- Email: [email protected]
Control exerted by ligaments
CONTROL CHAIYONG
EXERTED
BY LIGAMENTS
WONGCHAISUWAT
Department
of Electrical
and
Engineering.
HOOSHAKG
HE~IAN~
The Ohio State University
and
MARGARET J. HEXES Department
of Anatomy.
The Ohio State University.
OH 43210, U.S.A
Columbus,
Abstract-The function of the ligaments as local controllers, independent of the central nervous system. in maintainicg the integrity of the joir,t is demonstrated by modslling the human knee in the sagittal plane. and studying its anterior-posterior motion. In addition to the ligaments, the model includes the characteristic geometry of the joint surfaceand some muscle groups. The connecting reaction forces at the point ofcontact between the tibia and the femur are considered to be constraint forces due to three din‘erent surface motions-gliding. rolling and combined gliding and rolling. It is demonstrated that the ligamentous structure maintains these holonomic and nonholonomic constraints that describe the joint motion, and that stability of the knee joint is provided mainly by ligaments. Muscular structures further stabilize and contribute to joint movement. Computer simulation of rolling movement of the knee is presented to illustrate the importance of the ligaments for joint integrity and stability.
(11. (1973). A mathematical
IN’I’RODUC-I’ION
analysis (Lew and
Lewis,
1978; Lewis and Lew. 1978) and a Roentgen stereoThe function of human joints is to secure the union of
photogrammetric
the articulating
lating the ligament length patterns have been reported
bones in the body and to permit, even
facilitate. their movements The maintenance
in relation
to one another.
of function and preservation
of the
method (Dijk
while the stil‘fness of primary were studied in (Markolf
er al., 19791 for calculigaments
at the knee
ef ul., 1976; Trent et til.. 1976).
integrity of the joints are relegated to the coordinated
The extension and Rexion movements of the knee are
action of ligamcnts,capsule,
said
muscles, surrounding
tissues and the characterstic surfaces. The
individual
this composite
geometry
mechanical
structure
soft
of the joint
to
be
(Morrison,
controlled
by
1970b)--the
of
and quadriceps. These muscles have been proposed as
of
a model consisting of an active contractile
to obtain better under-
passive series elastic element
of
visco-elastic element (Glantz,
prostheticdevices.
in
forces transmitted
Specifically, the human kneejoint
the past has been selected as the object because it is anatomically
of various complicated
and functions during movement to maintain The
or range of motion functions
of the
primary
ligaments during
medial and lateral displacements
L’I u/. ;198Oa) and in varus-valgus and
internal
and
external
either rhe
of the lower limb.
human knee have been investigated posterior.
groups
gastrocnemius
components
standing of the motor control and to improvedesign
stability
muscle
have been the subject
numerous studies undertaken
investigations
three
hamstrings,
axial
femoral
of the
activities Smidt
rotations,
tibia1 rotations
by
by
1977). The
during
Morrison
these
ll970a).
Perry er al. (1975) and Bishop (1977).
studies
(Crowninshield
t’~
a/..
1976:
Moeinzadeh,
1981; Wismans et 01.. 1980) make use of
mathematical
models of the analysis of the knee joint.
anterior, by Piziali
1977; Hatze,
the knee joint
have been studied
(1973).
Other
at
element, a
and a passive parallel
In this paper, the role of the primary system ofcontrollers-for
ligaments as a
ensuring stability and main-
taining the integrity of the joint is described by means of a mathematical
model. The ligaments, the geometry
Scering ZI (11. (1980). Piziali et (11. (1980b) determined
of the joint surface and the disposition of some muscle
the contribution
groups are taken into account in the study of the gross
displacement
of the cruciatc ligaments to the load-
characteristics. The mechanical behavior
of the primary
ligaments
has been specified by vis-
movement
of the free-hanging
plane. The motion
knee in the sagittal
in this plane is so great that it
coelastic properties (Fung, 198 I). Wismans er al. (1980)
accounts for much of the joint motion. The non-linear
neglected the viscous properties and approximated
the
differential
equations
springs. The
considered
as a constrained
ligaments as quadratic primary
ligament
force-elongation
length patterns at various angles of
flexion, in neutral,and
in external and internal rotation
have been evaluated by Trent
et al. (1976) and Wang P[
reaction
I Srptembzr
1983;
,;:;_E
are
derived
dynamic
ling. The
and
system. The
forces at the point of contact
exerted during three different
between
the
forces
surface motions of the
rolling and combined
gliding and rol-
result of this study demonstrates
ligament functions as a local controller 525
H?,
motion
femur and tibia are considered to be constraint knee--gliding,
Rrcric& 24 June 1982; in reciscdform rrceiwd for publication 29 March 1984.
of
that a
that maintains
516
CHAIYONGWONGCHAISCMT, HOOSHANGHEMAMIand MARGARETJ. HIRES
the integrity of the joint. Computer simulations of rolling movement of the knee are presented to support this point of view. THE WEE
JOIST
IS THE SAGIITAL
PLASE
The present study is limited to the analysis of the anterior-posterior motion of the knee. Only the forces that act between the tibia and the femur in the sagittal plane are considered. The model includes the primary ligaments, the muscles crossing the joint and the geometry of the joint surfaces. In order to simplify the complex structure of the knee, several assumptions (Drillis and Contini, 1966; Wongchaisuwat, 1981) are made: 1. The stl-faces of the tibia and the femur are identical on the medial and lateral side. The outside rigid surface of contact is a straight line for the tibia and an ellipse for the femur (Fig. 1). 2. The femur is stationary and horizontal and the tibia moves relative to the femur. 3. The cartilage layer and the menisci between the tibia and the femur are neglected, so there is no clearance between the surfaces-i.e. the contact is at one point and it traverses along the perimeter of the ellipse. 4. The three primary ligaments affecting the kneethe anterior cruciate (ACL). the posterior cruciate (PCL), and the combined medial and lateral collateral (CLL)--are assumed to be represented by lines of action connecting respective femoral and tibia1 origin and insertion points. 5. Three main muscle groups-the hamstrings (HM), the gastrocnemius (GM) and the quadriceps femoris (QM)---are considered in the model. The
Fig. 1. The knee joint model showing the ligament forces and the contact reaction forces.
