Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements
A theoretical approach is proposed to define the force and position singular points (FSPs and PSPs) in the circular, ellipsoidal, and linear planar two-joint movements produced under steady loadings directed along the movement traces. The FSPs coincide with changes in the direction of the force m...
Gespeichert in:
| Veröffentlicht in: | Нейрофизиология |
|---|---|
| Datum: | 2016 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізіології ім. О.О. Богомольця НАН України
2016
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148336 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements / A.I. Kostyukov // Нейрофизиология. - 2016. - Т. 48, № 4. - С. 315-325. — Бібліогр.: 20 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-148336 |
|---|---|
| record_format |
dspace |
| spelling |
Kostyukov, A.I. 2019-02-18T10:19:03Z 2019-02-18T10:19:03Z 2016 Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements / A.I. Kostyukov // Нейрофизиология. - 2016. - Т. 48, № 4. - С. 315-325. — Бібліогр.: 20 назв. — англ. 0028-2561 https://nasplib.isofts.kiev.ua/handle/123456789/148336 159.946:611.737 A theoretical approach is proposed to define the force and position singular points (FSPs and PSPs) in the circular, ellipsoidal, and linear planar two-joint movements produced under steady loadings directed along the movement traces. The FSPs coincide with changes in the direction of the force moments acting around the joints; the PSPs show the locations of the extrema at the joint angle trajectories. The force synergy (defined by the location of FSPs) provides a strong influence on the activation synergy; the latter is largely described by correlations between the activities recorded from the muscles participating in the movement. The position synergy (defined by the location of PSPs) is responsible for a hysteresis-related modulation of the activation synergy. Geometrical procedures are proposed to define positions of the FSPs and PSPs along various movement traces; this can provide a general description of the force and position synergies for the movements. The force synergies in the circular movements cover four sectors with diverse loading combinations of the flexor and extensor muscles belonging to different joints. The variability of the synergy effects for changes in the size and position of the circular trajectories is analyzed; the synergy patterns are also considered for ellipsoidal and linear movement traces. A Force Feedback Control Hypothesis is proposed; it allows one to explain the decrease in the number of controlled variables during real multi-joint movements. This work was supported by the grant 0024/RSA2/2013/52 from the Rozwój Sportu Akademickiego, Poland. en Інститут фізіології ім. О.О. Богомольця НАН України Нейрофизиология Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements Теоретичний аналіз силових та позиційних синергій у двосуглобових рухах Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements |
| spellingShingle |
Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements Kostyukov, A.I. |
| title_short |
Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements |
| title_full |
Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements |
| title_fullStr |
Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements |
| title_full_unstemmed |
Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements |
| title_sort |
theoretical analysis of the force and position synergies in two-joint movements |
| author |
Kostyukov, A.I. |
| author_facet |
Kostyukov, A.I. |
| publishDate |
2016 |
| language |
English |
| container_title |
Нейрофизиология |
| publisher |
Інститут фізіології ім. О.О. Богомольця НАН України |
| format |
Article |
| title_alt |
Теоретичний аналіз силових та позиційних синергій у двосуглобових рухах |
| description |
A theoretical approach is proposed to define the force and position singular points (FSPs
and PSPs) in the circular, ellipsoidal, and linear planar two-joint movements produced under
steady loadings directed along the movement traces. The FSPs coincide with changes in
the direction of the force moments acting around the joints; the PSPs show the locations
of the extrema at the joint angle trajectories. The force synergy (defined by the location of
FSPs) provides a strong influence on the activation synergy; the latter is largely described by
correlations between the activities recorded from the muscles participating in the movement.
The position synergy (defined by the location of PSPs) is responsible for a hysteresis-related
modulation of the activation synergy. Geometrical procedures are proposed to define positions
of the FSPs and PSPs along various movement traces; this can provide a general description
of the force and position synergies for the movements. The force synergies in the circular
movements cover four sectors with diverse loading combinations of the flexor and extensor
muscles belonging to different joints. The variability of the synergy effects for changes in
the size and position of the circular trajectories is analyzed; the synergy patterns are also
considered for ellipsoidal and linear movement traces. A Force Feedback Control Hypothesis
is proposed; it allows one to explain the decrease in the number of controlled variables during
real multi-joint movements.
|
| issn |
0028-2561 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148336 |
| citation_txt |
Theoretical Analysis of the Force and Position Synergies in Two-Joint Movements / A.I. Kostyukov // Нейрофизиология. - 2016. - Т. 48, № 4. - С. 315-325. — Бібліогр.: 20 назв. — англ. |
| work_keys_str_mv |
AT kostyukovai theoreticalanalysisoftheforceandpositionsynergiesintwojointmovements AT kostyukovai teoretičniianalízsilovihtapozicíinihsinergíiudvosuglobovihruhah |
| first_indexed |
2025-11-26T13:22:21Z |
| last_indexed |
2025-11-26T13:22:21Z |
| _version_ |
1850622586855096320 |
| fulltext |
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4 315
UDC 159.946:611.737
A. I. KOSTYUKOV1, 2
THEORETICAL ANALYSIS OF THE FORCE AND POSITION SYNERGIES
IN TWO-JOINT MOVEMENTS
Received June 6, 2015
A theoretical approach is proposed to define the force and position singular points (FSPs
and PSPs) in the circular, ellipsoidal, and linear planar two-joint movements produced under
steady loadings directed along the movement traces. The FSPs coincide with changes in
the direction of the force moments acting around the joints; the PSPs show the locations
of the extrema at the joint angle trajectories. The force synergy (defined by the location of
FSPs) provides a strong influence on the activation synergy; the latter is largely described by
correlations between the activities recorded from the muscles participating in the movement.
