How does the central nervous system address the kinetic redundancy in the lumbar spine?...
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Engineers, Part H: Journal of Engineering in Proceedings of the Institution of Mechanical
http://pih.sagepub.com/content/224/3/487The online version of this article can be found at:
DOI: 10.1243/09544119JEIM668
2010 224: 487Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in MedicineE Rashedi, K Khalaf, M Reza Nassajian, B Nasseroleslami and M Parnianpour
L5 level−−Three-dimensional isometric exertions with 18 Hill-model-based muscle fascicles at the L4How does the central nervous system address the kinetic redundancy in the lumbar spine?
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How does the central nervous system address the kineticredundancy in the lumbar spine? Three-dimensionalisometric exertions with 18 Hill-model-based musclefascicles at the L4–L5 levelE Rashedi1*, K Khalaf2, M Reza Nassajian1, B Nasseroleslami3, and M Parnianpour1,4
1School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran2Department of Mechanical Engineering, American University of Shadjeh, Sharjeh, United Arab Emirates3Bioengineering Unit, University of Strathclyde, Glasgow, UK4Department of Information and Industrial Engineering, Hanyang University, Ansan, Gyeonggi-do, Republic of Korea
The manuscript was received on 30 May 2009 and was accepted after revision for publication on 16 September 2009.
DOI: 10.1243/09544119JEIM668
Abstract: The human motor system is organized for execution of various motor tasks in adifferent and flexible manner. The kinetic redundancy in the human musculoskeletal system isa significant property by which the central nervous system achieves many complementarygoals. An equilibrium-based biomechanical model of isometric three-dimensional exertions oftrunk muscles has been developed. Following the definition and role of the uncontrolledmanifold, the kinetic redundancy concept is explored in mathematical terms. The null space ofthe kinetically redundant system when a certain joint moment and/or stiffness are needed isderived and discussed. The aforementioned concepts have been illustrated, using a three-dimensional three-degrees-of-freedom biomechanical model of the spine with 18 anatomicallyoriented Hill-type-model muscle fascicles. The considerations of stability and its consequenceon the internal loading of the spine and coactivation consequences are discussed in bothgeneral and specific cases. The results can shed light on the interaction mechanisms in muscleactivation patterns seen in various tasks and exertions and can provide a significantunderstanding for future research studies and clinical practices related to low-back disorders.Alteration of recruitment patterns in low-back-pain patients has been explained on the basis ofthis biomechanical analysis. The higher coactivation results in higher internal loading whileproviding higher joint stiffness that enhances spinal stability, which guards against spinaldeformation in the presence of any perturbations.
Keywords: central nervous system, kinetic redundancy, lumbar spine, biomechanical model,Hill-model-based muscle fascicles
1 INTRODUCTION
In order to evaluate the loads acting on interverteb-
ral joints in the spine to prevent injuries from
overloading, the muscle forces should be deter-
mined. While in-vivo muscle forces are not measur-
able non-invasively, using mathematical models to
estimate these forces is unavoidable. The challenge
of predicting the muscle recruitment patterns for
maintaining biomechanical balance in the trunk is
due to redundancy in the neuromusculoskeletal
system, and how the central nervous system (CNS)
manages this problem is ill understood. There are
three approaches to addressing this issue: using
optimization methods [1, 2], electromyography
(EMG)-driven models [3], and kinematic-based finite
element models [4]. In the study by Pomero et al. [5],
which utilized some similar formulae to the math-
ematical equations in this paper, a model is used
*Corresponding author: Biomechanics Laboratory, School of
Mechanical Engineering, Sharif University of Technology, Azadi
Avenue, Tehran, Iran.
email: [email protected]
487
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which avoided spinal joint overloading by maintain-
ing spine loads within proper bounds of its tolerance
limits, while considering a predetermined schema
for muscle coactivations.
Kinetic redundancy in the human musculoskeletal
system gives considerable advantages, as it provides
greater methods of achieving a requested task by the
CNS. From a computational standpoint this is
usually a mathematical challenge to describe how
these extra degrees of freedom are managed for a
single task (which usually means how the internal
muscle forces are specified to balance the external
moments while satisfying the equilibrium condi-
tions) [7]. For simple joints with a pair of agonist and
antagonist muscles, antagonistic activation can be
discussed qualitatively from experimental data, and
quantitative analysis requires biomechanical models
[2]. However, for complex joints with multiple
muscles, there is difficulty even in assigning qualita-
tively the role of synergistic and antagonistic action
to muscles [2, 8]. In practice, there are many other
objectives associated with each task besides main-
taining equilibrium, such as the required accuracy,
stiffness, and necessary adaptation to muscular
fatigue among others [1].
