# Integrated Autopilot Guidance Based on Zero-Effort-Miss Formulation for Tail-Controlled Missiles
## 4 Numerical Simulation  
To demonstrate the proposed guidance and control method, we investigate the performance of the proposed guidance law through numerical simulation using two different missile models: canard-controlled and tail-controlled dynamics, respectively. The dual missile model used in [12] is adopted for the simulation, and the input channel of the tail or canard is ignored to implement the canard- or tail-controlled missile model, respectively. Specifically, the aerodynamic lift and pitch moment of the missile model are governed by  
\[\frac{L}{m}= L_{\alpha}^{B}f_{1}(\alpha)+L_{\delta_{c}}f_{2}(\alpha+\delta_{c} )+L_{\delta_{c}}f_{3}(\alpha+\delta_{t}) \tag{29}\] \[\frac{M}{I_{yy}}= M_{\alpha}^{B}f_{4}(\alpha)+M_{q}q+M_{\delta_{c}}f_{5}(\alpha+ \delta_{c})+M_{\delta_{t}}f_{6}(\alpha+\delta_{t}) \tag{30}\]  
The canard-controlled model is implemented by ignoring the tail fin deflection \(\delta_{t}\), and conversely, the tail-controlled model is realized with the canard deflection \(\delta_{c}\) as zero. The other expressions and specific values of the parameters can be found in [12]. As a means to evaluate the performance of the proposed method, a separate method consisting of the ZEM-based guidance law and Lyapunov-based autopilot is used, as shown in [9]. \[a_{M}^{c}= N\frac{z+zt_{go}}{t_{go}^{2}}=-N\dot{r}\lambda \tag{31}\] \[\delta_{c}= \delta-\frac{\tau_{\delta}C_{m_{\alpha}}}{C_{m_{\delta}}}\bigg{(} q-\frac{a_{M}}{V_{M}}\bigg{)}-M(a_{M}-a_{M}^{c}) \tag{32}\]  
where \(\tau_{c}\) is the time constant of the actuator dynamics of the control fin. \(N\) and \(M\) are the controller gains whose values are selected to be positive. The ZEM-based guidance law in (31) is structured as well-known proportional navigation guidance and the Lyapunov-based autopilot in (32) is designed to only stabilize the dynamics of \(a_{M}\) at \(a_{M}=a_{M}^{c}\). Now, the performance of the proposed integrated method in (27) can be compared with this separate method consisting of (31) and (32) through the simulation results under two different missile models. Figures 3 and 4 illustrate the simulation results of both methods under the canard-controlled missile model with the settings given in Table 1. Figures 3a and 4a show that both methods  
Figure 2: Guidance and autopilot structures. enable the canard-controlled missile to achieve interception against maneuvering targets of \(a_{T}\in\{-3,-5,-10\}\) g with a miss distance less than \(0.3\) m. The upper parts of Figures 3b and 4b also provide the results where both methods cause ZEM to converge to zero according to the design purpose. However, Figure 3b,c show that the separate method generates fluctuated commands for the canard deflection, causing the state variables to be vibrated. This is because the acceleration controller of the separate method in (32) is configured to only aim at the guidance command, without considering internal dynamics, such as the pitch rate. For reliable autopilot performance, it is necessary to stabilize the rotational dynamics as well as generate the required acceleration. To achieve such requirements, the proposed method in (27) is designed in a three-loop structure that can stabilize the internal dynamics. The results presented in Figure 4b,c show that the proposed integrated method prevents the state variables from fluctuating by generating smoother commands than when using the separate method. In addition, it is shown that all the methods exhibit diverging responses \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Missiles** & **Target** \\ \hline Initial position (km) & \((0,0)\) & \((2,0)\) \\ Initial flight path angle (\({}^{\circ}\)) & 0 & 150 \\ Speed (m/s) & 500 & 300 \\ Control input & \(|\delta_{c}|\leq 25^{\circ}\) & \(a_{T}\in\{-3,-5,-10\}\) g \\ Termination condition & \(r\leq 0.3\) m or \(\hat{r}>0\) m/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation settings. Figure 4: Simulation results of the integrated method under the canard-controlled missile model. (**a**) Flight trajectories.