Page 187 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

164 BIOMECHANICS OF THE HUMAN BODY

been shown that Kane’s formulation of the dynamical equations of motion is computationally more
efficient than its counterpart, the Newton-Euler method (Kane and Levinson, 1983).
The governing equations of motion for any multijoint system can be expressed as

2
M()qq + C()qq + G () + R() F MT + E (, ) = 0 (7.1)
q
q
q
q
where qqq,, are vectors of the generalized coordinates, velocities, accelerations, respectively; M()q
is the system mass matrix and M()qq is a vector of inertial forces and torques; C()qq 2 is a vector
of centrifugal and Coriolis forces and torques; Gq() is a vector of gravitational forces and torques;
R() is the matrix of muscle moment arms (see Sec. 7.4.3), F MT is a vector of musculotendon
q
forces, and R()qF MT is a vector of musculotendon torques; and Eq q(, ) is a vector of external
forces and torques applied to the body by the environment.
If the number of dof of the motor task is greater than, say, 4, a computer is needed to obtain Eq. (7.1)
explicitly. A number of commercial software packages are available for this purpose, including
AUTOLEV by On-Line Dynamics Inc., SD/FAST by Symbolic Dynamics Inc., ADAMS by Mechanical
Dynamics Inc., and DADS by CADSI.

7.4 MUSCULOSKELETAL GEOMETRY

7.4.1 Modeling the Paths of Musculotendinous Actuators
Two different methods are used to model the paths of musculotendinous actuators in the body: the
straight-line method and the centroid-line method. In the straight-line method, the path of a muscu-
lotendinous actuator (muscle and tendon combined) is represented by a straight line joining the cen-
troids of the tendon attachment sites (Jensen and Davy, 1975). Although this method is easy to
implement, it may not produce meaningful results when a muscle wraps around a bone or another
muscle (see Fig. 7.10). In the centroid-line method, the path of the musculotendinous actuator is rep-
resented as a line passing through the locus of cross-sectional centroids of the actuator (Jensen and
Davy, 1975). Although the actuator’s line of action is represented more accurately in this way, the
centroid-line method can be difficult to apply because (1) it may not be possible to obtain the loca-
tions of the actuator’s cross-sectional centroids for even a single position of the body and (2) even if
an actuator’s centroid path is known for one position of the body, it is practically impossible to deter-
mine how this path changes as body position changes.
One way of addressing this problem is to introduce effective attachment sites or via points at spe-
cific locations along the centroid path of the actuator. In this approach, the actuator’s line of action
is defined using either straight-line segments or a combination of straight-line and curved-line seg-
ments between each set of via points (Brand et al., 1982; Delp et al., 1990). The via points remain
fixed relative to the bones even as the joints move, and muscle wrapping is taken into account by
making the via points active or inactive depending on the configuration of the joint. This method
works quite well when a muscle spans a 1-dof hinge joint, but it can lead to discontinuities in the
calculated values of moment arms when joints have more than 1 rotational dof (Fig. 7.10).

7.4.2 Obstacle-Set Method
An alternate approach, called the obstacle-set method, idealizes each musculotendinous actuator as
a frictionless elastic band that can slide freely over the bones and other actuators as the configura-
tion of the joint changes (Garner and Pandy, 2000). The musculotendinous path is defined by a series
of straight-line and curved-line segments joined together by via points, which may or may not be
fixed relative to the bones.
To illustrate this method, consider the example shown in Fig. 7.11, where the path of the actu-
ator is constrained by a single obstacle, which is fixed to bone A. (An obstacle is defined as any