An integrated mathematical description for simul-
taneous force and stiffness control of a joint is
explained and relevant considerations about the
controlled and uncontrolled biomechanical para-
meters are discussed after the introductory explana-
tion of synergistic muscles actions, force generation,
and stiffness formulation of the trunk are derived.
The solution for a redundant system of equations is
obtained by optimization, while the redundancy
concept is explained by the null space of the system
of equations and the uncontrolled manifold. This
approach gives computational advantages compared
with the EMG-driven model, as it does not restrict
the domain of results to a few experimentally
observed patterns and allows classification of results
based on the theoretically defined concepts and
parameters. The possibility of comparing the results
of simulating various objective functions allows
further understanding of the normal and abnormal
patterns of trunk muscle recruitments (i.e. higher
coactivation by low-back patients). The solutions for
a simplified trunk model are used as illustrative
examples in three different tasks and specified
strategies. The generated moment, trunk stiffness,
and joint reaction forces are compared for these
three cases.
2 METHODS
2.1 Force equilibrium
A free-body diagram including all relevant muscles
could be analysed to give the mathematical relation-
ship of the equilibrium condition. All physiological
and geometrical properties such as the line of action
and cross-sectional area of the muscles will be
considered (Table 1 [5, 9]). The convention sug-
gested by the International Society of Biomechanics
is followed to describe the spine’s coordinate system
[10]. The y axis is the line passing through the
centres of the vertebra’s upper and lower end plates,
Table 1 The origin and insertion coordinates, the lengths, and the PCSAs of 18 muscle fascicles [5, 9]. The order inwhich muscles are arranged in the vector a is the same order in which they appear in the rows of this table
Musclenumber
Abbreviation of themuscle name*
Origin (m) Insertion (m)
Length (m) PCSA (m2)x y z x y z
1 RA-R 0.090 20.124 0.030 0.096 0.176 0.070 0.300 0.001 02 EO1-R 0.034 0.012 0.130 — 0.126 0.125 0.168 0.001 03 EO2-R 0.096 20.124 0 0.031 0.141 0.105 0.183 0.000 94 IO1-R 20.004 0.041 0.125 0.056 0.116 0.070 0.132 0.000 95 IO2-R 0.066 20.014 0.120 0.096 0.206 0 0.115 0.000 86 IL-R 20.080 20.008 0.068 — 0.216 0.084 0.181 0.001 17 LT-R 20.080 20.009 0.033 — 0.266 0.050 0.171 0.001 688 LD1-R 20.046 0.041 0.060 — 0.296 0.120 0.252 0.000 29 LD2-R 20.058 0.018 0.030 — 0.296 0.120 0.235 0.000 2
10 RA-L 0.090 20.124 20.030 0.096 0.176 — 0.300 0.001 011 EO1-L 0.034 0.012 20.130 — 0.126 — 0.168 0.001 012 EO2-L 0.096 20.124 0 0.031 0.141 — 0.183 0.000 913 IO1-L 20.004 0.041 20.125 0.056 0.116 — 0.132 0.000 914 IO2-L 0.066 20.014 20.120 0.096 0.206 0 0.115 0.000 815 IL-L 20.080 20.008 20.068 — 0.216 — 0.181 0.001 116 LT-L 20.080 20.009 20.033 — 0.266 — 0.171 0.001 6817 LD1-L 20.046 0.041 20.060 — 0.296 — 0.252 0.000 218 LD2-L 20.058 0.018 20.030 — 0.296 — 0.235 0.000 2
*RA, rectus abdominus; EO, external oblique; IO, internal oblique; LT, longissimus thoracis; IL, iliocostalis lumborum; LD, latissimus dorsi.
488 E Rashedi, K Khalaf, M Reza Nassajian, B Nasseroleslami, and M Parnianpour
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and pointing to the cephalad. The z axis is the line
parallel to a line joining similar landmarks on the
bases of the right and left pedicles, and pointing to
the right. Finally, the x axis is the line perpendicular
to the y and z axes, and pointing anteriorly. The
present model studies isometric exertions and/or
efforts in the upright standing position, while the
model structure is general and can address dynamic
three-dimensional (3D) exertions [2]. Figure 1 sche-
matically depicts the spinal features, muscle place-
ment, muscle forces and their directions, and the
axes used to define the force and torque directions.
