Internal models of reaching and grasping ∗Claudio Castellini1

Francesco Orabona1
1
2

Giorgio Metta1,2

Giulio Sandini2

LIRA-Lab, University of Genoa, Italy

Italian Institute of Technology, Genoa, Italy

contact:

Giorgio Metta, email:pasa@liralab.it

Abstract
One of the most distinguishing features of cognitive systems is the ability to predict the future
course of actions and the results of ongoing behaviours, and in general to plan actions well inadvance. Neuroscience has started examining the neural basis of these skills with behavioural oranimal studies, and it is now relatively well understood that the brain builds models of the physicalworld through learning. These models are sometimes called “internal models”, meaning that they
are the internal rehearsal (or simulation) of the world enacted by the brain.
In this paper we investigate the possibility of building internal models of human behaviours witha learning machine that has access to information in principle similar to that used by the brain
when learning similar tasks. In particular, we concentrate on models of reaching and grasping and
we report on an experiment in which biometric data collected from human users during grasping
was used to train a Support Vector Machine.
We then assess to what degree the models built by the machine are faithful representations of
the actual human behaviours. The results indicate that the machine is able to predict reasonably
well human reaching and grasping, and that prior knowledge of the object to be grasped improves
the performance of the machine, while keeping the same computational cost.

Keywords: Manipulation and Grasping, Computational Intelligence, Teleoperation, Cognitive Robotics.

1

INTRODUCTION

One of the most distinguishing features of cognitive systems is the ability to learn to predict the future
course of actions and the results of ongoing behaviours, and in general to plan actions well in advance.
It is now relatively well understood that the brain builds models of the physical world through learning.
∗ The work presented in this paper has been supported by the European projects ROBOTCUB (IST-2004-004370),

NEUROBOTICS (FP6-IST-001917) and CONTACT (NEST-005010). The authors would also like to also Luciano Fadiga
and Thierry Pozzo for the ongoing discussion on the topics of this paper.

1

These models are sometimes called “internal models”, meaning that they are the internal rehearsal (orsimulation) of the world enacted by the brain.
Interestingly, internal models are built not only because they are required to control movements, butalso, as it has been determined more recently, to interpret the movements of others [1, 2, 3, 4]. There is
now a large body of literature that links the observation of actions to action execution, like for example
the study of the motor system conducted by Rizzolatti and colleagues in relatively recent years [5, 6, 7].
It seems then that building internal models of actions is a key feature of intelligent living systems.
Moreover, it has been shown in the context of object grasping, that the efficiency of an internal
model can be improved by priors on the object to be grasped, i.e., the presence of a target object andits geometrical properties strongly constrain the type of grasp and the approach to the object, and, asa consequence, the brain might need to include this information when planning an appropriate courseof action.
In this paper we set forth to investigate whether a computer, equipped with enough sensory information about human movements, namely grasping, could acquire something like an internal model using
machine learning methods. In particular we ask (a) whether the final configuration of the hand, i.e., at
the very moment an object is grasped, could be predicted from the initial part of the movement; and(b) whether the knowledge of the object to be grasped could improve the model efficiency, leading to asmaller error in prediction.
To shed light on these questions, we have set up an experiment in which several able-bodied subjects
have performed a highly repetitive grasping task on various daily life objects, and we have collected
data about their hand position, orientation and posture. Then we have tried to put a computer in thesame situation a human observer would be if he were to see only the initial part of a grasping action,
the final part being occluded by a screen: a sub-sequence of each grasping sequence, namely the initial
segment a human observer would be able to see, was used to train an efficient machine learning system
based on Support Vector Machines.
We have then analyzed the error in predicting the final hand configuration; and we have analyzed
whether the a priori knowledge of the grasped object makes a difference in performance as it should
intuitively do. The results we present here, albeit still in a preliminary form, indicate that the machineis able to predict reasonably well human reaching and grasping, and that prior knowledge of the object
to be grasped improves the performance of the machine, while keeping the same computational cost.
Once actually realised, optimised and implemented, such internal models could potentially be usedin various ways including the control of semi-autonomous teleoperated / prosthetic robotic artifacts, theinterpretation and possibly mimicry of human movements [8]. For example, in controlling or teleoperating an anthropomorphic robotic platform, such models would be able to guess the user intention andask the robot to complete the action autonomously. Predicting the user intention finds its natural rolein building man-machine interfaces and possibly into the control of prosthetic devices.
The paper is structured as follows: after a brief review of related work, we describe the methods andthe experimental setup in section 2 and the results obtained in section 3; lastly we discuss them and2

comment on future development in section 4.

