Abstract This paper presents a new method to estimate both musculo-tendon forces anddetailed joint reactions during gait, using an original 3D lower limb musculo-skeletal modelwith 5 degrees of freedom: spherical joint at the hip and parallel mechanisms at both kneeand ankle. This can be realized by employing a typical set of natural coordinates into athree-steps process. First, the kinematic constraints associated with the kinematic modelsare applied through a global optimization method on the marker-based kinematics. Consis-tent time derivatives of the positions are computed by projecting the velocities and accel-erations in the null space of the Jacobian matrix. Then, a Lagrangian formulation of theequations of motion is proposed, introducing Lagrange multipliers and allowing a straightaccess to the musculo-tendon forces. Thanks to a parameter reduction procedure, the La-grange multipliers are cancelled and the musculo-tendon forces can be computed directly,using a static optimization algorithm with a typical cost function. Finally, the equationsof motion are rewritten with the Lagrange multipliers to compute detailed joint reactions(since they represent directly joint contact and ligament forces). Results show that the esti-mated musculo-tendon forces are consistent with measured EMG signals. Moreover, the useof “anatomically” consistent kinematic models allows computing total joint reaction at hipjoint and detailed joint reactions at both knee and ankle joints that are temporally consis-tent with the forces measured on the subject (i.e., knee joint contact forces) and the forcespublished in the literature (i.e., hip joint contact forces). Next step will be to optimize si-multaneously musculo-tendon forces and joint reactions to investigate and understand theinteractions acting into the musculo-skeletal system during gait.
F. Moissenet (�) · L. Chèze · R. DumasLaboratoire de Biomécanique et Mécanique des Chocs, Université de Lyon, UMR T9406, UniversitéLyon 1, IFSTTAR, 69622 Lyon, Francee-mail: [email protected]
Bring musculo-skeletal simulations to clinical applications is a real challenge that can befilled by introducing more detailed models and new hypotheses during the computation.A first step could be to include “anatomically” consistent kinematic models with jointcontacts and ligaments in order to allow a deeper understanding of the forces distributionthrough the different active and passive structures and a better view of the interactions actinginto the musculo-skeletal system during gait [8, 10, 33].
An application of musculo-skeletal simulations is the study of musculo-tendon forcesand, most of the time, this can be done using a generic musculo-skeletal model [14] and astatic optimization process [22] defining criteria and constraints to solve the muscular redun-dancy problem. Many criteria have been proposed [22] to estimate these forces, but in thecase of human gait, it is common to minimize the square of musculo-tendon forces, musculo-tendon stresses or muscular activations. Anyway, the main problem remains to validate theresults of these simulations. Even if electromyographic (EMG) signals can give a qualitativeidea of the muscular activity, they do not allow validating the estimated musculo-tendonforces and consequently the simulation. However, recent in vivo studies allow measuringjoint contact forces during a movement such as gait using instrumented implants [5, 16,26, 29, 38, 41, 44]. Based on these data, it is now possible to validate the estimated jointcontact forces and so to estimate the quality of the simulation. Unfortunately, the classicalapproaches only compute a whole joint reaction at the joint centre [14, 29, 37, 41] since thisjoint reaction, in the typical joint models (i.e., spherical, hinge, universal), represents boththe joint contact and the ligament forces. Consequently, a new approach, based on moredetailed kinematic models (e.g., including detailed contacts and ligaments), is necessary toaccess detailed joint reactions.
This study aims to fill three objectives. The first one is the development of an origi-nal 3D lower limb musculo-skeletal model consisting of extended kinematic models (i.e.,parallel mechanisms) [15, 23] and a widely used musculo-skeletal geometric model [14].In particular, these parallel mechanisms correspond to more “anatomical” constraints (e.g.,sphere-on-plane contacts, isometric ligaments) than the typical joint models (e.g., spherical,hinge or universal joints). They have been proposed for the ankle joint [15] and the kneejoint [23] and can provide a more “physiological” kinematics during gait when employ-ing a global optimization method [20]. The second objective consists of the introduction ofan original methodology for both musculo-tendon forces and joint reactions computation.By using a Lagrangian approach based on generalized coordinates [17] and a parameterreduction [25], the musculo-tendon forces are computed in a first phase by using a staticoptimization algorithm with a typical cost function [11]. In a second phase, the Lagrangemultipliers corresponding to the “anatomical” constraints (e.g., sphere-on-plane contacts,isometric ligaments) are derived. We show that they directly represent the joint contact andligament forces. The third objective is to use the developed framework to explore the impactof the use of these “anatomical” constraints on the musculo-tendon forces and joint reactionscomputation during gait.
In the next sections, the 3D lower limb musculo-skeletal model and the computation pro-cess will be presented. Then, based on the gait data of one subject, the estimated musculo-tendon forces will be compared to measured EMG signals and the estimated joint reactionscomponents (i.e., joint contact and ligament forces) will be detailed and compared to themeasured data (i.e., knee joint contact forces) and the literature data (i.e., hip joint contactforces).
wi ]T , allvectors expressed in the inertial coordinate system, ICS), joint kinematic constraints [20] (i.e., spherical jointat the hip joint and parallel mechanisms at both knee and ankle joints) and musculo-tendon geometry [14]
2 Material and methods
2.1 Kinematic and musculo-skeletal models
An original 3D lower limb musculo-skeletal model, consisting of pelvis, thigh, shank andfoot segments, was developed to perform this study (Fig. 1). This model was adapted fromgeneric models [14, 15, 23] scaled to our problem to describe the joint kinematic constraintsand the musculo-skeletal geometry.
