Problem Set 2 - Report

Problem Set 2 - Report Task 1: Experimentally determine all the necessary trims for the Bixler 2 aircraft as a function of airspeed. (Hugh, Steve, Kareem, Pat, Josh, Chris;; written by Kareem and Hugh) Mechanics of Constant Altitude and Airspeed Flight Amidst the plethora of dynamics present in flight, there are typically four main force contributions to describe the motion of an aircraft. The free body diagram below shows these factors: lift, thrust, drag and weight. With respect to the forward direction, (x-direction) the first line of the equation above shows the opposing nature of thrust and drag, and expresses the required thrust as dependent on the aircraft drag. In the same regard but in terms of the upward direction (y-direction), necessary lift relates to configuration weight. Considering constant altitude and airspeed flight, the acceleration in both axes must be zero. Thus, according to the simple equations of motion, thrust and lift must be equivalent to drag and weight respectively. However, the four forces are not necessarily independent of each other. Observing a definition of the lift force exhibited below and given an aircraft with a wing reference area S ref and lift coefficient C L, lift is strongly related the airspeed squared, U 2. As thrust relates to airspeed, the thrust and lift are closely coupled. Increasing/decreasing throttle translates to an increase/decrease in both airspeed and lift, and as a result the aircraft 1

will climb/descend. Consequently, another factor must be introduced to in attempt to achieve level flight. One such possible variable is pitch. By altering the pitching angle of the aircraft, the lift term is no longer solely acting to carry the aircraft, but also opposing the thrust to make flight at constant altitude and speed feasible. The force balance now reads as follows. Now, zero x- and y-acceleration can be achieved despite variations in lift or thrust with the addition of pitching created by a deflection of the elevator. The value of the pitching angle θ needed to achieve level flight can be determined by solving the modified equations of motion above. An important trend to be cognizant of is that typically the lift coefficient will increase with angle of attack-- which in a level flight case is similar to the pitch angle. Thus, for a given airspeed, the required lift to maintain altitude can be achieved by pitching the aircraft up to increase/decrease the lift coefficient. To maintain airspeed, forward acceleration must be zero. The analysis above assumes other flight dynamic motions such as roll and yaw are non-existent. During flight tests, this condition was emulated by trimming the aircraft appropriately prior to initiating constant altitude and airspeed maneuvers for each throttle setting. Initially, the Bixler was trimmed to a given elevator deflection, but producing the throttle input resolution necessary to match the pitched angle proved extremely difficult. Thus, the throttle setting was established first. Once a constant heading was achieved at a given throttle input, the elevator was slowly deflected to maintain altitude and airspeed for at least 3-5 seconds. Results of the flight tests are described in the next section. To determine trim values as a function of airspeed, the Bixler 2 was originally flown at constant elevator pitch values. The throttle was then manually adjusted in efforts to achieve level flight, which was ensured by monitoring attitude, vertical speed, heading, and airspeed from the ground 2

station. However, as it was found to be extremely difficult to pilot the aircraft to maintain steady, level flight by adjusting throttle, it was decided to set constant throttle values and to trim the elevator, rudder, and ailerons using the trim tabs on the transmitter. When these trims were set and level flight was achieved, Hugh (the pilot) would quickly switch from manual mode to autopilot mode then immediately back to manual mode, so as to make the appropriate flash data easier to recognize upon the data processing phase. For these experiments, we were able to achieve 3-6 seconds of steady level flight for each given throttle setting without needing to adjust the aileron or rudder whatsoever. It can be shown in the plotted results below that aileron and rudder trim conditions hardly changed with airspeed. To retrieve control surface deflection angle data, the max deflections (upper and lower bounds) were measured on the Bixler 2. The pulse width modulation ranges with respect to each control surface were then measured (easily displayed on the Configurations tab of the APM MAVlink user interface). The reported PWM values were converted to percentages and translated to their corresponding deflection angles using a linear relationship. Similarly, PWM values for CH3 (throttle) were converted to percentages before presenting the data. The plot below shows an example of a segment of steady, level flight data taken for a throttle setting of 48.4%. The spikes in elevator deflection shown in the figure indicated the transition between manual-autopilot-manual modes that were used to identify the start of each separate level flight test. The plots below show the results of trim conditions during several different segments of steady, level flight at different airspeeds. The sign convention that was used to determine control surface deflections should be noted: Aileron: (+): right wing down (roll left), (-): right wing up (roll right) Rudder: (+): left rudder (yaw left), (-): right rudder (yaw right) Elevator: (+): downward deflection (pitch down), (-): upward deflection (pitch up) 3

