9], this assumption is easy to justify
for a large class of robots that have high gear transmission rates.
Furthermore, the gear elasticities of industrial robots are commonly
weakly damped [14], and D = 0 is considered in the sequel.
Applying feedback linearization [6] or nonlinear decoupling [7] to
the system of (1) and (2) for D = 0 leads to the control law
H = A 2 G C - t ~ g A 2 R
h = -H* - P(i. + g) + (PN,’ + 1)CAq
(3)
(4)
( 5 )
Manuscript received December 19, 1988; revised September 18, 1989. where
This work was supported by the Volkswagen-Stiftung (U61 394).
The author is with Fachgebiet Mess-und Regelungstechnik, University of
Duisburg, West Germany.
IEEE Log Number 8933040.
1042-296X/90/0600-0373$01 .OO O 1990 IEEE
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