Hip was modelled using a spherical joint [2, 7, 31, 36]. Knee and ankle were modelledusing parallel mechanisms made, respectively, of 2 sphere-on-plane contacts (medial andlateral contacts) with 3 isometric ligaments (anterior cruciate ligament (ACL), posteriorcruciate ligament (PCL) and medial collateral ligament (MCL)) [23] and a spherical jointwith 2 isometric ligaments (medial (TiCaL) and lateral (CaFiL) collateral ligaments) [15](Fig. 1). The basic idea of these extended kinematic models is that joint contacts and liga-ments guide passive joint flexion. Under assumptions based on evidence and experimentalresults, a first human knee joint was modelled as a one degree of freedom parallel mech-anism by Wilson and O’Connor [43]. Indeed, it has been observed that fibres within theanterior cruciate ligament, the posterior cruciate ligament and the medial collateral ligamentcan be considered as isometric during the flexion of the unloaded knee. Moreover, the femurand tibia condyles can be approximated in a wide region by spherical and planar surfaces,respectively, which are in a single contact point in the medial and lateral compartments dur-
128 F. Moissenet et al.
ing motion [15]. This first knee model has been extended [23] and adapted to the anklejoint [15].
Then, in order to introduce musculo-tendon forces in the computation, a widely usedgeneric musculo-skeletal model [14] was adapted to our model (Fig. 1). The resulting model,consisting of 4 segments (pelvis, thigh, shank and foot) and 43 muscles, allows us to in-clude the muscular geometry and so the possibility to compute the muscle moment arms.Each segment of the model was scaled to the subject anthropometry using a simple homo-thetia.
2.2 Generalized coordinates
To compute kinematics and kinetics of the model, our approach is based on generalizedcoordinates [17] that consist, for each segment i, in two position vectors (the proximal Pi
and distal Di endpoints) and two unitary direction vectors (ui and wi ) (Fig. 1):
Qi = [ui rPirDi
wi
]T(1)
These parameters correspond to a typical set of natural coordinates [25]. All vectorsare expressed in the inertial coordinate system and each joint centre can be computed byregression methods [18, 35] or functional methods [21] from cutaneous marker trajectories.
For the pelvis, P4 is the lumbosacral joint centre, D4 is the midpoint between the hipjoint centres, u4 is the unit vector defined from middle of the posterior iliac spines to themiddle of the anterior iliac spines and w4 is the unit vector normal to the sagittal plane ofthe pelvis. For the thigh, P3 is the hip joint centre, D3 is the knee joint centre defined asthe middle of medial and lateral femoral epicondyles, u3 is the unit vector normal to theplane defined by the hip joint centre and the medial and lateral femoral epicondyles and w3
is the unit vector defined from the medial to lateral femoral epicondyles. For the shank, P2
is the knee joint centre, D2 is the ankle joint centre defined as the middle of medial andlateral maleoli, u2 is the unit vector normal to the plane defined by the fibula head and themedial and lateral maleoli and w2 is the unit vector from medial to lateral maleoli. For thefoot, P1 is the ankle joint centre, D1 is the middle of first and fifth metatarsal, which canbe described as the metatarso-phalangian joint centre, u1 is the unit vector defined fromcalcaneous to the middle of first and fifth metatarsal and w1 is the unit vector defined fromfirst to fifth metatarsal. These axes represent a non-orthonormal coordinate system for thesegments [17].
We defined then the joint kinematic constraints �k and the associated Jacobian matrixKk [20]. Finally, as 12 parameters represent the 6 degrees-of-freedom of each segment,rigid body constraints �r (i.e., constant distance between the points Pi and Di , unit lengthcondition of the vectors ui and wi and constant angles into the non-orthonormal coordinatesystem defined previously [25]) have to be considered in addition to the joint kinematicconstraints with the associated Jacobian matrix Kr (details can be found in [20]).
2.3 Equations of motion
It is well known that skin motion artifacts are a significant source of error in determiningjoint kinematics using marker-based motion capture systems [28, 31, 40]. In order to reducethese errors, a constrained global optimization [4, 20, 27, 31] was performed in order toobtain consistent segment positions Q. This method consists in searching the optimal poseof the overall model, by minimizing in a least squares sense the difference between the
Anatomical kinematic constraints 129
experimental markers and the virtual markers attached to the segments in the model, and byrespecting the kinematic constraints defined in the model.
Moreover, segment velocities and accelerations are usually obtained by using timederivatives of the positions Q. However, the kinematic constraints are not employed and ve-locities and accelerations are consequently non-consistent with the kinematic model [1, 40].In other words, the constraints � = [�k �r ]T are respected, but not their derivatives. Inorder to make these derivatives consistent, the following system was defined:
⎧⎪⎪⎨
⎪⎪⎩
� = ∂�
∂t= ∂�
∂QdQdt
= KQ = 0
� = ∂2�
∂t2= ∂
∂t
(∂�
∂QdQdt
)= KQ + KQ = 0
leading to two zero equality problems:⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
KQ = 0[
K 0
K K
]
︸ ︷︷ ︸AK
[Q
Q
]
︸ ︷︷ ︸xQ
=[
00
]
(2)
with K = [Kk Kr ]T . To ensure the equality KQ = 0, Q must be a vector of the null spaceof K. A trivial solution could be Q = 0, but it is also possible to use the projection of Q on
the null space of K. Consequently, the new vector ˜Q is built:
˜Q =∑
i
(Q ·ZKi
)ZKi (3)
where ZK is the orthogonal basis of the null space of K consisting of the i vectors ZKiand
obtained from a singular value decomposition. In this equation, the scalars obtained by theproduct (Q ·ZKi
) are the coordinates of Q about the basis vectors ZK.
Then ˜Q is used to compute K and the procedure is repeated for the null space of AK:
xQ =∑
i
((xQ ·ZAKi
)ZAKi
)(4)
where ZAK is the orthogonal basis of the null space of AK.
