Appendix A — Derivation of Nomoto Equations

The lienarized equations of motion derived in Chapter 3 are shown in \(\eqref{eq-steering-eom-3}\) and \(\eqref{eq-steering-eom-4}\).

\[\begin{align} (m - Y_{\dot{v}}) \dot{v} - (Y_{\dot{r}} - m x_G) \dot{r} - Y_v v - (Y_r - mU) r = Y_{\delta} \delta \label{eq-steering-eom-3} \end{align}\]

\[\begin{align} (I_z - N_{\dot{r}}) \dot{r} - (N_{\dot{v}} - m x_G) \dot{v} - N_v v - (N_r - mx_GU) r = N_{\delta} \delta \label{eq-steering-eom-4} \end{align}\]

Taking the Laplace transform of the above equations results in

\[\begin{align} (m - Y_{\dot{v}}) sv(s) - (Y_{\dot{r}} - m x_G) sr(s) - Y_v v(s) - (Y_r - mU) r(s) = Y_{\delta} \delta(s) \label{eq-steering-eom-5} \end{align}\]

\[\begin{align} (I_z - N_{\dot{r}}) sr(s) - (N_{\dot{v}} - m x_G) sv(s) - N_v v(s) - (N_r - mx_GU) r(s) = N_{\delta} \delta(s) \label{eq-steering-eom-6} \end{align}\]

Note that we are interested only in the steady state response and hence the dependence on initial condition when taking Laplace transform is ignored. Expressing it as a matrix equation results in \(\eqref{eq-steering-matrix-eom-7}\).

\[\begin{align} \begin{bmatrix} (m - Y_{\dot{v}})s - Y_v & - (Y_{\dot{r}} - m x_G) s - (Y_r - mU) \\ - (N_{\dot{v}} - m x_G) s - N_v & (I_z - N_{\dot{r}}) s - (N_r - mx_GU) \end{bmatrix} \begin{bmatrix} v(s) \\ r(s) \end{bmatrix} = \begin{bmatrix} Y_{\delta} \\ N_{\delta} \end{bmatrix} \delta(s) \label{eq-steering-matrix-eom-7} \end{align}\]

This can be solved as shown below:

\[\begin{align} \begin{bmatrix} v(s) \\ r(s) \end{bmatrix} = \begin{bmatrix} (m - Y_{\dot{v}})s - Y_v & - (Y_{\dot{r}} - m x_G) s - (Y_r - mU) \\ - (N_{\dot{v}} - m x_G) s - N_v & (I_z - N_{\dot{r}}) s - (N_r - mx_GU) \end{bmatrix}^{-1} \begin{bmatrix} Y_{\delta} \\ N_{\delta} \end{bmatrix} \delta(s) \end{align}\]

\[\begin{align} \begin{bmatrix} v(s) \\ r(s) \end{bmatrix} &= \frac{1}{\begin{vmatrix} (m - Y_{\dot{v}})s - Y_v & - (Y_{\dot{r}} - m x_G) s - (Y_r - mU) \\ - (N_{\dot{v}} - m x_G) s - N_v & (I_z - N_{\dot{r}}) s - (N_r - mx_GU) \end{vmatrix}} \times \nonumber \\ &\begin{bmatrix} (I_z - N_{\dot{r}}) s - (N_r - mx_GU) & (Y_{\dot{r}} - m x_G) s + (Y_r - mU) \\ (N_{\dot{v}} - m x_G) s + N_v & (m - Y_{\dot{v}})s - Y_v \end{bmatrix} \begin{bmatrix} Y_{\delta} \\ N_{\delta} \end{bmatrix} \delta(s) \end{align}\]

\[\begin{align} \begin{bmatrix} v(s) \\ r(s) \end{bmatrix} &= \frac{1}{As^2+Bs+C} \times \nonumber \\ &\begin{bmatrix} ((I_z - N_{\dot{r}}) Y_{\delta} + (Y_{\dot{r}} - m x_G) N_{\delta}) s - (N_r - mx_GU) Y_{\delta} (Y_r - mU) + Y_{\delta} \\ ((N_{\dot{v}} - m x_G) Y_{\delta} + (m - Y_{\dot{v}}) Y_{\delta}) s + (N_v Y_{\delta} - Y_v Y_{\delta}) \end{bmatrix} \delta(s) \end{align}\]

Note that \(A\), \(B\) and \(C\) are given by \(\eqref{eq-A-1}\), \(\eqref{eq-B-1}\) and \(\eqref{eq-C-1}\).

\[\begin{align} A &= (m - Y_{\dot{v}})(I_z - N_{\dot{r}}) - (Y_{\dot{r}} - m x_G)(N_{\dot{v}} - m x_G) \label{eq-A-1} \end{align}\]

\[\begin{align} B &= -(I_z - N_{\dot{r}})Y_v - (m - Y_{\dot{v}})(N_r - mx_GU) \nonumber\\ &- (Y_r - mU) (N_{\dot{v}} - m x_G) - (Y_{\dot{r}} - m x_G) N_v \label{eq-B-1} \end{align}\]

\[\begin{align} C &= (N_r - mx_GU) Y_v - (Y_r - mU) N_v \label{eq-C-1} \end{align}\]

From this it can be seen that the transfer function for \(r(s)\) is given by \(\eqref{eq-nomoto-laplace}\)

\[\begin{align} r(s) = \frac{K(1+T_3)}{(1+T_1s)(1+T_2s)} \delta(s) \label{eq-nomoto-laplace} \end{align}\]

where

\[\begin{align} K = \frac{N_v Y_{\delta} - Y_v Y_{\delta}}{C} = \frac{N_v Y_{\delta} - Y_v Y_{\delta}}{(N_r - mx_GU) Y_v - (Y_r - mU) N_v} \end{align}\]

\[\begin{align} T_3 = \frac{(N_{\dot{v}} - m x_G) Y_{\delta} + (m - Y_{\dot{v}}) Y_{\delta}}{N_v Y_{\delta} - Y_v Y_{\delta}} \end{align}\]

\[\begin{align} T_1T_2 = \frac{A}{C} &= \frac{(m - Y_{\dot{v}})(I_z - N_{\dot{r}}) - (Y_{\dot{r}} - m x_G)(N_{\dot{v}} - m x_G)}{(N_r - mx_GU) Y_v - (Y_r - mU) N_v}\\ T_1 + T_2 = \frac{B}{C} &= \frac{-(I_z - N_{\dot{r}})Y_v - (m - Y_{\dot{v}})(N_r - mx_GU)}{(N_r - mx_GU) Y_v - (Y_r - mU) N_v} \nonumber \\ &+ \frac{- (Y_r - mU) (N_{\dot{v}} - m x_G) - (Y_{\dot{r}} - m x_G) N_v}{(N_r - mx_GU) Y_v - (Y_r - mU) N_v} \end{align}\]

Taking an inverse Laplace transform of \(\eqref{eq-nomoto-laplace}\) yeilds the Nomoto second order model shown in \(\eqref{eq-nomoto-second-order-model-1}\).

\[\begin{align} T_1T_2\ddot{r} + (T_1 + T_2) \dot{r} + r = K \delta + KT_3 \dot{\delta} \label{eq-nomoto-second-order-model-1} \end{align}\]