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Techniques in the
Dynamic Modeling of
Human Joints with a
Special Application to
3
the Human Knee
3.1
3.2
Representation of the Relative Positions • Joint Surfaces and
Contact Conditions • Ligament and Contact Forces • Equations
of Motion • Numerical Solution Procedure
3.3
Solution Methods
Model Description • Alternative Methods of Solution
3.4
Joint Models
Numerical Results and Discussion
Ali E. Engin
3.5
Knee Joint
University of South Alabama
3.1 Introduction
Joints in the human skeletal structure can be roughly classified into three categories according to the
amount of movement available at the joint. These categories are named synarthroses (immovable),
amphiarthroses (slightly movable) and diarthroses (freely movable). The skull sutures represent examples
of synarthrodial joints. Examples of amphiarthrodial joints are junctions between the vertebral bodies
and the distal tibiofibular joint. The main interest of this chapter is the biomechanic modeling of the
major articulating joints of the upper or lower extremities that belong to the last category, the diarthroses.
In general, a diarthrodial joint has a joint cavity which is bounded by articular cartilage of the bone ends
and the joint capsule.
The bearing surface of the articular cartilage is almost free of collagen fibers and is thus true hyaline
cartilage. From a biomechanics view, articular cartilage may be described as a poroelastic material
composed of solid and fluid constituents. When the cartilage is compressed, liquid is squeezed out, and,
when the load is removed, the cartilage returns gradually to its original state by absorbing liquid in the
process. The time-dependent behavior of cartilage suggests that articular cartilage might also be modeled
as a viscoelastic material, in particular, as a Kelvin solid.
© 2001 by CRC Press LLC
 The joint capsule encircles the joint and its shape is dependent on the joint geometry. The capsule
wall is externally covered by the ligamentous or fibrous structure (fibrous capsule) and internally by
synovial membrane which also covers intra-articular ligaments. Synovial membrane secretes the synovial
fluid which is believed to perform two major functions. It serves as a lubricant between cartilage surfaces
and also carries out metabolic functions by providing nutrients to the articular cartilage. The synovial
fluid is non-Newtonian in behavior, i.e., its viscosity depends on the velocity gradient. Cartilage and
synovial fluid interact to provide remarkable bearing qualities for the articulating joints. More informa-
tion on properties of articular cartilage and synovial fluid can be found in a book chapter written by the
author in 1978.
8
Structural integrity of the articulating joints is maintained by capsular ligaments and both extra- and
intra-articular ligaments. Capsular ligaments are formed by thickening of the capsule walls where func-
tional demands are greatest. As the names imply, extra- and intra-articular ligaments at the joints reside
external to and internal to the joint capsule, respectively. Extra-articular ligaments have several shapes,
e.g., cord-like or flat, depending on their locations and functions. These types of ligaments appear
abundantly at the articulating joints. However, only the shoulder, hip, and knee joints contain intra-
articular ligaments. The cruciate ligaments at the knee joint are probably the best known intra-articular
ligaments. Further information about the structure and mechanics of the human joints is available in
Reference 1.
Returning to the modeling aspects of the articulating joints, in particular kinematic behavior of the
joints, we can state that in each articulating human joint, a total of six degrees-of-freedom exist to some
extent. One must emphasize the point that degrees-of-freedom used here should be understood in the
sense the phrase is defined in mechanics, because the majority of the anatomists and the medical people
have a different understanding of this concept; e.g., both Steindler
and MacConaill
26
imply that the
maximum number of degrees-of-freedom required for anatomical motion is three.
Major articulating joints of the human have been studied and modeled by means of joint models
possessing single and multiple degrees-of-freedom. Among the various joint models the hinge or revolute
joint is probably the most widely used articulating joint model because of its simplicity and its single
degree-of-freedom character. When the articulation between two body segments is assumed to be a hinge
type, the motion between these two segments is characterized by only one independent coordinate which
describes the amount of rotation about a single axis fixed in one of the segments. Although the most
frequent application of the hinge joint model has been the knee, the other major joints have been treated
as hinge joints in the literature, sometimes with the assumption that the motion takes place only in a
particular plane, especially when the shoulder and the hip joints are considered.