The muscles are modelled as straight Hill-type
actuators [11] that generate force along their line of
action (origin to the insertion path). It is assumed
that the moment about the considered L4–L5 joint is
generated by the muscles only. Now, considering the
force–length and velocity–force relationships, and
also the passive force of muscles, the suggested
equations for the muscle force are [2, 3, 12, 13]
f
fmax~af lð Þf _ll
� �zfp lð Þ, fmax~smax PCSA ð1Þ
f lð Þ~5:1{29l
l0
� �z56
l
l0
� �2
{41l
l0
� �3
z10l
l0
� �4
ð2Þ
fp lð Þ~ exp {10:671z7:675l
l0
� �ð3Þ
In the above equations, f is the muscle force, fmax is
the maximum muscle force, fp is the muscle passive
force, l is the length of the muscle, l0 is the initial
length of the muscle, f(l) represents the force–length
relation, f(l̇) represents the force–velocity relation,
PCSA is the physiological cross-sectional area of the
muscle, a is the dimensionless muscle activation
level (bounded between 0 and 1), and smax is the
maximum muscle stress with acceptable values
reported to be between 350 kPa and 100 kPa in the
literature. smax5 550 kPa is applied in this work [2].
Using these data for the m muscles considered in
an equilibrium model, the system of equations is
derived as
A3|mam|1~b3|1 ð4Þ
In the above equation the ith column A matrix is
Ai~ fmaxf lð Þf _ll� �h i
ri|ti� �
, i~1, 2, :::,m ð5Þ
where
fmaxi~smax PCSAi
L5muscle length (m)
l̇5 change rate of muscle length (m/s)
PCSAi5physiological cross-sectional area of the
ith muscle (m2)
ri~ rix, riy, r
iz
� �5moment arms (m)
ti~F i�Fi�� ��~ tix, t
iy, t
iz
� �5unit muscle force com-
ponents of the ith muscle
ai5 activation of the ith muscle
and b is given by
b~bexternal{bpassive ð6Þ
bexternal is the external moment and bpassive is the
moment generated by passive (osteoligamentous
and disc) structures and muscle forces. (Without
loss of generality the passive resistance of muscles
was taken to be represented by bpassive~Pmi~1 fmax fp(l)(r
i|ti).)
For the 18 muscle fascicles included in this model,
equation (4) will result in the equilibrium equation
(m5 18)
A3|18a18|1~b18|1 ð7Þ
This model governs during any general 3D exertions,
but for simplicity the upright isometric exertions as
encountered during various experimental studies
[14–17] have been assumed. The moments are
Fig. 1 A schematic diagram of spinal features, muscleplacement, muscle forces and their directions,and the axes used to define the force and torquedirections
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obtained about the origin of the coordinate system
at the L4–L5 joint.
2.2 Optimization method
To find the muscle activations in equation (7) for a
desired moment, the optimization method is used
with minimization of the norm of the muscle
activations; the cost function is represented in
reference [2] as
f að Þ~minX18i~1
a2i
!ð8Þ
The cost (or objective) function minimizes the sum
of squared muscle activations subject to equality and
inequality constraints of the system. The former
represents the equations of motion or equilibrium
conditions, and the latter are physiological con-
straints to keep possible activations positive and
bounded according to
Aa~b, 0¡ai¡1, i~1, . . . , 18 ð9Þ
The activation levels are determined by the required
moment, and their values are bounded between 0
and 1 [18].
A3618 is the matrix which is computed on the basis
of the physiology and geometry of the muscles about
the L4–L5 joint (i.e. the muscle origin and insertion
points, the muscles’ line of action, the PCSA, the
maximum stress, and other features that are not
considered in this study such as tendon mechanics,
fascicle fibre type, and muscle fibre orientation). It
has three rows and 18 columns. Each row of the A
matrix is a representation of the maximum con-
tributions of muscles for generation of a specific
moment component. Equivalently, each column
represents the magnitude and direction of maximum
moments that the corresponding muscle can con-
tribute.
For this model with the data provided in Table 1,
the maximum moment generation vectors for each
muscle at upright posture in 3D moment space are
shown in Fig. 2. It is clear that almost all muscles
generate moments about all three axes.
2.3 Null space and physiological constraints
The above optimization procedure will give a unique
set of muscle activations for each cost function. It
may be the objective of the CNS to maintain the net
moment while altering the muscles’ activation
pattern to avoid muscle fatigue, although it may be
achieved at higher values of the cost function [19]
(equation (8)).
In other words, the following question can be
asked: what changes in muscle activations result in
no moment change? This issue can be addressed by
the concept of ‘null space’. The uncontrolled mani-
fold (a subspace of the muscles’ activation space) is
constructed by principal muscle activation vectors
that cause no variation in the output moment, while
allowing variation in the muscle activations within
Fig. 2 Moment generation capacity of each muscle in 3D space
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the subspace. Null space depends on just the
physical characteristics of the model, which are
represented in the matrix A. To span the uncon-
trolled manifold, it is necessary solve the set of
equations [20]
A3|18e18|1~0 ð10Þ
This set of homogeneous linear equations can be
considered as a matrix and vector equation; e can be
characterized as a right singular vector correspond-
ing to a zero singular value. A non-negative real
number c is a singular value for A if and only if there
exist unit-length vectors u and u such that
Av~cu ð11Þ
The vectors u and u are called left-singular and right-
singular vectors respectively for c.
