Modular Neural Net System for Inverse Kinematics Learning Eimei OYAMA
Susumu TACHI
Robotics Department
Faculty of Engineering
Mechanical Engineering Laboratory
The University of Tokyo
Namiki 1-2, Tsukuba Science City
Hongo 7-3-1, Bunkyo-ku
Ibaraki 305-8564 Japan
Tokyo 113-8656 Japan
[email protected]
Abstract
and control method for trajectory tracking.
Inverse kinematics computation using an arti cial neural network that learns the inverse kinematics of a robot arm has been employed by many researchers.
1
Introduction
However, conventional learning methodologies do not
Human beings working in complex and unstructured
pay enough attention to the discontinuity of the inverse
environments generate adaptive behaviors by learning.
kinematics system of typical robot arms with joint lim-
Implementation of similar function in robots is neces-
its. The inverse kinematics system of the robot arms
sary in order to make future robots that will work
is a multi-valued and discontinuous function. Since it
in unstructured environments [1][2]. The task of cal-
is dicult for a well-known multi-layer neural network
culating all of the joint angles that would result in
to approximate such a function, a correct inverse kine-
a speci c position/orientation of an end-e ector of a
matics model for the end-e ector's overall position and
robot arm is called the inverse kinematics problem.
orientation cannot be obtained by using a single neural
An inverse kinematics solver using an arti cial neural
network. In order to overcome the discontinuity of the
network that learns the inverse kinematics system of
inverse kinematics function, we propose a novel mod-
a robot arm has been used in many researches; how-
ular neural network system for the inverse kinematics
ever, many researchers do not pay enough attention
model learning. We also propose the on-line learning
to the discontinuity of the inverse kinematics function
of typical robot arms with joint limits. The inverse
the inverse kinematics learning method that a neu-
kinematics function of the robot arms, including a hu-
ral network learns each solution branch calculated by
man arm with a wrist joint, is a multi-valued func-
the global searches in the joint space[6]. However, the
tion and a discontinuous function. It is dicult for
method is a purely o -line learning method and is not
a well-known multi-layer neural network to approxi-
applicable for on-line learning, i.e. simultaneous or al-
mate such a function. A correct inverse kinematics
ternate execution of the robot control and the inverse
solution for the end-e ector's overall position and ori-
model learning. Furthermore, the method is not goal-
entation cannot be obtained by the inverse kinematics
directed.
model consisting of a single neural network. Therefore a novel modular neural network architecture for
In this paper, we propose a novel modular architec-
the inverse kinematics model learning is necessary.
ture of the neural network and the on-line incremental learning method for the architecture. We also propose
Jacobs et al.[3] proposed a modular neural network
the on-line learning and control method for trajectory
architecture. Gomi and Kawato applied the modular
tracking. In order to evaluate the proposed architec-
architecture neural networks to the object recognition
ture, numerical experiments of the inverse kinematics
for manipulating a variety of objects and to inverse
model learning were performed.
dynamics learning[4]. Kawato et al. proposed Multiple Pairs of Forward and Inverse Models as a computational model of the cerebellum[5]. However, the input-output relation of their networks is continuous and the learning method of them is not sucient for the nonlinearity of the kinematics system of the robot arm. Their architecture is not suitable for the inverse kinematics model learning. The inverse kinematics function decomposes into a nite number of solution branches and each solution branch can be described as the product of an oneto-one inverse map and a set of free parameters describing the redundancy[6]. DeMers et al. proposed
2
Modular Neural Net System
Let be the m 2 1 joint angle vector and x be the n 2 1 position/orientation vector of a robot arm. The relationship between and x is described by x = f ( ). f is a C 1 class function. Let J ( ) be the Jacobian of
the robot arm, de ned as J ( ) = @ f ( )=@ .When a desired hand position/orientation vector xd is given, an inverse kinematics problem that calculates the joint angle vector d satisfying the equation xd = f (d ) is considered. In this paper, a function g (x) that satis es x = f (g (x)) is called an inverse kinematics system of f (). The acquired model of the inverse kinematics system g (x) in the robot controller is called an inverse
8im(x) be the output of the
kinematics model. Let
inverse kinematics model.
