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EU – FP7
AMARSi Adaptive Modular Architectures for Rich Motor Skills
ICT-248311
D 1.1
October 2010 (6 months)
Comparative evaluation of notions of modularity in human motor control and of existing algorithms for the
identification of motor primitives
Martin Giese1, Andrea d’Avella2, Tamar Flash4 , Yuri P. Ivanenko2, Thomas Schack3, Enrico Chiovetto1, Albert Mukovskiy1.
1- Section for Computational Sensomotorics, Department of Cognitive Neurology, Centre for Integrative Neuroscience, University Clinic Tübingen, Tübingen, Germany.
2- Department of Neuromotor Physiology, Santa Lucia Foundation, Rome, Italy. 3- Neurocognition and Action Research Group, Faculty of Psychology and Sport
Science, University of Bielefeld, Bielefeld, Germany. 4- Department of Computer Science and Applied Mathematics, Weizmann
Institute of Science, Rehovot, Isreal. Due date of deliverable 1st September 2010 Actual submission date 15th October 2010 Lead Partner Tübingen University Revision Final Dissemination level Public
Comparative evaluation of notion of modularity in human motor control and of existing algorithms for the identification of motor primitives
There exists a broad range of biological approaches to movement primitives that covers different
functional levels and different methodological approaches. Many of the existing approaches are
represented in the AMARSI consortium. This deliverable will, on the one hand, provide a brief
overview of the approaches towards movement primitives that have been established by the different
partners. Possible relationships that might form a basis to establish collaborations between the groups
and specifically with the technical groups in robotics will be briefly discussed. On the other hand, in the
second part of this deliverable, some quantitative work will be presented that compares between
methodologically closely related approaches of different groups. Such comparisons are possible only
between approaches that conceptualize movement primitives in similar ways, and at the same
representation level.
1 Overview of different biological approaches towards motor primitives The concept of movement primitives is quite popular in research in neuroscience. At the same time,
the use of this concept is very heterogeneous, and many different conceptualizations of motor
primitives appear in the biological literature. Some of these conceptions address phenomena at similar
levels, but using different methodologies. Others signify phenomena at very different representation
levels, such as muscle synergies or cognitive action plans. The AMARSI consortium comprises groups
that have worked on different approaches for the characterization of movement primitives, and which
cover a large part of the existing approaches. Some of these approaches are closely related, while
others address completely different representation levels. Table 1 gives an overview of the
approaches and conceptualizations of motor primitives that are represented within the AMARSI
consortium.
Partner People involved Level Theoretical / experimental
approaches Section
UniBi
T. Schack, B.Land, A. Krause, B. Blaesing
Cognitive representations
Volitional initiation control strategies
1.1
WI T. Flash, A. Barliya, R. Fuchs, Y. Meirovich
Kinematics, strokes / trajectory segments
Differential geometry, invariants, kinematic analysis
1.2
SFL A. d’Avella, Y. Ivanenko, G. Cappellini
Neuromuscular synergies / kinematics
Unsupervised learning, EMG, motion capture
1.3
UniTu
M. Giese, E. Chiovetto, A. Mukovskiy, N. Taubert
Kinematics, movement sequences
Unsupervised and supervised learning, dynamical systems, motion capture
1.4
Table 1. Biological approaches for the investigation of movement primitives within the AMARSI
consortium.
In the following, we will first sketch the definition and approaches for the identification and modelling of
motor primitives provided by the different partners that are involved in the analysis of biological data.
Finally, we will provide a comparison that highlights similarities and complementarities that might help
to establish fruitful interactions between the different partners.
1.1 Neurocognitive perspective on motor primitives
A very general definition of motor primitives is underlying the work by the partner UniBi. This
conception includes a variety of levels form low to high-level representations, and it addresses
specifically the cognitive aspects of the representation and control of complex movements. A particular
focus of this approach is the planning and memory of complex movements and changes associated
with skill learning and during development.
1.1.1 Notion of Motor Primitives (MP)
Motor Primitives are conceptualized as basic building blocks in a modular motor control architecture
(cognitive architecture of human motion). These building blocks are functionally relevant elementary
components or transitional states of (complex) movements. In our point of view it is possible to
understand a movement as a serial and functional order of significant and goal related body postures.
Such body postures and motion sequences are bi-directional linked with perceptual (movement-)
effects and typically stored in memory in a hierarchical order.
A main topic of our research is the cognitive architecture of human action, showing how it is organized
over several levels and how it is built up. Alongside Bernstein's (1947) approach to the construction of
action, there have been several formulations of the idea that movement control is constructed
hierarchically. In contrast to most of these approaches, the model proposed here views the functional
construction of actions on the basis of a reciprocal assignment of performance-oriented regulation
levels and representational levels (see Table 2). These levels differ with respect to their central tasks
on different regulation and representation levels. Each level is assumed to be functionally
autonomous.
Code Level Main function Subfunction Means
IV Mental Control
Regulation Volitional initiation control strategies
Goals; goal posture representation; strategies
III Mental representation
Representation Effect-oriented adjustment
Basic action concepts
II Sensorimotor representation
Representation Spatial-temporal adjustment
Perceptual effect representations
I Sensorimotor control
Regulation Automatization Functional systems; Transitional states
Table 2. Levels of Complex Motion in Humans (Schack, 2004).
We identify Basic Action Concepts (BACs) as major building blocks at the representation level. BACs
are based on the cognitive chunking of body postures and movement events concerning common
functions in realisation of action goals. Their characteristic set of features results from the perceptive
and functional properties of action effects: They tie together their functional and sensory features.
These functional features are derived from action goals (goal postures), and this connects BACs to
Level IV. However, BACs also integrate sensory features of sub-movements, for example, through
chunking. As a result, they also refer to the perceptual effects of movements. This connects BACs with
Level II. All together BACs can be viewed as the mental counterparts of functionally relevant
elementary components or transitional states of complex movements. They are characterized by
recognizable perceptual features. They can be described verbally as well as pictorially, and can often
be labelled with a linguistic marker. "Turning the head" or "bending the knees" might be examples of
such basic action concepts in the case of, say, a complex floor exercise. Based on our research we
learned that such movement representations might provide the basis for action control in skilled
voluntary movements in the form of cognitive reference structures.
With respect to robot control, current work is largely focused on a low level of abstraction that is very
close to the sensors and actuators. In contrast, human actions are heavily informed by huge amounts
of representations about the goal and the characteristics of the anticipated movements, the
encountered objects, and about how to counteract the numerous disturbances and mistakes that
usually occur during even moderately complex movements. Therefore one major goal of our research
is to design experiments and simulations concerning the formation and cooperation of motion building
blocks at a cognitive level (memory) and to learn about the integration of biomechanical and motor
constraints into cognitive motor representations and cognitive motor hierarchies.
1.1.2 Methods and experimental evidence
1.1.2.1 Experiments to study the cognitive representation of MP’s (BACs) in memory
We used different experimental methods to study:
cognitive structures in long term memory in different types of movements
movement based chunking in short term memory
representations of power- and precision grasps in memory
representations of manual action after stroke
the overlap between kinematical and cognitive movement structures
Summarizing the expertise studies, we found that in high-level experts the representational
frameworks were organized in a distinctive hierarchical tree-like structure, with remarkable similarities
between individuals. These frameworks were well matched with the functional and biomechanical
demands of the task. In comparison, action representations in low-level athletes and non-athletes
were organized less hierarchically, were more variable between persons, and were less well matched
with functional and biomechanical demands.
Figure 1. Representation structures for two chosen tennis expertise groups, respectively experts and
non-players (A and B), based on the hierarchical cluster analysis of basic action concepts (BACs) in the
tennis serve. The horizontally aligned numbers denote the BACs (for the code, see text); the vertical numbers
specify the Euclidean distances. The lower the numbers, the lower the distances between BACs in long term
memory. For all groups holds n = 11; p = 0.05; dcrit = 3.46 (from Schack & Mechsner, 2006).
