## 3 Attitude Indicator (Analogue Instrument and Computer Simulation)  
The attitude indicator shows the pitching and rolling movements in one single instrument. It is one of the oldest flight instruments, and it is considered to be a basic requirement for navigation. It has horizontal lines, with a dot between them representing the actual wings and nose movements [1; 2; 3; 4; 5; 6; 7; 14; 15; 16; 17; 18]. The attitude indicator uses gyroscopic physical principles to provide an inertial platform. It is set to a display with 2 degrees of freedom. The display is colored to indicate that the horizon is the division between the two colored segments (typically, blue is used for the sky and the green color is used for the ground). It has been designed to be the most intuitive possible [24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 41; 42; 43]. The actual banking angle is calibrated around the circumference of the instrument, and the pitching angle is indicated by moving the instrument along horizontal lines. The analogue instrument may develop small precession errors in pitch indication during long periods of acceleration or deceleration. Moreover, errors can also be developed during long periods of turning due to the erection of the mechanism that keeps the gyrorotor. Nevertheless, these errors are frequently considered as not significant because they are no more than 2 or 3 degrees per minute. Unfortunately, some attitude indicators can only support a defined range of bank angles if the aircraft rolls too steeply or achieves an extreme pitch attitude while performing. Moreover, the attitude indicator can \"tumble\" or \"topple\" and can be temporarily unstable; this is a frequent situation for acrobatic aircrafts. Thus, many attitude indicators are fitted with a caging mechanism; this device was developed to restore the gyroscope to an erect position. However, most recent instruments have been designed to tolerate 360\({}^{\circ}\) of rotation in pitch-and-roll scenarios without tumbling. Figure 2: Concordance between simulated terrain and assumptions for magnetic compass used to represent heading angles. Nowadays, attitude and heading reference systems (AHRSs) are capable of providing 3-axis information, shared with multiple devices in the aircraft, such as \"glass cockpit\" primary flight displays. AHRSs have been tested to provide a high degree of realism and are used worldwide in commercial and business aircraft [29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43]. The attitude indicator developed in this study was drawn in a rounded dial. Along the horizontal axis, there are two numbers (0) that indicate this condition. Along the vertical axis, there are also two numbers (90) used to represent the vertical aircraft orientation. There are 4 more principal divisions between these at (45\({}^{\circ}\)) and 5 more secondary subdivisions are included, being printed with shorter lines to improve visual perception for the pilot [15; 16; 17; 18; 24; 25; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39]. The rounded dial is fulfilled, using a bubble (1/2) blue to represent the sky and (1/2) a green color to represent the field, as it is in a real instrument, and the values for the pitching (\(\beta\)) and rolling (\(\chi\)) movements are calculated at every step time (\(\Delta t\)) during the simulation using Equations (4) and (5), respectively. Here, the superindexes (\(n\)) and (\(n-1\)) refer to the actual and previous maneuvers. The latest term is the step time calculated, as was explained for Equations (2) and (3).