network in Figure 1 approximates the continuous region of the inverse kinematics function. The expert selector selects one appropriate expert according to
Although the inverse kinematics function of the robot
the desired position/orientation of the end-e ector of
arms is a multi-valued and discontinuous function, the
the arm, as described in Section 2.2. The extended
inverse kinematics function can be constructed by the
feedback controller calculates the inverse kinematics
appropriate mixture of continuous functions[6]. We
solution based on the output of the selected expert
propose a novel modular neural network architecture
by using the output error feedback. When no precise
that can learn a discontinuous inverse kinematics func-
solution is obtained, the controller performs a type of
tion by the appropriate switching of multiple neural
global search, as shown in Section 2.3. The expert
networks[7].
generator generates a new expert network based on the inverse kinematics solution.
Expert Selector with Forward Model Predicted Error e (γ) Square of +
Expert Net γ
|e|
+
-
I. Iterative Method II. Initial Value Search
In order to cover the overall work space, each expert
S |e|
xp(α)
Extended Feedback - e(k) Controller
lection by the forward model
Φ (xd )
Square of e(α) Error Norm
Forward Model
+
Square of Error Norm S
(β) im
has its representative posture. The representative pos-
(α) (xd ) Φim
Expert Generator
xd
|e (γ) |
|e|
xp(β)
Expert Net α
(γ) (xd ) Φim
e(β) -
Forward Model
2.2 Con guration of the expert and se-
2
x(pγ) +
Expert Net β
Error Norm S
-
Forward Model
+
θ(k) +
ture is the inverse kinematics solution obtained in the
Robot Arm
(i)
θ (0)
global searches by the extended feedback controller
z -1 x(k)
(i)
θ (k+1) (i) = θ (k) + K(θ(k))(xd -x(k))
x=f(θ )
θ(k)
Figure 1 Inverse Kinematics Solver with Modular Architecture Networks
2.1 Con guration of proposed inverse kinematics solver
when the expert is generated. Let (ri) be the representative posture of the i-th expert and x(ri) be the end-e ector position/orientation that corresponds to (ri) . Let
8imi (x) be the output of i-th ( )
expert when
the input of the expert is x. Each expert is trained to satisfy the following equation:
8
i) (i) x(ri) = f ( (im (xr )))
(1)
Figure 1 shows the con guration of the inverse kine-
By changing the bias parameters of the output layer
matics solver with the modular architecture networks
of the neural network, the above equation can easily
for inverse kinematics model learning. Each expert
be satis ed. Each expert approximates the continuous
region of the inverse kinematics function in which the
tive procedure from the output of the selected expert
reaching motion can move the end-e ector smoothly
cannot reach a correct solution, the extended feedback
from its representative posture.
controller repeats the iterative procedure from a number of initial values until a correct solution is obtained.
The predicted position/orientation error is used as the performance index of the expert. When the desired
The proposed inverse kinematics solver solves an in-
end-e ector position xd is given, the i-th expert calcu-
verse kinematics problem according to the following
lates the output
8imi (xd). The expert selector selects ( )
an expert with the smallest predicted error among all
8fm () be the output of the forward i kinematics model and 8im (xd ) be the output of the experts. Let
( )
i-th expert. The predicted error of the i-th expert pi
8 8
i) (xd ))j. is calculated as pi = jxd 0 fm ( (im
Let 80fm () be the desired output for
8fm().
procedural steps: (1) When the desired end-e ector position xd is given, the expert selector selects the expert with the minimum predicted error among all the experts. The extended feedback controller moves the arm to the posture that corresponds to the
The
learning of the forward kinematics model is conducted
8
as 0fm ( ) = x = f ().
output of the expert and then improves the ende ector position/orientation by using the output error feedback, as described in Section 2.4. (2) When no precise inverse kinematics solution is ob-
2.3 Extended feedback controller
tained in step (1), the other expert is selected in
The conventional on-line inverse model learning meth-
iterative improvement procedure by the output
ods, such as Forward and Inverse Modeling proposed
error feedback is conducted.
by Jordan and Feedback Error Learning proposed by Kawato, are based on the local information of the forward system near the output of the inverse model. The desired output signal provided by these methods is not always in the direction that nally reaches the correct solution of the inverse problem. An extended feedback controller avoids that drawback by employing a
increasing order of the predicted error and the
(3) When no solution is obtained in steps (1) and (2), an expert is randomly selected and a reaching motion from the representative posture of the selected expert is conducted. The above procedure is repeated until the reaching motion is successfully conducted or all the experts are tested.