Results from two different lines of research addressing the mental representation level showed that
not only the structure formation of representations in long-term memory but also chunk formation in
working memory are built up on BACs and relate systematically to movement structures. Experiments
were designed to assess both the structure of mental representations in LTM (determined with the
SDA-M) and chunking in working memory (determined with Cognition and Movement Chronometry,
CMC). Results confirmed the interaction between long-term memory and short-term memory,
demonstrating that cognitive systems interact to produce complex movements. Our experiments have
shown that both, the order formation in LTM and the chunking in working memory build on the
topological (spatiotemporal) structure of the movement. This provides experimental evidence that
structures in movement and memory mutually overlap.
1.1.2.2 Experiments to study the relationship of cognitive motor representation and
biomechanical constraints
To accomplish a better understanding of the cognitive architecture of complex movements, it is not
only interesting to know whether LTM and working memory cooperate horizontally on, for example,
one level of mental representations. It is, for instance, also crucial to know whether there is a vertical
cooperation between the level of mental representations and the level of sensorimotor control. One
could ask whether biomechanically relevant features can be found in the structure of mental
representations. Experimental studies (Schack, 2004; Schütz et al., 2009) showed that
representational frameworks were organized in a hierarchical tree-like structure and revealed a good
match with the biomechanical demands of the task. After measuring kinematic parameters, we
investigated the relationship between the structure of motor representation and the kinematic
parameters of different movements. Our studies have revealed significant correlations between
kinematic parameters (time structure, angles according to the take-off-phase, tilt angle, angular
velocities, etc.) of movement and the corresponding parts of mental representations.
Other studies are done to find out the relationship between anticipated goal states (end-state-comfort)
and the representation of functional grasp constellations in children (Weigelt & Schack, 2010; Stöckel,
& Schack, 2010). All together our experimental results support the hypothesis that voluntary
movements are executed and stored in memory directly through representations of their anticipated
perceptual effects. We are combining experiments concerning the cognitive representation structure in
motor memory with PCA or a new kind of spatiotemporal analysis (Bläsing & Volchenkov, 2010) to
identify MP’s and to learn about the relationship of cognitive and biomechanical motor constraints
(MP’s).
1.1.2.2 Simulations and neural network architectures
In order to model the learning and generation of complex movements, a hierarchical, modular
neuronal network architecture is currently under development. The architecture will be able to learn
demonstrated movements (kinaesthetic teaching) and - after training - trigger adaptive movement
execution based on anticipated sensory data from the robot and the environment.
The idea is to have - on a lower level - a battery of recurrent networks controlling the execution of
automatically learned movement primitives in a tight sensory motor loop. The dynamics of those
networks is controlled through bifurcation by inputs (similiar to the work by Jun Tani regarding RNNPB
with parametric bias units) by a higher level network that selects- and interpolates between patterns
and generates long motor sequences. On the top level of the architecture, one or multiple hierarchical
self-organising maps (HSOM) represent the cognitive structures of the complex movements (BAC's,
see Schack & Mechsner, 2006). The HSOMs can analyse and cluster sensory data, detect proper
situations when to perform suitable movements and adjust task affordances (Krause et al., 2010).
In addition, other computational approaches for the simulation of the structure of mental
representations are being developed (Tscherepanow et al., 2010). Such biologically inspired models
of movement representations are to be contrasted with other, less biologically plausible models, such
as discussed in WP4 for motor primitives.
1.1.2 Implications for robotics
Because the production of manual actions is affected by a number of factors, such as biomechanical
constraints (Weigelt et al., 2009), a line of our studies focuses on the question of how structures of
sensory-motor representations are established and changed in a stepwise manner under the
consideration of task constraints (Bläsing et al. 2010b; Maycock et al., 2010). In these studies it is of
interest to learn about the relationship between the structure of mental representations and the
performance in manual actions, especially in situations in which actions result in errors. To this end,
we combined methods from cognitive psychology, biomechanics, and computer sciences (i.e.
augmented reality / virtual reality scenarios) to assess the structure of sensory-motor representations,
to introduce experimental manipulations for manual actions, and to examine the resulting performance
changes (including changes in erroneous performance). The insights gained in these first experiments
will be implemented on robotic platforms (i.e. a 7 DOF robot arm set-up) within a longer research
perspective.
Our research concerning the cognitive architecture of human action (Bläsing et al., 2010a; Schack,
2004; Schack, 2010) is of interest for the construction of robot architectures and vice versa. In different
research project, we translated our findings in studies of cognitive factors in human motor control into
models that can guide the implementation of cognitive robot architectures. Focusing on the issue of
manual action control, we illustrated some results in the context of grasping with a five-fingered
anthropomorphic robot hand (Schack & Ritter, 2009).
A research project about cognitive planning of action sequences and sensorimotor adaptation
(cooperation with the CoR-Lab and HRI Europe) addresses the cognitive and perceptual dimensions
of goal posture planning in humans and is related to grasp optimization and trajectory planning in
robots. The aim here is to integrate the research results in humans (e.g. Weigelt & Schack, 2010) into
a comprehensive framework, and posture based movement representation, that allows an efficient
realisation of a fluent robot grasping.
Our studies about cognitive representation of complex motor actions (e.g. Schack & Hackfort, 2007;
Schack & Mechsner, 2006), are of relevance for a control of full body movements in robots. We
created a research lines from an experimental analysis to a computational modelling of full body
movements (Krause et al., 2010). In this research line, we want to investigate complex movements of
the whole human body, as they occur in sports, dance or other expert tasks (Bläsing et al., 2010a), but
also in every-day life, in order to understand how these movements are controlled, learned and
reproduced under changing conditions. Our aim is to apply different biomechanical and psychological
methods to analyse the kinematics, force profiles, muscle activity, mental representations and other
cognitive control structures, and to develop a computational model that integrates the results of the
different measures into a simulation of the movement. Results of these studies could be helpful on
many levels for the general understanding of human movement control, but also for movement control
in artificial agents and robots, especially if we are interested in “human-friendly” interaction between
humans and technical systems.
Cognitive representations are reference frames in the implementation and control of human motor
action (Schack & Ritter, 2009; Zentgraf et al., 2009). For a better understanding of the structure and
functionality of such reference frames we planned in cooperation with Rolf Pfeiffer and Rüdi Füchslin
(ai-Lab, Zürich) different experiments about principles of tool use and tool manipulation in humans. In
a next step we are going in direction of a computational modelling and a technical implementation of
functionally structured motor representations in robotic platforms. The aim is here to address in a
longer research perspective the “frame of reference problem” in robotics from a new point of view.
1.2 Motor primitives at the level of kinematic invariants
A number of groups in the AMARSI consortium address the problem of movement primitives at the
level of kinematics. A very generic approach is followed by the partner WI who characterizes
movement primitives by their geometrical invariants. This approach works at a very basic level and
does not require higher cognitive representation levels. This conception is particularly suitable for the
segmentation and modelling of action streams and of complex movements that are modelled by
sequencing of simpler segments.
1.2.1 Notion of Motor Primitives (MP)
In dealing with the modularity of the motor control system the focus of the Weizmann group is on the
level of trajectory planning, but studies are also carried out which are dealing with motion planning at
the joint level. The term motor (or motion) primitive refers to elementary movements or building blocks
from which more complicated movements are constructed. In studying the issue of motor
compositionality one of the main goals is to be able to infer motion primitives from movements that are
apparently continuous. Hence, the main questions in investigating the nature of motion primitives and
compositionality are as follows. a) Can we identify an alphabet of motor primitives from which more
complex behaviors are constructed? b) What is the nature and internal representation of such
primitives? c) What generation rules are used to generate or span an entire motor repertoire of either a
single or several motor tasks from a limited set of elementary movements? d) What syntactic rules are
used in joining together motor elements?
