## 2 Parametric Analysis of a Steady Motion of Underwater Glider in a Vertical Plane  
For design and mission designation purposes, it is important to have a clear understanding of how geometrical, kinematical properties of the vehicle and its lifting system, as well as the relative excess of buoyancy, are related to speed on the trajectory and range for a given depth of operation. Consider a steady mode of descent/ascent of an underwater glider (UG), equipped with a buoyancy engine (BE). Theoretically, such a mode can be realized when the projection of the excess of buoyancy force (positive or negative) onto the direction of the trajectory equals the drag force. For example, this is possible when the depth-wise gradient of density and the pressure hull compression are negligibly small, and for the case of compensational control of the buoyancy by means of the BE. Consider a steady planing of a underwater glider in a vertical plane. Assuming the absence of acceleration, the equations of a dynamic equilibrium of the UG are written down, relating such parameters of ascent/descent as the vertical and horizontal components of the speed of the vehicle, angles of pitch and angle of trajectory with the configuration of the UG, position and characteristics of its lifting system, axial displacement of the battery and differential buoyancy volume. The position of the UG during ascent/descent is shown in Figures 1 and 2. Therewith, the axes \(x\) and \(y\) of the earth's coordinate system are directed to the right and down, correspondingly. Additionally considered are: a body coordinate system \(x_{1}0y_{1}\), whose axes are rotated with respect to \(x0y\) by a pitch angle \(\beta\) and a flow coordinate system \(x_{U}0y_{U}\). The following designations are introduced: \(\alpha\): angle of attack, \(\gamma=\alpha+\beta\): angle of trajectory, \(R_{W}\): weight force, \(R_{B}\): (Archimedes) buoyancy force, \(R_{x}\): drag force, \(R_{y}\): lift force, \(x_{p}\): abscissa of the point of application of the wing lift in the body coordinate system, \((x_{1W},y_{1W})\): coordinate of the center of gravity of the vehicle, \(x_{1B}\): abscissa of the point of application of the buoyancy force, \(x_{1Y}\): abscissa of the point of application of the lift force. Projecting the forces acting on the vehicle in steady motion mode onto the earth's system axes \(x\) and \(y\), we obtain:Horizontal axis projection  
\[R_{y}\sin(\alpha+\beta)=-R_{x}\cos(\alpha+\beta), \tag{1}\]  
Vertical axis projection  
\[R_{y}\cos(\alpha+\beta)+R_{W}-R_{B}+R_{x}\sin(\alpha+\beta)=0. \tag{2}\]  
Equilibrium equation for the moment with respect to the origin of the coordinate system. \[R_{B}x_{1B}\cos\beta-R_{W}(x_{1W}\cos\beta+y_{1W}\sin\beta)-x_{1Y}\cos\alpha \cdot R_{y}=0. \tag{3}\]  
It follows from Equation (1) that  
\[tg(\alpha+\beta)=tg\gamma=\varepsilon(\alpha), \tag{4}\]  
from where  
\[\gamma=\alpha+\beta=arctg[\varepsilon(\alpha)], \tag{5}\]  
\[\beta=arctg[\varepsilon(\alpha)]-\alpha, \tag{6}\]  
Figure 1: Steady ascent of the underwater glider. Scheme of acting forces. Figure 2: Steady descent of the underwater glider. Scheme of acting forces. where \(\varepsilon(\alpha)=R_{x}/R_{y}=1/K\): inverse hydrodynamic quality. It follows from Equation (5) that the angle of trajectory of the UG is inversely proportional to the arctangent of hydrodynamic quality of its lifting system. In other words, with an increase of the hydrodynamic quality of the UG's lifting system, its trajectory becomes flatter. Accounting for the known relationships  
\[ \begin{array}{l}\cos(\alpha+\beta)=\frac{1}{\sqrt{1+\varepsilon^{2}(\alpha)} },\\ \sin(\alpha+\beta)=\frac{\varepsilon(\alpha)}{\sqrt{1+\varepsilon^{2}(\alpha)} },\end{array}  \tag{7}\]  
one can derive from (2) the following equation  
\[R_{B}-R_{W}=\sqrt{R_{x}^{2}+R_{y}^{2}}, \tag{8}\]  
which shows that for the steady motion of the UG, its buoyancy (positive or negative) is countered by a resultant of the lift and drag force (see Figures 1 and 2).