# Underwater Glider Propulsion Systems VBS Part 1: VBS Sizing and Glider Performance Analysis
## 4 Glider Performance Analysis
### Energy Balance and Geometric Analysis  
The energy balance of the underwater glider flying at constant velocity \(U\) with a glide angle \(\alpha\) is obtained by the power developed by the gravitational force \(P_{g}=Bw\) and the power needed to overcome the drag \(P_{e}=DU\) generated by the hydrodynamic response of the vehicle [15; 16]. Due to the fact that in the steady state \(R=B\), and \(D=R\sin\gamma\) the net buoyancy \(B\) could be defined by Equation (10):  
\[B=D/\sin\gamma. \tag{10}\]  
To evaluate the glider performance, the concept of specific energy consumption \(E_{e}\) was used by [PERSON] et al. [15; 16]. It is desired for the specific energy consumption to be minimal in order to obtain the highest rate between lift and drag in a plane, as is shown in Equation (11):  
\[E_{e}=\frac{DU}{Bu}=\frac{D}{L}=\tan\gamma. \tag{11}\]  
It can be observed in Figure 7 that to achieve the balance of forces needed for steady state, the hydrodynamic components utilize the glide angle \(\gamma\) defined in Equation (11).  
Substituting the drag and lift forces from (8) and (9) into (11), obtains the specific energy consumption of the glider as a function of the hydrodynamic coefficients and the angle of attack:  
\[E_{e}=\frac{K_{D0}+K_{D1}\alpha+K_{D2}\alpha^{2}}{K_{L0}+K_{L1}\alpha}=\tan\gamma. \tag{12}\]  
To obtain the minimal value of \(E_{e}\), Equation (12) is derived with respect to the angle of attack \(\alpha\left(\partial E_{e}/\partial\alpha=0\right)\), obtaining the limit values of the angle of attack as follows:  
\[\alpha_{1,2}=\frac{-K_{D2}K_{L0}\pm\sqrt{K_{D2}^{2}K_{L0}^{2}-K_{D1}K_{D2}K_{L 0}K_{L1}+K_{D0}K_{D2}K_{L1}^{2}}}{K_{D2}K_{L1}}. \tag{13}\]  
Substituting the solutions from (13) into (12), two specific energy consumption values are obtained, which represent the minimum energy consumption in ascent state \(E_{e}(\alpha_{1})\) and descent state \(E_{e}(\alpha_{2})\). With the minimum energy consumptions, the next values of the glide angle are obtained:  
\[\gamma_{1}=\tan^{-1}[E_{e}(\alpha_{1})]\quad\text{ for ascent state,} \tag{14}\]  
\[\gamma_{2}=\tan^{-1}[E_{e}(\alpha_{2})]\quad\text{ for descent state,} \tag{15}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) represent the minimum glide angle values for which the balance of hydrodynamic forces with the buoyancy force is valid.  
Figure 7: Free body diagram of the underwater glider in descent state.  
Considering that the glide angle \(\gamma\) is an input parameter to calculate other flight parameters of the vehicle, the attack angle \(\alpha\) is obtained by solving Equation (12) as a function of the glider angle \(\gamma\) as is shown in the solution (16):  
\[\alpha_{3,4}(\gamma)=\frac{K_{L1}\tan\gamma-K_{D1}\pm\sqrt{\left[K_{L1}^{2} \right]\tan^{2}\gamma+\left[4K_{L0}K_{D2}-2K_{D1}K_{L1}\right]\tan\gamma+\left[ K_{D1}^{2}-4K_{D0}K_{D2}\right]}}{2K_{D2}}, \tag{16}\]  
where \(\alpha_{3}(\gamma)\) and \(\alpha_{4}(\gamma)\) are the angles of attack functions in ascent state and descent state respectively, considering that there are two limits or discontinuities that are defined by (13).  
Thus, the pitch angle \(\theta\) will be obtained by subtracting the value of the angle of attack from the glide angle:  
\[\theta=\gamma-\alpha. \tag{17}\]