# Autonomous Balloon Controls for Protection against Projectiles with Known Destinations
## 4 The Defense Strategy Utilizing Multiple Balloons  
Until now, we derived how to estimate the initial shooting angle \(\theta_{0}^{D}\) so that the flight range of the EP is \(L\). The proposed defense strategy is to position multiple balloons on the EP's viable trajectories so that they block the incoming EP effectively. Before discussing the case of utilizing multiple balloons, we first consider a case where a single balloon is utilized for blocking the EP's path. Let \(h\) denote the designated height of a balloon. See Figure 1 for an illustration of a single balloon at height \(h\). The balloon is located to block the EP's destination path. Considering discrete-time systems, we next present how to simulate the path of the EP, as the initial shooting angle \(\theta_{0}^{D}\) is given. At time-step 0, the EP is at the origin. Let \(dt\) denote the sampling interval. At each time-step \(k\), the EP's velocity and pitch angle are updated utilizing  
\[V(k+1) = V(k)+dt*(-g*s(\theta(k))-g*\kappa*V(k)^{2}),\] \[\theta(k+1) = \theta(k)-dt*g*c(\theta(k))/V(k). \tag{6}\]  
These equations are derived by discretizing (1). At time step 0, (6) uses \(V(0)=V_{0}\) and \(\theta(0)=\theta_{0}^{D}\). Let \((x(k),y(k),z(k))\) denote the location of the EP at time-step \(k\). Using (1), the position of the EP is updated as  
\[x(k+1) = x(k)+(V(k)*c(\theta(k)))*c(\psi^{D})*dt,\] \[y(k+1) = y(k)+(V(k)*c(\theta(k)))*s(\psi^{D})*dt,\] \[z(k+1) = z(k)+(V(k)*s(\theta(k)))*dt. \tag{7}\]  
Here, \(\psi^{D}\) presents the yaw angle of the destination with respect to the origin. In other words, \(\psi^{D}\) is given as  
\[\psi^{D}=atan2(y^{D},x^{D}). \tag{8}\]  
Figure 1: A balloon at height \(h\). The balloon blocks the EP’s path for reaching its destination. In (8), \(atan2(y^{D},x^{D})\) is utilized to return a correct and unambiguous value for the phase angle in converting from Cartesian coordinates to polar coordinates. Equivalently, \(atan2(y^{D},x^{D})\) is the argument (also called phase or angle) of the complex number \(x^{D}+j*y^{D}\). While simulating the EP's motion utilizing (7), we have the time moment \(k_{d}\) when  
\[z(k_{d}) >h,\] \[z(k_{d}+1) \leq h. \tag{9}\]  
_Control of Multiple Balloons_  
We may have errors in the prediction of \(\kappa\), \(V\), or \(M\) of the EP. In practice, we cannot predict the EP's path accurately before the EP is launched. Recall that (6) and (7) are utilized to build the EP's path with yaw angle \(\psi^{D}\). Furthermore, at time step 0, Equation (6) uses the velocity \(V(0)=V_{0}\) and the shooting angle \(\theta(0)=\theta_{0}^{D}\). Suppose that the true EP's path is built with yaw angle \(\psi^{D}+n_{\psi}\) where \(n_{\psi}\) is a Gaussian noise with zero mean and variance \(var_{\psi}\). Suppose that the true EP's path is built with the initial velocity \(V_{0}+n_{V}\), where \(n_{V}\) is a Gaussian noise with zero mean and variance \(var_{V}\). Furthermore, suppose that the true EP's path is built with the initial shooting angle \(\theta_{0}^{D}+n_{\theta}\), where \(n_{\theta}\) is a Gaussian noise with zero mean and variance \(var_{\theta}\). We assume that noise factors \(var_{\psi}\), \(var_{V}\), and \(var_{\theta}\) are known a priori. Our strategy is to deploy multiple balloons so that they can block the EP's feasible paths considering the variance mentioned in the previous paragraph. Suppose we have \(N\) balloons in total. In this case, we generate \(N\) feasible paths for the EP. We then deploy \(N\) balloons for blocking each feasible path, respectively. In order to block the true EP's path, Algorithm 1 is utilized. In this algorithm, one builds \(N\) feasible EP paths considering the noise factors \(var_{\psi}\), \(var_{V}\), and \(var_{\theta}\). In this algorithm, \(\eta\geq 1\) is used as a tuning constant. As \(\eta\) increases, we disperse the balloons to cover a wider space. We then deploy \(N\) balloons for blocking each feasible path, respectively.