## 3 Tracking Control Laws
### Derivation of Control Laws  
We present two theorems that are used to derive our control laws. **Theorem 1**. : _Using the trim input in the full cavity state, the rotation of the vehicle about the CG is zero._  
Proof. : As the vehicle is in the full cavity state, all forces applied to the vehicle are the fin force \(\rm F_{fin}\), the cavitator force \(\rm F_{cavitator}\), the thruster force \(\rm F_{thrust}\), and the gravity of the vehicle \(\rm F_{gravity}\). See (2) and (3). The skin friction drag, the plan force, and the buoyancy force are not generated, since the body of the vehicle does not intersect water. In the trim condition, \(\rm M_{gravity}\) (the moment generated by the gravity) is zero, since the gravity force \(\rm F_{gravity}\) applies to the CG. Moreover, \(\rm M_{thrust}\) (the moment generated by the thruster) is zero, since the thruster force \(\rm F_{thrust}\) applies along the body axis. \(\rm M_{fin}\) (the moment generated by the fin) is nullified by \(\rm M_{cavitator}\) (the moment generated by the cavitator). Otherwise, the vehicle cannot move straight while not rotating about the CG. See Figure 2 for an illustration of the case where we apply \(\rm u_{0}\) while the pitch is not zero. In Figure 2, the lifting force (the normal component of \(\rm F_{fin}\) such that the component is normal to the body axis) generated by the fins is depicted as a red dotted arrow. Also, the lifting force (the normal component of \(\rm F_{cavitator}\) such that the component is normal to the body axis) generated by the cavitator is depicted as a green arrow. These two lifting forces do not generate a moment, since \(\rm M_{fin}\) is nullified by \(\rm M_{cavitator}\)-\(\rm F_{gravity}\) which is applied to the CG is depicted as a bold arrow in Figure 2.  
Therefore, using the trim input in the full cavity state, the rotation of the vehicle about the CG is zero. \(\square\)  
Figure 2: We apply \(u_{0}\) while the pitch is not zero. Consider the case where the vehicle moves with a constant velocity by applying \(u_{0}\) as depicted in Figure 2. By changing the cavitator input in \(u_{0}\), we can change the lifting force (the normal component of \(F_{\text{cavitator}}\) such that the component is normal to the body axis) generated by the cavitator. Let us assume that we change the cavitator input in \(u_{0}\) slightly so that we reduce the lifting force generated by the cavitator. See Figure 3 for an illustration. Compared to Figure 2, the lifting force generated by the cavitator decreased. In this case, a moment is generated and the body rotates in the clockwise direction as depicted in Figure 3. \(M_{\text{gravity}}\) (the moment generated by the gravity) is zero, since the gravity force \(F_{\text{gravity}}\) applies to the CG. Our depth control laws are designed using the moment generated by changing the cavitator input slightly. Theorem 2 presents the relationship between the deflection of rudder and the roll of the vehicle. Beforehand, we need to introduce several definitions. Consider the case where the vehicle moves with a constant velocity in the full cavity state. The fin force generated by deflecting fin 2 by a certain angle \(\delta_{\text{f}}\) can be decomposed into three forces \(F_{\text{x}}^{2}\), \(F_{\text{y}}^{2},F_{\text{z}}^{2}\) in the body-fixed frame. Here, the upper subscript 2 indicates fin 2. Similarly, the fin force generated by deflecting fin 4 by \(\delta_{\text{f}}\) can be decomposed into three forces \(F_{\text{x}}^{4}\), \(F_{\text{y}}^{4},F_{\text{z}}^{4}\) in the body-fixed frame. See Figure 4 for an illustration. This figure shows the vehicle seen from the nose of the vehicle. In Figure 4, the dotted circle indicates the cavity intersecting the rudders. The pivot points of fins 2 and 4 are depicted with two crosses. Let \(L^{i}\) denote the distance between the pivot point of i-th fin and the principal axis of the vehicle. Under the configuration of our vehicle, \(L^{i}=L^{j}=L\) for all \(i,j\leq 4\). **Theorem 2**.