a neural network model for limb trajectory formation

Biol. Cybern. 61,417-425 (1989)
Biological
Cybernetics
9 1989
A Neural Network Model for Limb Trajectory Formation
L. Massone and E. Bizzi
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, MIT-Building, E25-526,
77 Massachusetts Avenue, Cambridge, MA 02139, USA
Abstract. This paper deals with the problem of repre-
senting and generating unconstrained aiming move-
ments of a limb by means of a neural network
architecture. The network produced time trajectories
of a limb from a starting posture toward targets
specified by sensory stimuli. Thus the network perform-
ed a sensory-motor transformation. The experimen-
ters trained the network using a bell-shaped velocity
profile on the trajectories. This type of profile is
9 characteristic of most movements performed by bi-
ological systems. We investigated the generalization
capabilities of the network as well as its internal
organization. Experiments performed during learning
and on the trained network showed that:
(i)
the task
could be learned by a three-layer sequential network;
(ii)
the network successfully generalized in trajectory
space and adjusted the velocity profiles properly;
(iii)
the same task could not be learned by a linear network;
(iv)

after learning, the internal connections became
organized into inhibitory and excitatory zones and
encoded the main features of the training set;
(v)
the
model was robust to noise on the input signals;
(vi)
the
network exhibited attractor-dynamics properties;
(vii)
the network was able to solve the motor-
equivalence problem. A key feature of this work is the
fact that the neural network was coupled to a mechan-
ical model of a limb in which muscles are represented
as springs. With this representation the model solved
the problem of motor redundancy.
1 Introduction
This paper deals with the problem of representing and
generating unconstrained aiming movements of a limb
by means of a neural network architecture.
Aiming movements are present in biological sys-
tems at different levels of complexity, from accurately
planned movements to reflexes (Georgopoulos 1986).
The present work focuses on unconstrained limb
movements elicited by sensory stimulation. They are
meant to mimic the wiping movements made by the leg
of spinal frogs when the frog's skin is stimulated by an
irritant (Berkinblitt et al. 1986; Giszter et al. 1989).
Scratch reflexes of spinal cats (Shadmehr and Lind-
quist 1988) represent another example of this class of

movements. Opto-electrical recordings of frogs' wiping
movements (Giszter et al. 1989) show that the motor
strategy remains basically the same in both intact and
spinal animals. This result suggests that the basic
motor programs for this particular task are generated
at the spinal cord level and not explicitly planned by
higher brain structures. Starting from this observation,
we adopted a non-hierarchical neural network to
represent such movements.
The neural network's task in this work involved
generating a trajectory of a limb from a starting
posture toward a target specified in terms of a sensory
stimulus. Hence, the network performed a sensory-
motor transformation. Aiming movements were as-
sumed to be planar (as in the aiming phase of the
movement made by the frog when it wipes its back), but
there is no theoretical limitation to the dimensionality
the network could deal with.
Surprisingly, a number of kinematic studies of arm
movements (Morasso 1981; Abend et al. 1982; Atke-
son and Hollerbach 1985; Howarth and Beggs 1981)
have shown that the integrative action of thousands of
sensors, neurons and skeleto-motor units result in
velocity profiles whose global shape is invariantly bell-
shaped "over a wide range of movements sizes and
speeds. Flash and Hogan (1985) showed that a
minimum-jerk model predicts both the qualitative
features and the quantitative details observed experi-
mentally in planar, multi-joint arm movements.
Accordingly, in the present work, the experimen-

ters used a bell-shaped velocity profile for the training
trajectories. The duration of movements was assumed
420
of the bell-shaped velocity profile. As far as stiffness is
concerned, the model does not allow, at this stage of
development, direct control over the stiffness values
during the transformation from end-point positions to
muscle activations. We employed the inverse trans-
formation to compute the output patterns necessary to
train the network. The direct transformation (from
muscle activation to end-point position) was used
during the testing phase.
To train the network we used a standard back-
propagation algorithm which makes use of a momen-
tum term; the learning rate was interactively lowered
during the training sessions. All trajectories used
during the training phase had a duration of six time
steps: initial posture, target posture and four inter-
mediate postures. All postures were equilibrium po-
sitions as defined in 2.
One of our major concerns about the training
phase was how many and which sequences the network
had to learn to correctly generalize the task. We started
with four sequences which corresponded to sensory
stimuli in the four quadrants into which the limb
workspace is ideally divided by the initial end-point
position. Figure 3 shows the four trajectories as they
were generated by the network after learning. It is
worth noting that the bottom-left trajectory contained
a joint reversal on the shoulder joint. We gradually

