Without any matrix multiplication required, we obtain the fully simplified relation involving e3 only:
(p2-a22-a 2-d32-d42)/2 = a2 (d4 S3 + a3 C3).
The last equation yields at most two solutions for e3. After applying trigonometric identities for angle sums to (9.2), we get
Pz = a2 S2 + a3 (S2C3+S3C2) d4 (C2C3-S2S3), and grouping terms, we obtain
(a2+a3C3+d4S3) S2 + (a3S3-d4C3) C2 = Pz"
With each value of 83, 2 values for e2 can be obtained from this last equation.
With e2 and e3 known, (9.1) and (9.3) become functions of 84 and e5 only. Although this system of 2 equations in two unknowns can theoretically be solved, its solution is not obvious. A simpler solution exists if Eqs. (4.13) to (4.16) are considered,
p x = R1(R2(R314 + 13) + 12) x = px, (9.5) p y = R1(R2(R314 + 13) + 12) y = py, (9.6) or
(d4S23+a3C23+a2C2) C1 + d3 S1 = Px
-d3 C1 + (d4S23+a3C23+a2C2) S1 = Py.
..