Steady, Level flight at 48.4% throttle Aircraft Trim Results The relationship between throttle and airspeed was essentially linear, which comes as no surprise. In general, this will not be a perfectly linear relationship, as other factors such as wind gusts, non-constant heading, and slight maneuvers contribute to varying airspeeds at a given throttle. The second plot shows the trimmed control surface deflections at different airspeeds. It should be noted that the trim conditions for rudder remained constant at every airspeed, while the aileron deflections only slight varied. It is not surprising that the ailerons may need to be slightly adjusted and deflected at different airspeeds (which correspond to different throttle values), as they may need to help counter a torque on the body of the aircraft induced by the propeller motion. The elevator deflection showed a more prominent trend with airspeed. As airspeed is increased, the elevator needs to be deflected less and less upward (defined as negative deflection). This trend makes sense from a throttle standpoint, in which airspeed and therefore wing lift are increased with throttle (see explanation of mechanics of flight above). This data is certainly useful in developing control algorithms for control surfaces for future iterations of the aircraft with respect to airspeed. From a piloting standpoint, it was found to be 4

easier to set a constant throttle and manually adjust the elevator trim to achieve steady level flight. However, from a controls point of view, in which the autopilot exhibits greater control resolution capabilities, it makes more sense to set a constant elevator pitch and adjust the throttle accordingly to maintain level flight. By doing this, the short period oscillations that may result from setting a given throttle and constantly adjusting elevator deflection to maintain altitude can be avoided. 5

Task 2: Aircraft System ID XFLR5 was used to estimate the aerodynamic performance and stability characteristics of Iteration 1. Additionally, further analysis of the Bixler 2 was performed in order to verify the craft s aerodynamic characteristics and classify its stability. Weights and Moments of Inertia (Hubert, Brian;; written by Brian) The weights of the different components on each of the aircraft were determined. This was a useful step in that it allowed for the crafts to be accurately modeled using Solidworks, which led to the calculation of moments of inertia to be used for stability analysis. Weight Table - Bixler 2 Parts Mass (g) LiPo Battery (11.1 V, 1500mAH) 119 3DR Antenna 17 GPS Sensor 17 Ardupilot (with velcro) 8 BRW Jumper Cable 5.5 Speed/Motor Controller 40 Main Wing 318 Horizontal Tail 30 Vertical Tail 11 Body 110.5 TOTAL WEIGHT 668 Weight Table - Iteration 1 Parts Mass (g) LiPo Battery (7.4 V, 1100mAh) 71 3DR Antenna 17 GPS Sensor 17 Ardupilot (with velcro) 28 6

BRW Jumper Cable 5.5 Speed/Motor Controller 23 Main Wing 306 Horizontal Tail 28 Vertical Tail 13.5 Body 105 Propeller 22 Motor 53.5 Servo 8x4 = 32 Receiver 10 TOTAL WEIGHT 731 The table below shows these values for the Bixler 2 and Iteration 1. Moments of Inertia - Bixler 2 and Iteration 1 Bixler 2 (kg-m 2 ) Iteration 1 (kg-m 2 ) I xx 0.042 I xx 0.048 I yy 0.029 I yy 0.077 I zz 0.066 I zz 0.034 I xz 0.001 I xz 0.00004 Aerodynamic Performance (Alex;; written by Alex) The aerodynamic performance of Iteration 1 and the Bixler 2 was studied through the use of XFLR5. The modeled geometries for each of the planes are pictured below. 7

XFLR5 Model - Iteration 1 XFLR5 Model - Bixler 2 The main wing was composed of a NACA 23015 airfoil, the horizontal tail was a flat plate, and the vertical tail utilized a NACA 0012 airfoil. The aerodynamic performance of the plane is heavily dependent on the main wing. For this reason the characteristics of the NACA 23015 are of special interest. Presented below are the lift curves for the airfoils present on the main wings of Iteration 1 and the Bixler 2. Data for iteration 1 will be shown in green and data for the Bixler 2 will be presented in red throughout the aerodynamic performance section. 8

Cl vs Angle of Attack - NACA 23015 (left) and S3021 (right) It was found that the Iteration 1 airfoil generated a coefficient of lift of 0.34 when the angle of attack was zero degrees. The maximum coefficient of lift was 1.035 at the given flight condition. This corresponded to an angle of attack of 8.9 degrees. It could be seen that moving away from the S3021 airfoil used on the Bixler 2 resulted in a lower maximum coefficient of lift. The benefit of the NACA 2015 is that it had its maximum lift at an angle of attack that was not near the stall region of the craft. This was unlike the Bixler 2 where it was found that the maximum lift coefficient of 1.175 occurred at an angle of attack of 11 degrees. This angle of attack was greater than the Bixler 2 s stall angle of 10.66 degrees. The tails for both aircraft had symmetric airfoils and were not considered to be significant lifting surfaces. The aerodynamic characteristics of the two planes were determined, based solely on the wings, without inclusion of the body geometry as suggested by XFLR5 (Guidelines, 2010). The following figure shows the relationship between L/D and angle of attack for Iteration 1 and the Bixler 2. L/D vs Angle of Attack - Iteration 1 (Left) and Bixler 2 (right) 9