In particular in attempting to identify and characterize motor primitives our approach is based on taking
advantage of the existence of motor invariants and templates at the level of trajectory planning. These
invariants consist, for example, of the invariance of hand paths and velocity profiles during reaching
movements or of the two-thirds power law observed for curved and drawing movements. These
kinematic laws of motion can also be accounted for by several optimization models, specifically the
minimum jerk model (Flash and Hogan, 1985), but also by more general maximum smoothness
models (MSD models). Earlier it was suggested that the observation of a piecewise segmented
power law and the change in the value of the velocity gain factor can be considered as evidence for
motor segmentation. This claim, however, is problematic (Flash and Hochner, 2005) and is not
consistent with the findings reported in Richardson and Flash (2002) who have shown that global
optimization gives rise to a similar piecewise constant relationship between the logarithms of speed
and curvature without any assumption about segmented control. More recent studies, however, have
suggested a somewhat alternative explanation for the power law and hence an alternative approach to
segmentation, i.e., the approach based on equi-affine differential geometry and specifically on the
observation that the two-thirds power law is equivalent to moving at a constant equi-affine speed
(Flash and Handzel, 2007). In particular, when this new framework was applied to the analysis of hand
trajectories it has led to a new focus on new geometric metrics and invariants which have included the
equi-affine arc-length and equi-affine curvature which remain invariant under equi-affine
transformations.
This approach was then extended to 3D drawing and curved movements by showing that the
hypothesis that equi-affine speed is kept piecewise constant also during 3D hand trajectories leads to
the formulation of a generalized power law whereby hand speed depends both on movement
curvature and torsion (Pollick et al., 2009; Maoz et al., 2009).
The equi-affine analysis has also served as a basis for developing a more general group theoretical
approach to the study of motor invariants and segmentation and to the suggestion of several new
ideas (Flash and Handzel, 2007; Polyakov et al., 2009a, 2009b): 1) Movement states within the motor
space might be characterized by their equi-affine differential invariants 2) A large variety of
movements might be achieved by applying equi-affine transformations on such limited number of
states and by combining them.
This theoretical approach has also led to a series of combined behavioral, neurophysiological and
computational studies that gave some validity to this idea (see below, Polyakov et al, 2009a, 2009b).
An important regularity observed during human movement which was not accounted for by the equi-
affine framework is the isochrony principle, i.e., movements of different lengths (amplitudes) having
nearly the same movement duration (Viviani and Flash, 1995). Also no principled rule for the selection
of the velocity gain factor was suggested. Hence more recently we have generalized the equi-affine
description to a new and broader theory based on geometrical invariance. The new notion is that
movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full
affine geometries where each geometry possess a canonical measure of distance along curves, an
invariant arc-length parameter (Bennequin et al., 2009).
The theory was mathematically formulated and its predictions were tested on three data sets:
drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms. This new
theory suggests new notions about motion compositionality and segmentation.
Figure 2. Equi-affine parameters for a parabolic-like recorded movement segment. a. Equi-affine velocity
(dots) and curvature (asterisks) for a scribbling segment. b. The actual drawing made by the monkey. Since
parabolas are characterized by zero equi-affine curvature, the motion segment can be well-fitted by parabolas,
(dashed lines). Figure taken from Polyakov et al., 2009b.
1.2.2 Methods and experimental evidence
The following pieces of experimental evidence support the discussed theory:
a) Earlier evidence for the existence of motion primitives at the kinematic levels comes from work
concerning the principle of superposition of elementary point-to-point to point movements from
a study of trajectory modification (Flash and Henis, 1991; Henis and Flash, 1995).
b) A co-articulation phenomenon was observed in a study involving continuous practice during
the generation of hand-writing like trajectories. The kinematic properties of well-practiced
movements were successfully accounted for by the maximum smoothness principle and newly
formed curved primitives were shown to emerge following sufficient practice (Sosnik et al.,
2004).
c) In relation to the above geometric approach to movement segmentation and compositionality
based on the equi-affine framework (Handzel and Flash, 1999; Flash and Handzel, 2007;
Pollick and Sapiro, 1997) we have derived a necessary mathematical condition on paths for
which the predictions of both the 2/3 power-law and the constrained minimum-jerk model
coincide (Polyakov et al., 2009a). Such paths must obey the following relation 06 rr
(differentiation w.r.t. the equi-affine arc-length). Using this condition and requiring the
invariance of these paths with respect to equi-affine transformations, we have shown that only
for parabolic shapes, the 2/3 power law and the criterion of smoothness maximization are
mathematically reconciled on the geometric level yielding trajectory elements that are invariant
to equi-affine transformations. We have also demonstrated how complex piece-wise parabolic
trajectories can be generated from a single parabolic template – based on equi-affine
geometric transformations and uniform scaling. We have then examined the validity of our
theoretical analysis by fitting free monkey scribbling movements with basic parabolic strokes
(see Figure 2) and found that following practice, these drawing movements could be
decomposed into only 3-4 well separated clusters of parabolic segments (see Figure 3).
Defining a movement primitive as an elementary stroke that cannot be intentionally stopped after its
initiation, we also found that when the monkey's motor performance was disrupted by giving a reward
at certain locations, the monkey indeed tended to decelerate and stop their movements but not before
the completion of parabolic-like path segments. In additional studies, the neural activities of multiple
single-units, underlying scribbling movements, were recorded in parallel from M1 and PMd during 8
recording sessions and have been segmented in an unsupervised way based on a Hidden Markov
Modeling analysis (as in Abeles et al., 1995). In many cases, the movements corresponding to the
identified states of neural activities formed clusters of similar geometric shapes; some clusters
consisting of parabolic-like segments. By applying partial cross-correlation analysis (Stark et al.,
2007), we have found a stronger representation in the activities of several cells of equi-affine speed
rather than of Euclidian speed.
1.2.3 Implications for robotics
Obviously, all the above computational problems and approaches may apply also to robotic systems
where algorithms and approaches for the selection of motion primitives at the end-effector and joint
levels and their blending arise also in relation to robot arm movements, locomotion and multi-effector
movements especially for robot humanoids, but also to more conventional manipulators or mobile
robots. An example is work in WP4 that models human motion using both, joint-level and end-effector
levels of control resulting in more effective reaching (see Hersch et al., 2008; Calinon et al., 2010).
Figure 3. Emerging parabolic clusters and dimensionality reduction. A. Typical histograms for the fitted
parabolic segments. In the one-dimensional histogram (left), the segments are counted according to their
orientation. In the color histogram (right), they are counted in distinct bins according to the orientation and focal
parameter of the parabola. B. Location of the vertex and orientation of the parabola for every 10th parabolic
segment for the recording sessions. Locations of the vertices of the similarly oriented parabolas are also
clustered. The clusters are marked by ellipses and the mean orientations of the parabolas within each cluster are
depicted by arrows (Taken from Polyakov et al., 2009b).
Another important application for robotics is the segmentation of natural movements into simpler
segments, which then can be supplied to learning algorithms and as basic segments for the design of
controllers. The proposed methods have the advantage that they do not require any prior knowledge
or training data, such as supervised segmentation algorithms, as the ones discussed in section 1.3.4.
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1.3 Motor primitives at the neuromuscular level
One of the lowest levels for the definition of movement primitives is the level of muscle synergies.
Such synergies can be identified by application of dimension reduction methods to muscle activity
patterns or to kinematic data. The partner SLF has been leading in the research within this field within
the last decade. For selected classes of movements this work has accurately characterized then
organization of the associated muscle activation patterns and their changes associated with different
tasks. In some sense, this characterization of movement primitives in biological systems is closest to
their implementation at the level of biomechanics and the control at the muscle level.
1.3.1 Notion of Motor Primitives (MP)
Two notions of motor primitives have been used by the SLF group to capture the modular organization
of the muscle patterns observed during human locomotion and reaching movements. The first notion
is that of a basic temporal component in the muscle activity patterns (Ivanenko et al., 2004, 2006). The
second notion is that of a time-varying muscle synergy (d'Avella et al., 2003; d'Avella et al., 2006;
d'Avella et al., 2008).
Both notions are based on the assumption that the central nervous system (CNS) generates the
muscle patterns appropriate for performing a task by superposition of a few basic motor programs.
However, each notion emphasizes different invariant features of these motor programs.
According to the first notion, the time-varying muscle activation vector observed in a specific task
condition is generated by the combination of a few basic temporal components each associated
with the synchronous activation of groups of muscles through a constant weighting vector :
∑
(1)
As the task conditions change, the temporal components are invariant while the weighting matrix is
adjusted appropriately.
