In the paper:
https://journals.sagepub.com/doi/pdf/10.1177/0278364914558129?casa_token=d355ZLeI3zEAAAAA:gNiS1aqxfrnLOoRinjoPZoE8G21AyOlH6lBFZSjWoos3DTPcYNZIw92kC1bYNoWm9N2EWKMU2zo2zQ
on page $97$, the derivative of $(16)$ is $(17)$.
I don't understand how the derivative was taken. Was it taken with respect to $t$? If so, can you explain more thoroughly how the result was reached at?
It looks like they differentiate $J$ with respect to $b_P$. $$\frac{\partial J}{\partial b_P}=2b_F^TR_{FP}+2b_P^TR_{PP}=0$$ From here $$b_P^TR_{PP}=-b_F^TR_{FP}$$ Transposing both sides yields $$R_{PP}^Tb_p=-R_{FP}^Tb_F$$ Assuming $R_{PP}^T=R_{PP}$ is invertible, multiply on the left side with $R_{PP}^{-1}$ and you get the answer.