Using the singular value decomposition method to
generate the null space, vectors A will be decom-
posed into three matrices according to
A3|18~U3|3S3|18VT18|18 ð12Þ
The diagonal entries of S are necessarily equal to the
singular values of A, with other elements of S equal
to zero. The columns of the matrices U and V are left
and right singular vectors respectively for the
corresponding singular values. The superscript T
indicates the matrix transpose.
There are two subspaces in the muscles’ activation
space which are represented by V: the null space and a
space perpendicular to this null space. Moving along
the unit vectors of the null space results in no change
in the producedmoment, while moving along the unit
vectors of the perpendicular subspace to the null
space results in changes in the produced moment.
The number of vectors in the null space is equal to
the degrees of redundancy in the system. It should
be noted that, as there are three equations with 18
unknowns, the degrees of redundancy will be 15, and
consequently there are 15 unit vectors in the null
space. Each of these vectors in the null space
represents a key direction of the aforementioned
redundancy in the space of the muscle activations.
Because of the physiological constraints (e.g. the
activation range and muscle maximum stress), not
all vectors in the null space may be permissible (i.e.
some directions or values yielding negative muscle
activation or supra-maximal activation). This may
reduce the real degree of redundancy.
To address this issue, it should be noted that using
the optimization method will give a point in the
space of the muscle activations that minimizes the
cost function and satisfies the equilibrium and
physiological constraints (9). This point will be in
an 18-dimensional unit hypercube which covers the
entire workspace here. There are 15 lines that go
through the mentioned point in the directions
provided by the unit vectors of the null space. Only
the sections of the lines inside the workspace
hypercube correspond to feasible results. Any com-
bination of permissible values of muscle activation
can be recruited without any concern about indu-
cing change in the desired moment. To apply this
concept, the required line equation is
x1{a1e1, j
~x2{a2e2, j
~ � � �~ xi{aiei, j
~tj
i~1, . . . ,18, j~1, . . . ,15 ð13Þ
In the above equation, a1, a2,…, a18 are the muscle
activations obtained by the above-mentioned opti-
mization procedure, e1, j, e2, j,…, e18, j are the Cartes-
ian components of the 15 unit vectors ej of the null
space, and x1, x2,…, x18 are the variables in the
muscle activation space. The range of tj determines
the permissible recruitment along the vectors of the
null space without changing the net moments. The
purpose is to find any possible range for the variable
tj which is obtained by considering the boundedness
of independent variables xi to the [0, 1] range. If
there is such a range for tj, any coefficient chosen
from this interval can be multiplied by ej and added
to the activation vector obtained by the optimization
method. Using the same procedure for all the vectors
in the null space will clarify the possibility of
changing the muscle activation in positive activation
space without changing the required moment.
To illustrate these concepts further, the uncon-
trolled manifold computed for the following ex-
ample is plotted in Fig. 3. Suppose that the model
has just the first three muscles to generate the
moment about the L4–L5 level, as mentioned in
Table 1. The CNS goal is to achieve the generation of
a flexion moment of 280Nm in the x direction. In
this case, there is just one equilibrium equation with
three unknown muscle activations, which is
A1|3a3|1j j~{80Nm ð14Þ
Figure 3 shows that various optimal solutions lie on
the uncontrolled manifold surface. It shows that the
choice of optimal solution a* in the construction of
the uncontrolled manifold is irrelevant. In this
example the three cost functions were the minimum
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sum of the muscle stresses, the minimum norm, and
the minimum sum of cubed activations. In short,
points on this surface are all the infinite solutions
that satisfy equilibrium conditions and keep the
muscle activation positive. Any points outside this
manifold violate the equilibrium conditions. The
characteristics of the uncontrolled manifold surface
(its normal vector direction in space) is just
dependent on A163. Different values of the moment
will shift this surface up or down along its normal
vector, but the normal vector of the uncontrolled
manifold surface will remain invariant.
2.4 Angular stiffness
The assumption that the muscle stiffness can be
effectively estimated as a function of muscle force
and length has been well established in spine
stability studies [21, 22]. Some studies have con-
cluded that the muscle stiffness is approximately
linearly proportional to the muscle force [23, 24].
One approach taken to reduce the redundancy is to
add new equations in terms of the same unknowns
(i.e. ai in our case) to the original system of
equations. This can be achieved by adding joint
stiffness equations as functions of the muscle
stiffness to the former model (9). To prepare these
equations, the desired angular joint stiffness is used
to provide three more equations, and the new
system of equations for a general case of m muscles
in the model will be
A3|m
B3|m
am|1~
b3|1
d3|1
ð15Þ
In equation (15) there are six equations and m
unknowns, and so the degree of redundancy for the
system is m2 6, and the null space has m2 6
dimensions. Similar formulae for simultaneous mo-
ment equilibrium and joint reaction force has been
used [25], but not for stiffness.