According to the second notion, the same time-varying muscle activation vector is generated by the
combination of a few time-varying muscle synergies, i.e. time-varying muscle weights ,
appropriately scaled in amplitude shifted in time:
∑
(2)
where is the amplitude scaling coefficient and ti the onset delay for the i-th synergy. In this case, the
time-varying synergies are invariant across task conditions and the changes in the muscle patterns are
captured by changes in the scaling and timing coefficients.
1.3.2 Methods and experimental evidence
Experimental evidence supporting both notions of motor primitives rely on unsupervised learning
(decomposition algorithms) to identify either temporal components and synchronous muscle
weightings or time-varying synergies and combination coefficients from multi-muscle
electromyographical recordings obtained during the performance of a motor task in many different
conditions.
Muscle patterns recorded during human walking at different speeds, with different body weight
unloading, can be reconstructed by five basic temporal components identified with factor analysis (see
Figure 4 and Ivanenko et al., 2004, 2006). Similar components are identified by other algorithms such
as non-negative matrix factorization or independent component analysis (Ivanenko et al., 2005), see
Figure 5. Finally, five temporal components also capture the temporal organization of running muscle
patterns with only one of these components significantly different from those for walking (Cappellini et
al., 2006), see Figure 6.
The spatiotemporal organization of the phasic muscle patterns for fast reaching movements in
different directions in vertical planes (d'Avella et al., 2006) can be reconstructed by a small number of
time-varying muscle synergies identified by an iterative optimization algorithm developed for this
purpose (Figure 7). The reconstruction of muscle synergies is robust against changes in posture and
load, and the amplitude coefficients show cosine directional tuning.
Moreover, modulation of phasic and tonic time-varying muscle synergies captures the variations in the
muscle patterns observed in reaching movements in different directions and with different speeds, as
shown in Figure 8 (d'Avella et al., 2008). These results suggest that the central nervous system might
be using a simple scaling strategy for generating the joint torque profiles appropriate for moving along
a given trajectory with different speeds. As the equation of motion for an articulated arm are invariant
for changes in the movement time scale (r) if the dynamic components of the joint torques are scaled
in amplitude by r2 and anti-gravity torque components are not changed. Thus, if a torque profile
adequate for reaching a given spatial target at one speed is known, a simple scaling rule allows to
generate the torque profiles for reaching that target, along the exact same path, at different speeds. A
low-dimensional representation of the muscle patterns for reaching in terms of phasic and tonic
muscle synergies would greatly simplify the implementation of such a control strategy. Indeed, the
amplitude coefficients of the phasic synergies were found to scale with speed to a relationship close to
quadratic.
Figure 4. Locomotion program as a characteristics timing of muscle activation (Ivanenko et al., 2004).
Similar five activation components, identified with factor analysis, account for about 90% of variance in the leg
EMG patterns during walking at different speeds and loads.
95%
75%
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stance swing
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comp 1
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speed body weight support
LocomotionMotor Program
LocomotionMotor Program
comp 1 32 4
CPG
5 1
TD TDLO
time
…comp 1 32 4
CPG
5 1
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…
Figure 5. Basic components. Basic temporal components identified by different statistical approaches (factor
analysis, independent component analysis, non-negative matrix factorization and fitting by Gaussian
components) from normalized EMG patterns during normal walking (Ivanenko et al., 2005).
1.3.3 Implications for robotics
Most models of adaptive modules proposed in the robotics literature or currently being developed as
building block for the AMARSI architecture use a representation of kinematic variables, since robot
dynamics is usually taken care of by low-level feedback control loops. In contrast, the notions of motor
primitives employed by the SLF group are mainly at the muscular, thus dynamic, level. While the
fundamental differences in the nature of the actuators and sensors between biological and robotic
systems might suggest that a notions of motor primitive at the dynamic level are not immediately
relevant for robot control, such statement might not apply to novel biologically inspired robot platform
with compliant actuators. Indeed, one of the features of the muscle-based actuator systems is that
they are naturally compliant and that they allow for adjusting the mechanical impedance by regulating
the amount of co-contraction. How the control of the impedance of multi-joint systems is regulated by
the nervous system and whether the representation and control of impedance relies on motor
primitives (such as, for example, muscle synergies which generate zero output force/torque but control
impedance of the limb along specific directions) are novel and open questions in human motor control
that might be highly relevant for the control of robots with compliant actuators.
Along these lines, a first attempt to combine estimation of the dynamics of reaching movements
through dynamical system with estimation of the impedance parameters required to adapt to
uncertainties during transport was conducted by the technical partner EPFL in Gribovskaya et al.,
2011 (see also AMARSI Deliverable 4.1: Ch.8, Stable Estimator of Dynamical Systems).
77EMGs
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Figure 6. Five basic temporal components for human running (Cappellini et al., 2006). The timing of the
main peak of components is similar for walking and running except for the second component that is shifted in
running.
1.4 Motor primitives defined by learned kinematic components
The partner UniTu takes the viewpoint of machine learning. The work has mainly focused on the
analysis and modelling of kinematics of body movements by the application of learning methods,
which are inspired by the concept of motor primitives form neuroscience. The derived primitive-based
representations have been exploited, on the one hand, for the analysis of motor behaviour and the
perception of body movements. On the other hand, we have developed methods that transform such
primitive-based representations into algorithms that are suitable for the offline and online synthesis of
body movements. In addition, we have used primitive-based representations for the study of
perception of body motion.
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Figure 7. Reconstruction of muscle patterns for fast-reaching movements by combinations of time-
varying muscle synergies (adapted from d’Avella et al., 2006). Five time-varying synergies (A) extracted from
the phasic muscle patterns recorded during point-to-point reaching movement to targets in different directions on
frontal and sagittal vertical planes explain the variation in the muscle patterns across directions (B) as due to the
selection of a small number of amplitude coefficients (represented by the height of the rectangles below the EMG
traces) and time-shifts (horizontal position of the rectangles). The amplitude coefficients (C, polar representation)
show a simple dependence to movement direction, approximated by a cosine function.
Figure 8. Reconstruction of muscle patterns for reaching movements in different directions and speeds
by combinations of phasic and tonic time-varying muscle synergies (adapted from d’Avella et al., 2008).
Thee phasic and three tonic time-varying synergies (A) extracted from the muscle patterns recorded during point-
to-point reaching movement to targets in different directions on the frontal plane and with different speeds capture
the variation in the muscle patterns (B) by modulation in amplitude and timing of the phasic synergies and in
amplitude only the tonic synergies. The cosine directional tuning of the amplitude coefficients for both synergies
(C) is modulated by speed for the phasic synergies.
1.4.1 Notion of Motor Primitives (MP)
In their learning-based approach UniTu distinguishes temporal primitives in the sense of individual
actions in longer sequences, and spatial primitives that permit the modelling of complex movements
by superposition of a small number of source components, which might include only subsets of the
available degrees of freedom.
1.4.1.1 Temporal primitives
Temporal primitives have been investigated in the context of complex action sequences, such as
forms in martial arts. A key problem in practice is the automatic segmentation of natural action
sequences in temporal segments that correspond to meaningful single actions. This problem has been
addressed by supervised learning approaches. For this purpose, the trajectories where characterized
by a sparse set of robust trajectory features (e.g. zeros of the velocity in individual degrees of
freedom). These features were optimized for segmentation. Individual temporal primitives were then
characterized by training examples and were modelled as a discrete sequence of key features that
corresponded to such vectors of such robust features. Such feature sequences form templates which
then could be matched in a robust manner to the feature sequences extracted from the trajectories by
a special dynamic programming technique (Figure 9). This algorithm made the matching process
robust against missing or additional key features in the matched sequence compared to the template.
After the automatic segmentation the individual temporal primitives were modelled by spatio-temporal
morphable models. This technique permits the very accurate modelling of complex movement
trajectories by interpolation in space-time. Inspired by image morphing techniques that are meanwhile
common in computer graphics, the underlying idea is to characterize classes of similar trajectories by
their space-time shifts against a reference trajectory (like the average of a set of prototypes). Such
space-time shifts can be computed by dynamic time warping. The correspondence shifts of a set of
prototypes relative to a reference trajectory form a basis of a vector space. This permits to model
intermediate movement trajectories by linear combinations of such correspondence shifts (Figure 10).
