a neural network model for limb trajectory formation

61,417-425 1989 Biological Cybernetics A Neural Network Model for Limb Trajectory Formation L.. The network produced time trajectories of a limb from a starting posture toward targets

Biol Cybern 61,417-425 (1989) Biological

Cybernetics

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

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;

space and adjusted the velocity profiles properly; (iii)

the same task could not be learned by a linear network;

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;

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

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

r i ' u u i J l l l u a u l i i

I I I I I I I l I R i

I J l l l l l ~ l l l l l l I I I I l l l l l i N I I

I I I I I I I I I I~11~11

I I I I I I I I I I I I I I I

I I I I I I I II II II II ~

I I I I I l l l l l l l l l l

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

a m o u n t 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

existence of a possible relationship between the task

linearity and the length of the trajectories T o this

purpose, we tried to teach to the linear network a

shorter trajectory in the direction of the b o t t o m 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 O r g a n i z a t i o n

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 o u t p u t 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 G e o r g o p o u l o s (1986), which show that aiming movements involve the wrist joint in only a very marginal way M o r e o v e r (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

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

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

m a n y point attractors In other words these networks

naturally exhibit certain dynamic properties In our

case, the network was instructed during training to

produce certain o u t p u t 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 a b o u t 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:

d l = d 0 + 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 p a p e r - 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

muscles (Mussa Ivaldi et al 1988b) Moreover, V I T E

cannot be easily applied to multi-joint m o v e m e n t s and

does not address learning

In our model the information a b o u t the actual

position of the end-point was not explicitly computed

(It is only implicitly available t h r o u g h 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 h u m a n experi-

ments M o r a s s o (1981) showed that the information

a b o u t the end-point position plays a crucial role at the

planning level If planning were to be incorporated in

o u r model, its architecture ought to be expanded to

include explicit c o m p u t a t i o n of the end-point

position

As far as task representation is concerned, we

merged the kinematic p r o b l e m and the velocity profile

into a single three-layer network, but this was not the

only possible choice; the two p r o b l e m s could as well

have been separately addressed and represented by

means of two interconnected networks The latter

possibility has been investigated by J o r d a n (1989b)

T h e w o r k described here has relevance to the

robotics research since it m a y 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 m o v e m e n t s , in which trajectories are

affected by the surrounding environment T o this

purpose, it is necessary to model and represent the

e n v i r o n m e n t and the interactions with it (Massone and

M o r a s s o 1986; M a s s o n e 1986, 1987) This topic raises

several interesting p r o b l e m s which have often been

addressed b y the artificial intelligence research O u r

future goals include building a neural-network archi-

tecture capable of providing a uniform representa-

tional f r a m e w o r k for environment and m o v e m e n t s

( H o g a n 1984) The analogical nature of neural net-

works might provide significant insights into those

problems, as well as useful suggestions for h o w such

p r o b l e m s are addressed by biological systems

As far as neuroscience is concerned, the relation-

ship between the research covered in this p a p e r and the

organization of biological systems is an open p r o b l e m

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

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