The position synergy (defined by the location of PSPs) is responsible for a hysteresis-related
modulation of the activation synergy. Geometrical procedures are proposed to define positions
of the FSPs and PSPs along various movement traces; this can provide a general description
of the force and position synergies for the movements. The force synergies in the circular
movements cover four sectors with diverse loading combinations of the flexor and extensor
muscles belonging to different joints. The variability of the synergy effects for changes in
the size and position of the circular trajectories is analyzed; the synergy patterns are also
considered for ellipsoidal and linear movement traces. A Force Feedback Control Hypothesis
is proposed; it allows one to explain the decrease in the number of controlled variables during
real multi-joint movements.
Keywords: motor control, two-joint movements, muscle synergy, central commands,
electromyogram.
1 University of Physical Education and Sport, Gdansk, Poland.
2 Department of Movement Physiology, Bogomolets Institute of Physiology,
National Academy of Sciences, Kyiv, Ukraine.
Correspondence should be addressed to A. I. Kostyukov
(e-mail: kostyuko@biph.kiev.ua).
INTRODUCTION
Three interdependent types of muscle synergies
are usually considered when describing human
movements. Both anatomical and neural factors
are combined in coordinated joint movements, thus
participating in various forms of the kinematic synergy
that is displayed in simultaneous covariations during
independent changes of the joint angles [1] and in
various tasks of manual exploration [2]. The kinetic
synergy, described usually by covariation of the forces
(torques), has also been observed during grasping
movements [1], in forced interaction of various
fingers [3, 4], or during handwriting [5]. The muscle
synergy, based on spatial and temporal coordination of
multiple muscle activities, has been observed during
static hand efforts [6] or in active force interactions of
muscles of the digits [7, 8].
The anatomy of the human limbs usually does not
allow experimenters to control all essential parameters
defining the synergy effects; at least partly, this is
related to practical impossibility of EMG recording
from deeply located muscles. As a result, not all
fundamental synergies can be identified experimentally
in multi-joint movements of the limbs. In previous
experimental studies of our group [9-11], we proposed
an approach allowing one to analyze quantitatively
the simplest form of the synergy effects in circular
movements of the subject’s right arm. The obtained
results were used to find the functional relationships
between basic mechanical parameters of two-joint
movements and the related central commands. In order
to determine inter-joint muscle interactions for these
movements, we proposed a simplified classification of
the synergy effects [11]; the same classification is used
in the present study. In accordance with definitions
accepted in the above study, we will further use the
terms of position, force, and activation synergies,
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4316
A. I. KOSTYUKOV
which are based on temporal changes in the following
parameters, correspondingly: (i) joint angles, (ii)
force moments at the joints, and (iii) activities of the
muscles participating in a given movement. The terms
force and position singular points, FSPs and PSPs,
respectively, have also been introduced.
As was shown in [11], waveforms of the averaged
EMG activities of the elbow and shoulder muscles
are closely related to the correspondent sectors of
the movement trajectories between neighboring
FSPs, in which the force moments acting around the
correspondent joints change their directions. Waves of
the activity in these sectors are alternated in flexors
and extensors; the activation patterns are reversed
with changes in the loading direction. On the other
hand, the EMG intensities are also dependent on the
movement direction; such hysteresis-related effects
are defined by sets of the PSPs (i.e., positions of the
extrema at the joint angle traces).
The locations of the FSPs and PSPs were defined in
[11] by computation of the time courses of the force
moments and joint angles. At present, graphical methods
begin to be widely used for the analysis of the synergy
effects (see, e.g., [12]). In this study, we propose a
graphical method for a theoretical definition of the
force and position synergies for planar two-joint arm
movements. In accordance with the methods described
in [11], the axis of the proximal joint in our model is
assumed to be in a fixed position, while the distal end
of the other limb segment moves with a small constant
velocity along the circular trajectory; the movements
are produced under the action of constant loadings
directed tangentially with respect to the trajectory.
Afterwards, we extrapolated this consideration on the
ellipsoidal and linear movement traces.
Hypothesis. Central commands to the muscles in
two-joint movements and the related synergy effects
are largely dependent on the relative positions
of the FSPs belonging to different joints; muscle
hysteresis participates in modulation of the commands
in accordance with the location of the PSPs. An
assumption has been put forward that, in order to
decrease the number of the controlled variables in
multi-joint movements, the CNS may use the force
feedback channels from the antagonist muscles of
different joints (a Force Feedback Control Hypothesis).