The technique of spatio-temporal morphable models provides an efficient low-dimensional
parameterization of movement style. It was shown to be efficient for the analysis as well as for the
synthesis of simple movements (Poggio and Giese, 1999; Giese and Poggio, 2000) and also for
movement parameterization form the study of movement perception in psychophysical experiments
(Giese and Lappe, 2002; Giese et al., 2008).
Figure 9. Supervised segmentation of action streams. The original movement is scanned with a time
window. The trajectory segment within the window is compared with template sequences (template primitives)
using an algorithm that is based on dynamic programming. (a) Trajectory segment with robust features s
i . The
segment can be characterized by a sequence of these key features (b) The prototypical primitive can be
characterise by a sequence of corresponding features m
i . Dynamic programming makes the matching robust
against additional or missing key features. (See Ilg et al., 2004 for details).
Figure 10. Modelling of trajectories by linear combination in space-time. (a) Spatio-temporal
correspondence is established between the individual prototype trajectory and a reference trajectory. Each
prototype is characterized by the functions ζn(t) and τn(t) that characterize the spatial and the temporal shifts
against the reference trajectories. (b) New trajectories can be generated by linear combination of the spatial and
temporal shifts, and space-time warping of the reference trajectories with the resulting shifts. (c) Generation of an
action sequence for a humanoid agent by linear combination of a neutral and a happy style. (For details see
Giese & Poggio, 2000; Ilg et al. 2004).
1.4.1.2 Spatial primitives
The approach for the extraction of spatial primitives from trajectory data of UniTu is closely related to
the approaches developed by SLF. The developed methods aim at an approximation of joint angle
trajectories by a minimum set of learned components or source signals that are combined using a
nonlinear mixing model, corresponding to an anechoic mixture. This model has been found to be
most appropriate by a detailed analysis of the approximation of sets of periodic and non-periodic body
movements, comparing different blind source separation algorithms (Omlor and Giese, 2006a; Giese
et al., 2009). The applied model is characterized by the equation:
∑
(3)
Here denotes the angle trajectory, and are the source functions or mixture components.
These functions are interpreted as invariants over different types of movements. The mixing weights
and the delays are the model parameters which are movement specific and which are modified
as part of movement planning and control. The weights determine the amplitude (gain) of the
corresponding source and the delays between the components highlight the importance of timing
for the coordination of joint movements. Mixture models of the type (3) are called anechoic. Detailed
analyses showed that various classes of complex body movements could be approximated by these
models with high accuracy and very few source components, typically less than five (Omlor and Giese,
2006a; Giese et al., 2009).
(a) (b)
(c)
The described model mainly aims at a minimal mathematical parameterization of the trajectories. The
interpretation of model components as primitives is, therefore, less direct than in some biologically
motivated methods:
The source functions components form a function basis that spans the trajectories. For
periodic movements these functions are similar to a Fourier basis. It remains to be clarified
how such functions are related to primitives in terms of controlled components.
Both, weights and delays are movement specific and therefore are modified as part of
movement control. Approximating sets of trajectories by the model (3) and analyzing the
model parameters shows that typically these parameters cluster for individual movement types
and styles (e.g. Roether et al. 2009). In addition the algorithm can be modified in order to
enforce identical parameter values within individual clusters. A primitive is thus best modeled
by a combination of the source function and specific motion-type specific changes of the
parameters and the delays . This approach has been successfully applied for the
modeling and analysis of emotional movement styles and the identification of emotion-specific
movement primitives (Omlor and Giese, 2006b, 2007; Roether et al., 2009 see Figure 10a).
It is relatively easily possible to modify the algorithm in a way that embeds additional priors
and sparseness constraints. This might make it possible to extend the approach for the
extraction of more sources which map directly onto physiologically relevant movement
primitives or physiological modules. Which exact priors are required for maximum
physiological interpretability remains to be discovered. One possibility is priors that enforce
primitives that are spatially localized on the moving face or skeleton of moving bodies.
In an additional set of studies we have extended the described approach for movement synthesis in
real-time (Figure 11). The underlying idea is that solutions of nonlinear dynamical systems, called
dynamic primitives in the following, are mapped onto the source signals identified by unsupervised
learning from the trajectories (Park et al., 2008a, 2008b, 2009). The resulting architecture consists of a
number of dynamic primitives, which fulfill a similar function as CPGs in biological organisms. The
coordination of the generated patterns in presence of noise can be stabilized by introduction of
dynamic couplings between the dynamic primitives. In this context it turns out that compact
representations, based on a small number of sources, show more controllable dynamical behavior
than models, e.g. based on PCA, that include a relatively large number of coupled components or
dynamic primitives (Park et al., 2008a, 2008b; Giese et al., 2009). In addition, we investigated for
simple examples the integration of non-periodic and periodic movement primitives within the same
architecture.
The same architecture can be extended by introduction of dynamical approaches for navigation or the
synchronization between multiple interacting agents in a scene (Giese et al., 2009). This makes the
approach suitable for an online synthesis of the body movements of individuals and also of the
coordinated movements of groups of agents.
Due to the simplicity of the resulting architecture, the framework is accessible for the analysis of the
dynamical stability of the resulting architecture. This has been exploited for the design of the dynamic
couplings between the individual primitives (Park et al., 2008a, 2008b) as well as in recent work for the
analysis and the design of the stability of collective order formation scenarios realized by groups of
human agents (Park et al. 2009; Mukovskiy et al., 2008, 2010). Present work focuses on the analysis
and design of the dynamical stability properties exploiting concepts from contraction analysis (e.g.
Lohmiller and Slotine, 1998; Pham and Slotine, 2007; Mukovskiy et al., 2010).
Figure 11. Application of the extraction of movement primitives by blind source separation. (a) Automatic
extraction of emotion-specific features form gait trajectories, using a blind source decomposisiton model
combined with sparse regressions (from Roether et al., 2009). Colors indicate the joint angle amplitudes
compared to normal walking for angry, fearful, happy and sad walking for the joints indicated along the vertical
axis. Each emotion is characterized by a characteristic profile of changes in joint angle amplitudes. The extracted
features closely match results from psychophysical studies (indicated by the black and white signs). (b)
Architecture for the online synthesis of complex body movements based on learned primitives. Based on the
learned mixing model that is determined by blind source decomposition of a set of training trajectories movements
are generated online. The source signals are generated online by mapping of the solutions of nonlinear dynamical
systems (dynamical primitives) onto the source signals using kernel methods. (c) By coupling of the dynamic
primitives of different agents coordinated crowd behaviour can be self-organized. This example shows the
autonomous formation of a crowd with agents that synchronize their step phases. Taken from Giese et al., 2009.
1.4.2 Methods and experimental evidence
A hierarchical model based on temporal primitives combined with a modelling of individual primitives
by spatio-temporal morphable models has been successfully applied for the modelling and imitation
learning of movements, such as writing or martial arts techniques (Ilg et al., 2004; Mezger et al.,
2005). The approach has been successful not only for movement synthesis but also for the analysis of
movement styles, such as the estimation of skill level from Karate sequences (Ilg et al., 2004). The
low-dimensional parameterization of movement styles by spatio-temporal morphable models has been
shown to be very useful in psychophysical and fMRI experiments on body movements (Giese and
Lappe, 2002; Giese et al., 2008; Jastorff et al., 2006, 2009), as well for the study of the perception of
facial movements (Knappmeyer et al., 2004).
Modelling of movements in terms of spatial primitives has been successfully applied for the study of
the execution and perception of emotional body movements (Roether et al., 2008, 2009; Omlor and
Giese, 2006b, 2007). The synthesis of trajectories based on dynamic primitives derived from learned
spatial primitives has been applied for the online synthesis of human locomotion and non-periodic
movements (Park et al., 2008a, 2008b). This work includes integration of primitives for non-periodic
(a) (b)
(c)
and periodic motion. Also this work demonstrates that the modelling of body movements in terms of
dynamic primitives is highly suitable for the modelling of the behavior of groups of human agents for
example locomoting groups (Figure 11C) or dancing (Giese et al., 2009). More recent work has started
to apply the framework of contraction theory (Lohmiller and Slotine, 1998, Wang and Slotine, 2005) for
the development of stability bounds for systems based on dynamic primitives (Park et al., 2009;
Mukovskiy et al., 2010). This work suggests that this approach might be suitable for guaranteeing the
stability of systems including large numbers of nonlinear dynamic primitives.
