SYNTHESIS OF HUMAN MOVEMENT—FORWARD SOLUTIONS - Biomechanics and Motor Control of Human Movement - page 210

Figure 8.2 Sliding mass example where mass m 1 moves horizontally and mass m 2
moves along the top of the sloped surface of m 1 .

8.2.7 Displacement and Velocity Vectors
In the following discussion, the notations r , v , x , y , z , ... areusedforLRS
while R , V , X , Y , Z , ... are used for GRS. The letters c and s are used to
denote cosine and sine, respectively. The derived displacement and velocity
vectors can be saved in two groups of lists, disp and velo, where an entry (i )
is coded as follows:

disp (i ) = [ x i , y i , z i ] (8.13)

velo( i ) = [ x i , ˙ y i , ˙ z i ] ˙ (8.14)

Equations (8.13) and (8.14) represent the displacement and the velocity,
respectively, of pt(i ) . Saving the derived components rather than deriving
them when they are needed will save considerable time, especially in the
case of complex systems. The angular velocity vectors of LRSs can also be
saved in a group, omga. An entry (j ) in this group represents components of
the angular velocity of LRS (j ) :

Omga (j ) = [ ω jx , ω jy , ω jz ] (8.15)

8.2.7.1 Two-Dimensional Systems. Let pt(a) be a moving point in the
LRS (i ) (Figure 8.3). Let R a be the absolute position vector for the same
point transformed to the directions of the GRS. From vector algebra,

R a = R i + [ φ ] r ia (8.16)

where φ is the angle between x i and X .
The same equation in expanded form is:
X
Y

X
Y

i +

c X

Y

s
sc

−

(8.16 )

a = ia