# Bifurcation Flight Dynamic Analysis of a Strake-Wing Micro Aerial Vehicle
## 6 Numerical Verification of Predicted MAV Behaviour
### \"Cobra\" Manoeuvre Simulation  
\"Cobra\" is one of aerobatic figures. The manoeuvre was first executed at the turn of the 1950s and 60s, by the pilots of the Swedish Air Force (_Svenska Flygyapnet_) flying J-35 Draken fighters. It has a Swedish name of \"kort parad\"--\"short parade\" and was part of standard short-range manoeuvring combat training for Swedish fighter pilots [48,49]. In the 1980s, the manoeuvre was executed by OKB Sukhoi test pilot, [PERSON], during Su-27 spin tests. The Cobra was first demonstrated in public by [PERSON] at the Le Bourget air show in 1989, on a MiG-29 fighter. This manoeuvre can be executed with an aircraft exhibiting excellent manoeuvrability and low thrust load or equipped with thrust vector control engines. Besides the spectacular impression it makes at air shows, enabling to demonstrate the manoeuvrability and turning abilities of modern combat aircraft, this manoeuvre, used in direct air combat, is primarily aimed at forcing the chasing foe to overtake through rapid deceleration, thus providing the chased aircraft with a convenient position to open fire. The effect of sudden deceleration is achieved owing to rapidly increasing aircraft drag resulting from vertical positioning of the aircraft body in the upward direction (perpendicular to the previous flight direction). Figure 10: Diagram of bifurcation control matching, which enables recovery from a spin (skipping through the so-called “cusp catastrophe”) [25] with the waveform of elevon deflection angle changes. The \"Cobra\" is conducted at supercritical angles of attack, significantly exceeding the value under normal operating conditions. Figure 12 shows a diagram of the \"Cobra\". This manoeuvre is divided into three basic stages:  
1. Transition from horizontal flight into the phase of increasing the aircraft pitch angle, due to very rapid increase in the elevator deflection angle (sudden pulling of the control stick to a maximum), while simultaneously throttling the engine or engines,
2. manoeuvre phase in which, as a result of such action by the pilot, the aircraft nose rapidly rises up, until it reaches a very high angle of attack (even up to 120\({}^{\circ}\)),
3. the exit phase, which involves increasing the thrust and releasing the control stick, leading to the aircraft rapidly increasing its pitch angle, while simultaneously accelerating and returning to horizontal flight, with a minor altitude loss. The \"Bee\" MAV has a stake-wing and its dynamics is similar to the dynamics of modern high-manoeuvring combat aircraft. Based on in-flight tests, it was concluded that it was able to execute a \"Cobra\" manoeuvre (Figure 12) [61]. The \"Cobra\" manoeuvre simulations were conducted based on aerodynamic and mass data of the \"Bee\" micro aerial vehicle. Simple analyses show that executing the \"Cobra\" manoeuvre will be possible without changing the flight altitude, when the vertical projection of the sum of external forces acting on the MAV in the course of the manoeuvre should be equal to zero. However, due to the fact that the computational thrust force value was initially negative, it was  
Figure 11: Spin simulation. Angle waveforms for selected flight parameters. Figure 12: “Cobra” manoeuvre phases (**top**), “Bee” MAV during a “Cobra” manoeuvre, time-lapse photos (**bottom**) [61]. assumed that the MAV starts the manoeuvre with the engine off (zero thrust). It was also assumed that, in the course of executing a manoeuvre, the thrust depends on the difference in the aircraft's angles of pitch \(\Theta\) and attack \(\alpha\), and the flight velocity \(V\)[62; 63]. Due to the fact that aircraft equipped with stroke wings are characterized by wing-rock instability and are not spirally stable, it was impossible to obtain a fully symmetrical flight parameter waveform. This is associated with the occurrence of Hopf bifurcation and torus creation bifurcation on G, E and F branches (Figures 5 and 6). The \"Cobra\" manoeuvre is initiated with a sudden downward displacement of elevons (\(\delta_{e}=-35^{\circ}\)).