When the degrees-of-freedom allowed in a joint model are increased from one to two, one obtains a
special case of the three-degrees-of-freedom spherical or ball and socket joint. Two versions of this
spherical joint which have received some attention in the literature. In the first version, no axial rotation
of the body segment is allowed and the motion is determined by the two independent spherical coordi-
nates
φ
and
θ
φ
, which represents the axial rotation of one of the body segments,
is introduced. The planar joint model, as the title suggests, permits the motion on a single plane and is
characterized by two Cartesian coordinates of the instantaneous center of rotation and one coordinate,
θ
, a third independent coordinate,
ψ
appears
to be the first to apply the instant centers technique to the planar motion study of the knee joint.
The six-degrees-of-freedom joint (general joint) allows all possible motions between two body seg-
ments. A good example of a general joint is the shoulder complex, which exhibits four independent
articulations among the humerus, scapula, clavicle, and thorax. Of course, at the shoulder complex, the
six degrees-of-freedom refer to the motion of the humerus relative to the torso. If one considers the total
6
© 2001 by CRC Press LLC
34
as shown in
In the second version, the axial rotation is allowed but the motion is
restricted to a particular plane passing through the center of the sphere. Again, most of the major joints
have been modeled by the two-degree-freedom spherical joint models by various investigators.
If we increase the degrees-of-freedom to three, we get the two obvious joint models, namely, the ball
and socket joint model and the planar joint model. For the ball and socket joint model in addition to
and
θ
, defining the amount of rotation about an axis perpendicular to the plane of motion. Dempster
Spherical or ball and socket joint is illustrated. Figure displays both versions of the two degrees-of-
freedom as well as the most general three-degrees-of-freedom spherical joint.
However, the selection of objective functions appears to
be arbitrary, and justification for such minimization criteria is indeed debatable.
Another technique dealing with the indeterminate nature of joint modeling considers the anatomical
and physiological constraint conditions together with the equilibrium or dynamic equations. These
constraint conditions include the fact that soft tissues only transmit tensile loads while the articulating
surfaces can only be subjected to compression. Electromyographic data from the muscles crossing the
joint also provide additional information for the joint modeling effort. The different techniques used by
various researchers mainly vary as to the method of applying these conditions. At one extreme, all
unknowns are included in the equilibrium or dynamic equations. A number of unknown forces are then
assumed to be zero to make the system determinate so that the reduced set of equations can be solved.
This process is repeated for all possible combinations of the unknowns, and the values of the joint forces
are obtained after discarding the inadmissible solutions.
32,33
At the other extreme, first the primary functions
of all structures are identified and equations are simplified before they are solved.
5
A combination of
these two techniques was also used to solve several quasi-static joint modeling problems.
30
3,4
All models described in the previous paragraph are quasi-static. That is, the equilibrium equations
together with the inertia terms are solved for a known kinematic configuration of the joint. Complexity
of joint modeling becomes paramount when one considers a true dynamic analysis of an articulating
joint structure possessing realistic articulating surface geometry and nonlinear soft tissue behavior.
Because of this extreme complexity, the multisegmented models of the human body, thus far, have
employed simple geometric shapes for their joints.
© 2001 by CRC Press LLC
FIGURE 3.1
number of degrees-of-freedom for the motions executed by the various bones of the shoulder complex,
one can easily reach a number higher than six, even with the proper consideration of various constraints
present in the joint complex.
A substantial difficulty in theoretical modeling of human joints arises from the fact that the number
of unknowns are usually far greater than the number of available equilibrium or dynamic equations.
Thus, the problem is an indeterminate one. To deal with this indeterminate situation, optimization
techniques have been employed in the past.
 cites mathematical joint models that consider both the geometry of the joint
surfaces and behavior of the joint ligaments, these models are quasi-static in nature, and employ the so-
called inverse method in which the ligament forces caused by a specified set of translations and rotations
along the specified directions are determined by comparing the geometries of the initial and displaced
configurations of the joint. Furthermore, for the inverse method utilized in Reference 40, it is necessary
to specify the external force required for the
40
equilibrium configuration. Such an approach is
applicable only in a quasi-static analysis. For a dynamic analysis, the equilibrium configuration preferred
by the joint is the unknown and the mathematical analysis is required to provide that dynamic equilibrium
configuration.