RESULTS
Curvilinear System of Coordinates. The present
model is based on the two-joint planar upper limb
movements produced under conditions of the fixed
positions of the trunk and shoulder joint (see Methods
in [11]). Naturally, the proposed consideration may
be applied to the movements of the lower limbs, wich
are realized, in particular, during a bicycle ride. The
test movements, according to natural positioning
patterns of the limb segments, can be analyzed within
the framework of the curvilinear coordinate system
shown schematically in Fig. 1. Four parameters, Rs,
Re, α, and β, completely define the “hand” position
F i g. 1. Operational space of two-joint pla-
nar movements and definition of the cur-
vilinear coordinate system. Parameters:
α and β) elbow and shoulder joint angles
changing within the following ranges:
αmax ≤ α ≤ 0; βmax ≤ β ≤ 0; Rs and Re) lengths
of the arm segments; S and E) positions of the
axes of the shoulder and elbow joints; H) “hand”
position, i.e., position of the distal end of the
second (distal) arm segment that can be called
provisionally the “forearm”; εi(α) and σk(β)) ba-
sic elements of the curvilinear coordinate system
(isolated movement traces in one of the joints
when another joint is fixed). The curve a-b-e-
d-c-a indicates the boundary of the operational
space Ω.
Р и с. 1. Операційний простір двосуглобових
рухів у площині та визначення криволінійної
системи координат.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4 317
THEORETICAL ANALYSIS OF THE FORCE AND POSITION SYNERGIES IN TWO-JOINT MOVEMENTS
(point H) within the operational space W. The first
two of them, the shoulder and forearm lengths, Rs
and Re, are characterized by fixed values for a given
subject, whereas the two other parameters, namely
angles α and β (in the elbow and shoulder joints),
change as independent coordinates within the ranges
0 ≤ α < αmax and 0 ≤ β < βmax. Any target point within
the operational space W can be presented as a function
of the two variables, W(α, β), which is based on simple
trigonometric relationships (for details see [11]).
Boundaries of the operational space W are as follows:
W(0, β) (arc ac in Fig. 1); W(α, βmax) (arc cd); W(α, 0)
(arc ab); W(αmax, β) (arc be). Additionally, the curve
de is defined by a natural trunk border of the subject.
Finally, the curvilinear coordinate system may be
presented graphically by two sets of the intersected
arc lines, e i(α) = W(α; iDβ) and sk(β) = W(kDα; β),
where Dβ and Dα define the extent of discretization of
the coordinates (Fig. 1). In the former case, arcs of
radius Re are distributed with a constant density; in the
latter one, arcs of the concentric circles with center S
show an increased density with a shift toward the outer
boundary of the operational space.
Force Singular Points Related to the Elbow
Joint. A scheme of the hypothetical test movements
is presented in Fig. 2A. During the test, the subject
was asked to produce a slow circular-form steady
movement; the movement trajectory is shown in
F i g. 2. Definition of the force and position singular points (FSPs and PSPs) and synergy sectors during circular movements of the hand.
A) Definition of the FSPs at the elbow joint; Me
(1, 2); Ze) virtual trajectory of the elbow joint positions at a zero moment; Et) trajectory
of possible spatial shifts of the elbow joint; E) elbow location defined by crossing of the Ze and Et curves. B) Definition of the FSPs at
the shoulder joint; Ms
(1,2); auxiliary circles Zs
(1, 2) have the centers in Ms
(1,2) and radii Re; E1 and E2) elbow locations defined by crossing the Ze
(1, 2)
and Et curves. C) Elbow PSP, Le
(1, 2) (open triangles), and shoulder PSP, Ls
(1, 2) (closed triangles), coinciding with the extremal positions of
the circle in the curvilinear coordinate system; the auxiliary curves are Ze
(1, 2) and Zs
(1, 2) (dashed lines). D) Location of the FSPs and PSPs, and
of the force synergy sectors I–IV, in which the correspondent combinations of the elbow and shoulder muscles obtain the external loading;
symbols f and e designate flexor and extensor muscles; subscripts e and s show muscles belonging to the elbow and shoulder joints; Mcw and
Mccw are clockwise and counterclockwise directions of the external loading.
Р и с. 2. Визначення силових та позиційних сингулярних точок і секторів синергії у кругових рухах руки.
A
B
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4318
A. I. KOSTYUKOV
Fig. 2A by the circle with center O and radius r. It
is assumed that test movements are produced under
the action of a constant external loading applied
tangentially with respect to the movement trace in
clockwise or counterclockwise directions.
The force moment acting around the elbow joint
attains a zero value when the external force vector
is going via the joint axis; therefore, in this case the
“forearm” should be directed tangentially with respect
to the movement trace. The term “forearm” is used
here in quotation marks to designate the distance
from the elbow joint axis to the point of the external
force application; it is assumed that the subject’s
wrist is rigidly fixed along the line connecting
centers of the wrist and elbow joints. The scheme
in Fig. 2A demonstrates the geometrical procedure
used to determine the FSPs at the circular movement
trajectories. The equilibrium elbow joint position E is
defined as the point of intersection of the auxiliary
curve Ze and elbow trace Et. The arc Ze is the part of
the circle passing via ends of the “forearm” length
segments placed tangentially with respect to the
movement trajectory. The point E is connected with
two different “forearm” positions corresponding to two
FSPs (Me
(1, 2)) in the movement trajectory. These FSPs
divide the circle into two unequal segments differing
from each other by the sign of the force moments
applied to the elbow joint muscles. For the clockwise
directions of both external loading and movement,
the force action at the elbow joint is changed from
extension to flexion during transition via point Me
(1); a
further passing via point Me
(2) evokes a reverse action.
Therefore, for the clockwise-directed loadings, the
elbow joint will undergo the action of extending/
flexing forces during the movement along the longer/
shorter segments of the circle divided by points Me
(1, 2);
for the counterclockwise loadings, the force moments
will change in the opposite direction.