increased the number of learned trajectories with the
purpose of achieving a generalization capability such
that the error on the end-point position 3 for each point
on the trajectory did not exceed the grid step. This
requirement was equivalent to demanding that the
3 Errors were measured, for each end-point position, as the
euclidean distance between the end-point position produced by
the network and the expected end-point position produced by the
mechanical model of muscles
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Fig. 3. Trajectories towards 4 stimuli in different quadrants of the
limb workspace. These trajectories are generated by the network
after learning. Points along the trajectories are equispaced in time
but not in space because of the bell-shaped velocity profile
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Fig. 4. Generalization capability after learning 15 trajectories.
The top-left trajectory contains a generalization of the joint
reversal on the shoulder. The rightmost trajectory in the second
row is a particular case of generalization in which the stimulus
was positioned right on the limb end-point. Although the
network has not been explicitly taught about the initial posture, it
has "understood" how the limb is positioned at the beginning of
each trajectory
network behave well at the resolution imposed by the
discretization of the limb workspace. This level of
performance was achieved after the network was
taught 15 sequences uniformly distributed over the
workspace. Figure 4 shows some generalized sequenc-
es: the end-point position is correct along the whole
trajectory, and the velocity profile is properly adjusted.
In addition, the network generated patterns of mus-
cular activation which corresponded to equilibrium
positions of the limb and could produce joint reversals
when necessary. A more detailed account of the
learning task is given in Massone and Bizzi (/989).
Three further experiments were performed during
the learning phase. First, the learning procedure was
repeated by making use of local coding instead of
coarse coding (one plan unit for each pixel for a total of
225 plan units). After learning the same 15 sequences,
the network was not able to generalize and behaved
like a look-up table. In the second experiment, the

learning procedure was repeated for a lower resolution
on the workspace, which was obtained by doubling the
grid step. This doubling led to a 7 x 7 array of pixels
coarse-coded by a 3 x 3 array
of plan units. In this
case
the network could learn the task (producing errors
lower than the grid step) with fewer learned trajectories
-8 as compared to 15. In the third experiment we
421
repeated part of the learning procedure with a linear
network obtained by removing the hidden layer. The
purpose of this experiment was that of investigating the
amount of non-linearity present in the input-output
transformation. We tried to teach to the linear network
the four trajectories shown in Fig. 3, first separately
and then jointly. We observed the following behavior:
9 The linear network could learn the trajectories
towards the top-left target and towards the top-right
target separately.
9 The same two trajectories could not be learned
jointly. This fact shows that the linear network could
not handle the interferences between the two trajec-
tories, while the non-linear network could.
9 The linear network could not learn the trajectories
towards the bottom-left target and towards the
bottom-right target, neither separately nor jointly.
We concluded that the task is highly non-linear,
except in a few peculiar cases. Furthermore, we
observed that the trajectories that the linear network

could learn were much shorter than those that could
not
be learned. Hence, we also investigated the
existence of a possible relationship between the task
linearity and the length of the trajectories. To this
purpose, we tried to teach to the linear network a
shorter trajectory in the direction of the bottom right
target. The linear network could not learn that trajec-
tory. We concluded that no relationship exists between
the trajectory's length and the extent to which the
linear approximation holds.
4 Internal Organization
We analyzed the connections of the trained network in
order to understand the organization produced during
learning. Interesting patterns were found in the con-
nections from hidden to output units; Table 1 shows
the values of the connections after the task was learned.
We observed that, whenever one hidden unit sends an
excitatory connection to a flexor, the same unit sends
an inhibitory connection to the corresponding ex-
tensor (negative correlation.) Similarly, whenever an
inhibitory connection is sent to a flexor, an excitatory
connection is sent to the corresponding extensor. The
network has represented in the connectivity pattern
the rule of reciprocal inhibition of agonist-antagonist
pairs. Inhibition and excitation were more marked for
shoulder, elbow and double-joint muscles than for
wrist muscles. This result agrees with the experimental
data of Georgopoulos (1986), which show that aiming
movements involve the wrist joint in only a very