1.1

Related work

In the monkey, premotor area F5 has been particularly well studied and it is in fact the location where“mirror neurons” were first identified. In this respect, mirror neurons are the quintessential correlateof internal models since they are activated both when executing a specific grasping action and whenobserving a congruent action being executed by another individual (or the experimenter) [9].
In a study by Umiltà et al. [10] the response of mirror neurons to the observation of actions thatterminate behind a screen has been investigated. In this case, the authors analyzed mirror neurons insituations where the final part of the action was occluded by an opaque screen with the monkey knowingof the presence/absence of an object to be grasped. As long as the object was shown to the monkey, thebrain could easily supply the missing visual information by rehearsing the internal model of the action.
The control experiment, in this case, was that of an identical hand kinematics, an identical screen but
the absence of the target object, that is, identical visual stimulation apart from the knowledge of thepresence of the object. Elsewhere it has been also shown that the presence of an object is required toelicit the mirror neurons response in the monkey [6].
A posteriori, given these results, it is easy to see how the presence of a target object and its geometrical properties strongly constrain the type of grasp and the approach to the object, and that, as aconsequence, the brain might need to include this information when planning an appropriate course ofaction. In the monkey these constraints are so strong that mirror neurons do not fire unless the goalof the action is clearly perceivable. The brain codes for the object-motor identity in part via anotherclass of F5 visuomotor neurons called “canonical neurons” (for a discussion see for example [11]). To
complete the picture, the work of Graziano, Hu, and Gross [12] has shown that the presence of objects
is coded in the ventral premotor cortex and maintained even when the object is no longer visible as longas there is evidence for its presence at a particular location.
Relevant to this discussion, the work of Fogassi et al. [13] contributed to the identification of mirror
neurons in the parietal cortex (inferior parietal lobule), which are thought to be related to the decodingof the intentions of others. Contextual information which links the enacted action to its final goal seems
to be implicated in this type of neural response. The presence of objects is a clear contextual cue. In
humans, it has been demonstrated that the activation of brain areas correlated to action observation is
not simply a perceptual effect but rather the activation of a precise sensorimotor model which includes
for example the hand kinematics [14].
Accordingly, Fadiga et al. [15, 16] have shown that motor imagery changes the excitability of thecortico-spinal connections specifically to the imagined action, that is, imagining a motor task causes
the under-threshold activation of the same neural pathways required to execute the task. This underthreshold activation was revealed by transcranial magnetic stimulation. In a conceptually similar experiment [17], the excitability of cortico-spinal pathways was also examined as a consequence of the actual3

sensory input. In summary, the motor system is similarly activated when acting in first person, when
imagining an action, or when watching somebody else’s action. Jeannerod [18], for example, goes to agreat length in showing how plausible is the fact that mental imagery uses the same internal models
used by actual action generation. It is known in this respect that the time required to simulate an actionis the same that is required to execute that action [19]. For a review refer to [20].
As far as gesture / hand configuration recognition is concerned, in a previous experiment we haveanalyzed the problem of recognizing hand gestures visually by incorporating a generative approach thatused motor information explicitly [21, 11]. In that case we showed that an action recognition systemthat uses motor information in a preprocessing step can perform better (97% recognition rate versus
80% on the test set) than a traditional classifier built directly in terms of visual information. Thisjustifies the fact that as a preprocessing step we can consider a visuo to motor mapping that transforms
the available visual information into motor data. This procedure is consistent in that it can be trained
through self-observation. We can imagine that the brain can exercise its control and simultaneously
acquire both the motor commands and the corresponding visual information and learn such a mapping.
In the following, we will only consider motor information since we can safely assume that the visuo-motor
map can always incorporated in the global internal model.

2

MATERIALS AND METHODSIn this Section we detail the process of gathering data from human subjects and the processing thatmakes them suitable for analysis by a machine learning system. In particular, we address the problem ofbuilding a training set, that is, a set of data effectively representing, for each user and object considered,
the grasping process, that could be used to train the system.