Force Singular Points Related to the Shoulder
Joint. The force moment acting at the shoulder joint
attains a zero value when the vector of the external
force is going via the joint axis; therefore, the FPSs
are shown by points of touching of the movement
circle by two tangent lines Zs
(1, 2) passing via the
joint axis (Fig. 2B). The shoulder FPSs are placed
symmetrically with respect to the line connecting the
center of the movement circle and the joint axis. The
definition of the elbow joint positions E1 and E2 for
the given FPSs can be derived as points of intersection
of the elbow trace Et with the circles of the radius Re
that are centered in Ms
(1, 2). Similarly with the elbow
joint, the shoulder FPSs define changes in the force
moments during the movement. When both external
force and movement have the clockwise direction,
the force moment changes its action from extension
to flexion at point Ms
(1); further movement via point
Ms
(2) will produce an opposite effect. Therefore, the
shoulder joint will undergo the action of extending/
flexing forces during the movement along longer/
shorter segments of the circle.
Position Singular Points. The PSPs at the elbow
and shoulder joints coincide with the points of the
movement trace where directions of the length changes
of the proper muscles are inverted (Fig. 2C). Pairs of
the PSPs, Le
(1, 2) (open triangles) and Ls
(1, 2) (closed
triangles), correspond to extremal positions at the
circle for the respective joint angles α and β; the
extrema are defined by points where the corresponding
coordinate traces Ze
(1, 2) and Zs
(1, 2) are touching the
movement circle. Note that the elbow PSPs Le
(1, 2)
are
on the line passing via the axis of the shoulder joint S
and center of the movement trace O.
Singular Points and Force Synergy Sectors.
Figure 2D describes the summarized location of FSPs
and PSPs defined above (Figs. 2A-C) and illustrates
the definition of the force synergy sectors (I-IV). The
force correlations between different functional muscle
groups belonging to different joints are most important
for the treatment of the processes of central activation;
therefore, it may be useful to define sectors at the
movement trajectory with different combinations of
the loadings on various muscle groups acting at
different joints. During the movement in sectors I
and II, muscles of the same modality, flexors (f) or
extensors (e), are loaded at both joints; the choice
between the combinations fefs and eees (subscripts e
and s denote elbow and shoulder) within the sectors
I and II depends on the direction of the external force
moment (Mccw and Mcw). Combinations of the opposite
modalities of the muscle loadings are observed in
sectors III (fees and eefs) and IV (eefs and fees); these
sectors are noticeably smaller, as compared with
sectors I and II.
Force Synergy Sectors for the Movement Circles
of Different Radiuses. The nonlinear character of the
system under study creates obvious prerequisites for
complex reordering in relative positions of singular
points at the circular movement traces with changes
in their radiuses. During such changes (Fig. 3A, B),
steady angle positions are typical only of elbow joint
PSPs (open triangles), whereas other singular points
at both joints shift along specific smooth curves.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4 319
THEORETICAL ANALYSIS OF THE FORCE AND POSITION SYNERGIES IN TWO-JOINT MOVEMENTS
Changes in the positions of the FSPs are accompanied
by reordering of the synergy sectors as well. When
the radii are decreased, a certain smoothing of the
differences between dimensions of the synergy sectors
I and II is observed, while weights of the sectors III
and IV remain almost invariable (Fig. 3B). It seems
that these effects may be related to the varying degree
of curvature of movement traces.
Change in the Placement of Movement Traces.
Patterns of singular points in the identical circular
movement traces remain unchanged for all traces
placed at the same distance from the shoulder joint axis
(Fig. 3C). In Cartesian coordinates, the set of singular
points turns in the clockwise/counterclockwise
direction in accordance with rightward/leftward
turning of the line connecting the axis of the shoulder
joint and the center of the movement circle. The
relative weights of the force synergy sectors are also
not changed in this case. On the contrary, shifts of the
movement traces in the distal direction along the line
passing via the axis of the shoulder joint (Fig. 3D)
lead to increases in sectors I and II and corresponding
decreases in sectors III and IV. It has been noted above
that the distribution of the synergy sectors and their
weights may depend on the degree of curvature of
movement traces. In this case, the latter parameter is
invariable; thus, it can be assumed that the observed
changes depend on the curvature of the “shoulder”’
coordinate traces (concentric circles of different radii).
On the other hand, these effects may also be related
to turning of the “elbow” coordinate traces for more
distal movement trajectories.
Ellipsoidal Movement Traces. Ellipsoids may be
used for the description of more complex movement
trajectories. Experimental setups in this case should
be significantly more complicated, as compared with
those used for studying circular movements [11].
These movements, however, may also be considered
theoretically using the methods proposed above
(Fig. 4). Determination of the “shoulder” FSPs would
not differ from that described above for the circular
traces. For the “elbow” FSPs, however, it is necessary
to introduce two different curves, Ze
(ccw) and Ze
(cw),
describing opposite movement directions separately
(Fig. 4A). These curves are constructed using (i)
several defined points at the ends of the tangential
segments Re (five points for each loading direction are
shown in Fig. 4A), and (ii) any kind of the nonlinear
F i g. 3. Analysis of the differences between
the patterns of singular points and force syn-
ergy sectors depending on the magnitude of
the movements and on their location within
the operational space. A) Singular points and
force synergy sectors for the movement traces
corresponding to concentric circles of different
radiuses. B) Comparison of the force synergy
sectors for circles of the maximal and mini-
mal radiuses shown in A. C and D) Transitions
of the movement circle center along a fixed
“shoulder” coordinate line O1–O2 (C) and per-
pendicularly with respect to “shoulder” coordi-
nate lines O3–O4 (D).