In the above equation, d361 is the desired triplet of
joint stiffness (Nm/rad). In general, the angular joint
stiffness can be described by
K~
LM1
Lh1LM1
Lh2LM1
Lh3LM2
Lh1LM2
Lh2LM2
Lh3LM3
Lh1LM3
Lh2LM3
Lh3
26666664
37777775
ð16Þ
K is the stiffness matrix of the three-degrees-of-
freedom musculoskeletal system, where, in
Kij5 LMi/Lhj, Mi and hj are the corresponding
moment and joint angle. Suppose that dm63 is the
moment arm matrix and can be described as
d~
Ll1Lh1
Ll1Lh2
Ll1Lh3
Ll2Lh1
Ll2Lh2
Ll2Lh3
Ll3Lh1
Ll3Lh2
Ll3Lh3
..
. ... ..
.
LlmLh1
LlmLh2
LlmLh3
26666666666666664
37777777777777775
ð17Þ
where, in dij5 Lli/Lhj, li and hj are the corresponding
muscle length and joint angle. According to the
Bergmark equation kmm5q(fm/lm) (where kmm is the
linear scalar stiffness of the mth muscle and is given
by kmm5 LFm/Llm)
k~
k11 0 0 � � � 0
0 k22 0 � � � 0
0 0 k33 0
..
. ... P
0 0 0 kmm
266666664
377777775
ð18Þ
The stiffness matrix K of the joint as a function of the
Fig. 3 The uncontrolled manifold and the optimiza-tion results for three cost functions: the mini-mum norm, the minimum sum of musclestresses, and the minimum sum of cubedactivations. In this example, the case of threemuscles acting on a single joint is modelled toallow visual inspection of the results. Hence thematrices involved have the following structure:A163 and a361
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muscle’s stiffness and geometrical configuration is
given by
Kij~LMi
Lhj~
LP
m Fmdmi
� �Lhj
ð19Þ
where i and j are the indices to describe the stiffnessin the body coordinates system and m is the indexfor the mth muscle (F is the muscle force). Thus
Kij~Pm
LFm
LhjdmizFm
Ldmi
Lhj
� �ð20Þ
Assuming that the second-order term Ldmi
�Lhj5
L2lm�(Lhi Lhj) can be ignored, as the second-order
terms are infinitesimal compared with the first-order
terms, especially in the static case
Kij~Pm
LFm
Lhjdmi~
Pm
kmmdmjdmi ð21Þ
Using the above matrix, K can be defined by
K~dTkd ð22Þ
As a simplified approach, the diagonal elements of
the joint stiffness matrix K are assumed to be the
controlled stiffness elements, denoted by d in
equation (15). These are the dominant components
according to the muscular configuration of the sys-
tem. d can be written as
di~Xmj~1
d2jikjj ð23Þ
which gives the relationship for B in equation (15)
when combined with equations (1) to (4) and the
Bergmark relationship for muscle stiffness. For the
purpose of simulations, the constant q of proportion-
ality in the Bergmark formula is assumed to be q58.
3 RESULTS
3.1 Muscle strength and redundancy
The trunk maximum voluntary capacity or strength
is characterized by a surface in the 3D space of
moments (Fig. 4). Any point outside this strength
surface is infeasible as it would require muscle
activation with a value above 1. Points on the
strength surface mean that at least one of the muscle
activations has reached its maximum value of 1. All
points interior to the surface are feasible and many
recruitments can achieve the required moments.
For example, suppose an external load of 50Nm
pure extension at L4–L5 level is being balanced (i.e.
b~ 0 0 50½ �T as a desired triplet moment). This
triplet moment is in the feasible space of the
muscles, as shown in Fig. 4. It is noteworthy that
Fig. 4 Feasible moment space for the muscles in the model known as the strength limits of themodel in clinical settings
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countless different types of muscle activation can
provide this moment. As a solution, the CNS can use
only RA-R and RA-L to produce the desired moment
a~ 0:33 0 0 0 0 0 0 0 0 0:33 0 0½0 0 0 0 0 0 �T
Alternatively, the CNS may use only EO2-R and EO2-L to do the same task (Fig. 5(a)). Using this strategy,the activation vector is
a~ 0 0 0:57 0 0 0 0 0 0 0 0 0:57½0 0 0 0 0 0 �T
The other alternative is to consider simultaneousactivation of all four muscles for the desired resistive
moment (Fig. 5(b)). In this case, one of the manyresultant muscle activations may be
a~ 0:18 0 0:26 0 0 0 0 0 0 0:18 0½0:26 0 0 0 0 0 0 �T
Employing the optimization method in equation (9)for various pure and combined resistive momentsusing all 18 muscles the optimal solution has beenobtained (Table 2). The minimum norm solution forbalancing the 50Nm extension moment using allmuscles will be
a~½ 0:20 0:08 0:12 0 0:10 0 0 0 0
0:20 0:08 0:12 0 0:10 0 0 0 0 �T
Fig. 5 Alternative solutions for resisting an extension of 50Nm of extension by either the rectusabdominis (RA) or external oblique (EO) muscles: (a) two different solutions forgenerating the desired moment; (b) an alternative solution for generating the desiredmoment with four muscles
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Determination of the agonist, synergist, and antago-
nist actions of muscles during complex exertions
requires detailed analytical reasoning. This issue will
be dealt with elsewhere, although preliminary dis-
cussions have been provided by Potvin and Brown
[25], which will be illustrated by an example.