1.4.3 Implications for robotics
The concept of temporal primitives has been successfully implemented for the realization of imitation
learning on a Mitsubishi robot arm (Ilg et al., 2004). The concept of learned dynamic primitives is
suitable for an online synthesis of complex full-body movements and the design of complex networks
of dynamic primitives. This makes it potentially interesting for the modelling of complex motor
behaviour on robot platforms. However, this framework at the moment lacks the integration of sensory
feedback signals. This step seems crucial for a transfer of this methodology to a variety of robot
systems and is a central problem of present work. In addition, it remains to be explored in how far
primitives learned from kinematic data can be approximated by control primitives on individual robot
platforms. Contraction theory as a tool for stability analysis in complex systems is very general and
has been applied extensively in the context of robotics and nonlinear control (Lohmiller and Slotine,
1998; Lohmiller and Slotine, 2000; Slotine, 2006; Chung and Slotine, 2009).
1.5 Relationships between the different approaches In order to address possible correspondences between the architecture of movement representations
in humans and in robots, it seems helpful to investigate MPs using maximally similar tasks.
A very general conception of movement primitives (MPs) is given by the approach of UniBi, that
envisions different levels of MPs, some of which overlap with the levels of description of the other
partners. Which levels are relevant depends on the task. For example, particular tasks might primarily
address the higher cognitive levels, while others more require a treatment on the level of muscle
synergies. In addition, the complexity of MPs might change dependent on the level of action
organisation. It seems possible that interactions exist between MPs at a more biological (motor-
driven), perceptually driven and cognitive (intentionally driven) levels of action organisation.
The general approach of UniBi offers many opportunities to interact with the other partners. For
example, the work on MPs based on the perception and the kinematics of body postures of UniTu can
be nicely combined with the approaches of UniBi for the analysis of kinematics and the cognitive
representation of movement keypoints in memory. A combination of these different approaches might
help to learn more about the relationships between kinematic, perceptual and the memory side of
MPs. The same question is related to the kinematic and muscle-based analysis of primitives by SLF.
Specific questions of such interactive research may be how body postures are represented in memory,
or what is the relationship between muscle-activation (SLF), body postures and kinematics (as
modelled by UniTu), and functional-biomechanical constraints in the memory representations of MPs.
A further set of important questions in this context is the interplay of learning processes and the role of
expertise, dependent development and structure formation of MP's in the memory (UniBi) and at the
neuromuscular levels (as treated by SLF).
In general, the approach of UniBi offers a tool for the comparison of representation structures for
complex movements in the long term memory of experts and novices on the basis of different sets of
building blocks, or motor primitives, defined via different methods. Similar approaches can likely be
applied to motor primitives that are defined via trajectory planning or movement planning on joint level,
as analyzed by Weizmann, to muscle activity patterns and muscle synergies, as analyzed by SLF, or
kinematic primitives such as extracted by the methods of UniTu.
Another set of connections exists in the domain of motion segmentation. Differential geometrical
approaches by Weizmann for the determination of invariants might be combined by methods for
supervised and unsupervised segmentation, such as developed by UniTu and also by UniBi. A related
interesting topic, presently being investigated at UniBi, is how human observers spontaneously
segment “abstract”, not object-related, complex movement sequences in different conditions, and how
their decisions are influenced by learning and expertise (Bläsing et al., 2010a). Results of such
experiments could be compared to results of algorithmic methods for movement segmentation in order
to compare human and statistical methods for movement segmentation.
A further problem is the development of methods for measuring of the similarity of different movement
and clustering. UniBi has compared motion segments by applying Procrustes analysis, which is widely
used in shape matching. The resulting movement segments can be applied to the SDA method was
used to analyze the mental representations of movements in long term memory. Other methods, e.g.
based on spatio-temporal correspondence, have been proposed by UniTu. Further similarity measures
might be developed exploiting concepts from differential geometry, such as applied in the work by
Weizmann.
Close connections also exist between the work of Weizmann and UniTu. Motion primitives at the level
of hand or COM trajectories during locomotion can be defined based on the hypothesis that kinematic
strokes at the task level are represented in terms of non-Euclidian variables, and that motion primitives
are invariant under certain group of non-linear transformations. Earlier work, partially carried out in
collaboration between Weizmann and UniTu (Dayan et al., 2007; Casile et al., 2010), has indicated
that similar invariant properties of movement apply also to motion perception. At the same time, work
by SLF (Ivanenko et al., 2010) suggests that internal models of automatic postural responses might
influence perception as well as motor control. This supports that primitives are represented at different
levels, such as formulated in the theory of UniBi.
Work involving compositionality principles at the joint level has also been carried out by Weizmann
group in collaboration with UniTu (Barliya et al., 2009) showing that using a relatively simple oscillatory
model at the joint level it is possible to account for the inter-segmental law of coordination observed
during human locomotion (Borghese et al., 1996; Ivanenko et al., 2008). This work is discussed in
more detail in section 2.2.
Tight connections exist also between the work of SLF and UniTu. The central algorithms for the
unsupervised learning of primitives from EMG and trajectory data are very similar, and a presently
ongoing computational study investigates how the underlying models are mathematically related, and
what are the advantages and disadvantages of the underlying algorithms for practical applications,
and especially for the analysis of data in motor control. Preliminary esults from this study are
presented in section 2.1. Another important aspect is the investigation of the relationship between the
neuromuscular and the kinematic level of MPs. Basic temporal components might be linked to specific
kinematic events, such as the onset of foot lift in walking (Ivanenko et al., 2006). Moreover, scaling in
amplitude and duration of phasic time-varying muscle synergies might underlie the path invariance
observed for reaching movements with different speeds (d'Avella et al., 2008). Here, interesting
connections between the approaches of SLF, UniBi, Weizmann and UniTu might be established.
The approach for the online synthesis of trajectories by nonlinear dynamical systems by UniTu is
closely related to the primitive-based control approaches developed at EPFL (e.g. Ijspeert et al., 2002;
Buchli et al., 2006). Challenges to be solved are to embed the proposed learned dynamic primitives for
complex movements into control architectures and to constrain them in a way that makes them
suitable for the embedding in the existing robot platforms. Here, learning-based approaches and the
concept of stability design in modular systems modelling highly complex motor behaviour, such as
investigated by UniTu, might form a basis for fruitful interactions with the approaches of EPFL that are
based on dynamic primitives that are implementable on the available robot platforms. (C.f. AMARSI
Deliverable 4.1: Ch.2, Dynamical Movement Primitives; Ch.8, Stable Estimator of Dynamical
Systems). Several discussed biological models were based on the notion of dynamical systems in
order to drive the generated motion. Notions of stability and tractability that are easily created by the
non-biological models such as DMP and SEDS (see AMARSI Deliverable 4.1: Ch.2, Ch.8). However,
such models cannot be easily transferred to the analysis of complex biological movements, as some of
the ones described in the previous sections. While this may be an impediment for robotics application,
further development of biologically plausible models may nonetheless contribute to improving the
complexity of the more mathematically oriented approaches in robotics. Such developments would
represent a nice desirable outcome of the AMARSI project.
2 Quantitative comparisons between selected approaches
Beyond the general considerations in the first part of this deliverable, some of the approaches are so
closely related that it makes sense to compare them mathematically or quantitatively based on
empirical or simulated data sets. This applies specifically to the approaches for the identification of
primitives by unsupervised learning by SLF and UniTu, and to the characterization of invariants of
locomotion patterns as studied by WI and UniTu. In the following we give a short report about these
more quantitative comparisons.
