262
C. Morrow et al.
can be in order to achieve a specific torque. The cost function adds a constant
value for each muscle depending on the diameter calculated. These constants
appear as G 40 , G 20 , and G 10 in Eq. 5. The value for these constants are larger
for larger diameters and smaller for smaller diameters.
C Diameter =
n
Muscle=1
⎧
⎪ ⎨
⎪ ⎩
G 40 , φ Muscle = 40 mm
G 20 , φ Muscle = 20 mm
G 10 , φ Muscle = 10 mm
(5)
The starting PAM locations for the muscle placements come from Bolen and
Hunt [2]. The algorithm creates a starting cost value based on these previously
determined positions. The algorithm then conducts an exhaustive search iteration for eight points around the starting locations. It chooses new locations for
the muscle attachment points that are before a joint and after a joint. Eight
new locations are created for each of these attachment points, by adding or subtracting a small distance, , to the original location in three directions. These
points define a cube around the original point for the algorithm to conduct its
search. Then the algorithm begins to calculate the updated cost function for
each combination of these new points. Once all calculations have completed, the
algorithm updates the new starting points with the points that correspond to the
minimum cost value. The algorithm then repeats the generation of new points
until it has found a local minima. Once this has occurred, the epsilon value used
to determine new via point locations is decreased by multiplying it by a refinement rate term, γ. This tightens the range of analyzed points about the current
minimum. When this process has been repeated and epsilon has been reduced
to a negligible value, the algorithm ends.
3 Results
This section shows the results of the algorithm performing optimization about
the set of muscles on the right side of the lumbrosacral joint. This joint was
chosen as a test case for the algorithm, as it only includes three uniarticular
muscles: the erector spinae, internal oblique, and external oblique. By focusing
on this joint, the algorithm can be tuned to perform better for joints with larger
sets of muscles. Only the muscles on the right side of the body are considered,
as the left side of the body will mirror the results of the right side. The axes of
interest that the torque acts about are the pelvis x and z axis, found in Fig. 3.
Rotation about these axes are lateral bending of the back (x axis) and back
flexion and extension (z axis). Figure 4 shows torque surfaces constructed from
the human model and the generated model. Each point along the surface shows
the combined torque of all muscles about the joint when the joint is rotated
to the specified degrees. While the new model doesn’t match the human model
completely, it is an improvement over the previously hand placed model. This
improvement can be seen in Fig. 5, which shows the absolute torque differences
between the previous model and the human model as well as the absolute torque
C. Morrow et al.
can be in order to achieve a specific torque. The cost function adds a constant
value for each muscle depending on the diameter calculated. These constants
appear as G 40 , G 20 , and G 10 in Eq. 5. The value for these constants are larger
for larger diameters and smaller for smaller diameters.
C Diameter =
n
Muscle=1
⎧
⎪ ⎨
⎪ ⎩
G 40 , φ Muscle = 40 mm
G 20 , φ Muscle = 20 mm
G 10 , φ Muscle = 10 mm
(5)
The starting PAM locations for the muscle placements come from Bolen and
Hunt [2]. The algorithm creates a starting cost value based on these previously
determined positions. The algorithm then conducts an exhaustive search iteration for eight points around the starting locations. It chooses new locations for
the muscle attachment points that are before a joint and after a joint. Eight
new locations are created for each of these attachment points, by adding or subtracting a small distance, , to the original location in three directions. These
points define a cube around the original point for the algorithm to conduct its
search. Then the algorithm begins to calculate the updated cost function for
each combination of these new points. Once all calculations have completed, the
algorithm updates the new starting points with the points that correspond to the
minimum cost value. The algorithm then repeats the generation of new points
until it has found a local minima. Once this has occurred, the epsilon value used
to determine new via point locations is decreased by multiplying it by a refinement rate term, γ. This tightens the range of analyzed points about the current
minimum. When this process has been repeated and epsilon has been reduced
to a negligible value, the algorithm ends.
3 Results
This section shows the results of the algorithm performing optimization about
the set of muscles on the right side of the lumbrosacral joint. This joint was
chosen as a test case for the algorithm, as it only includes three uniarticular
muscles: the erector spinae, internal oblique, and external oblique. By focusing
on this joint, the algorithm can be tuned to perform better for joints with larger
sets of muscles. Only the muscles on the right side of the body are considered,
as the left side of the body will mirror the results of the right side. The axes of
interest that the torque acts about are the pelvis x and z axis, found in Fig. 3.
Rotation about these axes are lateral bending of the back (x axis) and back
flexion and extension (z axis). Figure 4 shows torque surfaces constructed from
the human model and the generated model. Each point along the surface shows
the combined torque of all muscles about the joint when the joint is rotated
to the specified degrees. While the new model doesn’t match the human model
completely, it is an improvement over the previously hand placed model. This
improvement can be seen in Fig. 5, which shows the absolute torque differences
between the previous model and the human model as well as the absolute torque