At a cruise speed of 12 m/s it was found that (L/D) max was 18.4 for Iteration 1. This corresponded to an angle of attack of six degrees. This value was lower than that of the Bixler 2 and occurred at a higher angle of attack. The relationship between the coefficient of lift and the angle of attack was is shown below. Cl vs. Angle of Attack - Iteration 1 (Left) and Bixler 2 (right) The maximum coefficient of lift was 1.215. It was found that the main wing would enter stall at an angle of attack of 12.6 degrees. This surpassed the Bixler 2, which had a maximum lift coefficient of 1.169 that occurred at 10.9 degrees angle of attack. It was seen that Iteration one was able to reach a higher maximum lift coefficient and fly at higher angles of attack without stall. The drag polars are featured below. Drag Polar - Iteration 1 (left) and Bixler 2 (right) The drag polars above were generated by performing a sweep on angles of attack from -1 degree to stall angle of attack. The aircraft speed was set to 12 m/s. The parasite drag for the Iteration 1 wings was slightly below 0.02. This value is smaller than what would be expected for a full model that would include the body. The Bixler 2 wings generally experienced less drag than Iteration 1 at fixed coefficients of lift. 10

The stall velocity of Iteration 1 was 6.7 m/s. This is an important parameter in that it would theoretically set a threshold for how slow the craft would be able to fly. This can be seen along with how the L/D changes with airspeed in the following plots. L/D vs Velocity - Iteration 1 (left) and Bixler 2 (right) Iteration 1 had a maximum L/D at an airspeed of 8.8 m/s, and the Bixler 2 had a maximum L/D at 8 m/s. The table below provides some of the significant aerodynamic performance characteristics for the Bixler 2 and Iteration 1. Comparison of Aerodynamic Performance - Bixler 2 vs. Iteration 1 Bixler 2 Iteration 1 Main Wing Airfoil/Cl max S3021/1.175 NACA 23015/1.035 (L/D) max at 12 m/s 18.95 18.4 Cl max at 12 m/s 1.148 1.215 V stall (m/s) 6.61 6.7 V max L/D (m/s) 8 8.8 α stall (deg) 10.9 12.6 It was seen that Iteration 1 had performance values that were comparable to the Bixler 2. These performance characteristics were obtained for Iteration 1 with a relatively simple design scheme. It is expected that in future iterations the aerodynamic performance of the designed craft will be able to surpass those of the Bixler 2. Stability of the Bixler 2: The stability of the Bixler 2 was studied through the use of XFLR5. During this part of the 11

analysis the model utilized the entire aircraft, including the body. The eigenvalues corresponding to the longitudinal natural modes of the aircraft are provided in the table below. Longitudinal Stability Natural Eigenvalues - Bixler 2 Eigenvalues Mode Frequency (Hz) Damping lambda 1-12.1326-8.4023i Short Period 1.337 0.822 lambda 2-12.1326 + 8.4023i Short Period 1.337 0.822 lambda 3-0.0096-0.6637i Phugoid 0.106 0.15 lambda 4-0.0096 + 0.6637i Phugoid 0.106 0.15 A root locus defining the impact of the elevator on longitudinal stability was generated by studying the change in the eigenvalues as the elevator was deflected from -2 to 2 degrees. The selection of the range of deflections was somewhat arbitrary, but was sufficient to show how the eigenvalues moved with control surface deflection. The root locus corresponding to the elevator deflection impact on the longitudinal mode of the aircraft is presented below. Root Locus - Longitudinal Mode - Elevator Deflection - Bixler 2 The natural modes (zero control surface deflection) are shown in green and the modes due to elevator deflection are shown in yellow. It was seen that a positive deflection (downward) in the elevator caused the eigenvalues of the short period mode to move further into the left hand plane and for the magnitude of the imaginary part of the roots to increase. As the elevator was deflected upward the eigenvalues of the short period mode moved inward, decreasing in magnitude for both the real and imaginary parts. There was very little change in the position of the phugoid mode. Additionally, the lateral stability characteristics were studied. eigenvalues for the natural modes of the craft. The table below shows the 12