hamstrings and gastrocnemius muscles are represented by lines of action between their origin and insertion points. The line of action of the quadriceps muscle acting on the tibia is coincident with the direction of the patellar ligament. In addition, the ankle joint is assumed to be at the position where the gastrocnemius muscle force is very small. 6. The length of each ligament (or muscle) essential for developing the ligament (or the muscle) force is taken to be the distance between two sites of attachment. With these assumptions. the knee joint model studied is shown in Fig. 2. The normal and the tangential component of the reaction forces which act on the tibia at the point of contact with the femur are represented by I- = [F,, Fr]? The muscle force vector U = [F,, F1, F,]’ has three components, one each for the hamstrings, gastrocnemius and quadriceps muscle groups. The collateral, anterior cruciate and posterior cruciate ligament force vector is represented by V = [F,, F,, F,]‘. The angles of the muscle force (U) and the ligament force (V) with the horizontal axis are labeled S,, S1, S,, .Sj, S, and S6 respectively. The dimensions and the coordinate of the attachments of the ligaments and muscles on the femur and the tibia are given by Wongchaisuwat (1981). Most of these numerical values are selected from Drillis and Contini
FQ
Fig. 2. The knee joint model showing the muscle forces and gravity force.
Control
Fig. 3. Block diagram
of a feedback model of the
exerted by hgaments
knee joint.
(1966) with some values which are estimated from Crowninshield et al. (1976), Dijk et al. (1979) and Lew and Lewis (1978). The mechanical behavior of the primary ligaments can be summarized by their viscoelastic properties (Fung, 1981). The lengths of the primary ligaments expressed as the function of the position of tibia are provided in Basmajian (1978), Wismans ef al. (1980) and Wongchaisuwat (198 1). Newtonian dynamicsare used to write theequations of motion. The surface motions between the femur and the tibia are described by three idealized segmentsgliding, rolling and combined gliding and rolling. Due to these surface motions, both holonomic and nonholonomic constraints (Hemami, 1980) appear in the model (Wongchaisuwat, 1981). The reaction forces at the joint are combined in a vector of forces of constraint. It was shown in (Hemami, 1980) that the forces of constraint can be derived as functions of the state (position and velocities) and the inputs when the constraints are maintained. The above analysis provides the feedback model of the knee joint (Fig. 3). LINEARIZATION
ASD STATIC
ANALYSIS
The equations of the model are non-linear and complex. The problems of control and stability of these non-linear equations are considered here by linearization. The stiffness of the ligaments for the three dimensional case as given in the literature (Markolf et ol., 1976; Trent cf al.. 1976) with the ratio 5.x0 kp -cannot be used for this study because here the 7 Z&Y ligaments are projected onto a plane. In order to achieve the proper stiffness values for the planar study the following assumptions are made: 1. At the free hanging position, it is assumed that the tangential reaction force is zero and the three muscle groups crossing the knee joint are inactive. 2. From the free-hanging position to full flexion, the tangential reaction force resists the movement, and is positive. 3. During knee extension, in order to resist the movement, the tangential reaction force should be
1
wre
2.
comb,red
3.
pu4
>!ldwl gltdlr<
I”d
rollIn?
r,lll”g
Fig. 4. The analysis of the tangential reaction force for three surface motions.
negative. The resulting force pattern is sketched in Fig. 4. 4. It is assumed that the ankle joint is at the position where the gastrocnemius muscle force acting on it is very small, about twenty times smaller than the hamstrings muscle force exerted at the knee. In order to obtain the tangential reaction force which behaves as stated above, the stiffness and the unstrained length of the collateral, anterior cruciate and posterior cruciate ligaments are estimated as 279891 Pa, 0.043 m; 92726 Pa, 0.038 m; and 375567 Pa, 0.023 m, respectively. The stiffness values are different from those reported in Trent et al. (1976) because these are planar studies. The reaction forces and the muscle forces are plotted with the angle I) in Figs 5 and 6 respectively, It is shown that in the freehanging position where the muscle forces are zero is at U ‘c - 0.16 radian. The hamstrings and gastrocnemius muscle forces are maximum when the knee is at full llexion, and then steadily decrease to zero as the leg returns to the free-hanging position. The quadriceps muscle force gradually increases from the free-hanging to the full extension position of the tibia. In the combined gliding and rolling section shown in Fig. 6 the muscle forces may be estimated as dotted lines. The contact force on the tibia in the combined rolling and gliding movement can be calculated from the estimated muscle force. STABILITY
AND THE MUSCLE
MODEL
The unstable knee-movement system proposed in the previous sections can be stabilized when muscles are added to the model (Fig. 7). The hamstrings, gastrocnemius and quadriceps are three groups of skeletal muscle which cross the knee joint. They perform the movement and the stabilizing action for the knee (Gowitzke and Milner, 1980; Henneman, 1980; Rosse and Clawson, 1980). During the knee extension the quadriceps muscle functions as a prime
528
CH.MONG WOVXHAISW+T.
HOOSHAM HEWAM and MARGARETJ. HIXES
\
\
‘\
3
s
0
$j
‘--L-.-l+-_ ____.__
__m-me.-..
Cl
7
I
t
\ \
0
I ,.._
\
\\ ia?!
7
Ii= Ezu@Jeq
a&4,
!n
9: ____ ___.__ __------..._ __-
“qxq,
0
,,“,
.----
.