kind of global search technique based on the multiple
(4) If a precise solution is obtained in each iterative
starts of the iterative procedure[8][9]. When the itera-
computation, the solution is used as the desired
output signal for the expert, as shown in 2.4.
ity:
T 01
(5) When no solution is obtained in the above proce-
jx d 0 x s j < T rst
(3)
dural steps, the controller starts a type of global
The desired trajectory xd (k ) is a straight line from
search. The controller repeats the initial joint an-
xs = f ((0)) to xd calculated as follows:
gle vector generation by using a uniform random number generator and the reaching motion from
8 > < (1 0 Tk )xs + Tk xd xd (k) = > : xd
(0 k < T ) (k T )
(4)
the generated posture, as described in 2.4, until
When the orientation is represented by the Direction
a precise solution is obtained. When a precise so-
Cosine Matrix or the Quartanion formulation, the
lution is obtained, a new expert is generated and
components of xd (k ), which represents the orienta-
the solution is used as the representative posture
tion, must be normalized.
r of the expert.
2.4 Reaching motion and expert learning
Let J + () is the pseudo-inverse matrix (MoorePenrose generalized inverse matrix) of J ( ), which is calculated as J + ( ) = J T ( )(J ()J T ( ))01 . J ( ) can be calculated by the observation of the movement
An illustration of the iterative procedure follows. Let (0) be the initial posture of the iterative procedure, which is the output of the selected expert
8 i (xd), the representative posture of the selected ex( )
pert (ri) , or the randomly generated posture.
of the robot arm and the numerical di erentiation technique. Let (k ) be an approximite inverse kinematics solution at step k. When rst is small enough, (k ) can be calculated as follows: (k + 1) = (k ) + J + ((k ))(xd (k + 1) 0 f ( (k))) (5)
The extended feedback controller conducts a reaching motion from the posture (0) to xd . The reaching motion is conducted as the tracking control to the following desired trajectory of the end-e ector xd (k)(k = 0; 1; : : : ; T + 1).
matics model or the learning feedback controller [10] can also be utilized for the above iterative calculation. If a precise solution (k ) is obtained, the solution can be used for the desired output for the inverse kinemat-
Let xs be the initial end-e ector position de ned as follows: xs = f ((0))
The output error feedback through the forward kine-
(2)
Let T be an integer that satis es the following inequal-
8
ics model as 0im (xd (k)) = (k ). When the controller cannot nd a precise solution because of the singularity of Jacobian J ( (k)) or the
joint limits, the reaching motion is regarded as a fail-
output of which can reach xd (0) precisely as the
ure.
procedure described in Section 2.4 and the predicted error of which on the representative points
2.5 Discontinuity check
xd (kr ) is the smallest among all experts.
The initial status of the experts sometimes causes a situation where one expert approximates multiple solution branches of the inverse kinematics function. In the case that one region contacts the other region in the workspace coordinates, the result is that the expert tries to approximate a discontinuous function. This situation should be avoided. When the matrix
8
i) norm of @ (im (x)=@ xjx=xd is larger than an appropri-
ate threshold rjix , a new expert with a new representative posture r corresponding to xd is generated to
8
approximate the region near to xd . @ im (x)=@ xjx=xd (i)
is approximately calculated by using numerical di erentiation and checked when the i-th expert is selected according to step (1) in Section 2.3.
3 Tracking Control In this section, the tracking control to desired ende ector trajectories is considered. When the desired end-e ector position/orientation trajectory xd (k )(k = 0; 1; 2; : : : ; Td ) is given, the procedure of the tracking control by the proposed solver is as follows:
(2) We assume that the i-th expert which can reach xd (0) is selected and the precise inverse kinemat-
ics solution (0) is obtained. The tracking control to xd (k) is conducted according to the following control low.