2.1 Comparison between the unsupervised learning approaches by SLF and UniTu
2.1.1 Introduction As discussed in the first part of this deliverable, there are currently two main different definitions of
motor primitives at muscle level used by the SLF group: the first one is that of basic temporal
components in the muscle activity patterns (Ivanenko et al., 2004, 2006). The second notion is that of
time-varying muscle synergies (d'Avella et al., 2003; d'Avella et al., 2006; d'Avella et al., 2008). Both
notions rely on the decomposition of EMG datasets in terms of a linear combination of motor
primitives, scaled in amplitude by some scalar weights. In particular, the first one is based on the
following equation
∑
(4)
where is a vector of muscle activations, defining the basic temporal components (scalar
functions of time), being real vectors of scaling weights, and where n is the total number of
primitives.
In contrast, the second notion is based on the following equation
∑
(5)
for which and have the same meaning as above, and where the specify time delays relative
to the vectors . Note that this time the are time-dependent variables, whereas the weights are
not. Furthermore, the elements of the vectors and all scaling coefficients were always
constrained to be non-negative in both models, given the non-negativity of the elements of the vector
. Interestingly, the models in (4) and (5) can be seen as two special cases of the model used by
the UniTu group to study spatial primitives (Omlor and Giese, 2007). This model is described by the
equation
∑
(6)
with representing either a joint-angle time series or, in the other case, an EMG signal at instant t.
It is indeed straightforward to see that each row of (4) can be re-expressed in the form of equation (6)
by equating , , and by considering each coefficient as the i-th
element of the j-th vector . Similarly, each row of (5) can be re-expressed as a specific case of
equation (6) by taking , for , and . Note
however that, for the same dataset , to express eq. (5) with eq. (6) a much larger number of
source signal may be required with respect of the case of expressing eq. (4) with eq. (6). In general,
each muscle (index i) of a time-varying synergy might require a different source , so that
.
Therefore, given an EMG dataset is given, it can be decomposed according to one of the models
presented above. However, the number of the parameters to be identified and their identification
methods differ for model to model. Coefficients and primitives of model (4) were identified by using
standard methods such as Principal or Independent Components Analysis and Non-Negative Matrix
factorization (Ivanenko et al., 2005). Parameters of model (5) were identified by using an iterative
optimization algorithm that identifies shift-invariant multidimensional bases using Matching Pursuits
and NMF (see (d'Avella et al., 2003; d'Avella et al., 2006). For parameter extraction, Omlor and Giese
(2007) exploited an extraction algorithm based on a time-frequency transformation (Wigner-Ville
distribution). Also the computation time is dependent of the model and the identification method. For
all these reasons, it is hard to establish specific criteria that are suitable for choosing between these
models with respect to a given application. In order to establish a benchmark and for a comparison
between these approaches for primitive extraction we chose the following proceeding: Starting from
the generative models (4), (5) and (6), three different EMG-like datasets were generated.
Subsequently, different algorithms for dimensionality reduction based on these three models were
applied to the generated datasets and their results were compared. This quantitative comparison
might help to reveal non-obvious differences between the models and should suggest advantages and
disadvantages of the individual identification methods for synergy extraction from muscle activity.
2.1.2 Methods
2.1.2.1 Generation of artificial data sets
Generation of source signals: For the quantitative validation simulated datasets were generated
automatically. These data sets tried to replicate coarsely the properties of real EMG signals, recorded
from a number of muscles during NumT trials with executions of motor behavior. Different
generative models were tested (see below). All generative models derived the data from a set of
statistically independent EMG-like waveforms (referred to as source signals, or simply as sources, in
the following). For the generation of these signals we used an autoregressive moving average (ARMA)
model, which was based on the following equation
∑ ∑
(7)
where is the order of the autoregressive part of the model (AR), is the order of the moving
average part (MA), are the coefficients of the recursive linear filter, and where are the coefficients
of the non-recursive linear filter. The signal signifies white noise. The ARMA model can be
interpreted as a discrete linear system, with the white noise input and the output signal . The
AR and MA coefficients were first estimated (using the MATLAB function armax.m with , )
from real EMG data. This data was recorded during from the biceps muscle of a subject performing an
elbow flexion. Muscle activity was at 1 KHz for a period of 400 ms. The raw signals had been
amplified, rectified, low-pass filtered with a cut-off frequency of 5 Hz. In addition, it was resampled in
order to the fit a sample time window with time steps). output signals were then obtained
by varying the input noise of the model (MATLAB function sim.m). The noise input was given by a
normally distributed random vector of length . Positive Independent component analysis (a particular
version of independent component analysis with non-negativity constraints in the output) was applied
to the output signals of the ARMA model in order to obtain a set of non-negative statistically
independent waveforms. Finally, these signals were normalized by division by their maximum values.
Generative models: The generated source signals were subsequently used to generate datasets
simulating the EMG activities of muscles during hypothetical experimental trials. The simulated
EMG-like datasets were characterized by an intrinsic modular structure. The goal of the analysis using
dimension reduction methods was to extract the basic primitives, and the underlying structure from this
simulated data. The simulated datasets were generated by mixing together a set of primitives
derived directly from the source signals assuming different mixture models that reflected the structures
of the generative models which were underlying the different unsupervised learning methods. In the
following each algorithm will be described along with the model that underlies the corresponding
simulated EMG datasets.
2.1.2.2 Tested dimensionality reduction algorithms
1) Non-Negative Matrix Factorization (NMF): NMF is a standard algorithm for multivariate analysis
where a matrix is factorized into two matrices, and , so that
(8)
The key aspect of this method is that all the elements of the matrices and are imposed to be non-
negative. Therefore, if is thought to be a matrix in which each row represents an EMG signal.
According to equation (8) each EMG signal in results from the linear combination of the rows of the
matrix scaled by the elements of the rows of the matrix .
According to this mixing model, one simulated EMG dataset could be generated by multiplying a
matrix of dimension by (in which the rows were independent source signals) by a
by matrix of positive random numbers, drawn by a uniform distribution in the interval
(0,1). different datasets were then obtained by changing each time the elements of . Note
that in this case each primitive coincides with a source signal.
2) Blind source separation (or anechoic demixing, which from now on we will refer to as An) is an
unsupervised learning algorithm that approximates signals by linear superposition of components with
signal-specific time delays (Omlor and Giese, 2007). Given a simulated EMG matrix , each row
signal can be expressed as
∑
(9)
where are scalar weighting factors, each is a time source signal and the are time delays
between source signals and the signals . Note that no non-negativity constraints are imposed by
this algorithm and that also in this case each primitive coincides with a source signal with .
Based on the modular organization proposed by this model, simulated EMGs were generated by
combining linearly independent delayed sources weighted by weight coefficients .
Delays were chosen randomly and drawn from an uniform discrete distribution in the interval .
3) Positive anechoic demixing: The positive anechoic demixing (pAn) algorithm works exactly as An,
but additional non-negativity constraints are imposed to the coefficients and the sources .
4) Time-varying synergies: The time-varying synergies extraction algorithm (TV) is a method for
primitive extraction that was described in previous works of the partner SLF (d’Avella and Bizzi, 2005;
d’Avella et at., 2006). The method is based on the following generative model
∑
(10)
where is a vector of real numbers, each component of which represents a specific simulated EMG
activation at time . The vector signal represents the muscle activations for the primitive,
and is a time delay and a non-negative scaling coefficient. All parameters of the model are
constrained to be non-negative. Therefore, given a matrix of simulated EMG signals, the algorithm
identifies, through an iterative optimization process, all the non-negative parameters of the second
member of equation (10).
In our work, to generate simulated EMG dataset having modular structure of the TV model we
considered sources and pooled them in primitive matrices of dimension
by . In this case, a primitive coincided with one of such matrix. We then computed random
numbers that were used as weighting coefficients (whose values were taken form a uniform
distribution with comprised between 0 and 1) and random integer numbers form a uniform
discrete distribution in the interval that specified the time delays . Finally, the primitives were
combined according to equation (10).
2.1.2.3 Performance measures
The approximation quality of the models for the data was characterized by computing the explained
variance of the data that was captured by the fitted dimension reduction model. This measure was
given by the coefficient
‖ ‖
‖ ̅ ‖ (11)
where was the matrix of the actual dataset, the reconstructed values by the fitted model, and
where ̅ is a matrix with the mean values of the data over trials.