I
Control exerted by ligaments
Fig.
7.
Block diagram of the knee joint model including a state feedback part due to muscles.
mover or agonist in both maintaining and contributing to the desired movement. The hamstrings and gastrocnemius muscles function as the antagonists to oppose the quadriceps extension movement. On the other hand, the hamstrings and gastrocnemius are the prime movers and the quadriceps is an antagonist when the knee movement is reversed-i.e. in flexion. The force of gravity must be considered as the prime moving force from full Aexion to the free-hanging position as well as from full extension to the freehanging position of knee flexion. The movement of the knee in these periods is controlled by the antagonist muscles. Based on Hatze’s muscle model (Hatze, 1977). the active contractile element and the series elastic element are combined to form a simplified tension generator F. The muscle constants position feedback gain matrix K and the velocity feedback gain matrix E move the poles of the linearized system to the left half of the complex frequency plane. In this work, it is assumed that the matrices K and Bare not necessarily diagonal (Wongchaisuwat, 1981). The coupling between their elements is by way of higher control centers.
DIGITAL COMPUTER SIMULATION
The objective of the simulation is to test the effectiveness of the ligaments in maintaining the constraints of the joints. In the present work, in the free-hanging leg, the rolling movement of the tibia on the femur is simulated. The initial. final and reference state for the rolling movement are specified as X, = [0.239805, -0.027267, X,= [0.241000, -0.013732, x, = [.YfJ,Jo. 0,. 0.0, o-jr
1.508, 0, 0, O]r 1.571, 0, 0, Olr
where
I
0.063 ; + 1.508
0, =
I.571
:t ,< 10 :r > I,
x0= -0.051216 0: +0.140903 U; - 0.050667 Ue + 0.17 1422 y. = - 0.052216 0: + 0.234423 U; - 0.135630 0, - 0.176766. The initial, final and reference bias muscle forces are given as U, = [O,O, 182.7721r U,=
[O,O, 194.254]r
Uo = [O, 0, Fso]’
where
F,o =
1
:t
[O
<
10
1194.254 I.482 !- + 182.772 :I > 10.
With the above final state and bias muscle forces, the linearized equations are i=AX+Bi.
where
0 0 A=
0 9.9182 153.03 54.515
0 0
0 0
0 0 19.656 - 5.9873 339.01 -91.558 121.86 -32.902
10 0 01 0 00 I 0 0 .97656 0 0 -0.022888 0 0 -0.007629
530
CHAIYONG
WONGCHAISLWAT,
HOOWANG
HEMAW
and
F
0 B=
20
0
0
1
MARGARET
1. HINES
nt.,
i
1
0
0
0
-0.003052 - 0.032043 -0.012277
- 0.003052 - 0.039673 - 0.0 14496
0.005168 0.080856 0.029056
1
and
*
Let the desired eigenvalues associated with x, J and 8 be selected as (- 0.2, - 0.3), ( - 0.3, - 0.2) and ( - 0.3, - 0.3) respectively. The corresponding feedback gains K and B are K=
B=
i
- 35409 94085 48360
769742 - 1939944 - 573254
285292 - 713629 - 208464
- 503.27 1766.5 703.83
184404 -26216 59704
66963 -32154 I 10586
I
F3 (nt.)
:::+-
In the simulation I, is arbitrarily chosen as 0.8 s. The rolling condition is investigated by testing the constraint equations (Wongchaisuwat, 1981. ch. 5). The results of the simulation are shown in Figs 8-10. The tibia moves from its initial position to the specified final position (Fig. 8) the rolling constraint is satisfied. The muscle forces (Fig. 9) are high and become
1
ot-
80
0
_ 1
2
t (SCC)
3
Fig. 9. The muscle forces vs time.
negative-not a physically meaningful result. The reaction forces are plotted in Fig. 10. In the present simulation several points should be noted. The constants K and B are very large. To obtain smaller gains,
FN
(nt.j
t
o.-__ : 0 FT 0
0
-35
‘ -
1
2
3
1
2
3 !
(nt)
t--
-70
-105 b-
Fig. 8. The position of the tibia vs time.
t (XC)
Fig. 10. The reaction forces vs time.
t (set)
Control
exerted by ligaments
the desired eigenvalues associated with x, J and 0 should be selected closer to the imaginary axis. With a proper set of diagonal feedback gains, the simulations in the vicinity of the operating points for two other surface motions, gliding and combined gliding and roiling can be performed in the same manner as the rolling simulation.
DISCUSSIOS
ASD COSCLL’SIOSS
A model of the human kneejoint in thesagittal plane was presented as a constrained system. The primary ligamentous function is that ofa local controller, which maintains the integrity of the joint. in addition to the ligaments, the geometry of the joint surface is recognized as being important in this function. Three surface motions of the knee-gliding, rolling, and combined gliding and rolling-were considered. Holonomic and nonholonomic constraint equations describe these three movements. It was shown that the ligamentous structures maintain the system on the constrained surface. They preserve the joint function and relative motion while the muscular structures both contribute to the movement and further stabilize the system. Substantial geometric and kinesiologic simplifications were made. The primary ligaments affecting the knee were assumed to be represented by quadratic non-linear springs. The muscles were simply modeled as force generators controlled by a central system and the visco-elastic elements of the surrounding tissues. A kinematic study of three surface motions of the knee was performed to supply the lengths and the lines of action of the muscles and ligaments. The stiffness and the unstrained length of the ligaments were estimated to ensure the zero tangential constraint force in the combined gliding and rolling motion. Since the mode was formulated for planar studies only, the parameters used in this work were contrasted with those reported in the literature. The problem of stability of the knee was considered by properly selecting the stiffness and the unstrained length of the ligaments and determining the linear position and velocity feedback gains for the visco-elastic elements of the muscles. In this work, the assumption was made that the muscle constants K and B. i.e. position and velocity feedback gains, were coupled. Digital computer simulations of the rolling motion were carried out to test the effectiveness of the ligamentous structures in maintaining the constraints. The rolling movement was executed properly in spite of the inadequacy of the muscle models. With a better muscle model, more complete simulation for the whole sequence of motion between full flexion and full extension of the knee may be carried out. This work should provide a better understanding of the functional role of the ligaments of the knee joint. Since the actual movement in extension of the knee occurs simultaneously in three planes according to its
531
‘screw-home’ or locking mechanism, further study requires that the joint be changed into a threedimensional system. More studies should be undertaken which consider the geometry of the articular surfaces, including the menisci. a more precise modeling of the ligaments and the muscles, and the role of the articular cartilages. With more realistic models, using the principles enunciated here. one should be able to demonstrate that the knee joint stability is usually provided primarily by ligaments and the joint surface structures, and secondarily by muscles.
