r x S = So hS
Since r = H + EH, we can have the following equations:
(H + rH) x S = So hS (5.35)
or
rH x S + hS = So H x S (5.36)
The velocity of the center H of the platform can be
obtained as
YH = (rEH x S + hS) (5.37)
or
MH = u(So H x S) (5.38)
Then the linear displacement of H can be obtained as
8!H = 68(So H x S) (5.39)
As we consider differential motions, let us define the
rotation matrix of a small rotation angle 88 about unit
vector S. For small angles, sin6E = 69, cos68 = 1 and V68
= cos68 1 = 0. Thus, we can have the rotation matrix
with small 68 as
Sx2V68+C68 SxSyV6g-SS68 SxSzV88+SySS0 0
SG- SyV68+SzS68 S 2V68e+C86 SyS,V68-SS68 0
R[69, S] =
S,[SV68-SyS6Q SySzV6S+SxS68 S 2V66+C6e 0
0 0 0 1
(continued)
(5.34)