1fb(0) = (0) 0 8imi (xd(0)) (6) i (k + 1) = 8im (xd (k + 1)) + 1 fb (k + 1) ( )
( )
(7)
1fb(k + 1) = 1fb(k) + J
+
((k ))e(k) (8)
e(k ) = xd (k ) 0 f ((k ))
(9)
0< 1 In this paper, is set at 0:9. (3) When the solver cannot calculate the joint angle vector (k) that can realize xd (k ) due to the singularity of J ((k )) or the joint limits, the solver give up the tracking control. When there is enough time, another expert is selected or another initial value is generated by using a random numbers and the tracking control continues.
(1) Let xd (kr ) be the representative points of the desired trajectory xd (k ). In this paper, kr = 0; Td is used. First, the solver tries to nd an expert the
During the tracking control, the learning of the se-
8
(i) lected expert is conducted as 0im (xd (k)) = (k).
4 Simulations 4.1 Inverse kinematics learning of a 2DOF arm We performed simulations of the inverse kinematics model learning for a 2-DOF arm moving in the 2-DOF plane. The relationship between the joint angle vector
(a) RMS position error
= (1 ; 2 )T and the end-e ector position vector x =
(x; y )T is as follows:
x = x0 + L1 cos(1 ) + L2 cos(1 + 2 ) (10) y = y0 + L1 sin(1 ) + L2 sin(1 + 2 ) L1 is 0:30m, L2 is 0:25m, the range of 1 is [30; 150 ], and the range of 2 is [0150 ; 150 ].
(b) RMS position error of forward model
In the simulations, joint angle vectors were generated by using a uniform random number generator, and the end-e ector positions that correspond to the generated vectors were used as the desired end-e ector positions. In order to evaluate the performance of the solver, 1,000 desired end-e ector positions were generated for the estimation of the root mean square (RMS)
8
error of the end-e ector position e = xd 0f ( im (xd )).
(c) Percentage of successful rst selection
A 4-layered neural network was used for the simulations. The 1st layer, i.e., the input layer, and the 4th layer, i.e., the output layer, consisted of linear neurons. The 2nd and the 3rd layers had 15 neurons each. The back-propagation method was utilized for the learning. Before the learning, the inverse kinematics solver
(d) Number of experts Figure 2 Performance Change of Inverse Model by Learning
Figure 2 shows the progress of the inverse kinematy
ics model learning. Figure 2(a) shows the RMS error of the end-e ector position
pE [eT e] by using the in-
verse kinematics model. It can be seen that the RMS error decreases and the precision of the inverse model becomes higher as the number of trials increases. The x (a) Proposed Modular Neural Networks
RMS error became 1:50 2 1003 m after 5 2 107 learning trials. The precision of MNCFM is better than that of MNFM. However, there is not so much dif-
y
ference between MNFM and MNCFM. Figure 2(b) shows the RMS error of the forward kinematics model
q
E [eTfm efm ]. efm is the error vector of the forward
8
kinematics model de ned as efm = f () 0 fm (). x
Figure 2(c) shows the percentage of the trials in which the posture generated by the rst selected expert can
(b) Single Neural Network
successfully reach the desired position.
After 105
learning trials, the rst selection was always correct. Figure 3 Error Vectors of Inverse Kinematics Model
The percentage of MNCFM is better than that of MNFM. Figure 2(d) shows the number of the experts
had one expert the representative posture of which
which constructs the inverse kinematics model. The
was (0:0; 0:0)T . re was 0:001m, rst was 0:05m, and rjix
number of the experts which constructs the inverse
was 102 . For comparison, the solver with the complete
kinematics model became 3 after 6 times learning tri-
forward model which outputs f ( ) without error was
als.
also tested. Hereafter, MNCFM indicates the Modular Neural network system with a Complete Forward Model. MNFM indicates the Modular Neural network system with a Forward kinematics Model that consists of a neural network with no previous learning.
Figure 3 shows the position error vector of one example of an inverse kinematics model acquired by the proposed method. We were able to obtain a precise inverse kinematics model of the overall points that the robot arm can reach. The simulations of the inverse kinematics model learning, consisting of a single neural networks, by using the Forward and Inverse Model-
ing, were also performed. The learning was performed
desired end-e ector positions were generated by using
from 10 di erent initial states of the neural network.
a uniform random number generator for the evalua-
The minimum RMS error after 109 learning trials was
tion.