Р и с. 3. Аналіз розбіжностей патернів син-
гулярних точок та секторів силової синергії
залежно від величини рухів та їх розташу-
вання в операційному просторі.
A B
C D
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4320
A. I. KOSTYUKOV
smooth approximation procedure. Further positioning
of the ellipsoidal trace within the operational space
should be further considered with Ze
(ccw) and Ze
(cw)
curves fixed to the movement trace (Fig. 4B-D).
Transections of these auxiliary curves with the elbow
trace Et define the elbow joint positions (E1, E2), which
correspond to the required FSPs Me
(1, 2).
Ellipsoidal Movement Traces of Different Sizes.
Using the methods described above, we can define
singular points for a system of ellipsoidal movement
traces of different sizes (Fig. 4E). The distributions
of the FSPs and of the force synergy sectors is largely
similar to those of the concentric circular traces
presented in Fig. 6, although some elongation of the
traces leads to the respective shifts of the FSPs along
larger axes of the ellipses, thus changing the force
synergy sectors. It may be emphasized that sectors
III and IV decrease in this case, as compared with
those in the circular traces (Fig. 3A). Elongation
of the movement traces also evokes an interesting
phenomenon, namely the appearance of two additional
elbow PSPs (lines Z3 and Z4) at larger-size traces
(R4 – R6) in Fig. 4E. Most likely, this can be related to
differences in the curvature degree of different traces
and/or to the dependence on positioning of the ellipses
within the operational space.
Singular Points in the Linear Movement Traces.
Linear (or quasi-linear) traces may be considered
important elements of many real movements. A
method allowing one to define the “elbow” FSPs
for the linear movements is considered below.
Similarly to the circular and ellipsoidal movements,
F i g. 4. Analysis of ellipsoidal trajectories of the
movement. A) Definition of the force singular points
(FSPs) for the elbow joint; auxiliary curves Zccw and
Zcw coincide with virtual trajectories of the elbow joint
positions with a zero moment. B, C, and D) Definition
of the FSPs and proper elbow joint positions for
different placements of the ellipsoids within the
operational space. E) Location of the singular points
and force synergy sectors at the ellipsoidal movement
traces of various dimensions; note the complex pattern
of the “elbow” PSPs; in addition to the PSPs located
along the lines Z1 and Z2, in a part of the larger ellipses
(R4–R6), pairs of additional PSPs (Z3 and Z4 branches)
appear.
Р и с. 4. Аналіз еліпсоїдних траєкторій руху.
A B
C
E
D
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4 321
THEORETICAL ANALYSIS OF THE FORCE AND POSITION SYNERGIES IN TWO-JOINT MOVEMENTS
the loadings are in this case also directed along the
movement trajectory (Fig. 5). The main problem for
linear movements is related to the definition of the
“elbow” FSPs. For this purpose, we shall analyze two
traces, the circular R1 and the linear R2, going through
two points, A1 and A2. In accordance with the method
described earlier, for circle R1 we can define both the
position of the elbow joint E1 (as the point of crossing
of the auxiliary curve Ze1 with the elbow trace Et) and
the related FSPs Me1
(1, 2). On the other hand, the linear
trace R2 may be presented as a result of the endless
increase in the radius of circle R1 under conditions of
its going through points A1 and A2. In this case, the
auxiliary curve Ze2 will coincide with the movement
trace itself, i.e., with the R2 line. Therefore, since A1
and A2 are points of crossing of the curves Ze2 and
Et, they also coincide with the elbow positions E2
(1, 2)
where the forces are going through the joint axis, and
the “forearm” is oriented along line R2. The additional
condition for the appearance of the FSPs in the linear
movement trace is its crossing of (or, at least, touching
to) the elbow trace E t. On the other hand, there are
positions where possible FSPs could exit out of the
operational space.
Synergy Areas for Sets of Parallel Linear
Movement Traces. The above-described approach
for finding the “elbow” FSPs in the linear movement
traces may be used to define the synergy areas
for sets of parallel linear movement traces having
various slopes (Fig. 6). In the movement traces going
orthogonally with respect to the frontal plane (Fig.
6A), the “elbow” FSPs are defined using the auxiliary
curve Ze. This curve coincides with the elbow trace
E t shifted vertically for the distance Re along the
movement trace of a zero loading (line z in Fig. 6A).
The FSPs can be defined as points of crossing of curve
Ze with the movement traces located rightward from
line z. The “forearm” locations at these FSPs coincide
with the movement traces; the proper positions of the
shoulder segment are shown by dotted lines going
toward small circles at curve Et. Note that the FSPs
are absent on the left with respect to line z, because
points of crossing of the auxiliary curve Ze with the
movement traces do not obey to a natural condition
of positivity for the elbow joint angles (the forearm
cannot be placed along these traces without destroying
the joint).
The arc Ze and line z constitute natural boundaries
confining the synergy area III (marked in gray in
Fig. 6A), where muscles of different modality acting
on the elbow and shoulder joints are loaded. In two
other areas of synergy (I and II), muscle groups of
the same modality are loaded. Similarly to the circular
movements, the antagonistic muscles are loaded with
a change in the direction of the external loading. The
synergy areas may also be defined in a similar way for
other directions of the movement traces (Fig. 6B - D).