In this chapter, a formulation of a three-dimensional mathematical dynamic model of a general two-
body-segmented articulating joint is presented first. The two-dimensional version of this formulation
subsequently is applied to the human knee joint to investigate the relative dynamic motion between the
femur and tibia as well as the ligament and contact forces developed in the joint. This mathematical joint
model takes into account the geometry of the articulating surfaces and the appropriate constitutive
behavior of the joint ligaments. Representative results are provided from solutions of second-order
nonlinear differential equations by means of the Newmark method of differential approximations and
application of the Newton-Raphson iteration process. Next, to deal with shortcomings of the iterative
method, alternative methods of solution of the same dynamic equations of the joint model are presented.
With improved solution methods, the dynamic knee model is utilized to study the response of the knee
to impact loads applied at any location on the lower leg. The chapter also deals with the question of
whether the classical impact theory can be directly applied to dynamic joint models and its limitations.
In addition, the two-body segmented joint model is extended to a three-body segmented formulation,
and an anatomically based dynamic model of the knee joint which includes patello-femoral articulation
is presented to assess patello-femoral contact forces during kicking activity.
preferred
3.2 General Three-Dimensional Dynamic Joint Model
The articulating joint is modeled by two rigid body segments connected by nonlinear springs simulating
the ligaments. It is assumed that one body segment is rigidly fixed while the second body segment is
undergoing a general three-dimensional dynamic motion relative to the fixed one. The coefficients of
friction between the articulating surfaces are assumed to be negligible. This is a valid assumption due to
the presence of synovial fluid between the articulating surfaces.
31
Accordingly, the friction force between
the articulating surfaces will be neglected.
The main thrust of this section is the presentation of a mathematical modeling of an articulating joint
defined by contact surfaces of two body segments which execute a relative dynamic motion within the
constraints of ligament forces. Mathematical equations for the joint model are in the form of second-
order nonlinear differential equations coupled with nonlinear algebraic constraint conditions. Solution
of these differential equations by application of the Newmark method of differential approximation and
subsequent usage of the Newton-Raphson iteration scheme will be discussed. The two-dimensional
version of the dynamic joint model will be applied to the human knee joint under several dynamic
loading conditions on the tibia. Results for the ligament and contact forces, contact point locations
between the femur and tibia, and the corresponding dynamic orientation of the tibia with respect to
femur will be presented.
Representation of the Relative Positions
The position of the moving body segment 1 relative to fixed body segment 2 is described by two coordinate
systems as shown in
The inertial coordinate system (x, y, z) with unit vectors
ˆˆ
and is
k
connected to the fixed body segment and the coordinate system (
x
′
,
y
′
,
z
) with unit vectors
ˆˆ
and
k
′
is attached to the center of mass of the moving body segment.
© 2001 by CRC Press LLC
Although the literature
i, j
′
′ ′
i,j
ˆ
 A two-body segmented joint is illustrated in three dimensions, showing the position of a point, Q,
attached to the moving coordinate system (
x
′
,
y
′
,
z
).
) coordinate system is also taken to be the principal axis system of the moving body
segment. The motion of the (
x
′
y
′
,
z
′
) system relative to the fixed (x, y, z) system may be characterized
by six quantities: the translational movement of the origin of the (
x
′
,
y
′
,
z
′
x
′
,
y
′
,
z
′
) system in the x, y, and z
rotations with respect to the x, y, and z axes.
Let the position vector of the origin of the (
θ
,
φ
, and
ψ
x
′
,
y
′
,
z
′
) system in the fixed system be given by (
:
rxi yj zk
o
=++
ˆ
ˆ
ˆ
(3.1)
o
o
o
Let the vector,
ρ
Q
′
be
t
he position vector of an arbitrary point,
Q
, on the moving body segment in the
ˆˆˆ
ˆˆˆ
base (
i,j,k
). Let be the position vector of the same point in the base (
r
Q
i, j, k
). That is,
ρ
QQ Q Q QQ Q Q
= ′′+ ′′+ ′′ =++
xi y j zk, r xi y j zk
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(3.2)
Referring to
vectors and have the following relationship:
ρ
Q
′
r
Q
{}
=
{}
+
[]
r
r
T
T
{}
ρ
′
(3.3)
Q
o
Q
) system
with respect to the (x, y, z) system is specified by the nine components of the [T] matrix and can be
written as a function of the three variables,
×
3 orthogonal transformation matrix. The angular orientation of the (
x
′
,
y
′
z
′
θ
,
φ
, and
ψ
:
T = T(
θ
,
φ
,
ψ
)
(3.4)
© 2001 by CRC Press LLC
FIGURE 3.2
′
The (
,
directions, and
′ ′ ′
where [T] is a 3
,
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