Thereby, we are sure that the new vector xQ = [ ˜Q ˜Q]T is contained in the null spaceof AK and therefore that time derivatives of the positions are consistent with the kinematicmodel.
Then, the equations of motion can be defined. In contrast with the classical approach[6, 22] determining net joint moments (i.e., active and passive joint moments) beforemusculo-tendon forces, the following formulation allows to have a direct access to themusculo-tendon forces f by introducing the Jacobian matrix of both joint kinematic andrigid body constraints K = [Kk Kr ] and the Lagrange multipliers λ in the equations of mo-tion. The model is made of 4 rigid segments (Fig. 1) but the equations of motion are onlywritten for the right limb (i.e., right foot, shank and thigh) by applying the principle ofvirtual power with natural coordinates [25]:
130 F. Moissenet et al.
⎡
⎣G1 012×12 012×12
012×12 G2 012×12
012×12 012×12 G3
⎤
⎦
︸ ︷︷ ︸G
⎡
⎢⎣
˜Q1˜Q2˜Q3
⎤
⎥⎦
︸ ︷︷ ︸Q
+⎡
⎢⎣ [Kk
A]T [KkK ]T [Kk
H ]T[Kr
1]T 012×6 012×6
012×6 [Kr2]T 012×6
012×6 012×6 [Kr3]T
⎤
⎥⎦
︸ ︷︷ ︸[K]T
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
λkA
λkK
λkH
λr1
λr2
λr3
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
︸ ︷︷ ︸λ
=⎡
⎢⎣
[NC11 ]T m1g
[NC22 ]T m2g
[NC33 ]T m3g
⎤
⎥⎦+
⎡
⎣[NP1
1 ]T (−F0) + [N∗1]T (−M0 + (rP0 − rP1) × (−F0))
012×1
012×1
⎤
⎦
︸ ︷︷ ︸E
+ [L1 . . . Lj . . . L43 ]︸ ︷︷ ︸
L
⎡
⎢⎢⎢⎢⎣
f 1
. . .
f j
. . .
f 43
⎤
⎥⎥⎥⎥⎦
︸ ︷︷ ︸f
(5)
where Gi is the generalized mass matrix of the segment i,KkA,Kk
K and KkH the Jacobian
matrices of the joint kinematic constraints of, respectively, the ankle, the knee and the hip,Kr
i the rigid body constraints of the segment i, λk and λr the Lagrange multipliers associatedwith the joint kinematic and rigid body constraints, NCi
i the interpolation matrix of the centreof mass of the segment i,mi the mass of the segment i,g the gravity acceleration vector, NP1
1the interpolation matrix of the point P1 described previously, F0 and M0 the ground-reactionforces and moments, P0 the centre of pressure of the foot, N∗
1 a pseudo-interpolation matrix[17] associated with the segment 1, and Lj and f j the generalized moment arm and forceof the muscle j .
2.4 Musculo-tendon forces optimization using a parameter reduction
Equation (5) must be rewritten to identify the unknowns. By reorganizing this equation, thefollowing linear system is defined:
[L −KT
][ fλ
]= G ˜Q − E (6)
where the unknowns of the system combine the musculo-tendon forces and the Lagrangemultipliers. In order to keep musculo-tendon forces as only unknowns, Lagrange multipliersmust be cancelled. This can be done by using a parameter reduction [25] or, in other words,by using ZK, the orthogonal basis of the null space of K. Indeed, an interesting property ofthis basis is that KZK = 0 ⇔ ZT
KKT = 0.
Anatomical kinematic constraints 131
Thereby, the system (6) can be modified as follow:
ZTKG ˜Q + ZT
KKT λ = ZTK(E + Lf)
⇔ ZTKG ˜Q = ZT
K(E + Lf)
⇔ Aeqf = beq
with Aeq = ZTKL and beq = ZT
K
(G ˜Q − E
)
(7)
Using this method, the contribution of the multipliers λ is cancelled since the system isprojected in the subspace of possible motions.
Finally, an optimization problem is defined to minimize musculo-tendon forces in (7)and constrain them to respect the linear system (6) and to be positive:
minf
J = 1
2fT Wf
subject to
{Aeqf = beq
f ≥ 0
(8)
with J the objective function to minimize and W a diagonal weight matrix based on physi-ological cross-sectional areas (PCSA) [11] where Wjj = (1/PCSAj )
2.A pseudo-inverse method is used here [42] to solve this problem. It has been shown
[19, 32] that this method, when compared with traditional ones (e.g., Lagrange multipliersand penalty methods), is numerically simpler, presents a better suited cost function and socould be better adapted to the muscular redundancy problem. This method is simply basedon the change of variable f = A†
eqbeq + (1−A†eqAeq)f, where f are the musculo-tendon forces
to optimized and A†eq the pseudo-inverse of Aeq:
minf
J = 1
2
[A†
eqbeq + (1 − A†eqAeq
)f]T
W[A†
eqbeq + (1 − A†eqAeq
)f]
constraint to(1 − A†
eqAeq
)f − A†
eqbeq ≥ 0(9)
2.5 Joint reactions computation
Once the musculo-tendon forces have been computed by optimization, it is possible to de-termine the value of each Lagrange multiplier by rewriting (6) as follow:
λ = [KT]†(−GQ + E + Lf
)(10)
with λ = [λkA λk
K λkH λr
1 λr2 λr
3]T the Lagrangian multipliers associated with both joint kine-matic and rigid body constraints. Since KT is a nonsquare matrix, the use of the pseudo-inverse [KT ]† is needed in this case.
It can therefore be interesting to identify the Lagrange multipliers since they can havea mechanical meaning. This can be done by computing the external forces power on theconsidered segment. We demonstrate here that the sphere-on-plane joint contact force canbe obtained directly.