A~~norlrdyrmunt-This work was supported by National Science Foundation under grant ECS-8201240.
[he
REFERENCES Basmajian. J. V. (1978) I%fusclr A/it.e 4th Edn. William and Wilkins, Baltimore. Bishop, R. E. D. (1977) On the mechanics of the human knee. Enyny Mrd. 6. 46-52. Crowninshield. R.. Pope, M. H. and Johnson. R. J. (1976) An analytical model of the knee. J. Eiomechunics 9. 397-405. Dijk, R. van, Huiskes. R. and Selvik, G. (1979) Roentgen stereophotogrammetric methods for the evaluation of the three dimensional kinematic behaviour and cruciate ligament length patterns of the human knee joint. J. Biomdrunicx 12. 727-73 I. Drillis. R. and Contini, R. (1966) Body segment parameters. Tcxhnical report no. 1116.03. New York University, University Heights, New York. Fung. Y. C. (I98 I) Biomechunics: ~~lechunicul Propwrks o/ Liciny Tissue. Springer, Berlin. Glantz. S. A. (1977) A three-element description for muscle with viscoelastic passive elements. J. Biomrrhunics IO, 5-20. Gowitzke. B. A. and Milner, M. (1980) Undrrsranding rhr Shvrri/rrc- Buses ol Human Mocemrnr (2nd Edn.) William and Wilkins, Baltimore. Hatze. H. (1977) A myocybernetic control model of skeletal muscle. Bioloyicul Crbrrnrrics ZS, 103-l 19. Hemami. H. (1980) A feedback on-off model of biped dynamics. IEEE Trans. on Sysr. Xfun C_ybernrr. SIMC-10, 376383. Henneman. E. (1980) Skeletal muscle. Mrdid Phpsiolog) (Edited by Mountcastle. V. B.) 14thedn. Vol. I. p. 674.C. V. Mosby. St. Louis. Lew, W. D. and Lewis, J. L. (1978) A technique for calculating in cico ligament lengths with application to the human knee joint. J. Eiomrchunics 1 I, 365-377. Lewis. J. L. and Lew. W. D. (1978) A method for locating an optimal “fixed” axis of rotation for the human knee joint, J. biomech. Engng 100, 187-193. Markolf. K. L.. Mensch, J. S. and Amstutz, H. C. (1976) Stiffness and laxity of the knee-the contributions of the supporting structures. J. Bone Jr Surg. 58A, 583-593. Moeinzadeh, M. H. (1981) Two and three dimensional dynamic modeling of human joint structures with special application to knee joint. Ph.D. Dissertation, The Ohio State University, Columbus, Ohio. Morrison, J. B. (197Oa) The mechanics of the knee joint in relation to normal walking. J. Biomechunics 3, 51-61. Morrison, J. B. (1970b) The mechanics of muscle function in locomotion. J. Biomrchunics 3, 431-451. Perry, J., Antonelli. D. and Ford, W. (1975) Analysis of kneejoint forces during flexed-knee stance. J. Bone Jr Surg. S7A 961-967.
532
CHAIYONG
WONGCHAISCWAT.
HOOSHAHG HEYAW and
Piziali. R. L., Seering, W. P.. Nagel. D.A. and Schurman. D. J. (1980a) The function of the primary ligaments of the knee in anterior-posterior and medial lateral motions. 1. Biomechunics 13, 777-784. Piziali. R. L.. Rastegar, J., Nagel, D. A., and Schurman, D. J. (1980b) The contribution of the cruciate ligaments to the load-displacement characteristics of the human knee joint. J. biomrch. Enyny 102, 277-283. Rosse, C. and Clawson, D. K. (1980) The .\fusculoskelrrul System in Health and Dismsu. Harper and Row. Seering, W. P., Piziali, R. L., Nagel. D.A. and Schurman. D. J. (1980) The function of the primary ligaments of the knee in varus-valgus and axial rotation. J. Biomrchanics 13, 185-194.
MAWARET
1. HIRES
Smidt. G. L. (1973) Biomechanical analysis of knee flexion and extension. J. Biomrchunics 6, 79-92. Trent, P.S., Walker, P.S. and Wolf, B. (1976) Ligament length patterns, strength, and rotational axes of the knee joint. Clin. Orrhop. Rd. Rex 117. 263-270. Wang, C. J.. Walker, P. S. and Wolf. B. (19731 The effects of flexion and rotation on the length patterns of the ligaments of the knee. J. Biomrchunics 6. 587-596. Wismans, J., Veldpaus. F.. Janssen. J.. Huson. A. and Struben. P. (1980) A three-dimensional mathematical model of the knee-joint. J. Eiomrchanics 13, 677-685. Wongchaisuwat. C. (1981) Modeling and control exerted by ligaments of the human knee joint in the sagittal plane. ht. S. Thesis. The Ohio State University, Columbus, Ohio.
