the three rotational variables of the spherical joint B,
which play no part in the manipulation of the platform.
Postmultiplying both sides of Eq. (2.29) by A3-1A2-1
yields Eq. (2.30). Premultiplying both sides of Eq. (2.29)
by Al-1 and then postmultiplying both sides by A3-1 yields
Eq. (2.31).
Al = H0A3-1A2-1 (2.30)
A2 = A1-1HA3-1 (2.31)
Since ([Al](1,4))2 + ([A1](2,4) 2 = ([HOA3-1A2-11(1,4) 2 +
([HOA3-1A2-1](2,4))2 is true, we can obtain the following
equation:
(ax2 + ay2)dc2 + 2[(c +b)nxax + (c + b)nyay axpx
-aypy]dc + (c + b)2(nx2 + ny2) + px2 + py2 -
2(c + b)(nxPx + nypy) a2 = 0 (2.32)
Equation for 8, is obtained since [A1](2,4) / [A](1,4) =
[HOA3-1A2-1](2,4) / [HOA3-1A2-1]1,4) holds, and can be
expressed as
Sta -([ + b)ny ayd + y (2.33)
9a = tan-[ ] (2.33)
-(c + b)nx axdc + px
It is observed that [Al](3,4) = [HOA3-1A2-1](3,4) directly
implies the translation of joint A as
da = -(c + b)nz azdc + pz
(2.34)