As described in [20], contact components K in the KkK Jacobian matrix associated with
the sphere-on-plane contact constraints, can be written as follow:
KT λ =[ [Nni
i ]T (NVi+1i+1 Qi+1 − NVi
i Qi ) − [NVi
i ]T Nni
i Qi
[NVi+1i+1 ]T Nni
i Qi
]
λ (11)
132 F. Moissenet et al.
Fig. 2 Identification of the(a) joint contact (e.g., lateral kneejoint contact) and (b) ligament(e.g., medial collateral ligamentof the knee) forces
with λ the Lagrange multiplier associated with the considered contact, Nni
i the interpolation
matrix of the normal vector ni of the segment i,NVi+1i+1 the interpolation matrix of the virtual
marker Vi+1 corresponding here to the centre of the contact sphere of the segment i + 1 andNVi
i the interpolation matrix of the virtual marker Vi corresponding here to a point on theplane of the segment i. Interpolation matrices allow getting the coordinates r of any item inthe inertial coordinates system (ICS) using the relation r = NQ.
We can then consider this contact by studying external forces acting on each isolatedsegment (Fig. 2). We define then two forces fext→i+1 and fext→i that describe, respectively,the external forces acting on segment i + 1 and the external forces acting on segment i:
{fext→i+1 = f ni
fext→i = f (−ni )(12)
with f the amplitude of the force acting at the contact.We also define the two new virtual markers V and V ′ that are the points of application of
each force (Fig. 2), obtained, respectively, by projecting the sphere centre Vi+1 on the spheresurface along the normal vector ni and by projecting the point V on the contact plane alongthe normal vector ni :
{rV = rVi+1 − dni
rV ′ = rVi+1 − ((rVi+1 − rVi)ni
)ni
(13)
with rV the coordinates of a virtual marker V and d the radius of the contact sphere.The external forces power Pi+1 on the segment i + 1 can be written:
Pi+1 = fext→i+1 · rV
⇔ Pi+1 = ([rVi+1 − dni]T ni
)f
⇔ Pi+1 = ([NVi+1i+1 Qi+1 − dNni
i Qi
]TNni
i Qi
)f
(14)
Finally the partial derivatives of the external forces power can be written:⎧⎪⎪⎨
⎪⎪⎩
∂Pi+1
∂Qi
= (−d[Nni
i
]TNni
i Qi
)f
∂Pi+1
∂Qi+1
= ([NVi+1i+1
]TNni
i Qi
)f
(15)
Anatomical kinematic constraints 133
Similarly, the external forces power Pi on the segment i can be written:
Pi = fext→i · rV
⇔ Pi = (−[ni]T((
rVi+1 − ((rVi+1 − rVi)ni
)ni
)))f
+ (−[ni]T(−((rVi+1 − rVi
)ni
)ni − ((rVi+1 − rVi
)ni
)
︸ ︷︷ ︸d
ni
))f
⇔ Pi = −[Nni
i Qi
]T (NVi+1
i+1 Qi+1 − ([NVi+1i+1 Qi+1 − NVi
i Qi
]TNni
i Qi
)Nni
i Qi
)f
+ [Nni
i Qi
]T (([NVi+1
i+1 Qi+1 − NVi
i Qi
]TNni
i Qi
)Nni
i Qi + dNni
i Qi
)f
(16)
with the following partial derivatives:⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
∂Pi
∂Qi
= ([Nni
i Qi
]T (−[NVi
i
]TNni
i Qi
)Nni
i Qi
)f
+ ([Nni
i Qi
]T ([NVi+1
i+1 Qi+1 − NVi
i Qi
]TNni
i
)Nni
i Qi + [Nni
i Qi
]TdNni
i
)f
∂Pi
∂Qi+1
= (−[Nni
i Qi
]TNVi+1
i+1 + [Nni
i Qi
]T ([NVi+1
i+1
]TNni
i Qi
)Nni
i Qi
)f
(17)
By identifying the scalar and unitary items of these expressions, we can rewrite them asfollow:
∂Pi
∂Qi
= ((−[NVi
i
]TNni
i Qi
)
︸ ︷︷ ︸scalar
[Nni
i Qi
]TNni
i Qi︸ ︷︷ ︸
(ni )2=1
)f
+ (([NVi+1i+1 Qi+1 − NVi
i Qi
]TNni
i
)
︸ ︷︷ ︸scalar
[Nni
i Qi
]TNni
i Qi︸ ︷︷ ︸
(ni )2=1
+[Nni
i Qi
]TdNni
i
)f
∂Pi
∂Qi+1
= (−[Nni
i Qi
]TNVi+1
i+1 + ([NVi+1i+1
]TNni
i Qi
)
︸ ︷︷ ︸scalar
[Nni
i Qi
]TNni
i Qi︸ ︷︷ ︸
(ni )2=1
)f
(18)
We can then write the generalized joint contact force as the sum of the partial derivativesof the powers described previously:
∑
⎡
⎢⎢⎣
∂P∂Qi
∂P∂Qi+1
⎤
⎥⎥⎦=
[ [NVi+1i+1 Qi+1 − NVi
i Qi]T Nni
i − [NVi
i ]T Nni
i Qi
[NVi+1i+1 ]T Nni
i Qi
]
f
⇔∑
⎡
⎢⎢⎣
∂P∂Qi
∂P∂Qi+1
⎤
⎥⎥⎦=
[ [Nni
i ]T (NVi+1i+1 Qi+1 − NVi
i Qi ) − [NVi
i ]T Nni
i Qi
[NVi+1i+1 ]T Nni
i Qi
]
f
(19)
By identification, we have finally f = λ and so the sphere-on-plane joint contact forcescan be determined directly from the computation of the associated Lagrange multiplier inthe equation of motion. The same relation can be found for the joint contact forces in aspherical joint.
A similar demonstration can also be made for the ligament forces (Fig. 2), and we showthat finally, we have f = λ/2L with L the length of the isometric ligament.
