11.t [R2-1 12 R3 z] = t.p [R3-1 13 z]. (5.11) Equation (5.11) is also independent of 92 and 84.
Using Eq. (5.4), the inner-product p.p satisfies p.p = [R1 (R2 13 + 12) + 11].[R1 (R2 13 + 12) + 11']. Expanding the left hand side, using inner-product invariance of rotations where needed, and rearranging terms yield 13.R2- 1 12 + p.11 = [p.p + 11.11 12.12 13.13]/2. (5.12)
Equations (5.11), (5.2), (5.18), and (5.12) form a linear system in the variables S1, C1, S3 and C3. The four equations obtained are
alty S1 + altx C1 + a2C3 S3 02c3d2 C3 = r1 (5.13)
01tx S1 Olty C1 + 0203 C3 = r2 (5.14)
0lpx S1 alPy C1 a2a3 S3 = r3 (5.15)
alPy S1 + alpx C1 + a2a3d2 S3 + a2a3 C3 = r4 (5.16)
with
rI = t.p T3d3 dltz T23d2 (5.17)
r2 = 273 -rtz (5.18)
r3 = T(d tz) + d2 + T2d3 (5.19)
r4 (p.p + al2+d 2-a22-d22-a3 2-d32)/2
-dlPz r2d2d3. (5.20)
The linear system of equations formed by Eqs. (5.13)-(5".16) will be _referred to as the reduced system for a four-DOF
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