2.1

Experimental Setup

Devices
We collected data using a 22-sensors Immersion CyberGlove for the hand posture [22], an Ascension
Flock-Of-Birds (FoB) for the hand position [23] and a Force Resistor Sensor (FSR) to detect the contact
moment with the object. Figure 1 shows the devices, as worn by a subject.
The CyberGlove was worn by the subject on the right hand. The device returns 22 8-bit numbers
linearly related to the angles between the ends of the sensors and roughly indicating the angles between
the subject’s hand joints; the sensors are embedded in the glove in order for them to be adherent to thesubject’s skin. The resolution of the sensors is on average about 0.5 degree [22], but the noise associatedwith the sensors has been experimentally determined to be 1.1 on average and 3 at the maximum [24].
The sensors describe the position of the three phalanxes of each finger (for the thumb, rotation and two
phalanxes), the four finger-to-finger abductions, the palm arch, the wrist pitch and the wrist yaw.
The FoB was firmly mounted on the CyberGlove, just above the subject’s wrist, with the X/Y plane

4

(a)

(b)

Figure 1: The devices used for the experiment, as worn by a subject: (a) the CyberGlove, with the
Flock-of-Birds just above the subject’s wrist; (b) the Force Resistor Sensor attached to the subject’s
thumb.

being parallel to the palm plane in the resting position. The device returns 6 double-precision numbers
describing the position (x, y and z in inches) and rotation (azimuth, elevation and roll in degrees) of thesensor with respect to a magnetic basis mounted about one meter away from the subject. The FoB’s
resolution is 0.1 inches and 0.5 degrees [23].
Lastly, the FSR was mounted on the subject’s thumb. It returns a 32-bit number approximately
inversely proportional to the pressure applied to the surface of the sensor. We only used the FSR as an
on-off indicator of when the subject made contact with the object.
All data were collected, synchronised, and saved in real time at a frequency of 50Hz.
Subjects
Eleven subjects, four females and seven males aged 24 to 34 of different nationalities, joined the experiment. They were all right-handed and fully able-bodied, and were given initially some knowledge of theaim of the experiment.
MethodThe subjects were asked to sit comfortably in front of a clean workspace of about one square meter,at the center of which an object was placed, in a predefined position. The subjects were then asked towear the devices and choose a resting position for their right hand and arm. They were then instructed
to grasp the object with their right hand as they felt appropriate, not necessarily the same way eachtime, keeping a “natural” attitude. After grasping the object, they had to drop it somewhere else in
the workspace, and then return their right hand and arm in the initial resting position. Subsequently,
they had to use their left hand to reposition the object roughly in the same place it was before.
We first had the subjects do a trial run of the experiment, in order for them to gain confidence inthe setup. A beeping sound was heard each time the subject made contact with the object (that is, each5time the FSR signalled a significant change), and they were asked to try and hear the beep each timethey grasped the object. Although this ruled out grasps which made no use of the thumb, it enabled us
to better determine the contact points.
After the trial run, subjects were asked to repeat the grasp/drop/reposition procedure 120 times
for each object. We will call both this procedure and the data time sequence gathered during theprocedure, a session. We employed, in turn, three objects: a beer can, a duct tape roll and a mug(Figure 2 shows the objects). The objects were chosen so that each of them could be grasped in several
different ways, but with a certain degree of overlapping, e.g., both the beer can and the mug could begrasped cylindrically, but only the mug could be grasped using the handle.

(a)

(b)

(c)

Figure 2: The objects used in the experiment: a beer can (a), a duct tape roll (b) and a mug (c).

Each experiment employed one subject and consisted of six sessions: first the can, then the roll andthen the mug, all of them twice, for an approximate total of 720 grasps per subject, 240 per object. The
numbers are not precise since now and then the subjects would grasp without properly activating the
FSR. This problem has been corrected in the batch analysis of the data.
Each experiment lasted 35 to 56 minutes depending on the subject’s confidence and speed; althoughalmost no subjects reported tiredness, we allowed them to rest between sessions. It was reported byalmost every subject that the experiment became rapidly boring, which lets us claim that almost all
grasps were done in a natural, almost unconsciously. Figure 3 shows the main phases of the experiment.

2.2

Building the training set

Detecting grasps
In order to figure out when each single grasp starts and ends in a session, we first observed the values of
the FSR mounted on the subject’s thumb. We manually verified that the FSR correctly reacted in almost
all cases with a spike, signalling, whenever the subject made contact with the object, a significantly
different value from that recorded elsewhere shortly before the contact. The spike instants were taken
as the ending points of each grasp, and were gathered by checking when the first derivative of the FSR

6

(a)

(b)

(c)

Figure 3: The experiment. The subject sits confortably in front of a clean workspace, at the center ofwhich an object is placed (a), with his right hand in a resting position. He then grasps the object anddrops it somewhere else in the workspace (b), bringing then his arm and hand in the resting position.
Lastly, he repositions the object in the initial position using his left arm and hand (c).