EXERTED
BY LIGAMENTS
WONGCHAISUWAT
Department
of Electrical
and
Engineering.
HOOSHAKG
HE~IAN~
The Ohio State University
and
MARGARET J. HEXES Department
of Anatomy.
The Ohio State University.
OH 43210, U.S.A
Columbus,
Abstract-The function of the ligaments as local controllers, independent of the central nervous system. in maintainicg the integrity of the joir,t is demonstrated by modslling the human knee in the sagittal plane. and studying its anterior-posterior motion. In addition to the ligaments, the model includes the characteristic geometry of the joint surfaceand some muscle groups. The connecting reaction forces at the point ofcontact between the tibia and the femur are considered to be constraint forces due to three din‘erent surface motions-gliding. rolling and combined gliding and rolling. It is demonstrated that the ligamentous structure maintains these holonomic and nonholonomic constraints that describe the joint motion, and that stability of the knee joint is provided mainly by ligaments. Muscular structures further stabilize and contribute to joint movement. Computer simulation of rolling movement of the knee is presented to illustrate the importance of the ligaments for joint integrity and stability.
(11. (1973). A mathematical
IN’I’RODUC-I’ION
analysis (Lew and
Lewis,
1978; Lewis and Lew. 1978) and a Roentgen stereoThe function of human joints is to secure the union of
photogrammetric
the articulating
lating the ligament length patterns have been reported
bones in the body and to permit, even
facilitate. their movements The maintenance
in relation
to one another.
of function and preservation
of the
method (Dijk
while the stil‘fness of primary were studied in (Markolf
er al., 19791 for calculigaments
at the knee
ef ul., 1976; Trent et til.. 1976).
integrity of the joints are relegated to the coordinated
The extension and Rexion movements of the knee are
action of ligamcnts,capsule,
said
muscles, surrounding
tissues and the characterstic surfaces. The
individual
this composite
geometry
mechanical
structure
soft
of the joint
to
be
(Morrison,
controlled
by
1970b)--the
of
and quadriceps. These muscles have been proposed as
of
a model consisting of an active contractile
to obtain better under-
passive series elastic element
of
visco-elastic element (Glantz,
prostheticdevices.
in
forces transmitted
Specifically, the human kneejoint
the past has been selected as the object because it is anatomically
of various complicated
and functions during movement to maintain The
or range of motion functions
of the
primary
ligaments during
medial and lateral displacements
L’I u/. ;198Oa) and in varus-valgus and
internal
and
external
either rhe
of the lower limb.
human knee have been investigated posterior.
groups
gastrocnemius
components
standing of the motor control and to improvedesign
stability
muscle
have been the subject
numerous studies undertaken
investigations
three
hamstrings,
axial
femoral
of the
activities Smidt
rotations,
tibia1 rotations
by
by
1977). The
during
Morrison
these
ll970a).
Perry er al. (1975) and Bishop (1977).
studies
(Crowninshield
t’~
a/..
1976:
Moeinzadeh,
1981; Wismans et 01.. 1980) make use of
mathematical
models of the analysis of the knee joint.
anterior, by Piziali
1977; Hatze,
the knee joint
have been studied
(1973).
Other
at
element, a
and a passive parallel
In this paper, the role of the primary system ofcontrollers-for
ligaments as a
ensuring stability and main-
taining the integrity of the joint is described by means of a mathematical
model. The ligaments, the geometry
Scering ZI (11. (1980). Piziali et (11. (1980b) determined
of the joint surface and the disposition of some muscle
the contribution
groups are taken into account in the study of the gross
displacement
of the cruciatc ligaments to the load-
characteristics. The mechanical behavior
of the primary
ligaments
has been specified by vis-
movement
of the free-hanging
plane. The motion
knee in the sagittal
in this plane is so great that it
coelastic properties (Fung, 198 I). Wismans er al. (1980)
accounts for much of the joint motion. The non-linear
neglected the viscous properties and approximated
the
differential
equations
springs. The
considered
as a constrained
ligaments as quadratic primary
ligament
force-elongation
length patterns at various angles of
flexion, in neutral,and
in external and internal rotation
have been evaluated by Trent
et al. (1976) and Wang P[
reaction
I Srptembzr
1983;
,;:;_E
are
derived
dynamic
ling. The
and
system. The
forces at the point of contact
exerted during three different
between
the
forces
surface motions of the
rolling and combined
gliding and rol-
result of this study demonstrates
ligament functions as a local controller 525
H?,
motion
femur and tibia are considered to be constraint knee--gliding,
Rrcric& 24 June 1982; in reciscdform rrceiwd for publication 29 March 1984.
of
that a
that maintains
516
CHAIYONGWONGCHAISCMT, HOOSHANGHEMAMIand MARGARETJ. HIRES
the integrity of the joint. Computer simulations of rolling movement of the knee are presented to support this point of view. THE WEE
JOIST
IS THE SAGIITAL
PLASE
The present study is limited to the analysis of the anterior-posterior motion of the knee. Only the forces that act between the tibia and the femur in the sagittal plane are considered. The model includes the primary ligaments, the muscles crossing the joint and the geometry of the joint surfaces. In order to simplify the complex structure of the knee, several assumptions (Drillis and Contini, 1966; Wongchaisuwat, 1981) are made: 1. The stl-faces of the tibia and the femur are identical on the medial and lateral side. The outside rigid surface of contact is a straight line for the tibia and an ellipse for the femur (Fig. 1). 2. The femur is stationary and horizontal and the tibia moves relative to the femur. 3. The cartilage layer and the menisci between the tibia and the femur are neglected, so there is no clearance between the surfaces-i.e. the contact is at one point and it traverses along the perimeter of the ellipse. 4. The three primary ligaments affecting the kneethe anterior cruciate (ACL). the posterior cruciate (PCL), and the combined medial and lateral collateral (CLL)--are assumed to be represented by lines of action connecting respective femoral and tibia1 origin and insertion points. 5. Three main muscle groups-the hamstrings (HM), the gastrocnemius (GM) and the quadriceps femoris (QM)---are considered in the model. The
Fig. 1. The knee joint model showing the ligament forces and the contact reaction forces.