In these cases, the corresponding rotation of the same
synergy areas (I - III) is observed in accordance with
turning of line z. For positions of the movement traces
shown in panels B and D of Fig. 6, it is possible to
observe the disappearance of both line z and area II
due to natural limitations of the operational space.
In accordance with the general definition, the z lines
F i g. 5. Elbow force singular points (FSPs) at the
linear movement traces. The linear trace R2, passing
via points A1 and A2, corresponds to a limit passage
of the circle R1 passing via the same points during an
endless radius increase. The virtual trajectory of the
elbow joint positions with a zero moment for the R1
circle is presented by arc Ze1, whereas such a trajectory
for the line R2 (Ze2) coincides with the movement trace
itself. In this case, it is possible to define two joint
positions, E2
(1, 2), and corresponding FSPs, Me2
(1, 2),
for the movement line R2. An important condition for
the existence of the “elbow” FSPs is the necessity for
the linear movement trace to intersect (or, at least, to
touch) the trajectory of possible movements of the
elbow joint Et; the FSPs defined in this way cannot
exit out of the limits of the operational space.
Р и с. 5. Силові сингулярні точки для передпліччя
на траєкторіях лінійних рухів.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4322
A. I. KOSTYUKOV
in all sets of the parallel movement traces divide the
entire operational space in accordance with the sign of
the force moment at the shoulder joint. The absence of
the line z at a given movement direction signifies that
the force moment at the shoulder joint does not change
its sign within the entire operational space.
In addition to the FSPs and the respective synergy
areas, all panels in Fig. 6 include lines of distribution
of the PSPs, where the directions of the muscle length
change in the corresponding joints are reversed.
DISCUSSION
Force and Position Synergies. Our study is devoted
to a theoretical analysis of various types of planar
movements produced under different combinations of
directions of both the external force moment and the
movement itself. It has been shown earlier that central
commands to the muscles in two-joint movements
depend predominantly on the relative positions of the
FSPs where the force moments change their directions
[11]. At the same time, the commands are also dependent
on the PSPs connected with the extremal points at
the joint angle traces. In this study, the positions of
singular points in the movement traces were defined
graphically. This approach allows the researcher to
analyze not only circular traces, but separate elements
of more complex trajectories as well.
Formally, the force synergies may be classified
in accordance with the functional modality of the
muscles belonging to different joints, which are loaded
simultaneously. The coinciding synergy corresponds to
simultaneous loading of muscles of the same modality
(flexors-flexors; extensors-extensors); while the
opposing synergy belongs to combinations of muscles
A
C
B
D
F i g. 6. Force and position singular points (FSPs and PSPs) and force synergy areas defined for sets of the parallel linear movement
traces of different directions. A–D) Areas III (marked in grey) correspond to the opposing patterns of loading, when the elbow flexors are
loaded together with the shoulder extensors, and vice versa; areas I and II correspond to the coinciding patterns of loading for the muscles
belonging to different joints. Designations are the same as in Fig. 5; the load directions are shown by arrows in brackets. The lines marked
by arrow z in panels A and D designate traces of the zero moment at the shoulder joint.
Р и с. 6. Силові та позиційні сингулярні точки та зони силових синергій, визначені для серій паралельних траєкторій лінійних рухів
різного напрямку.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4 323
THEORETICAL ANALYSIS OF THE FORCE AND POSITION SYNERGIES IN TWO-JOINT MOVEMENTS
of different modality (flexors-extensors; extensors-
flexors). Muscle combinations within both types of
the synergy effects depend on the loading direction.
It is interesting that, in all the considered types of
movement trajectories, both closed (circular and
ellipsoidal) and open (linear) ones, the prevalence of
the coinciding synergies is manifested. In the circular
traces, the weight of the coinciding synergy sectors (I,
II) is greater, as compared with that of the opposing
synergy ones (III, IV). In ellipsoidal traces, these
differences seem to be expressed even more clearly
(compare Figs. 3A and 4E). In the general case of the
parallel movement traces going under different angles
to the subject’s frontal plane (Fig. 6), the operational
space is usually divided into three areas, two of which
are connected with the coinciding synergy (I, II), while
the third one is related to the opposing synergy (III).
Similarly, the weight of the coinciding synergy areas
seems to be relatively greater as compared to that of
the opposing synergy.
The force synergy patterns are changed for identical
circular movement traces going at different distances
from the shoulder axis (Fig. 3D). At the same time,
both relative distribution of singular points and
weights of the synergy sectors remain unchanged
for the same distances from the shoulder axis, and
the pattern of all synergy points is simply rotated in
the course of such a transition (Fig. 3C). At present,
preliminary analysis of the FSPs (Figs. 3 and 4)
allows us to conclude that the observed variabilities
of the force synergy patterns at various parts of the
operational space are likely related to the differences
in the curvature indices of both movement trajectories
and traces of the curvilinear coordinate system.
The position synergy is defined by the distribution
of PSPs along the movement trajectories, and its
influence is directly related to muscle hysteresis (for
review, see [13]). The effects of the position synergy
are often smoothed when PSPs are placed close to
the nearest FSPs, although the movement-dependent
differences in the EMG intensities may be also rather
significant, especially for distal muscles [11].