value dropped by more than 10% of its overall minimum value. Moreover, after each spike, we ignored
one second of the session, to avoid detecting possible spurious spikes which happened immediately after
the grasp, due to object slippage and/or blurred values coming from the FSR.
Subsequently, in order to detect the starting point of each action, for each ending point we observed
the hand speed and acceleration, averaged over 0.2 seconds, from the ending point backwards. Since we
had instructed the subjects to always return to the resting position before initiating a new grasp, when
the grasp starts, the speed must be close to zero and the acceleration must be negative (the subject’s
arm is moving toward the FoB’s reference point). Therefore, we set the grasp starting point at thenearest moment in time before the ending point in which the hand speed was close to zero and the handacceleration was negative. In order to avoid detecting spurious speed/acceleration glitches when the handmade contact with the object, we ignored 0.1 seconds just before the ending point; moreover, we ignored
grasps which resulted shorter than 280 milliseconds. All these values were determined experimentallyto be near optimal in order to catch as many grasps as possible while avoiding spurious ones.
Figure 4 (a) shows an example set of detected grasps. As one can see, the hand speed (green curve)
shows the well known bell-shaped profile of a planar reaching movement [25]: the hand acceleration
diminishes, changes sign and then goes back to zero at the end of the trajectory.
Overall, the procedure could recognise 716 ± 12 grasps for each subject, which matches the desired
result of 720, that is 120 per session, each user running six sessions (during two experiments, the FSR
sensor broke down, resulting in the recognition of only 550 and 649 grasps). All data were also parsed
by hand in order to verify that spurious detected grasps would be an insignificant fraction of the total
grasps.

7

1
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Figure 4: (a) Detecting the grasps. The Figure shows 10 seconds of a subject grasping an object. The
red bands indicate the start and end of a grasp; the black line is the FSR response; and the green
line is the hand speed. As one can see, the ending points are found near the FSR spike, indicating
contact; moreover, the hand speed shows the well known bell-shaped profile of planar reaching. (b)Grasps duration. The Figure shows the duraton of the grasps (moving average over 50 grasps) averaged
for all subjects. As the experiments advance, the duration becomes shorter.

Grasping speed
In general, in order to make time sequences suitable for a machine learning system, they all must havethe same length;1 in order to do this, since in general not all grasps have the same length, we stretchedeach sequence to a predefined length. The predefined length was chosen according to the average speedof the grasps. Figure 4 (b) shows the average grasp durations for all subjects over each experiment(moving average over 50 grasps); as one would expect, in general the subjects get rapidly used to thegrasp/drop/reposition task and the grasps become faster and faster. It must be remarked, though, that
this is not the case for all subjects when considered individually.
On average, the grasp duration was 0.62 ± 0.20 seconds. We decided then to stretch every grasp to 1
second by linear interpolation, obtaining fixed-length time sequences of 50 samples for each sensor andgrasp.

2.3

Support Vector Machines

Our machine learning system is based upon Support Vector Machines (SVMs). Introduced in the early90s by Boser, Guyon and Vapnik [27], SVMs are a class of kernel-based learning algorithms deeply rooted
in Statistical Learning Theory [28], now extensively used in, e.g., speech recognition, object classificationand function approximation with good results [29]. We now give a very quick account of SVMs; for an
extensive introduction to the subject, see, e.g., [30].
1 An alternative possibility appears, e.g., in [26]; this issue is the subject of future research.

8

We are interested here in the problem of SVM regression, that is: given a function whose value isknown only for a finite number of points in its input domain, find its best approximation f drawn froma suitable functional space F. In practice, let S = {xi , yi }li=1 , with xi ∈ Rm and yi ∈ R be a set of lpoints and output values of the unknown function (actually, the training set); then the resulting f (x)is a sum of l elementary functions K(x, y), each one centered on a point in S, and weighted by realcoefficients αi :f (x) =lX

αi K(x, xi ) + b

(1)

i=1

where b ∈ R. The choice of K, the so-called kernel, is done a priori and defines F once and for all; it istherefore crucial. According to a standard practice (see, e.g., [29]) we have chosen a Gaussian kernel,
which has one positive parameter σ ∈ R which is the standard deviation of the Gaussian functions used
to build (1).
Let C ∈ R be a positive parameter; then the αi s and b are found by solving the following minimisation
problem (training phase):min R(S, K, α) + C

l
X
i=1

L (xi , yi , f )!

