Proceedings
of
the American Control Conference
San
Diego,
California June
1999
Adaptive
ANN
Control
of
Robot Arm Using Structure
of
Lagrange Equation
Masaki YAMAKITA and Takashi SATOH Department
of
Control and Systems Engineering Tokyo Institute
of
Technology
Abstract
Several adaptive control algorithms for robot manipu lators using artificial neural network ANN) have been pro posed and an algorithm using physical structure of robot manipulators also has been proposed. But the structure of ANN should be modified according to the change of the de gree of freedom of the system. In this paper, we propose the algorithm in which structure of ANN is independent of the degree of freedom and compare it to some other algorithms by simulation and experiment.
1
Introduction
Adaptive control algorithms for robot manipulators have been studied for
a
decade, e.g.,
[2]
and they usually use a property that the dynamic equation of the system can be represented by
a
multiplication of
a
known regressor matrix and
an
unknown dynamic parameter vector. It
is
not easy, however, to obtain the regressor matrix if the kinematic pa rameters are unknown. Several adaptive control algorithms using Artificial Neural Network ANN) have been also pro posed to overcome the problem. The explicit calculation of the regressor matrix is not required in these algorithms
[l]
[3].
The learning capability of the network was very low be cause the structure of the dynamic equations was not taken into account. In
[4]
an adaptive control algorithm which considers the structure of the property has been proposed, however, since the allocation of the neural elements was fixed, the structure of the neural network should be modified according to the degree of the system and neural elements may not be used effectively. In this paper
a
new adaptive
ANN
controller which over comes the problem
is
proposed. It uses
a
structure of dy namic energy of the system. The efficiency
of
the proposed method is compared to other constructions of the neural net works by the numerical simulations and experiments.
2
Structure
of
ANN
For computational efficiency we use RBF Radia1 Basis Function
)
neural network which is known that it can ap proximate any continuous nonlinear function with any ac curacy for
a
bounded set of the d0main.A neural network
4~
which has
n
input, one output and p functions can be represented by where
B
E
Rp,<
E
Rp,p;
E
R .
Of course, multi output functions can be represented by
a
combination of the net works. We introduce the following assumption
as
in
[3].
[Assumption] Let
4 ~)
e a continuous function to be estimated and
as
sume that there exists
a
known matrix function
Y,
satisfying in the domain where
11 11
stands for
a
matrix induced norm or the Frobenious norm and
&
is
an unknown constant matrix. Please notice that this condition is satisfied by the RBF neu ral network and also by three layers neural networks if we assume bounds of weighting matrices
as
in
[3].
3
Design of Controller
We consider the system whose dynamic equation is repre sented by
M(q)i
+
(C(q7
4
+
D)i
+
G(q)
=
5)
where
q
is
a
generalized coordinate vector, is a general ized force vector and
M(q),
C q,
,
D,
G(q)
are inertia, Col ioris/centrifugal and damping, gravity terms, respectively. Let
qd,
qd,
i[d
E
L,
be desired trajectories and
e
:=
q
qd,
s
:=
i+Ae
(A
>
0),
r
:=
qd
he.
If we use
a
control input defined by where
B
=
r
eYT(q,
41
r)s,
re
>
0
8)
=
raYallSll,
r
>
0, 9)
and
8
is estimated function of
8,
then
q,
converges to the desired signal
qd,qd.
The convergence of the error can be shown as in
[l]
3]
under the assumption. That is proved in appendix.
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Figure 1:
Control
System The remained problem of the control design is how to con struct
4~.
n the following 4 alternative methods are com pared and Method
3
and Method 4 are methods proposed in this paper. Method1
[l]
4~
is constructed by
a
single
NN
with the knowledge of passivity structure of
4,
and it
is
constructed by and
d,
G
can be derived
as
in Method 2. Please notice that the allocation of the neural elements
is
not fixed in this Method.
k
s constructed by a multiplication of estimated ma trices and neural elements and the potential function
U
s constructed by the another
NN
as
Met ho d4 proposed)
n
M
=
xkits
i l
O
=
eG<C.
where
A?
is
a
matrix. The block diagram of .the, net work is equal to that of Method2.) Though
C,
can be derived
as
in Method 2, usually the number of the estimated parameters increases compared to Method3.
4
Simulation and Experiments
noted
shadow
area
in
Figure
1
has
no
structure in this Method. In the following Methods the knowledge of the structure,
JN
=
k
d
+
D)qr
+
G
is used for the construction of
ANN,
Method2 [4] Each element of
M(rn,,)
is approximated by each
NN
and potential function
U
s done by another
NN.
C q,
4
and
G q)
are computed by partial derivative operations of
M q)
and
V(q)
as Method3 proposed)
To
use
full
structure
of
the whole energy function
of
the system, first kinetic energy
h
and potential energy
U
are constructed by two
NNs,
and the required ele ments are calculated by partial derivative operations
4.1
Computer Simulation
Each algorithm in the previous section is applied to
a
two link mainpulator system in
a
vertical plain. In the following figures, the responses to the desired angle and anglar ve locity are shown. The center of the each
RBF
is specified randomly in the operation range of the desired signal and the total number of RBFs in each Method
is
100
n the sim ulations. From the figures it can be seen that Method2