It is emphasized that many muscles are involved
during a task requiring pure axial rotation or during
complex exertions, even though the minimum norm
solution by definition keeps the antagonist activity to
a minimum (Table 2). Any solution with higher
activation levels for the same b must have used
some combination of activations from the null space
which keeps moments invariant but affects the joint
reaction forces and its stiffness.
The joint stiffness could be added to the above
system of equations. In the model considered here,
the maximum stiffness can be approximated by
multiplying B by the maximum muscle activations,
resulting in
dmax~ 2600 566 3828½ �T Nm=rad
An arbitrary triplet joint stiffness can be chosen as
the submaximal desired angular stiffness according
to
d~ 1000 100 1000½ �T Nm=rad
Balancing the 50Nm external extension moment as
in the previous example and maintaining the desired
specified d, the new matrix for muscle activations by
the minimum norm optimization method will be
a~ 0:42 0 0:12 0:05 0:57 0:14 0:37 0:01½0:03 0:66 0:48 0:60 0 0:28 0:21 0:37
0 0 �T
Comparing the following activation with those of the
first row in Table 2, the additional activity levels can
now be observed as coactivation or antagonist acti-
vity that has provided the desired joint stiffness d.
Obtaining the null space of equation (15) as
explained before and using the same procedure to
Table 2 Muscle activations obtained by the minimum norm optimization method for various pure and combinedexertions
Moment type*Mx
{
(Nm)My
{
(Nm)Mz
{
(Nm)
Activation of the following trunk muscles{
RA-R EO1-R EO2-R IO1-R IO2-R IL-R LT-R LD1-R LD2-R
PureE 0 0 50 0.20 0.08 0.12 0 0.10 0 0 0 0F 0 0 250 0 0 0 0.03 0 0.10 0.15 0.01 0.01LB 50 0 0 0 0 0 0 0 0 0 0 0AR 0 50 0 0 0 0 0.35 0.18 0.06 0.11 0.03 0.03
CombinedE-LB 50 0 50 0.14 0 0.08 0 0 0 0 0 0F-LB 50 0 250 0 0 0 0 0 0 0.09 0 0.01E-AR 0 50 50 0.13 0 0 0.31 0.31 0 0 0.01 0.01F-AR 0 50 250 0 0 0 0.39 0.14 0.15 0.24 0.04 0.05LB-AR 50 50 0 0 0 0 0.24 0.08 0 0.06 0.02 0.03E-LB-AR 50 50 50 0.01 0 0 0.14 0.15 0 0 0 0F-LB-AR 50 50 250 0 0 0 0.28 0.04 0.06 0.19 0.03 0.04
Moment type*Mx
{
(Nm)My
{
(Nm)Mz
{
(Nm)
Activation of the following trunk muscles{
RA-L EO1-L EO2-L IO1-L IO2-L IL-L LT-L LD1-L LD2-L
PureE 0 0 50 0.20 0.08 0.12 0 0.10 0 0 0 0F 0 0 250 0 0 0 0.03 0 0.10 0.15 0.01 0.01LB 50 0 0 0.09 0.18 0.07 0.14 0.13 0.10 0.07 0.01 0.05AR 0 50 0 0.04 0.42 0.18 0 0 0.02 0.02 0 0
CombinedE-LB 50 0 50 0.23 0.20 0.15 0.13 0.24 0 0 0 0F-LB 50 0 250 0 0.18 0.02 0.13 0.04 0.22 0.25 0.03 0.02E-AR 0 50 50 0.21 0.36 0.24 0 0 0 0 0 0F-AR 0 50 250 0 0.43 0.14 0 0 0.12 0.17 0 0LB-AR 50 50 0 0.09 0.58 0.23 0 0 0.13 0.12 0 0E-LB-AR 50 50 50 0.26 0.56 0.30 0 0 0 0 0 0F-LB-AR 50 50 250 0.01 0.60 0.20 0 0 0.23 0.26 0.01 0
*F, flexion; E, extension; LB, right lateral bending; AR, left axial rotation.{Mx, lateral bending moment; My, axial rotation moment; Mz, flexion–extension moment.{RA, rectus abdominus; EO, external oblique; IO, internal oblique; LT, longissimus thoracis; IL, iliocostalis lumborum; LD, latissimus dorsi.