hamstrings and gastrocnemius muscles are represented by lines of action between their origin and insertion points. The line of action of the quadriceps muscle acting on the tibia is coincident with the direction of the patellar ligament. In addition, the ankle joint is assumed to be at the position where the gastrocnemius muscle force is very small. 6. The length of each ligament (or muscle) essential for developing the ligament (or the muscle) force is taken to be the distance between two sites of attachment. With these assumptions. the knee joint model studied is shown in Fig. 2. The normal and the tangential component of the reaction forces which act on the tibia at the point of contact with the femur are represented by I- = [F,, Fr]? The muscle force vector U = [F,, F1, F,]’ has three components, one each for the hamstrings, gastrocnemius and quadriceps muscle groups. The collateral, anterior cruciate and posterior cruciate ligament force vector is represented by V = [F,, F,, F,]‘. The angles of the muscle force (U) and the ligament force (V) with the horizontal axis are labeled S,, S1, S,, .Sj, S, and S6 respectively. The dimensions and the coordinate of the attachments of the ligaments and muscles on the femur and the tibia are given by Wongchaisuwat (1981). Most of these numerical values are selected from Drillis and Contini
FQ
Fig. 2. The knee joint model showing the muscle forces and gravity force.
Control
Fig. 3. Block diagram
of a feedback model of the
exerted by hgaments
knee joint.
(1966) with some values which are estimated from Crowninshield et al. (1976), Dijk et al. (1979) and Lew and Lewis (1978). The mechanical behavior of the primary ligaments can be summarized by their viscoelastic properties (Fung, 1981). The lengths of the primary ligaments expressed as the function of the position of tibia are provided in Basmajian (1978), Wismans ef al. (1980) and Wongchaisuwat (198 1). Newtonian dynamicsare used to write theequations of motion. The surface motions between the femur and the tibia are described by three idealized segmentsgliding, rolling and combined gliding and rolling. Due to these surface motions, both holonomic and nonholonomic constraints (Hemami, 1980) appear in the model (Wongchaisuwat, 1981). The reaction forces at the joint are combined in a vector of forces of constraint. It was shown in (Hemami, 1980) that the forces of constraint can be derived as functions of the state (position and velocities) and the inputs when the constraints are maintained. The above analysis provides the feedback model of the knee joint (Fig. 3). LINEARIZATION
ASD STATIC
ANALYSIS
The equations of the model are non-linear and complex. The problems of control and stability of these non-linear equations are considered here by linearization. The stiffness of the ligaments for the three dimensional case as given in the literature (Markolf et ol., 1976; Trent cf al.. 1976) with the ratio 5.x0 kp -cannot be used for this study because here the 7 Z&Y ligaments are projected onto a plane. In order to achieve the proper stiffness values for the planar study the following assumptions are made: 1. At the free hanging position, it is assumed that the tangential reaction force is zero and the three muscle groups crossing the knee joint are inactive. 2. From the free-hanging position to full flexion, the tangential reaction force resists the movement, and is positive. 3. During knee extension, in order to resist the movement, the tangential reaction force should be
1
wre
2.
comb,red
3.
pu4
>!ldwl gltdlr<
I”d
rollIn?
r,lll”g
Fig. 4. The analysis of the tangential reaction force for three surface motions.
negative. The resulting force pattern is sketched in Fig. 4. 4. It is assumed that the ankle joint is at the position where the gastrocnemius muscle force acting on it is very small, about twenty times smaller than the hamstrings muscle force exerted at the knee. In order to obtain the tangential reaction force which behaves as stated above, the stiffness and the unstrained length of the collateral, anterior cruciate and posterior cruciate ligaments are estimated as 279891 Pa, 0.043 m; 92726 Pa, 0.038 m; and 375567 Pa, 0.023 m, respectively. The stiffness values are different from those reported in Trent et al. (1976) because these are planar studies. The reaction forces and the muscle forces are plotted with the angle I) in Figs 5 and 6 respectively, It is shown that in the freehanging position where the muscle forces are zero is at U ‘c - 0.16 radian. The hamstrings and gastrocnemius muscle forces are maximum when the knee is at full llexion, and then steadily decrease to zero as the leg returns to the free-hanging position. The quadriceps muscle force gradually increases from the free-hanging to the full extension position of the tibia. In the combined gliding and rolling section shown in Fig. 6 the muscle forces may be estimated as dotted lines. The contact force on the tibia in the combined rolling and gliding movement can be calculated from the estimated muscle force. STABILITY
AND THE MUSCLE
MODEL
The unstable knee-movement system proposed in the previous sections can be stabilized when muscles are added to the model (Fig. 7). The hamstrings, gastrocnemius and quadriceps are three groups of skeletal muscle which cross the knee joint. They perform the movement and the stabilizing action for the knee (Gowitzke and Milner, 1980; Henneman, 1980; Rosse and Clawson, 1980). During the knee extension the quadriceps muscle functions as a prime
528
CH.MONG WOVXHAISW+T.
HOOSHAM HEWAM and MARGARETJ. HIXES
\
\
‘\
3
s
0
$j
‘--L-.-l+-_ ____.__
__m-me.-..
Cl
7
I
t
\ \
0
I ,.._
\
\\ ia?!
7
Ii= Ezu@Jeq
a&4,
!n
9: ____ ___.__ __------..._ __-
“qxq,
0
,,“,
.----
.
