# Bifurcation Flight Dynamic Analysis of a Strake-Wing Micro Aerial Vehicle
## 5 Bifurcation Flight Dynamic Analysis of a Micro Aerial Vehicle
Development of equilibrium position asymmetry can be observed at continued reduction in the elevator deflection angle, and at \(\delta_{e}\approx 4^{\circ}\), the micro aerial vehicle enters the range of equilibrium states marked with C, which corresponds to spiral motions with a high amplitude of bank angles (Figure 7). Stable position branches corresponding to spiral motions exist on both sides of the straight line defining unstable horizontal flight equilibrium positions. Due to the aerodynamic load asymmetry, the micro aerial vehicle banks to the left wing (with a negative banking angular velocity \(P\)), since the equilibrium branch  
Figure 6: Bifurcation diagram. Quasi-steady states for various elevator deflection angles; \(P(\delta_{e})\). representing the breaking away is connected with A (Figure 6). The equilibrium branch representing rightward MAV banking is practically detached from branch A. The bifurcations therein lead to the appearance of unstable equilibrium positions, where \(P\approx 0\) [\({}^{\circ}\)/s]. However, it should be noted that the spiral motion direction is determined in practice by the nature of the disturbance or the transient dynamics state of a micro aerial vehicle. As shown in Figures 5 and 6, equilibrium branches of dynamics regime C (within the physical range of elevator deflections), do not converge on the straight line corresponding to the horizontal flight conditions. The MAV will remain at one of two spiral branches only up to a value of \(\delta_{e}=-35^{\circ}\). Branch C, shown on the bifurcation diagrams (Figures 5 and 6), also represents undesirable steep spirals. Steep spirals can be deemed undesirable flight conditions, therefore, it is important to determine a recovery strategy. Based on a bifurcation diagram (Figure 5), a micro aerial vehicle can be easily recovered from such a flight state by reducing the angle of attack to a level below which no spiral branches are formed. This can be achieved by increasing the elevator deflection angle. The ability to find and determine branches, such as steep spirals, stresses the advantage of continuation and bifurcation analyses over classical linear methods. Although classical linear methods for analysing flight stability can identify stability changes, they are unable to find stable and unstable branches of quasi-steady flight states (Figure 8). The slight lateral instability is also present at low angles of attack (\(\alpha\leq 2^{\circ}\)), (branch D in Figures 5 and 6). In the case of the bifurcation analysis and continuation test pattern in question, which assumes constant thrust, flight velocity reaches the highest and not always realistic values at low, negative angles of attack, which entails an increase in the negative angle of MAV pitch, until inverted flight conditions. Thus, although regime D can be deemed an inverted spiral, it is not representative for realistic flight conditions and will not be discussed in more detail (Figure 8). Figure 7: Bifurcation diagram. Quasi-steady states for various elevator deflection angles; \(\alpha(\delta_{e})\). Equilibrium positions of quasi-steady flight states also appear at high angles of attack \(65^{\circ}<\alpha<85^{\circ}\) (bifurcation diagrams shown in Figures 5 and 6). There are unstable equilibrium position branches for negative deflection angular velocities and angles of attack within a range of \(80^{\circ}<\alpha<85^{\circ}\), and elevator deflections angles in the range of \(-35^{\circ}<\delta_{e}<10^{\circ}\). There is a certain area of stable equilibrium positions for positive values of deflection angular velocity. The differences arise from the asymmetry of MAV loads. Left- and right-handed spins differ in terms of the state of equilibrium of inertial and aerodynamic forces, which undoubtedly impacts the local stability of spin branches (just as in the case of lower angles of attack \(\alpha\)). Stable equilibrium positions, corresponding to spins executed at a constant angular velocity, are represented by branch E of the bifurcation diagrams.