(2)where R is a regularisation term and L is a loss functional. In practice, after the training phase, someof the αi s will be zero; the xi s associated with non-zero αi s are called support vectors. Both the training
time (i.e., the time required by the training phase) and the testing time (i.e., the time required to findthe value of a point not in S) crucially depend on the total number of support vectors; therefore, thisnumber is an indicator of how hard the problem is.
In (2), minimising the sum of R and L together ensures that the solution will approximate well
the values in the training set, at the same time avoiding overfitting, i.e., exhibiting poor accuracy on
points outside S. Smaller values of the parameter C give more importance to the regularisation termand vice-versa.
Moreover, in SVM regression, L (xi , yi , f ) = max(0, |yi − f (xi )| − ), where  > 0 controls the widthof an “insensitive band” around the output values, e.g., errors on the training set within this band are
not considered.
There are, therefore, three parameters to be tuned in our setting: C, σ and . In all our regression
tests, we found the optimal values of C and σ by grid search with 2-fold cross-validation, whereas  waschosen accordingly to the resolution of the sensors being examined (see next Section for a more detailed
discussion).
Notice, lastly, that the quantity to be minimised in Equation (2) is convex; due to this, as well as tothe use of a kernel, SVMs have the advantages that their training is guaranteed to end up in a global
solution and that they can easily work in highly dimensional, non-linear feature spaces, as opposed toanalogous algorithms such as, e.g., artificial neural networks. Our system employs LIBSVM v2.82 [31],a standard, efficient implementation of SVMs.
9

According to the procedure described in the previous parts of this Section, we decided to define R 50
as the input space of our machines, and to use, for each sensor in the setup, the stretched grasping
sequences as points; the target values would then be the values of the same sensor at the time of contact
with the object. This way we have obtained 28 SVMs, each one approximating the value of a sensor at
the time of contact.

3

RESULTS

We were mainly interested in answering two questions:
1. how far in the future can our system predict well?
2. how does the knowledge of the grasped object affect the error?
In order to answer the first question, we have checked how the error on regression changes as theblind fraction 0 ≤ B ≤ 1 of the grasp increases from 0.1 to 0.5. The blind fraction indicates whatpercentage of the grasp, from the contact point backwards, is hidden to the system (Figure 5 showsa typical situation). It was intuitively expected that larger values of B would smoothly lead to larger
errors.
160
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thumb rotthumb innthumb out

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Figure 5: The blind window (the grey zone in the Figure) indicates what fraction of each grasp, fromthe contact point backwards, is hidden to the SVM. The data shown is a typical trajectory of the thumb(rotation, inner phalanx, outer phalanx) during a grasp. In this case the grasp lasts 0.78 seconds andB = 0.375. The last sample (for t = 0.78) is the target value.

This procedure was repeated independently for each single sensor; the regression error obtained isthe error attained by 2-fold cross validation with the optimal C and σ, determined using the procedure
detailed in the previous Section. The errors for each sensor were then grouped and averaged accordingly
to their measurement unit and meaning: the position of the hand (3 sensors, the x, y, z from the FoB),
the hand orientation (3 sensors, the azimuth, elevation and roll from the FoB), and the posture of the10hand (22 sensors, the joint positions from the CyberGlove). According to the device resolutions (seethe previous Section), we set  to 0.1 inches for the hand position, 0.5 degrees for the hand orientationand 1 degree for the hand posture.
In order to answer the second question, we first compared the error obtained as described above
using all sessions for each single object, so to obtain an estimate of how complex it is to approximate
the grasp for the can, roll and mug, unbiased by the differences among the subjects. Subsequently weaveraged these three errors and compared the average with the overall error, obtained by joining all
sessions together in a single training set. Figure 6 shows the experimental results.
Consider Figure 6, left column: as one can see, as far as the hand position and orientation areconcerned, the three objects show a comparable error. On the other hand, there is a precise ranking in
the hand posture regression: the mug is more difficult than the scotch roll, which is in turn harder thanthe beer can. This is intuitively sensible, since it is possible to grasp the scotch roll in more ways thanthe can, and it is possible to grasp the mug in even more ways (especially, using the handle).
To determine how far in the future SVMs can predict well, we need to decide what an acceptableerror is. In general, this is application-dependent. In this case we decided to accept an error as large as 5times a minimum threshold, determined by taking into account the resolutions of the sensors as declaredin the devices manuals and related publications (see the previous Section). This lead us to 0.5 inches
for the hand position, 2.5 degrees for the hand orientation, and 7.5 degrees for the hand posture. As far
as the hand posture is concerned, it must be remarked that, in this paper, we have only considered theaverage of errors on all the 22 sensors, whereas in a more detailed analysis one should take into accountthat, e.g., an error on the wrist pitch would lead to a worse displacement of the hand, than an error ona phalanx would. This is subject of future research.
As one can see from the graphs, the acceptable error is attained for the hand position at B = 0.3,
for the hand orientation at B = 0.2 and for the hand posture at B = 0.15 (mug and scotch roll) and
B = 0.3 (beer can). Since the average grasp lasts on average 0.62 seconds, we can say that the system
can predict reasonably well
• something less than 200 milliseconds in advance the hand position,• about 120 milliseconds in advance the hand orientation, and• about 90 milliseconds in advance the hand posture while grasping the mug or the scotch roll, andabout 200 milliseconds in advance the hand posture while grasping the beer can.
This answers the first question.
As far as the second question is concerned, consider Figure 6, right column: the curve representing theerror on the single objects is always consistently smaller than the other one, indicating that a specific
SVM trained on a single object will on average be more precise than a SVM trained on all objects
altogether: the a priori knowledge of the object improves the performance. A further analysis of C (seeFigure 7), indicates that the optimal values found for C show a decreasing trend. This correctly suggests
11