I
Control exerted by ligaments
Fig.
7.
Block diagram of the knee joint model including a state feedback part due to muscles.
mover or agonist in both maintaining and contributing to the desired movement. The hamstrings and gastrocnemius muscles function as the antagonists to oppose the quadriceps extension movement. On the other hand, the hamstrings and gastrocnemius are the prime movers and the quadriceps is an antagonist when the knee movement is reversed-i.e. in flexion. The force of gravity must be considered as the prime moving force from full Aexion to the free-hanging position as well as from full extension to the freehanging position of knee flexion. The movement of the knee in these periods is controlled by the antagonist muscles. Based on Hatze’s muscle model (Hatze, 1977). the active contractile element and the series elastic element are combined to form a simplified tension generator F. The muscle constants position feedback gain matrix K and the velocity feedback gain matrix E move the poles of the linearized system to the left half of the complex frequency plane. In this work, it is assumed that the matrices K and Bare not necessarily diagonal (Wongchaisuwat, 1981). The coupling between their elements is by way of higher control centers.
DIGITAL COMPUTER SIMULATION
The objective of the simulation is to test the effectiveness of the ligaments in maintaining the constraints of the joints. In the present work, in the free-hanging leg, the rolling movement of the tibia on the femur is simulated. The initial. final and reference state for the rolling movement are specified as X, = [0.239805, -0.027267, X,= [0.241000, -0.013732, x, = [.YfJ,Jo. 0,. 0.0, o-jr
1.508, 0, 0, O]r 1.571, 0, 0, Olr
where
I
0.063 ; + 1.508
0, =
I.571
:t ,< 10 :r > I,
x0= -0.051216 0: +0.140903 U; - 0.050667 Ue + 0.17 1422 y. = - 0.052216 0: + 0.234423 U; - 0.135630 0, - 0.176766. The initial, final and reference bias muscle forces are given as U, = [O,O, 182.7721r U,=
[O,O, 194.254]r
Uo = [O, 0, Fso]’
where
F,o =
1
:t
[O
<
10
1194.254 I.482 !- + 182.772 :I > 10.
With the above final state and bias muscle forces, the linearized equations are i=AX+Bi.
where
0 0 A=
0 9.9182 153.03 54.515
0 0
0 0
0 0 19.656 - 5.9873 339.01 -91.558 121.86 -32.902
10 0 01 0 00 I 0 0 .97656 0 0 -0.022888 0 0 -0.007629
530
CHAIYONG
WONGCHAISLWAT,
HOOWANG
HEMAW
and
F
0 B=
20
0
0
1
MARGARET
1. HINES
nt.,
i
1
0
0
0
-0.003052 - 0.032043 -0.012277
- 0.003052 - 0.039673 - 0.0 14496
0.005168 0.080856 0.029056
1
and
*
Let the desired eigenvalues associated with x, J and 8 be selected as (- 0.2, - 0.3), ( - 0.3, - 0.2) and ( - 0.3, - 0.3) respectively. The corresponding feedback gains K and B are K=
B=
i
- 35409 94085 48360
769742 - 1939944 - 573254
285292 - 713629 - 208464
- 503.27 1766.5 703.83
184404 -26216 59704
66963 -32154 I 10586
I
F3 (nt.)
:::+-
In the simulation I, is arbitrarily chosen as 0.8 s. The rolling condition is investigated by testing the constraint equations (Wongchaisuwat, 1981. ch. 5). The results of the simulation are shown in Figs 8-10. The tibia moves from its initial position to the specified final position (Fig. 8) the rolling constraint is satisfied. The muscle forces (Fig. 9) are high and become
1
ot-
80
0
_ 1
2
t (SCC)
3
Fig. 9. The muscle forces vs time.
negative-not a physically meaningful result. The reaction forces are plotted in Fig. 10. In the present simulation several points should be noted. The constants K and B are very large. To obtain smaller gains,
FN
(nt.j
t
o.-__ : 0 FT 0
0
-35
‘ -
1
2
3
1
2
3 !
(nt)
t--
-70
-105 b-
Fig. 8. The position of the tibia vs time.
t (XC)
Fig. 10. The reaction forces vs time.
t (set)
Control
exerted by ligaments
the desired eigenvalues associated with x, J and 0 should be selected closer to the imaginary axis. With a proper set of diagonal feedback gains, the simulations in the vicinity of the operating points for two other surface motions, gliding and combined gliding and roiling can be performed in the same manner as the rolling simulation.
DISCUSSIOS
ASD COSCLL’SIOSS
A model of the human kneejoint in thesagittal plane was presented as a constrained system. The primary ligamentous function is that ofa local controller, which maintains the integrity of the joint. in addition to the ligaments, the geometry of the joint surface is recognized as being important in this function. Three surface motions of the knee-gliding, rolling, and combined gliding and rolling-were considered. Holonomic and nonholonomic constraint equations describe these three movements. It was shown that the ligamentous structures maintain the system on the constrained surface. They preserve the joint function and relative motion while the muscular structures both contribute to the movement and further stabilize the system. Substantial geometric and kinesiologic simplifications were made. The primary ligaments affecting the knee were assumed to be represented by quadratic non-linear springs. The muscles were simply modeled as force generators controlled by a central system and the visco-elastic elements of the surrounding tissues. A kinematic study of three surface motions of the knee was performed to supply the lengths and the lines of action of the muscles and ligaments. The stiffness and the unstrained length of the ligaments were estimated to ensure the zero tangential constraint force in the combined gliding and rolling motion. Since the mode was formulated for planar studies only, the parameters used in this work were contrasted with those reported in the literature. The problem of stability of the knee was considered by properly selecting the stiffness and the unstrained length of the ligaments and determining the linear position and velocity feedback gains for the visco-elastic elements of the muscles. In this work, the assumption was made that the muscle constants K and B. i.e. position and velocity feedback gains, were coupled. Digital computer simulations of the rolling motion were carried out to test the effectiveness of the ligamentous structures in maintaining the constraints. The rolling movement was executed properly in spite of the inadequacy of the muscle models. With a better muscle model, more complete simulation for the whole sequence of motion between full flexion and full extension of the knee may be carried out. This work should provide a better understanding of the functional role of the ligaments of the knee joint. Since the actual movement in extension of the knee occurs simultaneously in three planes according to its
531
‘screw-home’ or locking mechanism, further study requires that the joint be changed into a threedimensional system. More studies should be undertaken which consider the geometry of the articular surfaces, including the menisci. a more precise modeling of the ligaments and the muscles, and the role of the articular cartilages. With more realistic models, using the principles enunciated here. one should be able to demonstrate that the knee joint stability is usually provided primarily by ligaments and the joint surface structures, and secondarily by muscles.