134 F. Moissenet et al.
2.6 Experimental gait data
Data from the “Second Grand Challenge Competition to Predict in Vivo Knee Loads”, orga-nized by B.J. Fregly, D. D’Lima, T. Besier, D. Lloyd and M. Pandy, where used to performthis study (https://simtk.org/home/kneeloads) [24]. We present here the results obtained us-ing the data from one gait trial of a male subject (67 kg, 172 cm).
Kinematics of the subject was obtained using optoelectronic cameras sampled at 120 Hzand a markerset of 43 cutaneous markers. Ground-reaction forces and moments were mea-sured using forceplates sampled at 3840 Hz and then filtered using a fourth order Butter-worth filter with a cutoff frequency of 5 Hz. Electromyographic (EMG) signals were mea-sured on 14 muscles (gluteus maximus, gluteus medius, adductor magnus, vastus medialis,vastus lateralis, rectus femoris, semimembranosus, biceps femoris long head, tensor fas-cia lata, gastrocnemius medialis, gastrocnemius lateralis, soleus, tibialis anterior, peroneuslongus) and sampled at 1000 Hz. EMG signal processing follows the recommendations ofDe Luca [12, 13]. First, the signal offsets were removed and a second order high-pass But-terworth filter was applied at a cutoff frequency of 20 Hz. Then the signals were rectifiedand a “whitening” method was used to remove all data under two standard deviations ofthe baseline. Finally, a second order low-pass Butterworth filter at a cutoff frequency of6 Hz was used to draw the signals envelop in order to compare these data with simulatedmusculo-tendon forces. The amplitude of the EMG signal envelop was adjusted to the am-plitude of the simulated musculo-tendon forces to compare the pattern of these data. Inaddition, an instrumented knee prosthesis [16] had been implanted to the right leg of thesubject. This prosthesis, based on six sensors (i.e., respectively, for the anterior-posterior,superior-inferior and lateral-medial forces and moments), allows getting joint contact forcessampled at 200 Hz.
3 Results
The computation duration of the simulation was 54 s (30 s for global optimization, 13 s fordynamics equation writing and 11 s for forces computation) on a standard computer withIntel Centrino 2 processor and 4 Go of memory.
3.1 Musculo-tendon forces
Only force patterns are described here and compared with EMG signals, since the ampli-tude of these data cannot be compared directly and additional information (e.g., maximumvoluntary contraction and maximal isometric force values) would be necessary to normalizethese data.
On the whole, simulated musculo-tendon force patterns are in agreement with EMG sig-nals for most of the muscles during the gait cycle (Fig. 3). The best results are obtainedfor gluteus maximus, gluteus medius, vastus medialis and vastus lateralis where estimatedforces reproduced well the main muscle activity peaks. Estimated gastrocnemius, soleus andperoneus longus forces have similar patterns but fail in reproducing the first muscle activitypeak at 10% of gait cycle. Semimembranosus and biceps femoris long head forces presenta temporal shift of 10% of gait cycle with an activity that begins earlier than EMG signal.Rectus femoris force pattern is different than the EMG signal with 2 similar peaks at 20%and 50% of gait cycle instead of 1 peak at 30% of gait cycle for the EMG signal. Simulatedadductor magnus force pattern does not present the second peak at 70% of gait cycle thatappears on EMG signal. Finally, simulated tibialis anterior force presents a really differentpattern than EMG signal.
Fig. 3 Comparison between simulated musculo-tendon forces (lines) and EMG signals (areas) on a samemuscle during a gait cycle (%) (toe-off is indicated by a vertical dashed line)
136 F. Moissenet et al.
3.2 Joint reactions
Estimated hip total joint reaction (i.e., the hip contact force since no ligament is includedin the hip joint model) is expressed in the distal segment coordinate system (SCS) (i.e.,femur SCS) and compared with published hip contact force obtained from an instrumentedhip prosthesis [5]. On the whole, the estimated hip total joint reaction is consistent withpublished data for each direction (Fig. 4). First, two peaks appear during stance phase (17%and 50% of gait cycle). Second, forces are very low during swing phase. Third, S-I andM-L forces direction are in agreement with published data. However, amplitudes are moreimportant in estimated than in published data (e.g., 3.6 BW instead of 2.3 BW for the firstpeak of S-I force during stance phase) and A-P force remains positive during the whole gaitcycle whereas published A-P force is negative after 30% of gait cycle.
Estimated knee total joint reaction (i.e., the sum of the joint contact and ligament forcesof the knee joint model), detailed joint contact forces (i.e., medial and lateral joint contactforces) and ligament forces are expressed in the prosthesis coordinate system (PCS), whereS-I axis lies on the line crossing the knee joint centre and the ankle joint centre, M-L axis lieson the line crossing the virtual medial knee joint contact and the virtual lateral joint contactand A-P axis is obtained by the cross product of these two axes (Fig. 1). Estimated andmeasured [24] joint contact forces are then compared. On the whole, the estimated knee totalcontact force is consistent with measured data for each direction (Fig. 4). First, estimatedA-P and M-L forces are null, which is confirmed by very low measured A-P and M-L forces(i.e., <0.2 BW). Second, estimated S-I force pattern follow the double-peak shape of themeasured force during stance phase with a first peak at 17% of gait cycle and a second peakat 50% of gait cycle. It has to be noted that the estimated knee lateral joint contact force onlycontributes to the second force peak since it is almost null at the beginning of stance phaseand during the whole swing phase. Regarding ligament forces, they contribute to the kneetotal joint reaction for each direction (Fig. 4), even in the S-I direction. However, MCL andPCL forces are almost null, respectively, in A-P and S-I directions. Their highest impact onknee total joint reaction appears on A-P and M-L directions since joint contact forces arenull in these direction in our model. Finally, only MCL force is opposite to the knee totaljoint reaction in the S-I direction.