Activation Synergy. Despite strong experimental
support for the assumption of existence of connections
between force and activation synergies in real circular
movements [11], it has been demonstrated in the cited
study that EMG activities of the elbow and shoulder
muscles may be rather noticeable out of the zones of
their direct loading. This phenomenon may be related
to a more complicated arrangement of the joints,
as compared with that in a simple pivotal model.
An exhausting analysis of the complex geometry of
the rotation movements in the shoulder joint can be
found in [14]. The elbow joint biomechanics is highly
intricate as well; recently, it has been considered as an
assemblage of three interactive joints [15]. It seems
that such complex mechanical systems as the elbow
and shoulder joints can provoke indeterminacy in the
force moments acting around these joints.
It is quite clear that, with change in the movement
pattern, muscle activities are rearranged in correspon-
dence to new movement tasks. At the same time, it
should be noted that classification of the muscles as
belonging exclusively to the elbow or shoulder joints
is noticeably oversimplified; sites of the force applica-
tions can be fixed only for the monoarticular muscles,
while the procedure of their identification for the bi-
articular muscles is significantly more complex [16].
The set of efferent activities controlling two-joint
movements is often localized within separate time or
space zones within which programs of co-contraction
can predominate. The movement phases are primarily
accompanied by co-contractions of the antagonistic
muscles within the areas adjacent to the zones of
their direct loading. The co-contraction patterns can
distinctly reduce both the after-effects of the ongoing
residual movements at the apexes of movement and
the uncertainty effects related to muscle hysteresis
[13, 17]. Behavioral studies of postural tasks have
demonstrated that subjects use muscle co-contraction
as a strategy of stabilization of the limb joints in the
presence of external loadings [18]. Humans are also
able to modulate independently the relative balance of
co-contraction and limb stiffness in different spatial
directions [19] and at different joints [20]. At the same
time, co-contraction of the antagonistic muscles should
increase the energy costs of the real movements.
Force Feedback Control Hypothesis. The close
resemblance of the force and activation synergies
allows us to propose the force feedback control
hypothesis, which introduces a hypothetical mechanism
via which the CNS can regulate descending motor
commands in multi-joint movements. This mechanism
can be based on using feedback signals with
information on the presence or absence of loading of
the antagonistic muscles acting on joints participating
in a given movement. During the movement, any
crossings of the FSPs belonging to a given joint would
evoke “inversion” of the corresponding feedback
signals, thus informing the motor control system on the
necessity to redirect descending activity between the
groups of antagonistic muscles of a given joint. As a
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4324
A. I. KOSTYUKOV
result, muscles that were active earlier become silent,
while their antagonists are activated. At any moment,
predominant activation is directed toward the muscles
loaded by the force moment acting at the proper joint.
The Golgi tendon organs seem to be the best candidates
for providing this force afferent signals, although
the involvement of other proprioceptor types in this
process cannot be ruled out as well. Voluntary control
can also participate in this case for both co-contraction
of the antagonistic muscles and the desired shaping of
the movement trajectories. The co-contraction of the
antagonistic muscles is better seen within the trace
areas located near the correspondent FSPs [11]. The
force feedback control hypothesis can probably explain
the decrease in the number of controlled variables
during multi-joint movements due to the possibility
for the CNS to use simultaneously the complex of
proprioceptive information coming from all joints
involved in a given movement program. Moreover,
such “force information matrix” might be created
without any additional expense for the CNS on the
assessment of the real position and forces at the joints
involved in the movement. In this case, it is likely
that this automatically obtained information concerning
the force synergy is simply “transformed” into a
preliminary pattern of the activation synergy, which
is naturally needed in some voluntary modifications,
in order to obtain the desired parameters of the
movements.
Acknowledgment. This work was supported by the grant
0024/RSA2/2013/52 from the Rozwój Sportu Akademickiego,
Poland.
This study is devoted to theoretical analysis of the processes
of motor control; it was not directly related to experiments on
animals or tests on humans. Therefore, it does not require the
confirmation of compliance with the existing international
ethical standards for experimental studies.
The author, A. I. Kostyukov, confirms the absence of any
conflict related to the commercial or financial problems and to
the relations with organizations or persons, which could in any
way be associated with the investigation.
О. І. Костюков1,2
ТЕОРЕТИЧНИЙ АНАЛІЗ СИЛОВИХ ТА ПОЗИЦІЙНИХ
СИНЕРГІЙ У ДВОСУГЛОБОВИХ РУХАХ
1 Університет фізичного виховання та спорту, Гданськ
(Польща).
2 Інститут фізіології ім. О. О. Богомольця НАН України,
Київ (Україна).
Р е з ю м е
Пропонується теоретичний підхід для визначення силових
та позиційних сингулярних точок (ССТ та ПСТ відповід-
но) при циркулярних, еліпсоїдних та лінійних двосуглобо-
вих рухах у площині, які реалізуються в разі наявності по-
стійних навантажень, орієнтованих вздовж траєкторій рухів.