A~~norlrdyrmunt-This work was supported by National Science Foundation under grant ECS-8201240.
[he
REFERENCES Basmajian. J. V. (1978) I%fusclr A/it.e 4th Edn. William and Wilkins, Baltimore. Bishop, R. E. D. (1977) On the mechanics of the human knee. Enyny Mrd. 6. 46-52. Crowninshield. R.. Pope, M. H. and Johnson. R. J. (1976) An analytical model of the knee. J. Eiomechunics 9. 397-405. Dijk, R. van, Huiskes. R. and Selvik, G. (1979) Roentgen stereophotogrammetric methods for the evaluation of the three dimensional kinematic behaviour and cruciate ligament length patterns of the human knee joint. J. Biomdrunicx 12. 727-73 I. Drillis. R. and Contini, R. (1966) Body segment parameters. Tcxhnical report no. 1116.03. New York University, University Heights, New York. Fung. Y. C. (I98 I) Biomechunics: ~~lechunicul Propwrks o/ Liciny Tissue. Springer, Berlin. Glantz. S. A. (1977) A three-element description for muscle with viscoelastic passive elements. J. Biomrrhunics IO, 5-20. Gowitzke. B. A. and Milner, M. (1980) Undrrsranding rhr Shvrri/rrc- Buses ol Human Mocemrnr (2nd Edn.) William and Wilkins, Baltimore. Hatze. H. (1977) A myocybernetic control model of skeletal muscle. Bioloyicul Crbrrnrrics ZS, 103-l 19. Hemami. H. (1980) A feedback on-off model of biped dynamics. IEEE Trans. on Sysr. Xfun C_ybernrr. SIMC-10, 376383. Henneman. E. (1980) Skeletal muscle. Mrdid Phpsiolog) (Edited by Mountcastle. V. B.) 14thedn. Vol. I. p. 674.C. V. Mosby. St. Louis. Lew, W. D. and Lewis, J. L. (1978) A technique for calculating in cico ligament lengths with application to the human knee joint. J. Eiomrchunics 1 I, 365-377. Lewis. J. L. and Lew. W. D. (1978) A method for locating an optimal “fixed” axis of rotation for the human knee joint, J. biomech. Engng 100, 187-193. Markolf. K. L.. Mensch, J. S. and Amstutz, H. C. (1976) Stiffness and laxity of the knee-the contributions of the supporting structures. J. Bone Jr Surg. 58A, 583-593. Moeinzadeh, M. H. (1981) Two and three dimensional dynamic modeling of human joint structures with special application to knee joint. Ph.D. Dissertation, The Ohio State University, Columbus, Ohio. Morrison, J. B. (197Oa) The mechanics of the knee joint in relation to normal walking. J. Biomechunics 3, 51-61. Morrison, J. B. (1970b) The mechanics of muscle function in locomotion. J. Biomrchunics 3, 431-451. Perry, J., Antonelli. D. and Ford, W. (1975) Analysis of kneejoint forces during flexed-knee stance. J. Bone Jr Surg. S7A 961-967.
532
CHAIYONG
WONGCHAISCWAT.
HOOSHAHG HEYAW and
Piziali. R. L., Seering, W. P.. Nagel. D.A. and Schurman. D. J. (1980a) The function of the primary ligaments of the knee in anterior-posterior and medial lateral motions. 1. Biomechunics 13, 777-784. Piziali. R. L.. Rastegar, J., Nagel, D. A., and Schurman, D. J. (1980b) The contribution of the cruciate ligaments to the load-displacement characteristics of the human knee joint. J. biomrch. Enyny 102, 277-283. Rosse, C. and Clawson, D. K. (1980) The .\fusculoskelrrul System in Health and Dismsu. Harper and Row. Seering, W. P., Piziali, R. L., Nagel. D.A. and Schurman. D. J. (1980) The function of the primary ligaments of the knee in varus-valgus and axial rotation. J. Biomrchanics 13, 185-194.
MAWARET
1. HIRES
Smidt. G. L. (1973) Biomechanical analysis of knee flexion and extension. J. Biomrchunics 6, 79-92. Trent, P.S., Walker, P.S. and Wolf, B. (1976) Ligament length patterns, strength, and rotational axes of the knee joint. Clin. Orrhop. Rd. Rex 117. 263-270. Wang, C. J.. Walker, P. S. and Wolf. B. (19731 The effects of flexion and rotation on the length patterns of the ligaments of the knee. J. Biomrchunics 6. 587-596. Wismans, J., Veldpaus. F.. Janssen. J.. Huson. A. and Struben. P. (1980) A three-dimensional mathematical model of the knee-joint. J. Eiomrchanics 13, 677-685. Wongchaisuwat. C. (1981) Modeling and control exerted by ligaments of the human knee joint in the sagittal plane. ht. S. Thesis. The Ohio State University, Columbus, Ohio.