Estimated ankle total joint reaction (i.e., the sum of the joint contact and ligament forcesof the ankle joint model), joint contact force and ligament forces are expressed in the distalsegment coordinate system (SCS) (i.e., foot SCS). To our knowledge, no measured anklejoint contact force has been published to date. Therefore, results are given for information.On the whole, estimated ankle total joint reaction is mainly in the S-I direction (A-P and M-L maximum force values are under 1 BW and S-I maximum force value is next to 6 BW).Regarding contact forces, their main contribution remains in the S-I direction where twoforce peaks appear during stance phase at 20% and 46% of gait cycle. Regarding ligamentforces, they contribute to the ankle total joint reaction for each direction (Fig. 4), even in theS-I direction. However, TiCaL force is almost null in A-P direction and CaFiL in S-I andM-L directions.
4 Discussion
An original 3D lower limb musculo-skeletal model, composed of existing kinematic andmusculo-skeletal geometric models [14, 15, 23], has been introduced and allows the studyof musculo-tendon forces, hip total joint reaction and both knee and ankle detailed joint
Anatomical kinematic constraints 137
Fig. 4 Estimated joint reactions, published hip joint contact forces [5] and measured knee joint contact forces[24] expressed in body weight (BW) during 100% of gait cycle (toe-off is indicated by a vertical dashed line)
reactions during gait. To our knowledge, very few studies have included detailed joint re-actions, consisting of joint contacts and ligaments forces, in a whole lower limb musculo-skeletal simulation. Rather than developing complex models [30, 39], the idea is to intro-duce “anatomical” kinematic constraints in a Lagrangian approach and compute the valuesof the Lagrange multipliers to get joint contact and ligament forces. The main consequenceis that the computation time is short (i.e., 54 s) and thanks to the use of natural coordinates[25], a lot of constant matrices (e.g., interpolation and pseudo-interpolation matrices) can becomputed off-line, so it should be possible to develop a real-time framework based on thismodel. The use of kinematic constraints provides physiological kinematic coupling withoutusing functions of the degrees of freedom of the model [3, 14]. This is essential in order toobtain physiological active joint moments and muscle moment arms. Moreover, despite nu-
138 F. Moissenet et al.
merous joint biomechanics studies, the interactions between musculo-tendon, joint contactand ligament forces remain unclear, certainly because the measure of these forces is stillinvasive. Musculo-skeletal modelling, by providing a numerical framework allowing “what-if” simulations, seems to be an interesting tool for a deeper understanding of the forcesdistribution through the different active and passive structures and for a better view of theinteractions acting into the musculo-skeletal system during gait [8, 10, 33]. Since parallelmechanisms [15, 23] can be personalized (e.g., by modifying the joint contacts geometryand ligaments insertions or by removing a ligament), our model should provide insights interms of musculo-tendon forces, joint contact forces and ligament forces interactions.
Results show that most of the estimated musculo-tendon forces are in agreement with theEMG signals. The amplitude of these data cannot be compared directly and additional infor-mation (e.g., maximum voluntary contraction and maximal isometric force values) wouldbe necessary to normalize them. Anyway, our model is able to predict reasonably well themuscular temporal activity during a complex movement such as human gait. Regarding theestimated hip total joint reaction (corresponding in our case to the hip contact force), thejoint contact force measurements performed by Bergmann et al. in 2001 using an instru-mented prosthesis [5] provides interesting data to evaluate our model. Results show that,even if amplitudes are too high, force patterns are in agreement with the measured data andreproduce well the main force peaks. Then, regarding the estimated knee total joint reaction(corresponding in our case to the sum of the knee joint contact and ligament forces), themeasurements realized by Fregly et al. in 2011 [24], made of kinematic, kinetic, EMG andknee contact forces data, allow us to validate our model. Results show that a consequence ofthe use of a parallel mechanism for the knee joint modelling is that contact forces only lie onthe S-I axis. In fact, the direction of this 3D force is given by the normal of the contact plane(Fig. 2). Consequently, this force would lies on several axes (i.e., not only on the S-I axis) ifthe plane was oriented in a different manner (e.g., if the model is personalized the respect thejoint geometry of a subject). Anyway, this result is confirmed by very low measured A-P andM-L forces (i.e., <0.2 BW). Regarding the S-I axis, joint contact force is in agreement withmeasured data, but with a higher amplitude, like for the hip model. Another consequence ofthe use of a parallel mechanism is that detailed joint contact forces (i.e., medial and lateralcontact forces) and ligament forces (i.e., ACL, PCL and MCL forces) are directly accessi-ble. This information should provide insights in the forces repartition between the differentstructures of the knee joint during a movement. Finally, even if to our knowledge, no mea-sured ankle joint contact force has been published to date, results provide interesting datathat should allow a better understanding of the ankle joint.
But even if our model provides new opportunities for musculo-skeletal simulations, thisstudy still possesses a number of limitations. First, only one gait cycle from a single subjectwas analyzed. It is of course necessary to extend this study to multiple subjects and cyclesbefore generalizing our results. Moreover, the data used were recorded on a subject withinstrumented knee prosthesis and so cannot be exactly considered as normal gait data. Sec-ond, our model is based on different generic models that are scaled and so do not representthe real geometry of the subject. Furthermore, musculo-tendon, joint contact and ligamentforces may be affected by the choice of this geometry. Third, foot joints were limited toankle joint: tarsometatarsal and metatarso-phalangian were not modelled and consequentlyour model cannot be used to study muscles that cross these joints. Fourth, a consequenceof considering ligaments as isometric structures in our model is that ACL, PCL and TiCaLgenerate a force in the direction of the joint reaction (i.e., S-I direction). Consequently, evenif these forces are low at the knee joint, they reveal that ligaments are slightly pushing inthe S-I direction instead of pulling. This could be avoided by introducing deformable lig-aments in our model [25]. Fifth, the simulated knee lateral contact force only contributes
Anatomical kinematic constraints 139
to the total joint reactions during the second part of stance phase. Shelburne et al. [39] alsofound a knee lateral contact force next to zero between 10% and 15% of the gait cycle. It hasbeen suggested [39] that not only ligaments but also muscles primarily responsible for gaitpropulsion (quadriceps and gastrocnemius) are required to prevent this lateral compartmentunloading. By minimizing all musculo-tendon forces in our optimization, it is consequentlynot possible to simulate this muscular contribution to joint stability.