marginal way. Moreover (see again Table 1), we ob-
served
that:
9 units :~ 3 and ~ 10 exhibited a total positive corre-
lation between all flexors and between all extensors;
9 all other units exhibited a total positive correlation
between
- the shoulder flexor and the double-joint flexor;
-
the shoulder extensor and the double-joint extensor;
- the elbow flexor and the wrist flexor;
- the elbow extensor and the wrist extensor;
Hidden unit ~ 2 was an exception: the shoulder
and double joint exhibited a negative correlation.
These observations could be interpreted as follows.
First of all, there is evidence for a number of synergies
between all hidden units. These synergies are the
necessary condition for the network to exhibit good
generalization properties. Furthermore, the network
seems to have represented in the connectivity pattern
the main features of the set of patterns that was used as
Table
1. Hidden to output connections. Each row contains the connections from all hidden units to one particular output unit
Shoulder F1.
1.596938 -0.459756 0.238360 0.464764 -1.130529 1.225434 1.249900 -0.822495 0.946107 -1.160787
Shoulder Ex.
1.597220 0.458936 -0.238326 -0.464758 1.130295 -1.225155 -1.249560 0.822554 -0.946128 1.161238
Elbow F1.
2.205403 1.201014 0.990554 -0.725229 1.035587 -0.788169 -1.198095 0.046117 -1.464604 -0.762154
Elbow Ex.

-4.299550 -0.824471 -2.072892 1.214777 -1.433692 0.294094 1.581221 0.024668 2.398601 0.365059
Double J. F1.
-0.822154 1.664527 0.419754 0.069341 -0.511715 0.004101 0.556992 -0.617517 0.068661 -1.571512
Double J. Ex.
0.580872 -1.538881 -0.525151 -0.019188 0.490678 -0.050527 -0.561579 0.623930 0.010338 1.459015
Wrist FI.
0.260377 0.220753 0.122209 -0.092430 0.129453 -0.155435 -0.139240 -0.012050 -0.188938 -0.090522
Wrist Ex.
0.260409 -0.220748 -0.122226 0.092426 -0.129474 0.155452 0.139244 0.012054 0.188934 0.090537
422
the training set. In fact, the sign of muscle activations in
the training sequences was always the same for elbow-
wrist flexors and elbow-wrist extensors and almost
always the same for shoulder-double joint flexors and
shoulder-double joint extensors. The network encoded
that "almost" by means of a negative correlation at
unit ~2. Finally, the network devoted two hidden
units, 4~ 3 and # 10, to encode the synergies between all
flexors and between all extensors. Assuming that one
hidden unit corresponds to one family ofinterneurons,
these results suggest that interneurons may be orga-
nized into functionally overlapping groups (Edelman
1987).
5 Experiments
We performed three experiments with the trained
network.
The first experiment was concerned with the dura-
tion of the trajectories. Pineda (1987) showed that
arbitrary networks of logistic units typically have
many point attractors. In other words these networks

naturally exhibit certain dynamic properties. In our
case, the network was instructed during training to
produce certain output patterns for six time steps; no
instructions were given about what should be done
after the sixth time step. We tested the network for 15
time steps, and we observed that in about 80 percent of
the cases (i.e. in about 80 percent of the limb work-
space), the limb remained steady at the final posture
which corresponded to the position of the sensory
stimulus. In other words, in 80 percent of the cases, the
final posture of the limb acted as a point attractor. In
certain portions of the workspace, the limb became
unstable after the sixth time step. The portion Varied
with different learning sessions, depending on which
solution the network had settled into. There were also
cases in which the entire workspace was steady.
The second experiment aimed at testing the robust-
ness of the system when the sensory stimulus was
varied. The network was trained with stimuli coded as
gaussian distributions centered on the target with a
certain standard deviation do; we modified the value of
the standard deviation during testing as follows:
dl=d0+0.1 *d o ,
d 2 = d o + 0.2 * d o .
Both of the above cases correspond to a stimulus
which is flatter and more spread over the workspace.
We measured the average distance between the trajec-
tories generated with standard deviation d o and the
trajectories generated with standard deviation dl and
d z.