Lateral Stability Natural Eigenvalues - Bixler 2 Eigenvalues Mode Frequency (Hz) Damping lambda 1-34.8927-15.2529i Dutch Roll 2.427 0.916 lambda 2-34.8927 + 15.2529i Dutch Roll 2.427 0.916 lambda 3-22.3302 Roll Damping N/A N/A lambda 4 0.08216 Spiral Mode N/A N/A The lateral stability of the aircraft was managed with use of the ailerons and the rudder. The individual impacts of the two different control surfaces were studied separately. The rudder deflection was varied and the change in the location of the eigenvalues was studied. The rudder position was varied from 0 to 2 degrees. The root locus is presented below. Root Locus - Lateral Modes - Rudder Deflection - Bixler 2 It was seen that as the rudder was deflected from 0 degrees the eigenvalues corresponding to the dutch roll mode showed the most movement, with a tendency to move towards the imaginary axis as the rudder deflection was increased. The eigenvalue corresponding to roll damping moved further left along the real axis. The spiral mode showed very little change. The ailerons were asymmetrically deflected from 0 degrees through 5 degrees. The plot below shows the impacts of this deflection on the location of the eigenvalues for the lateral mode. 13

Root Locus - Lateral Modes - Asymmetric Aileron Deflection - Bixler 2 As the ailerons were deflected the eigenvalues corresponding to the Dutch roll moved further into the left hand plane and became more imaginary, the roll damping mode became more negative, and there was little change in the spiral mode. The plots below shows the relationship between the deflection of the elevator and the velocity/angle of attack of the aircraft. Velocity (left) and Angle of Attack (right) vs. Elevator Deflection - Bixler 2 As the elevator was deflected downwards the trim velocity of the aircraft increased and the angle of attack decreased. As the elevator was deflected upwards the trim velocity decreased and the angle of attack increased. From the elevator deflection study it was found that in order to obtain a trimmed velocity of 12 m/s, the elevator had zero deflection and an angle of attack of 2 degrees. The relationship between trimmed angle of attack and velocity followed intuition, leading to an increase in confidence of the stability analysis performed. Changes in the physical characteristics of the short period and phugoid modes in respect to 14

trimmed velocity are presented in the plots below. Short Period (left) and Phugoid (right) Frequency vs. Trim Velocity - Bixler 2 The above relationships were studied by varying the deflection of the elevator of the craft. An increase in the velocity would lead to an increase in the frequency of the short period and a decrease in the frequency of the phugoid mode. Plots showing the mode damping as a function of velocity are shown below. Short Period (left) and Phugoid (right) Frequency vs. Trim Velocity during Elevator Deflection - Bixler 2 Changes in the trimmed velocity had little impact on the short period damping, and the damping of the phugoid mode decreased with increases in velocity. The damping of this mode is very small to begin with, and it is believed that at such a low level of damping these changes would not have a strong impact on the actual flight of the aircraft. A similar analysis was performed for deflections of the rudder. The plots below show the relationship between the deflection of the rudder and the trimmed velocity/angle of attack. 15

Velocity (left) and Angle of Attack (right) vs. Rudder Deflection - Bixler 2 As expected, deflections of the rudder led to small changes in the trimmed angle of attack and velocity of the craft. From the study of the rudder deflections it was seen that for the desired trim velocity of 12 m/s the rudder would have no deflection and the craft would be flying at an angle of attack of 2 degrees. The relationship between the frequency and damping of Dutch roll mode are presented in the figures below. Dutch roll Frequency (left) and Damping (right) vs. Trim Velocity during Rudder Deflection - Bixler 2 Increases in the trimmed speed led to increases in the frequency and decreases in the damping of the Dutch roll mode. Lastly, the impact of the aileron deflections on the frequency and damping of the different modes were studied. The plots below show the relationship between the deflection angle of the aileron and the trimmed velocity/angle of attack of the craft. 16

Velocity (left) and Angle of Attack (right) vs. Aileron Deflection - Bixler 2 Deflection of the ailerons had little impact on the trimmed angle of attack of the craft. As the angle of the deflections was increased beyond three degrees the trimmed velocity showed signs of increasing quadratically. The figures below show the impact of the ailerons on the frequency and damping of the Dutch Roll. Dutch roll Frequency (left) and Damping (right) vs. Trim Velocity during Aileron Deflection - Bixler 2 Deflection of the ailerons led to an increase in the Dutch roll frequency and little change in the damping. The three independent studies on the impacts of the control surface deflection on the velocity and angle of attack showed that for the desired trim speed of 12 m/s, stable flight could be achieved with no control surface deflection if the craft was flying at an angle of attack of 2 degrees. The stability derivatives can then be determined at these conditions. The stability derivatives were taken with respect to the stability axis. This axis was defined with the x-axis representing the projection of the velocity vector onto the symmetric plane of the aircraft, the y-axis out of the right wing of the craft, and the z-axis pointing downwards completing the orthogonal directions. The table below presents the values corresponding to the stability derivatives. 17