1:20 2 1002 m. An inverse model which consists of a
Figure 6(a) shows the change of the RMS error of the
single neural networks learned by using Forward and
end-e ector position. Figure 6(b) the percentage of
Inverse Modeling cannot obtain a precise inverse kine-
the successful rst selection. After 107 learning trials,
matics model of the arm as shown in Figure 3(b).
8 experts were generated. After 107 learning trials, the
Figure 4 illustrates how the expert is selected. Be-
RMS end-e ector position error became 1:30 2 1002 m.
cause the representative posture of the rst expert was
Figure 7(a) shows the position error vectors of the pro-
(0:0; 0:0)T and the Jacobian of the posture is singu-
posed inverse kinematics model. Figure 7(b) shows
lar, the expert is rarely used. The second expert and
those of the invese kinematics model which consists of
the third expert covers almost all region. Figure 4(b)
ingle neural network learned by forward and inverse
shows the region where the predicted output error of
modeling. We concluded that the proposed method
the second expert is lower than 0:01m. The graphics of
succeeded in the inverse kinematics model learning of
the robot arm in Figure 4(b) shows the representative
a 7-DOF manipulator. However, almost 106 times
posture of the second expert. Figure 4(c) shows the
learning trials were necessary to decrease the RMS
region where the predicted output error of the third
error lower than 2:0 2 1002 m. The precise forward
expert is lower than 0:01m.
kinematics model of the robot arm must be obtained before the inverse kinematics learning.
4.2 Inverse kinematics learning of a 7DOF arm
4.3 Learning during tracking control
We considered the inverse kinematics model learning
The simulations of the inverse kinematics learning dur-
of a 7-DOF arm (Mitsubishi Heavy Industries, Ltd.'s
ing the tracking control were performed using the 2-
"PA-10" ). The con guration of the arm is illustrated
DOF arm used in Section 4.1.
in Figure 5. The simulations of the inverse kinematics learning as described in Section 4.1 were conducted.
A desired trajectory xd (k ) was generated as follows. First, an initial joint angle vector s was generated by using a uniform random number generator. The
The 2nd and the 3rd layers of the forward kinematics
end-e ector position xs that corresponded to s was
model and the experts had 40 neurons each. 16,384
regarded as the initial desired end-e ector position.
Second, the desired total change of the hand po-
were used. The percentage increased to more than
sition
90% after 105 times learning trials.
1xdt was generated by using a normal ran-
dom number generator, with the standard deviation of 0:25m. The nal desired hand position was calculated
5
as xe = xs + xdt . xd (k ) = (1 0 k=Td )xs + (k=Td )xe
We proposed a novel modular neural network archi-
was used as the desired trajectory. Td was xed at
tecture for the inverse kinematics model learning and
20. The trajectories for the learning includes trajecto-
tested it by numerical experiments. The performance
ries that do not have the inverse kinematics solution
of the proposed nerual net system was con rmed by
corresponding to all xd (k ). 1,000 desired trajectories
numerical experiments.
were generated for the evaluation of the inverse kine-
learning speed can be improved by using improved
matics solver. All of the trajectories for the evaluation
methods for neural network learning. The utilization
have the inverse kinematics solution. The trials of the
of the redundant degrees of freedom will be reported
tracking control de ned in Equation (8) to the gener-
in near future.
1
Conclusions
We believe that the slow
ated trajectories were performed with the inverse kinematics model learning as described in Section 3. When the hand position error became larger than 0:01m, the trial was regarded as a failure and the tracking control was given up. Figure 8 shows the progress of the proposed learning method. 3 experts were generated after 10 times
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Figure 4 Con guration of Learned Inverse Kinematics Model
y 1 0
Figure 5 Con guration of 7-DOF Robot Arm
-1 1
0.5
z 0
-0.5 -1 -0.5
0 0.5 1
x
(a) Proposed Modular Neural Networks y 1 0 -1 1
(a)RMS position error
0.5
z 0
-0.5 -1 -0.5
0 0.5 1
x
(b) Single Neural Network Figure 7 Position Error Vector of Inverse Kinematics
(b)Percentage of successful rst selection Figure 6 Performance Change of Proposed Inverse Kinematics Solver
Model of 7-DOF Arm
(a)RMS position error of inverse kinematics model
(b)RMS position error in tracking control trials
(c)Percentage of successful expert selection Figure 8 Performance Change of Inverse Kinematics Model