To assess the approximation qualities of the algorithms dependent on the compatibility of the
algorithm with the generative model of the data, we simulated three datasets according to equations
(4), (5) and (6). To each one of them we applied NMF, An, pAn and TV. We finally computed the
index for all combinations of simulated models and dimensionality reduction algorithms.
As a second measure, we assessed the similarity between original and extracted primitives. This was
done by computing the maximum of the scalar products between original and recovered primitive over
all possible time delays. In detail, we considered the two sets of original and reconstructed primitives
for a specific model (NMF, An or TV). Then, given two normalized primitives and , we computed
the maximum of their scalar product over all possible delays for the second primitive, that is
(12)
where indicates the second primitives delayed in time by j time steps. For NMF and An, and
were already vectors (of length ). Differently from NMF and An, for TV and were matrices of
dimension by . In this case, before computing the scalar product, and were rearranged
the entries in form of vectors by concatenating them in rows with a length of . By definition,
can only adopt values between 0 and 1. For all possible pairs of primitives in the two groups the
corresponding values were computed. The pair with highest similarity was selected and the
corresponding primitives were removed from the two groups of primitives. The similarities between the
remaining primitives was then computed, and the best matching pair of primitives was selected and
removed from the original and reconstructed model. This procedure was iterated until all primitives
were matched.
2.1.3 Preliminary results and discussion
In the following, we present results obtained with the dimensionality reduction algorithms applied to
datasets with , , and . Table 1 shows the R2 values that quantify the
reconstruction accuracy for the generated datasets with the different algorithms. Table 3. Approximation quality for the different generative models obtained with the different dimensionality reduction algorithms. Rows correspond to the generative models and columns to the algorithms.
Data simulated with the NMF generative model was perfectly reconstructed by NMF. However, this
failed for the accurate reconstruction of datasets derived from generative models with specific
temporal structure (An and TV). The anechoic algorithms (An and pAn) resulted in general in the
highest performance. However, for the dataset derived from an anechoic mixture model (second row
in Table 3) pAn performed better than An. This makes sense since this algorithm implies non-
negativity constraints for the estimation of the parameters that match the constraint of the generating
model. The TV algorithm performed perfectly for the reconstruction of the corresponding generated
dataset, but failed for the other datasets. The most interesting aspect is whether the algorithms are
able to retrieve the original primitives by applying dimensionality reduction to the simulated datasets.
This was evaluated by assessing the similarities between the primitives that were used to generate a
dataset, based on a specific modular structure (i.e NMF, An or TV), and the primitives extracted by the
corresponding algorithm that is based on the same generative model. One example is shown in Figure
12A, showing that NMF could perfectly retrieve the original primitives (corresponding to the very high
value of the measure S). A different situation arises for the An dataset, for which the sources signals,
weights and delays where all non-negative. In this case the An algorithm, which does not exploit this
positivity constraint results in reasonable approximation of the original data, but a small similarity
between the original and the recovered primitives (Figure 12B). This is different for the algorithm pAn,
which results in high approximation (Table 3) and in addition perfectly retrieves the original source
components (Figure 12C). The same is true if this algorithm is applied to the dataset TV (Figure 12D).
While the presented results are still preliminary, they seem to represent a useful starting point for a
further validation of the different algorithms. In addition, this comparison points to interesting
NMF An pAn TV
NMF 1,00 0,99 1,00 0,63 An 0,74 0,92 0,99 0,43 TV 0,84 0,97 0,98 0,99
theoretical questions, such as why An, and in particular pAn, always perform well for many of the
tested datasets. As expected from statistical learning theory, a good capability of reconstructing the
original dataset is not always a guarantee a good recovery of the original primitives. A good
understanding of these issues seems essential for any analysis of biological data using such statistical
methods.
Some works based on the study of muscle synergies have demonstrated that scaling and temporal
parameters associated with muscle synergies can be task-dependent and differentiate between
different behaviors (Overduin et al, 2008), or healthy and impaired motor function (Cheung et al,
2009). Therefore, an thorough understanding whether such considerations can be made, independent
on the applied model for muscle synergies, It seems thus to be of essential importance for scientist in
the motor control to assess which claims can be derived from the parameters of such learned models,
Figure 12. Comperison between actual and extracted primitives. A. Primitives extracted by NMF from the
NMF generated dataset. B. Primitives extracted by An from the dataset generated by the An model. Remark
that all generated sdources were constreained to be non-negative. The algorithm An extracts malso negative
signals because no constraints were imposed on the the extracted sources. C. Primitives extracted by pAn from
the An generated dataset. D. Primitives extracted by TV from the TV the generated dataset.
2.2 Inverse kinematics and computational constraints at the joint level (joint work of WIS and UniTu)
Work involving compositionality principles at the joint level conducted by the Weizmann group in
collaboration with UniTu (Barliya et al., 2009) shows that based on the use of a relatively a simple
oscillatory model at the joint level, it is possible to account for the inter-segmental law of coordination
observed during human locomotion (Borghese et al., 1996; Ivanenko et al., 2008). Application of the
blind source separation analysis developed by UniTu to gait trajectories resulted in models for
locomotion that closely resemble the mathematical model developed by the group at the WI. Earlier work shows that the redundancy in the control of walking can be efficiently parameterized if the
trajectories of the lower limb are characterized by elevation angles, i.e. the angles between the
segments and the cardinal axes in the external frame of reference. In this case, the trajectories of the
elevation angles of thigh, shank and foot are lying within a two-dimensional plane, effectively
eliminating one available degree of freedom (Borghese et al., 1996; see Figure 13). The analytical model for the temporal variation of the elevation angles is described in detail in Barliya
et al. (2009). It turns out that these angles were well approximated by the simple sinusoidal time
dependence:
tAa sin (13)
In addition, it was shown that under the assumption that the natural frequencies fulfill the condition
FST (T=thigh, S=shank, F=foot),the orientation of the plane is very well approximated by
2121
3131
3232
sin
sin
sin
AA
AA
AA
n (14)
The above expression for the normal vector to the plane is a function of the amplitudes and phases of
the sinusoids that describe the elevation angles, but not their frequencies. This predicted normal to the
plane differs from the actual normal as computed by using PCA by less than 3°. Figure 14 illustrates
this result.
Applying the blind source separation algorithm developed by UniTu (Omlor and Giese, 2006a) it was
found that the sources used to describe joint rotations during human locomotion are similar to the
Fourier based description of joint rotations of the different leg segments applied by Barliya et al.
(2009).
A very important question that is still open in the motor control area is: given the kinematic redundancy
that exists in the mapping from task to joint levels, how are elementary primitives or strokes identified
at the task level map into possible sets of MP or elementary motions at the joint level. In another study
currently carried out in collaboration between Weizmann and UniTu (Barliya et al., 2010) it was found
that the mapping from hand to joint spaces can be relatively simple and may involve the anechoic
mixture of similar sources both at the joint and task spaces, but only when using quite specific
kinematic representations of the joint coordinates which is described in an absolute extrinsic frame of
reference. Moreover, such similarity between task and joint related sources was not found for
alternative representations. The fact that two spaces share the same set of sources might be
exploited.
One direction that is currently explored is how this set of sources can serve as a mediator between the
task and joint spaces. The observation that both spaces share one set of sources only when the joint
space is represented in a particular manner has led us to further investigate another old question, that
of representations. The tendency for “laws” to emerge when biomechanical systems are represented
in an extrinsic frame of reference is consistently observed. A concrete example is the law of
intersegmental coordination which was just described above and holds only for elevation angles that
Figure 13. Elevation angles of the different leg segments during one gait cycle as function of time (left),
and as angle-angle plot (right). The trajectories lies in a two-dimensional plane, reducing the effective number
of degrees of freedom. (From Barliya et al., 2009.)
Figure 14. Two planes illustrating the average difference (2.73°) between the plane constrain derived
from the original data by a PCA analysis and the plane predicted by the analytical model (from Barilya et
al., 2010).
are described in an external frame of reference. This work has laid the basis for future studies
addressing the question what are the relations between motor primitives at the task and joint levels.
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