Stability Derivatives for Trim Conditions (velocity = 12 m/s;; angle of attack = 2 degrees) - Bixler 2 Longitudinal Derivatives Lateral Derivatives Cx u -0.01188 CY b -7.4768 Cx a 0.17096 CY p 1.235 Cz u -0.00026706 CY r 4.5857 CL a 4.607 Cl b 0.04167 CL q 5.4725 Cl p -0.51797 Cm u -0.0047567 Cl r 0.026502 Cm a -0.55512 Cn b 2.7393 Cm q -12.271 Cn p -0.52931 Neutral Point 0.1464 (m) from LE Cn r -1.6803 The control derivatives for the trimmed flight condition were obtained and are presented below. Control Derivatives for Trim Conditions (velocity = 12 m/s;; angle of attack = 2 degrees) - Iteration 1 Elevator Rudder Aileron CX de -0.0052828 CX dr -0.0011257 CX da -0.11522 CY de 0 CY dr 0.057743 CY da -0.28166 CZ de -0.33396 CZ dr -0.0011168 CZ da 0.027903 CL de 0 CL dr 0.0053012 CL da 0.16626 CM de -0.87216 CM dr 5.4972e-06 CM da 8.933e-07 CN de 0 CN dr -0.020931 CN da 0.10293 Stability of Iteration 1: The stability of Iteration one was also studied using XFLR5. It was initially found that in order for the craft to be trimmed at a feasible speed the elevator would need to be deflected. This initial deflection was set to 2.5 degrees, which corresponded to the desired trim speed. This was then set as the default elevator setting during the evaluation of the other control surfaces. The table below provides the natural longitudinal eigenvalues when there are no additional deflections of 18

the control surfaces. Longitudinal Stability Natural Eigenvalues - Iteration 1 Eigenvalues Mode Frequency (Hz) Damping lambda 1-17.4302 N/A N/A N/A lambda 2-4.808 N/A N/A N/A lambda 3-0.8339 N/A N/A N/A lambda 4 0.6044 N/A N/A N/A All of the natural longitudinal modes were non-oscillatory, one of which, was unstable. A root-locus plot was generated by varying the deflections of the elevator from its set position of 2.5 degrees. The plot below shows the movement of the roots as the elevator s deflection was changed from 1.5 degrees to 5 degrees (relative to the horizontal). At elevator deflection angles less than 1.5 degrees from the horizontal the lift was negative and was not able to balance the weight of the craft for any speeds. These elevator deflection angles were not considered. Root Locus - Longitudinal Mode - Elevator Deflection - Iteration 1 As the elevator was deflected beyond 4 degrees, lambda 2 and lambda 3 met on the real axis and broke into the imaginary plane becoming oscillatory. Lambda 1 moved further into the left hand plane with positive deflections of the elevator. Lambda 4 changed little with the deflections of the elevator. The eigenvalues corresponding to the natural lateral stability state are tabulated below. 19

Lateral Stability Natural Eigenvalues - Bixler 2 Eigenvalues Mode Frequency (Hz) Damping lambda 1-4.0529-10.639i Dutch Roll 1.693 0.356 lambda 2-4.0529 + 10.639i Dutch Roll 1.693 0.356 lambda 3-30.9085 Roll Damping N/A N/A lambda 4-0.1017 Spiral Mode N/A N/A The values given above correspond to the initial state where the elevator is deflected by 2.5 degrees and the ailerons and rudder have zero deflection. The root-locus representing the changes in the roots locations with deflection of the rudder is presented below. The rudder position was varied from 0 degrees to 2 degrees. Root Locus - Lateral Modes - Rudder Deflection - Iteration 1 It was found that deflection in the rudder had no effect on the location of the eigenvalues corresponding to lateral stability. Lastly, the effect of asymmetric aileron deflection on the position of the lateral eigenvalues were studied by varying the deflection of the ailerons from 0 degrees to 5 degrees. 20

Root Locus - Lateral Modes - Asymmetric Aileron Deflection - Iteration 1 Again it was found that changes in the deflection of the ailerons had no impact on the location of the roots characterising the lateral stability of the craft. Further analyses were performed where the rudder and aileron were deflected up to 30 degrees. There was still little movement in the location of the eigenvalues. This is reason for concern in future designs as the rigidity of the lateral eigenvalues represents ineffectiveness in the ailerons and rudder to influence the flight of the craft. The plots below show the relationship between the deflection of the elevator and the velocity/angle of attack of the aircraft. The elevator deflection was varied from 1.5 degrees to 12.5 degrees (relative to the horizontal). Velocity (left) and Angle of Attack (right) vs. Elevator Deflection - Iteration 1 Increasing the deflection of the elevator led to a decrease in the trimmed velocity and an increase in the trimmed angle of attack. It should be noted that in the plots provided above a control value of zero corresponds to an elevator deflection of 2.5 degrees. 21