ССТ співпадають із точками зміни напрямку моментів сили,
які діють на суглоби, а ПСТ відповідають розташуванню
екстремумів на траєкторіях суглобових кутів. Силова синер-
гія, що визначається розташуванням ССТ, інтенсивно впли-
ває на синергію активації; остання в основному описується
кореляціями між активністю м’язів, залучених у реалізацію
руху. Позиційна синергія, що визначається розташуванням
ПСТ, відповідальна за пов’язану з гістерезісом модуляцію
активаційної синергії. Пропонуються геометричні процеду-
ри для визначення положень ССТ та ПСТ на траєкторіях різ-
них рухів; це може допомогти формуванню загального опи-
су силових та позиційних синергій для різних рухів. Силові
синергії в кругових рухах перекривають чотири сектори з
різними комбінаціями м’язів-флексорів та екстензорів, що
діють на різні суглоби. Аналізовано варіативність ефектів
синергії щодо величини та розташування кругових траєкто-
рій; патерни синергії розглядаються також для еліпсоїдних
та лінійних траєкторій рухів. Запропоновано гіпотезу контр-
олю сили на базі зворотного зв’язку; вона дозволяє поясни-
ти зменшення кількості контрольованих змінних величин у
перебігу реальних багатосуглобових рухів.
REFERENCES
1. M. Santello and J. F. Soechting, “Force synergies for
multifingered grasping,” Exp. Brain Res., 133, 457-467 (2000).
2. P. H. Thakur, A. J. Bastian, and S. S. Hsiao, “Multidigit
movement synergies of the human hand in an unconstrained
haptic exploration task,” J. Neurosci., 28, 1271-1281 (2008).
3. I. V. Grinyagin, E. V. Biryukova, and M. A. Maier, “Kinematic
and dynamic synergies of human precision-grip movements,”
J. Neurophysiol., 94, 2284-2294 (2005).
4. J. K. Shim, H. Olafsdottir, V. M. Zatsiorsky, and M. L. Latash,
“The emergence and disappearance of multi-digit synergies
during force-production tasks,” Exp. Brain Res., 164, 260-270
(2005).
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2016.—T. 48, № 4 325
THEORETICAL ANALYSIS OF THE FORCE AND POSITION SYNERGIES IN TWO-JOINT MOVEMENTS
5. A. W. Hooke, S. Karol, J. Park, et al., “Handwriting: three-
dimensional kinetic synergies in circle drawing movements,”
Motor Control, 16, 329-352 (2012).
6. C. Castellini and P. van der Smagt, “Evidence of muscle
synergies during human grasping,” Biol. Cybern., 107, 233-
245 (2013).
7. M. L. Latash, J. P. Scholz, and G. Schoner, “Toward a new
theory of motor synergies,” Motor Control, 11, 276-308
(2007).
8. B. Poston, A. Danna-Dos Santos, M. Jesunathadas, et al.,
“Force-independent distribution of correlated neural inputs to
hand muscles during three-digit grasping,” J. Neurophysiol.,
104, 1141-1154 (2010).
9. A. V. Lehedza, A. V. Gorkovenko, I. V. Vereshchaka, et al.,
“Comparative analysis of electromyographic muscle activity of
the human hand during cyclic turns of isometric effort vector
of wrist in opposite directions,” Fiziol. Zh., 61, No. 2, 3-14
(2015).
10. T. Tomiak, A. V. Gorkovenko, A. N. Tal’nov, et al., “The
averaged EMGs recorded from the arm muscles during
bimanual “rowing” movements,” Front. Physiol., 6, No. 349,
doi: 10.3389/fphys.2015.00349 (2015).
11. T. Tomiak, T. I. Abramovych, A. V. Gorkovenko, et al.,
“The movement- and load-dependent differences in the
EMG patterns of the human arm muscles during two-joint
movements (a preliminary study),” Front. Physiol., 7, No. 218,
doi: 10.3389/fphys.2016.00218 (2016).
12. P. D. Neilson, M. D. Neilson, and R. T. Bye, “A Riemannian
geometry theory of human movement: The geodesic synergy
hypothesis,” Human. Mov. Sci., 44, 42-72 (2015).
13. A. I. Kostyukov, “Muscle hysteresis and movement control: a
theoretical study,” Neuroscience, 83, 303-320 (1998).
14. A. M. Hill, A. M. J. Bull, A. L. Wallace, et al., “Qualitative
and quantitative descriptions of glenohumeral motion,” Gait
Posture, 27, No. 2, 177-188 (2008).
15. C. D. Bryce and A. D. Armstrong, “Anatomy and biomechanics
of the elbow,” Orthop. Clin. North Am., 39, No. 2, 141-154
(2008).
16. B. M. Van Bolhuis, C. C. Gielen, and G. J. van Ingen Schenau,
“Activation patterns of mono- and bi-articular arm muscles
as a function of force and movement direction of the wrist in
humans,” J. Physiol., 508, 313-324 (1998).
17. A. V. Gorkovenko, S. Sawczyn, N. V. Bulgakova, et al.,
“Muscle agonist-antagonist interactions in an experimental
joint model,” Exp. Brain Res., 222, 399-414 (2012).
18. T. E. Milner and C. Cloutier, “Damping of the wrist joint
during voluntary movement,” Exp. Brain Res., 122, 309-317
(1998).
19. E. Burde, R. Osu, D. W. Franklin, et al., “The central nervous
system stabilizes unstable dynamics by learning optimal
impedance,” Nature, 414, 446-449 (2001).
20. P. L. Gribble and D. J. Ostry, “Independent coactivation of
shoulder and elbow muscles,” Exp. Brain Res., 123, 335-360
(1998).
|