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find any common interval for tj from equation (13),
valid intervals for tj, represented by the lower
bounds tl and upper bounds tu, are computed as
t l~ 0 0 0 0 0 0 0 {0:20 {0:20½0 0 0 �
tu~ 0 0 0 0 0 0:50 0 0 0½0 0 0 �
This is a reasonable result, as the angular joint
stiffness provides a situation in which the sixth,
eighth, and ninth vectors of the null space can
change the muscle activations without any effect on
the desired moment while the positive activations
are guaranteed. This concept can be seen by the
following illustrative examples.
The effect of varying the required or demanded
stiffness while balancing an invariant 50Nm exter-
nal flexion moment in the upright position is shown
in Fig. 6. Higher stiffness is provided by higher
muscle activation as expected with all the negative
energy and loading consequences for the joints.
3.2 Simulation of three cases
The concept is further illustrated by a series of
simulations, assuming 18 muscles in the model
identified in Table 1. Three different tasks or
strategies used by the CNS can be assumed:
Case I: The CNS resists a flexion moment of only
50Nm (b~ 0 0 {50½ �T) and is not affected by
the angular stiffness of the joint.
Case II: The CNS goal is to set the angular joint
stiffness of d~ 1000 100 1000½ �T Nm=rad and
does not specify the moments.
Case III: The CNS is required to set both the
resisting moment and the angular stiffness of the
trunk simultaneously with the values specified in
cases I and II.
The null space basis vectors tl(i) and tu(i) are
computed accordingly and are presented in Table 3.
In case I, it is expected that different angular
stiffnesses and joint reaction forces will be experi-
enced, although the external moment is not altered,
as the CNS goal is to maintain this desired value
(Fig. 7). It should be noted that, since the muscle
activities dominate the joint reaction forces, the
constant value of body weight has been excluded in
joint reaction forces throughout this paper in all
results. There is one set for t in case I and,
consequently, just the correspondence unit vector
for null space is used in Fig. 7. In case II, all the null
space unit vectors take part in altering the muscle
activations, as can be seen in Table 3. In case III,
Fig. 6 Activations of different muscles for different levels of stiffness demands during thebalancing of a flexion moment of 50Nm (dmax~ 2600 566 3828½ �T Nm=rad)
496 E Rashedi, K Khalaf, M Reza Nassajian, B Nasseroleslami, and M Parnianpour
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only one of the basis vectors of the null space is
available. In case I, the CNS goal is to generate a
desired value for torque. The results of case II are
interesting since varying t yields large changes in
muscle activations, which affected the generated
torque significantly (Fig. 8(a)). The effects on joint
reaction force were more pronounced when varying
t in this case for some unit vectors of the null space,
i.e. the ninth unit vector (Fig. 8(b)). The last simula-
tion, case III, specified both the joint stiffness and
Table 3 Upper and lower limits of t(i) for the three simulated cases
CNScontrolobjective
Upper and lower limits of the null space basis vector
i5 1 i5 2 i5 3 i5 4 i5 5 i5 6 i5 7 i5 8 i5 9 i5 10 i5 11 i5 12 i5 13 i5 14 i5 15
Case I tl 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0tu 0 0 0 0 0 0 0 0 0 0 0 0 0 0.98 0
Case II tl 20.10 20.40 20.39 20.64 20.02 20.03 20.24 20.40 20.19 20.10 20.40 20.39 20.64 20.02 20.03tu 0.11 0.39 0.70 0.45 0.98 0.98 0.80 0.30 0.83 0.11 0.39 0.70 0.45 0.98 0.98
Case III tl 0 0 0 0 0 0 0 0 0 0 0 0 — — —tu 0 0 0 0 0.34 0 0 0 0 0 0 0 — — —
Fig. 7 Case I: maintaining b~ 0 0 {50½ �T Nm. Variations in (a) the joint stiffness and (b) thejoint reaction forces, as representative parameters of changes in the muscles’ activation(t14) in the uncontrolled manifold
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the required torque and hence limited the range of
predicted muscle activation and joint reaction forces
(Fig. 9).