The relationship between the trimmed flight velocities and the frequency/damping of the Phugoid mode is presented in the figures below. Short Period (left) and Phugoid (right) Frequency vs. Trim Velocity during Elevator Deflection - Iteration 1 Increases in the trimmed velocity led to decreases in the frequency of the Phugoid mode. This continued until a velocity of 10 m/s where the mode joined the real axis and was no longer oscillatory. The short period was non-oscillatory for all of the trimmed velocities studied. Short Period (left) and Phugoid (right) Damping vs. Trim Velocity during Elevator Deflection - Iteration 1 Increases in the trimmed velocity led to the damping of the Phugoid mode to increase until it became critically damped. Beyond this the damping decreased until the mode became non-oscillatory. The damping of the short period was consistently at zero corresponding to its non-oscillatory nature. It will later be shown that the rudder and ailerons had little impact on the trimmed velocity of the craft, and for this reason the elevator s influence on the frequency and damping of the Dutch roll 22

become important. These relationships are shown in the figures below. Dutch roll Frequency (left) and Damping (right) vs. Trim Velocity during Elevator Deflection - Iteration 1 Increases in the trimmed velocity initially led to a decrease in the Dutch roll frequency and an increase in the damping. The frequency then began to increase and the damping decreased as the trimmed velocity was increased beyond 5 m/s. Next, the relationship between the frequency and damping of the lateral modes and the deflection of the rudder were studied. As mentioned earlier, deflections in the rudder did not cause for much change in the locations of the eigenvalues. It was anticipated that the influence of the rudder on trim velocity and angle of attack would not be seen at small deflection angles. For this reason the study incorporated a sweep of rudder deflection from 0 degrees to 30 degrees. A deflection of 30 degrees corresponded to the largest possible deflection of the rudder. These relationships are shown below. Velocity (left) and Angle of Attack (right) vs. Rudder Deflection - Iteration 1 23

Significant changes in the trimmed velocity of the craft started to occur at rudder deflections greater than 20 degrees. The trimmed angle of attack was impacted by the deflection angle at lower speeds. There was a local minimum of 1.2 degrees at a deflection of 13 degrees and deflections greater than 18 degrees led to a quick drop in the trimmed angle of attack. The relationship of the velocity variation and the frequency and damping of the Dutch roll mode are presented below. Dutch roll Frequency (left) and Damping (right) vs. Trim Velocity during Rudder Deflection - Iteration 1 Increases in the velocity led to a linear increase in the Dutch roll frequency. The damping of this mode remained constant. Lastly, the effects of the ailerons on the trimmed velocity and angle of attack were studied. The ailerons were asymmetrically swept from 0 degrees to 30 degrees. The relationship between deflection angle and trimmed speed are shown below. Velocity (left) and Angle of Attack (right) vs. Aileron Deflection - Iteration 1 24

The ailerons did little impact on the trimmed velocities and angles of attack of the aircraft. As a result of this a relationship between the modal frequencies/damping and trimmed velocity was not able to be generated. From the previous studies it was found that for the desired trim velocity of 12 m/s and an elevator deflection of 2.5 degrees the craft would be flying at an angle of attack of 2.15 degrees. In this situation there is 0 degrees of deflection for the ailerons and rudder. Stability Derivatives for Trim Conditions (velocity = 12 m/s;; angle of attack = 2.15 degrees) - Iteration 1 Longitudinal Derivatives Lateral Derivatives Cx u -0.00627 CY b -0.40964 Cx a 0.19387 CY p -0.2527 Cz u -0.00024997 CY r 0.26464 CL a 5.2932 Cl b -0.14459 CL q 5.2909 Cl p -0.53448 Cm u 4.8468e-07 Cl r 0.063294 Cm a 0.58762 Cn b 0.088532 Cm q -20.91 Cn p -0.059117 Neutral Point 0.10912 (m) from LE Cn r -0.088206 Additionally, the control derivatives for the trimmed flight condition were obtained and are presented below. Control Derivatives for Trim Conditions (velocity = 12 m/s;; angle of attack = 2.15 degrees) - Iteration 1 Elevator Rudder Aileron CX de -0.012949 CX dr -2.3924e-05 CX da -0.00013399 CY de 0 CY dr 0.13251 CY da 0.034474 CZ de -0.36331 CZ dr 1.483e-05 CZ da 4.1666e-05 CL de 0 CL dr -0.0011537 CL da 0.25039 CM de -1.3752 CM dr 5.0264e-06 CM da -1.6359e-06 CN de 0 CN dr -0.053493 CN da 0.0073322 25