Nevertheless, the most significant limitation in the present study is that joint contact andligament forces are computed in a second time using the optimized musculo-tendon forces.Thereby, this formulation does not allow including the influence of joint reactions in themuscular redundancy problem [8, 9, 34]. For example, a subject with a knee pain wouldcertainly try to reduce his knee contact force and for sure musculo-tendon force patternswill be affected by this criterion. This could explain why estimated hip, knee and perhapsankle joint contacts always have higher amplitude than measured data. With the proposedmusculo-skeletal model, a cost function minimising simultaneously musculo-tendon forcesand joint reactions could be easily considered if the parameter reduction is not used to cancelthe Lagrange multipliers.
5 Conclusion
In conclusion, this paper presents a new method to determine both musculo-tendon forcesand detailed joint reactions, including joint contact and ligament forces, during gait usingan original 3D lower limb musculo-skeletal model. It has been demonstrated that this newapproach provides better insights about the structures involved in joint during gait (e.g.,musculo-tendon units, joint contacts and ligaments) and their interactions, without havingto introduce a complex knee model [24, 30, 33]. Thanks to the published data of Bergmannet al. [5] and to the measured data of Fregly et al. [24], our model has been, respectively,evaluated for the hip joint model and partially validated for the knee joint model.
Next step will be to compute simultaneously musculo-tendon forces and detailed jointreactions, allowing the introduction of joint reactions in the minimization criterion and theinvestigation of the consequences on each musculo-tendon force. Our model can be com-pleted by introducing passive moments, resulting from passive structures such as ligaments.Some recent works have already been proposed in this way [19].
References
1. Alonso, F., Cuadrado, J., Lugris, U., Pintado, P.: A compact smoothing-differentiation and projectionapproach for the kinematic data consistency of biomechanical systems. Multibody Syst. Dyn. 24(1),67–80 (2010)
2. Andersen, M., Damsgaard, M., Rasmussen, J.: Kinematic analysis of over-determinate biomechanicalsystems. Comput. Methods Biomech. Biomed. Eng. 12(4), 371–384 (2009)
3. Arnold, E., Ward, S., Lieber, R., Delp, S.: A model of the lower limb for analysis of human movement.Ann. Biomed. Eng. 38(2), 269–279 (2010)
4. Ausejo, S., Suescun, A., Celigeta, J.: An optimization method for overdetermined kinematic problemsformulated with natural coordinates. Multibody System Dynamics, 1–14 (2011)
5. Bergmann, G., Deuretzbacher, G., Heller, M., Graichen, F., Cuadrado, J., Alonso, F., Rohlmann, A.,Strauss, J., Duda, G.: Hip contact forces and gait patterns from routine activities. J. Biomech. 34(7),859–871 (2001)
6. Blajer, W., Czaplicki, A., Dziewiecki, K., Mazur, Z.: Influence of selected modeling and computationalissues on muscle force estimates. Multibody Syst. Dyn. 24, 473–492 (2010)
140 F. Moissenet et al.
7. Charlton, I., Tate, P., Smyth, P., Roren, L.: Repeatability of an optimised lower body model. Gait Posture20(2), 213–221 (2004)
8. Cleather, D., Bull, A.: An optimization-based simultaneous approach to the determination of muscu-lar, ligamentous, and joint contact forces provides insight into musculoligamentous interaction. Ann.Biomed. Eng. 39(7), 1925–1934 (2011)
9. Collins, J.: The redundant nature of locomotor optimization laws. J. Biomech. 28(3), 251–267 (1995)10. Collins, J., O’Connor, J.: Muscle-ligament interactions at hte knee during walking. Proc. Inst. Mech.