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Figure 6: Regression results as the blind fraction B increases from 0.1 to 0.5. In each row, representing
a different set of sensors (in turn, hand position, orientation and posture), the left-hand side picturescompare the errors on different objects, while the right-hand side pictures compare the average error on
single objects and the overall error.

12

that, as B increases, more and more information is missing from the training set (the regularisation
term in Equation (2) becomes increasingly important).
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Figure 7: The trend of the hyperparameter C as B is increased. C has a definite decreasing trend.

Let us now focus upon the number of support vectors found by the SVMs. It turns out that SVMs
trained on single objects have roughly the same number of support vectors as the one trained on theoverall sequence (figures show about 1% more in the first case). This means that the single-object
problems are computationally as hard as the overall one, but, as we have seen, they are more precise.
Summing up, we can say that if the problem is split into subproblems, each one regarding a single
object, performances are better and the total computational complexity of the solution remains roughly
the same. This answers the second question.

4

DISCUSSION

With this initial experiment we really pose further questions and sketch future research rather thandraw definite conclusions. The machine learning questions addressed in this paper do indeed have ananswer, albeit partial; on the other hand, it remains difficult to say something other than speculations
when comparing these results to neuroscience.
In short, the answer to the two questions posed in Section 2 is that we can predict well given that wehave access to motor information at least during learning, and that knowing the objects to be grasped
improves the ability to predict the outcome of an action. There are many caveats in this experiment,
as for example, the question on whether a pre-processing of the data through clustering could improve
performance further: i.e., given that objects afford certain grasping postures and they are executed withhigh probability. In humans the quality of the prediction of grasping is a function of the expectancies ofthe various possible grasp types which are in turn determined by the past experience of manipulationof the target object2 .
2 personal communication with Luciano Fadiga.

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The solution found by the SVMs detailed in the previous Section is optimal, since the dependence
from hyperparameters has been optimized out in our case by grid search and cross-validation that
although expensive is known to provide good results. An analysis of the solution should thus provide
an accurate characterization of the problem qua the data set that has been collected.
In this sense (and only in this sense) we have shown that by partitioning the training set per object
provides a general improvement of the quality of the solution and simultaneously of the training time(worst case O(l 3 ) versus O(3 · (l/3)3 ) in our case with 3 objects and l the total number of samples). Thiscan be an effective strategy when the world affords such an intuitive partitioning as for objects (seen as
discrete entities).
This is also true from what is known about the brain structures that control grasping where thepresence of a target object, its shape and affordance, and in general any contextual cue, are coded
separately by different populations of neurons and influence simultaneously the response of the neurons
that enact specific motor plans. After motor prediction is in place, the next step, that of recognizing
the action of another individual is conceptually simple since it amounts to building a classifier on highly
predictable motor trajectories.
Another interesting question that is left to future research is whether we can investigate the complexity of the controllers of reaching and grasping (which are known to develop separately in humans)from the complexity of the learned internal models or as a consequence of the prediction error.
Clearly, the fact that we can train such internal models is prone to be applied in various contexts, as
we mentioned, ranging from control of robots through interpretation and prediction of human behavior
in particular for man-machine communication.

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