SYNTHESIS OF HUMAN MOVEMENT—FORWARD SOLUTIONS - Biomechanics and Motor Control of Human Movement - page 215

8.3.1 Segment Energy
As was previously discussed, a segment is replaced by an arbitrary LRS. Let
pt(j ) be the origin of the segment LRS and pt(c) be its center of mass. The
segment has a mass M and inertia tensor [ J ]. The formula for calculating the
kinetic energy of the segment has the form (Wittenburg, 1977):

2 M v j t v j +

M v j t ˜ ω r jc +

1 1

2 [ ω ] t [ J ][ ω ]

KE = (8.23)

A closer look at the first and third terms in this formula reveals the
well-known kinetic energy formula I ω 2 ) . The second term reflects

the selection of the LRS origin at a point rather than the center of mass of
the segment. When pt(c) and pt(j ) coincide, the second term vanishes. The
inertia tensor [ J ]isa3

1 / 2 (mυ 2

+

3 matrix containing the mass moments of iner-
tia (I xx , I yy , I zz ) and the mass products of inertia (I xy , I xz , I yz ) of the segment
relative to its LRS. The inertia tensor has the general form:

×

⎡ ⎤

I xx − I xy − I xz

⎣ ⎦

[ J ] = − I yx I yy − I yz (8.24)

− I zx − I zy I zz

It is possible to select LRS in such a way that the products of inertia vanish
and the tensor [ J ] becomes a diagonal matrix. The axes of the selected LRS
are then called the principal axes of the segment .
If the Z axis of the GRS is selected in the direction of the gravity field and
its origin is taken as the zero level of the system, then the potential energy
of a segment j is defined as

PE j = MgZ c (8.25)

where g is the gravity constant and Z c is the z component of R c . It should
be noted that the first element of the pt(c) list is the segment LRS index
[Equation (8.11)]. Hence the pt( c ) list contains the necessary information
about segment kinematics. It follows that a segment i is completely specified
when its mass, inertia, and the index of its center point are given. This
information can be stacked in the seg (i ) list as follows:

seg (i ) = [ c , M , I xx , I yy , I zz , I xy , I xz , I yz ] (8.26)

There are many variations of this list. When segment LRS is a principal
system, the last three elements are not required. A segment in a 2D system
is completely specified by the first three elements in the list, while a particle is
specified by two elements only. It should be noted that a massless segment
(i.e., a rigid link that joins two points) need not be specified. Hence, its LRS
replaces it completely.