Eng., H J. Eng. Med. 205(1), 11–18 (1991)11. Crowninshield, R., Brand, R.: A physiologically based criterion of muscle force prediction in locomo-
tion. J. Biomech. 14, 793–801 (1981)12. De Luca, C., Gilmore, L., Kuznetsov, M., Roy, S.: Filtering the surface emg signal: Movement artifact
and baseline noise contamination. J. Biomech. 43(8), 1573–1579 (2010)13. De Luca, S.: The use of surface electromyography in biomechanics. J. Appl. Biomech. 13, 135–163
(1997)14. Delp, S., Loan, J., Hoy, M., Zajac, F., Topp, E., Rosen, J.: An interactive graphics-based model of the
lower extremity to study orthopaedic surgical procedures. IEEE Trans. Biomed. Eng. 37(8), 757–767(1990)
15. Di Gregorio, R., Parenti-Castelli, V., O’Connor, J., Leardini, A.: Mathematical models of passive motionat the human ankle joint by equivalent spatial parallel mechanisms. Med. Biol. Eng. Comput. 45(3),305–313 (2007)
16. D’Lima, D., Patil, S., Steklov, N., Slamin, J., Colwell, C.: Tibial forces measured in vivo after total kneearthroplasty. J. Arthroplast. 21(2), 255–262 (2006)
17. Dumas, R., Cheze, L.: 3d inverse dynamics in non-orthonormal segment coordinate system. Med. Biol.Eng. Comput. 45(3), 315–322 (2007)
18. Dumas, R., Cheze, L., Verriest, J.: Adjustments to McConville et al. and Young et al. body segmentinertial parameters. J. Biomech. 40(3), 543–553 (2007)
19. Dumas, R., Moissenet, F., Gasparutto, X., Cheze, L.: Influence of the joint models on the lowerlimb musculo-tendon and contact forces during gait. Proc. Inst. Mech. Eng., H, J. Eng. Med. (2011).doi:10.1177/0954411911431396
20. Duprey, S., Cheze, L., Dumas, R.: Influence of joint constraints on lower limb kinematics estimationfrom skin markers using global optimization. J. Biomech. 43(14), 2858–2862 (2010)
21. Ehrig, R., Taylor, W., Duda, G., Heller, M.: A survey of formal methods for determining the centre ofrotation of ball joints. J. Biomech. 39(15), 2798–2809 (2006)
22. Erdemir, A., McLean, S., Herzog, W., van den Bogert, A.: Model-based estimation of muscle forcesexerted during movements. Clin. Biomech. 22(2), 131–154 (2007)
23. Feikes, J., O’Connor, J., Zavatsky, A.: A constraint-based approach to modelling the mobility of thehuman knee joint. J. Biomech. 36(1), 125–129 (2003)
24. Fregly, B., Besier, T., Lloyd, D., Delp, S., Banks, S., Pandy, M., DLima, D.: Grand challenge competitionto predict in vivo knee loads. J. Orthop. Res. (2011, conditionally accepted)
25. Garcia de Jalon, J., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems. The Real-TimeChallenge. Springer, New-York (1994)
26. Kim, H., Fernandez, J., Akbarshahi, M., Walter, J., Fregly, B., Pandy, M.: Evaluation of predicted knee-joint muscle forces during gait using an instrumented knee implant. J. Orthop. Res. 27, 1326–1331(2009)
27. Koh, B.I., Reinbolt, J., George, A., Haftka, R., Fregly, B.: Limitations of parallel global optimization forlarge-scale human movement problems. Med. Eng. Phys. 31(5), 515–521 (2009)
28. Leardini, A., Chiari, L., Della Croce, U., Cappozzo, A.: Human movement analysis using stereopho-togrammetry. Part 3. Soft tissue artifact assessment and compensation. Gait Posture 21(2), 212–225(2005)
29. Lenaerts, G., De Groote, F., Demeulenaere, B., Mulier, M., Van der Perre, G., Spaepen, A., Jonkers, I.:Subject-specific hip geometry affects predicted hip joint contact forces during gait. J. Biomech. 41(6),1243–1252 (2008)
30. Lin, Y., Walter, J., Banks, S., Pandy, M., Fregly, B.: Simultaneous prediction of muscle and contactforces in the knee during gait. J. Biomech. 22, 945–952 (2010)
31. Lu, B., O’Connor, J.: Bone position estimation from skin marker co-ordinates using global optimisationwith joint constraints. J. Biomech. 32(2), 129–134 (1999)
32. Moissenet, F., Cheze, L., Dumas, R.: Musculo-tendon forces prediction by static optimization: a com-parative study between three methods. J. Biomech. Eng. (2011, under review)
33. Pandy, M., Andriacchi, T.: Muscle and joint function in human locomotion. Annu. Rev. Biomed. Eng.12, 401–433 (2010)
34. Raikova, R.: Investigation of the influence of the elbow joint reaction on the predicted muscle forcesusing different optimization functions. J. Musculoskelet. Res. 1(1), 31–43 (2009)
35. Reed, M., Manary, M., Schneider, L.: Methods for Measuring and Representing Automobile OccupantPosture. Society of Automobile Engineers, Warrendale (1999)
36. Reinbolt, J., Schutte, J., Fregly, B., Koh, B., Haftka, R., George, A., Mitchell, K.: Determination ofpatient-specific multi-joint kinematic models through two-level optimization. J. Biomech. 38(3), 621–626 (2005)
37. Seireg, A., Arvikar, R.: The prediction of muscular load sharing and joint forces in the lower extremitiesduring walking. J. Biomech. 8(2), 89–102 (1975)
38. Shelburne, K., Pandy, M., Anderson, F., Torry, M.: Anterior-Cruciate Ligament Forces in the Intact KneeDuring Normal Gait. American Society of Biomechanics, Calgary (2002)
39. Shelburne, K., Torry, M., Pandy, M.: Contributions of muscles, ligaments, and the ground-reaction forceto tibiofemoral joint loading during normal gait. J. Orthop. Res. 24(10), 1983–1990 (2006)
40. Silva, M., Ambrosio, J.: Kinematic data consistency in the inverse dynamic analysis of biomechanicalsystems. Multibody Syst. Dyn. 8(2), 219–239 (2002)
41. Stansfield, B., Nicol, A., Paul, J., Kelly, I., Graichen, F., Bergmann, G.: Direct comparison of calculatedhip joint contact forces with those measured using instrumented implants. An evaluation of a three-dimensional mathematical model of the lower limb. J. Biomech. 36(7), 929–936 (2003)
42. Terrier, A., Aeberhard, M., Michellod, Y., Mullhaupt, P., Gillet, D., Farron, A., Pioletti, D.: A muscu-loskeletal shoulder model based on pseudo-inverse and null-space optimization. Med. Eng. Phys. 32(9),1050–1056 (2010)
43. Wilson, D., Feikes, J., O’Connor, J.: Ligament and articular contact guide passive knee flexion.J. Biomech. 31, 1127–1136 (1998)
44. Zhao, D., Banks, S., D’Lima, D., Colwell, D., Fregly, B.: In vivo medial and lateral tibial loads duringdynamic and high flexion activities. J. Orthop. Res. 25, 593–602 (2007)