4 DISCUSSION
While there is a similarity between the present
approach and the formulae employed by Pomero et
al. [5], here fundamentally significant questions are
being raised and different results presented. The
CNS system can map muscle activation to the joint
moment and stiffness. In the past, a pseudo-inverse
method has been used to map the joint moment to
the muscle force in redundant neuromuscular
systems [7, 26, 27]. The shortcoming of that
technique was lack of consideration of the signifi-
cant physiological constraint on the sign of the
muscle forces. Muscles, in a similar way to ropes,
can only pull and cannot push on their environ-
ments. Hence, the following approach has resolved a
significant limitation on previous approaches using
the pseudo-inverse method. The augmentation of
stiffness to desired moments on the right-hand parts
of the system of equations (15) is based on the
concept that each muscle can act as an independent
agent. However, experimental results indicate that
muscles are recruited in synergistic groups whose
membership and their characteristics are entailed in
complex networks known as sensory-motor maps
(i.e. coordination) [28–30]. The existing limitation to
the present approach is that the functional anatomy
of the musculoskeletal system has been modelled
while the constraints of the neural system have been
Fig. 8 Case II: maintaining d~ 1000 100 1000½ �T Nm=rad. Variations in (a) the resistingmoment and (b) the joint reaction forces, as representative parameters of changes in themuscles’ activation (t1 to t15) in the uncontrolled manifold
498 E Rashedi, K Khalaf, M Reza Nassajian, B Nasseroleslami, and M Parnianpour
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ignored. In other words, each muscle is considered
as an independent agent, ignoring task-dependent
influences by higher cortical and spinal level
commands as well as by environmental factors
conveyed by the sensory system. An excellent paper
reviewing the evidence on these phenomena in
understanding the hand function has been pre-
sented elsewhere [31]. The desired stiffness based on
previous experience, environmental conditions (e.g.
vibrations), and fear of injury or perceived pain
would alter muscle activation to a given external
load. There are a number of theories and hypotheses
predicting the emergence of coactivation strategy
and altered recruitment patterns in low-back-pain
patients [32]. Whether these constraints reduce
intersegmental mobility to protect the injured
tissues, or the alteration in muscle recruitments are
a maladaptation to pain experiences, remains to be
proven [16, 17].
One potential direct use of the results of this study
is to understand the source of the recruitment
patterns of normal and abnormal muscles. Optim-
ization method results or EMG-driven methods are
the common approaches to study and to understand
the neuromechanics of the spine [33]. However,
when a deviation from a commonly observed or
predicted muscle recruitment pattern is observed,
most models fall short of assigning a specific
secondary objective or constraint as the main cause
of the altered pattern. The approach in this paper
makes it possible to attribute an altered pattern to
potential (secondary) task objectives such as an
increased stiffness demand in a specific direction or
a reduction in a specific joint reaction force. This can
be of special interest when abnormal patterns in
patients are observed and for diagnosing the
potential cause of the abnormal patterns.
There are a number of limitations that must be
addressed in future work. It is noteworthy that, for a
comprehensive biomechanical analysis, other bio-
mechanical modelling considerations should be
taken into account, e.g. intra-abdominal pressure
modelling, multi-segment modelling of the spine,
inclusion of passive tissues, inclusion of muscle via
points, more detailed musculotendon modelling,
and extension to dynamic tasks and fatigue.
The way that redundancy is resolved and why
invariant patterns are observed are still unresolved
problems in motor control [34]. What has been
learned may also lead to a gradual reduction in the
tight control on muscle activation so that muscles
can become less determined in terms of parameters
being assigned (i.e. joint stiffness), and so that the
system remains more flexible and capable of using
its null space as a result of the fatigue reduction
strategy and responding to altered environmental
conditions. The presented framework allows future
geometric and analytical methods to address bio-
mechanical concepts such as controllability, varia-
bility, complexity, and determinism, to name just a
few interesting concepts from the non-linear dy-
namics theory of motor control [35].
Fig. 9 Case III: maintaining b~ 0 0 {50½ �T Nm and d~ 1000 100 1000½ �T Nm=rad. Varia-tions in the joint reaction forces, as representative parameters of changes in the muscles’activation (t5) in the uncontrolled manifold
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5 CONCLUSIONS
In this study, different ways that the CNS may use
the redundancy in the musculoskeletal system are
discussed and the kinetic redundancy concept has
been explored in a model using 18 muscles at the
L4–L5 level. The null space of the kinetically redun-
dant system when a certain joint moment and/or
stiffness is required is derived and discussed.
The results show that the proposed formulae for
maintaining a constant moment or achieving a
desired trunk stiffness is functional, and the recruit-
ment from the null space of the system affects only
the uncontrolled parameters. This holds for cases
where either only one parameter or both the
moment and the stiffness are controlled. The
changes in uncontrolled parameters (e.g. joint
reaction forces) can range from negligible to con-
siderable, implying the importance and significance
of the redundant degrees of freedom in activation
space (null space). The results emphasize the role of
added controlled parameters for a redundant system
and establish a framework for further analysis of
complex multi-muscle models, together with rel-
evant theoretical and clinical considerations.
F Authors 2010
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