\begin{eqnarray} A_I &=& a_0 + a_1x_d + a_2 \theta_d\\ B_I &=& \sum_{i=0}^1\sum_{j=0}^1\sum_{k=0}^1\sum_{l=0}^1 b_{ijkl} x_d^i \theta_d^i y_d^k \phi_d^l + \sum_{i=0}^1 b_{Li} x_d^i y_L\\ \end{eqnarray}

By expanding the equation,

\begin{eqnarray} B_I &=& b_{0000} \\ && + b_{1000} x_d + b_{0100} \theta_d + b_{0010} y_d + b_{0001} \phi_d\\ && + b_{1100} x_d\theta_d + b_{1010} x_dy_d + b_{1001}x_d\phi_d + b_{0110}\theta_dy_d + b_{0101}\theta_d\phi_d + b_{0011} y_d\phi_d\\ && + b_{1110} x_d\theta_dy_d + b_{1101} x_d\theta_d\phi_d + b_{1011} x_dy_d\phi_d + b_{0111} \theta_dy_d\phi_d\\ && + b_{1111} x_d\theta_dy_d\phi_d\\ && + b_{L0} y_L + b_{L1} x_d y_L \end{eqnarray}