Given the stimulus d I the average distance was
lower than 0.4; in the second case the average distance
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Fig. 5. Double target experiment. The first target was turned on
for two time steps, and then it was turned off. The experiment was
performed on both learned and generalized trajectories
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Fig. 6. Double target experiment. The first target was turned on
for three time steps, and then it was turned off. The experiment
was performed on both learned and generalized trajectories
was higher (around 0.7), which resulted in trajectories
that were "noisy", but still acceptable. This experiment
showed that the architecture was reasonably robust in
the face of slight changes in the stimulus
representation.
The third experiment was performed with the aid of
a double target. A sensory stimulus was given to the
network. As the limb was moving in steps toward the
stimulus, the stimulus was turned off, and another
stimulus at a different location was turned on. When
this occurred, the limb switched direction towards
the new target. The experiment was repeated in the
following two cases:
1. with the duration of first stimulus correspond-
ing to the first two time steps made by the limb;
2. with the duration of first stimulus correspond-
ing to the first three time steps.

Figure 5 shows the resulting trajectories for the
first case, while Fig. 6 represents the trajectories for the
423
second case. Note that in both cases the limb reached
the second target. The results of this experiment show
that the network, having learned how to reach a set of
targets from a fixed initial position of the limb, was also
able to reach the same set of targets from a different
posture, the one in which the limb was positioned when
the second target was turned on. This result indicates
that the network was able to solve the so-called motor-
equivalence problem. Experiments on intact biological
systems (Georgopoulos et al. 1981 ; Massey et al. 1986)
clearly show that shape and length of the trajectories
generated by a double-target experiment depend upon
the duration of the first stimulus. The network in our
experiments seemed to be insensitive to this parameter
(compare Figs. 5 and 6). In a sense, the network
"forgot" everything about the first trajectory when the
second stimulus was turned on. These different
behaviors may indicate that a non hierarchical system
- like the neural network described in this paper- does
not contain any smoothing mechanism, whereas such
mechanisms are present when planning occurs. In
other words, trajectory smoothing is not a direct
consequence of the mechanical properties of the ac-
tuator, but the result of some specialized brain func-
tions. With neural networks, one could obtain a
smoothing behavior by enriching the network with
other blocks which somehow implement the above

mentioned brain functions. Alternatively, one could
add some dynamics (i.e. self-connections) at the output
units (like in Jordan's flow networks [1989a]) to make
the next output of the network a function of the past
outputs. The latter solution would correspond to
assuming that trajectory smoothing is a low-level
operation handled by motorneurons.
6 Discussion
In this paper we presented a model for the formation of
limb trajectories, based on a neural network architec-
ture. The task under consideration was that of reaching
a target defined in terms of a sensory stimuli. The
trajectories had a bell-shaped velocity profile. The
network produced trajectories in muscle space, which
were translated into end-point space by means of a
model which takes into account the elastic properties
of muscles. The inverse transformation, from end-
point space to muscle space, was used to generate the
training sequences as described in Sect. 3. The partic-
ular architecture used for producing time trajectories
was that proposed by Jordan (1986).
We have shown that:
I. The task could be learned by a three-layer
sequential network trained by a standard back-
propagation procedure.
2. The network successfully generalized in trajec-
tory space: the error of the generalized trajectories
measured at the end-point could be made lower than
the discretization step of the limb workspace.
Moreover, the velocity profiles generalized

