We define two nonlinear functions of 91 and E2 as differences between the inner-products uL qL, qL.qL and the inner-products u.q and q.q, respectively;
f(81,82) = uLqL u.q, (7.12)
g(81,82) qL-qL q.q. (7.13)
If the values of 81 and E2 used to compute pose matrix Q in Eq. (7.3) do correspond to a solution set, then Eq. (7.2) will hold and vectors UL and qL will be exactly equal to u and q forcing both functions f and g to be equal to zero. In other words solution sets of Eq. (7.1) correspond to zeros of the functions f and g defined in Eqs. (7.12) and (7.13).
Computing f(812) and q(91,82).
In order to compute the values of f and g for a given pair (81,82), the components of vectors u, q and the inner products u.q and q.q are needed to solve the 4-DOF equation (7.2) which in turn allows computation of inner products ULqL and qL.qL and finally the values of f and g.
Vectors u and q, computed from Eq. (7.3), are
u = R2-1 R -1 R z = R -1 R t (7.14)
u=2 1. 2 1
and
q = R2-1 [R1-1 (p 11) 12]. (7.15)
If we consider the components of vector t as expressed with respect to frame Fl,
..