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Task 3: Updated the mission plan / strategy. (Chris and Brian;; written by Chris and Brian) Strategy Overview The general mission strategy has not changed significantly from problem set 1 though it has been slightly simplified. It now is made of two distinct steps. Step 1 aims to locate the targets as quickly as possible and Step 2 aims to hone in on the targets. More detailed explanations of steps can be found in Problem Set 1 but below are brief discussions of the techniques and changes. Step 1: Sighting the targets The goal of Step 1 is to minimize the t sight portion of the score and to set up Step 2 with realized target locations. Initially the aircraft will climb to the maximum altitude of 400 ft. At 400 ft the field of vision is largest, allowing the aircraft to cover the most land area with the smallest flight time and distance. The tradeoff between power lost from the climb versus the t sight increase will be studied with simulations, but it is predicted that the reduction in t sight will be worth it. While in Step 1 the aircraft will fly in an inward spiral, as shown below. This spiral will allow for the most area to be covered with the least amount of distance flown. Optimization discussed below will ultimately determine the flight path though. Step 1 aircraft searching pattern Step 2: Honing in on Individual Targets Once all 3 targets are sighted the aircraft will immediately transition into Step 2. The goal of Step 2 is to reduce the error target location as much as possible. Initially the aircraft will glide down to the lowest altitude. This gliding technique will help to recover much of the battery power lost from 27

the climb. The order in which targets are visited will be determined by minimizing the aircraft flight distance. Once the aircraft is positioned over a target it will circle the target until a predetermined number of pictures are taken. This number will be optimized based on simulation results and will include a factor to ensure we are able to reach all three targets within the trial time. Aircraft encircling targets with initial locations determined by Step 1 The method for encircling targets is the main alteration from the previous problem set. It now simultaneously utilizes taking pictures with the target in the field of vision and taking pictures where the field of vision bisects the possible target region. The section technique used is shown in the figure below. The possible target region starts as both A and B based on previous pictures. The aircraft then circles the possible region at a turning radius equal to the field of vision. When a picture is taken, the results return either a found target or no target found. If no target is found then the target is guaranteed to be in region A and if a target is found, the target must be in region B. This technique, herein referred to as negative photographs (because it purposely excludes the target from the image) removes randomness that occurs with repeated target pictures. To use both negative photographs and to get more pictures of the target, Step 2 will encircle each target a radius equal to the field of vision. Some experimentation will be necessary to see the accuracy of the autopilot as a certain precision is required to use negative photographs. 28

Location Determination Algorithm To determine the location of the circle given the taken photographs, multiple techniques have been discussed. The discussed techniques each has a tradeoff between computational cost and accuracy. Due to limitations on the APM, it is likely that a less optimal algorithm will be used to allow for onboard processing. The problem is set up within the system as an array of successful images, each with a target number, location, and radius. There is also an array of pictures taken that includes the field of vision, aircraft location, and targets not found. The center method takes the successful image array and averages the location of the centers of the images for each target. This technique utilizes the fact that the returned circles are randomly placed around the target, so over several images they will encircle the correct target area. The figure below shows a simulation of the center method with 5 pictures. In this case, the guessed location was within 1m of the actual target location, well within the minimum miss of 3m. This algorithm does not guarantee a solution within the area of circle overlap, but has been shown to be quite close to the correct solution. Another disadvantage of this algorithm is that it will be difficult to implement negative photography s results. The major benefit of this algorithm is its speed for use on the APM. If processing speed proves to be a major issue for on board calculations, this algorithm perform decently. 29

Center method for target location with 5 pictures. Red represents centers of pictures, green cross represents the target, blue square represents the estimation, and the yellow circle represents a 3m error. The results of two versions of this method are shown below. The version without R weighting simply averages the centers of all the circles. The version with R weighting takes into account that with larger images, it is more likely the target will be farther away from the center than with a small image. If two circles are considered with different radii, the centroid of the overlap area will actually be closer to the smaller of the two circles, a result that this R weighting attempts to utilize. Based on the figure below, using an R weighting yields better results, if only slightly. Center method average target miss versus number of pictures with standard deviation shown in error bars. After 6 pictures the average error plus one standard deviation is less than 3m. The grid method forms a grid of potential target locations around the first successful image. All future images of the target are checked against each grid location. If a grid point is within the picture s circle radius, then that grid point s value is incremented by 1. The possible target 30