appropriately.
3. The same task could not be learned by a linear
network.
4. The internal connections became organized,
after learning, into inhibitory and excitatory zones; in
particular, connections from hidden units to output
units exhibited a number of positive and negative
correlations that encoded the main features of the
training set. Between hidden units, we observed a
number of synergies which are the necessary basis for
good generalization properties.
5. The model was robust with respect to the input
signals: slight changes to the stimulus coding did not
significantly affect the network overall performance.
6. The network spontaneously exhibited attractor-
dynamics properties. Final end-point positions
behaved like point attractors in the majority of the
limb workspace.
7. The network could solve the motor-equivalence
problem as shown by the double-target experiment.
The network did not exhibit smoothing properties, and
seemed to be insensitive to the duration of the first
stimulus.
Kawato et al. (1987) studied voluntary movements
and proposed a hierarchical, structured model for
generating motor commands (torques) from a desired
trajectory expressed in body-centered coordinates.
Moreover, Kawato et al. (1988) studied the coordinate-
transformation problem and proposed an iterative
control learning algorithm. Our research dealt with a

sensory-motor transformation based on a non-
hierarchical layered architecture which translated a
sensory stimulus directly into time-varying patterns of
muscular activation which corresponded to minimum
jerk trajectories. We did not face the coordinate
transformation problem since we made the hypothesis
that both target and movement were already expressed
in the same body-centered reference frame. We did
address the problem of trajectory formation based on a
constant sensory stimulus rather than a reference
trajectory. The issue of trajectory formation was also
faced, among others, by Bullock and Grossberg (1988)
who presented a model called VITE which produces
arm trajectories from a target position command
(TPC) and a GO command which defines the
movement's speed. Although VITE has nice general-
ization properties, it is worth noting that trajectories
are generalized in joint space. By contrast, our model
could generalize trajectories in muscle space and then
in end-point space through the mechanical model of
424
muscles (Mussa Ivaldi et al. 1988b). Moreover, VITE
cannot be easily applied to multi-joint movements and
does not address learning.
In our model the information about the actual
position of the end-point was not explicitly computed.
(It is only implicitly available through the muscles'
model.) This fact did not represent a limitation for the
task under consideration, since the task was per-
formed, as already pointed out, at the reflex level,

without any planning. However, in human experi-
ments Morasso (1981) showed that the information
about the end-point position plays a crucial role at the
planning level. If planning were to be incorporated in
our model, its architecture ought to be expanded to
include explicit computation of the end-point
position.
As far as task representation is concerned, we
merged the kinematic problem and the velocity profile
into a single three-layer network, but this was not the
only possible choice; the two problems could as well
have been separately addressed and represented by
means of two interconnected networks. The latter
possibility has been investigated by Jordan (1989b).
The work described here has relevance to the
robotics research since it may suggest some basic
principles for designing artificial limbs whose structure
is inspired by natural systems (De Rossi et al. 1988).
Moreover, we plan to extend this work to cope with
constrained movements, in which trajectories are
affected by the surrounding environment. To this
purpose, it is necessary to model and represent the
environment and the interactions with it (Massone and
Morasso 1986; Massone 1986, 1987). This topic raises
several interesting problems which have often been
addressed by the artificial intelligence research. Our
future goals include building a neural-network archi-
tecture capable of providing a uniform representa-
tional framework for environment and movements
(Hogan 1984). The analogical nature of neural net-

works might provide significant insights into those
problems, as well as useful suggestions for how such
problems are addressed by biological systems.
As far as neuroscience is concerned, the relation-
ship between the research covered in this paper and the
organization of biological systems is an open problem
and will be the object of further investigations.
Acknowledgements. The authors wish to thank Michael Jordan
for his constant support and valuable suggestions, and for
making available the basic software which implements sequential
networks. Thanks also go to (in alphabetic order): Chris Atkeson,
Simon Giszter, Joe Mclntyre, Ferdinando Mussa Ivaldi and
Tomaso Poggio. This work has been supported by the Office of
Naval Research Grant N00014/88/k/0372 and the Sloan
Foundation.
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Received: June 5, 1989
Accepted: June 6, 1989
Dr. Lina Massone
Department of Brain and Cognitive Sciences
Massachusetts Institute of Technology
MIT-Building, E25-526
77 Massachusetts Avenue
Cambridge, MA 02139
USA