give bet ter performance. The Method3 gives the best performance for the velocity tracking from the Figure
6

9.
4.2
Experiment
We applied each algorithm to an industrial robot arm under the same condition. It
was
observed that Method2,
3,
4 give better perfomance, however, the differences between Method2, 3 and
4
are not significant. The reason may be in fered
as
that dumping effect in the real robot arm was very big and dominant.
5
Concluding Remarks
In this paper an effective way to construt regression vec tor for an adaptive controller using a neural network has been proposed. The proposed Methods were compared with other Methods by numrical simulation and experiments, and it was convinced that the proposed Method gives good track ing performance. In the practical applications robustness of the adaptive scheme, e.g., sigma modification
so
on, should be combined. In the propoed Method RBF neural network was used for comutational simplicity, however, the perfomance of the
2835
network depends on the allcation of the center of each basis function,
pi.
Further study shoud consider adaptive alloca tion of the center, and also another type of neural network, e.g., three layered neural network shold be compared to the RBF neural network.
APPENDIX A.1 The Proof of Stability
where Choose a candidate of Lyapunov function
V
=
Vi
+
%,
where
=
8
,
=
&
.
The time derivative
of
VI
an be derived
as
2
=
sTMi+sTMs
2
=
sTM(i
&)
+
sTMs
1
2
=
sT(r
C
D)q
G
Mir)
+
sTMs
ST(&&
6
+
b)qr
sT(Y
DS
+
v
KdS)
sTDs
TKds
+
STV
STY
TI
= = =
DS
+
v
KdS)
[3] WeiDer Chang,
et.
al: “ADAPTIVE ROBUST NEURALNETWORK BASED CONTROL FOR SISO SYSTEM” IFAC,13th Triennial World Congress, San Fran cisco,USA, 1996)
[4]
Shuzhi
S
Ge: “ROBUST ADAPTIVE CONTROL WARKS” IFAC,13th Triennial World Congress, San Fran cisco,USA,1996. OF ROBOTS BASED ON STATIC NEURAL NET
Simulation results
The respances for desired angle Figure
2

5)
5
5
sTDs
TKds
+
sTv
Ty
+
~
sTDs
TKds
+
sTv
TY
+
BTYalls/l.
Figure
4:
Method3
Figure
5:
Method4 The respances for desired anglar velocity Figure 6

9)
he time derivative of
V2
can be
also
calculated
as
r2
=
iTrgfB+ Tr l
a
8Tre1j Trali.
~~~~~
,
?ad
sec
Q
ad
SCE
am
m
tm
>m
&m
Im
am
om
Im
,m
>m
uo
xm
m
method
4
10
Therefore the time derivative of
V
is
am
v
=
Vi+
,a
am am
sTDs
STKds
+
sTv
Traiii
+
‘YfiIIsll
1.4
I 4
=
sTDs
TKds
+
sTv
GTre’ii
+
&‘Y&lls11
Figure
6:
Method1
Figure
7:
Method2
+tiT
Yn
1311
=
sTDs
TKds
:R;R
d
rad
SCC
In
a
am
to the transfer function from
s
to
e.
Furthemore it can be
om
nm
zm
>m
<m
x
~0
om
~m
tm
rm
~m
~m
method
0.
4
O
Vis
negative semidifinite. Hense
s,
,
are bounded and
s
E
L2.
s

as
1

CO
because
e
E
L2
n
L,,
i
E
L2
accoding
4 1
11
[ 4
seen that
d
0
because
L
E
L,
from the boundedness of
i.
0
Figure
8:
Method3
Figure
9:
Method4
References
[I]
F.L.Lewis, et. al: “Neural Net Robot Controller: Structure and Stability proofs.” Proc. of IEEE CDC’93,
2785/2791
1993)
[2]
JeanJacques E.Slotine Weiping Li: “On the Adap tive Control of Robot Manipulators” Int. Jontal in Robotics Reserch, vo1.6
49/59 (1987)
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