locations will all share the highest value, which indicates all circles overlap there. The centroid is taken of all these locations and returned as the estimated position. The figure below illustrates this method. The gray area is the initial grid while the blue area is the possible target locations. The centroid of that area, the cyan square, is the estimated target location. In the figure on the left, the picture circles overlapped a small enough amount to give a resulting target location within 3m, but the picture on the right had such a large area, that the miss was about 4m. The third image shows how the values change for each grid point as more circles overlap. Grid method where the black points are the grid, the blue points are the possible locations, the green cross is the correct location, and the cyan circle is the estimated location. One benefit of the grid method is the ease of implementation of negative photography. By decrementing the value of points within the field of vision of a negative photograph, that area will be effectively eliminated. Because such large areas of uncertainty, like that shown in the above figure on the right, negative photographs to guarantee an area reduction will be extremely useful. The grid method s computational cost is its main downfall though. On the above 50x50 grid, each new picture requires 2500 rounds of calculations to be performed. It would be ideal to have a coarser grid, which may be achievable through refinement as more information is gathered. It is important to keep a grid that is significantly fine though, to reduce the chances of the only possible solutions being in between grid points. If this case were to happen, one option is to use 31

the next highest number of overlaps. Another option is to find the overlap of the first circle (with the grid around it) and the last circle. This overlap should be quite small, within the 3m, and could thus be returned as a solution. The final method implemented was an intersection method. The method finds the intersection locations between each pair of circles. These intersections are then downselected to only the intersections that are in every circle. The resulting locations are averaged to find the expected target location. Intersection method where the blue stars are all the intersection points and the green stars are corners of the overlap centroid. The intersection method guarantees a result within the possible area but like the grid method, it comes at a high computational cost. With a low number of pictures taken, the cost is reasonable, but cost goes up with the number of images squared, as shown in the below figure. This cost comes with minimal gains, if any, in the average accuracy. The intersection method also allows for easy implementation of negative photography. 32

Comparison of all locationing methods Simulation To test several different paths, we have set up a MATLAB-based simulation. The simulation takes in a trajectory and performs a statistical analysis on the results of a set of 50,000 simulations. The constraints on the path considered are a minimum turn radius, a maximum change in height between waypoints, and that it must provide full coverage of the area. Each of these paths is assumed to begin at 400 feet and remain at 400 feet. This was done as a first iteration such that we could get a high-level understanding of which paths to consider in the optimization stage. The time to climb to this altitude was estimated to be 16 seconds, which we have rounded to 20 seconds to get conservative results. In the future, we hope to have the aircraft taking pictures during the climb and to be able to use this information in our path planning. We also use the assumptions that the aircraft is flying at a speed of 12m/s and constantly taking pictures at 3Hz once reaching 400 feet. This allows us to calculate the location of the pictures taken by the aircraft. The radius of the turns was computed using the circumradius of three points in the path. Circumradius is the radius of the circle that has all three points along its circumference. The following table depicts the paths considered. 33

Inward Spiral Outward Spiral Concentric Circles A Concentric Circles B Modified Lawnmower Seven Sectors These paths vary in both length and time, and are defined by a varying number of 34

waypoints. Because of this, it is necessary to have a function which calculates the aircraft position as a function of time. The function we have written reformats the path such that the aircraft positions are interpolated for every 3 seconds. With this done, we can find the location of each camera picture and score the results based on a random set of target locations. This is done 5000 times to ensure that a sufficient number of cases are considered. After finding all three targets, we will loiter at the first until we have six pictures. We will then command the aircraft to travel to the estimated location of the next closest target and loiter there until we have six pictures in total for that target, and the same will be done for the final target. By having six pictures we can locate the target to within three meters over 60% of the time, thus maximizing the reward for accuracy. 35

Inward Spiral: PROS: Full area coverage Minimum turn radius satisfied Moderate efficiency CONS: Worst average score Second longest average time to sight We chose to use an Archimedean spiral since the radius increases linearly with theta. This is useful because we will get a more efficient area coverage rather than using a logarithmic spiral. To actually get full coverage, the aircraft must cross the border of the search area and fly an area which cannot have possible targets. This path is moderately efficient with the average number of pictures being 3.2 at each point in the search area. We expected this to be the highest scoring path. Including the second stage of the path, we can have an average score of about 49.9. Min Average Max Variance 1 Deviation Score 13.8 25.3 87.4 52.1 7.2 Time to Sight 26 120.4 155 559 23.6 Pictures of Targets 4 9.5 15 2.1 1.4 36

Outward Spiral: PROS: Full area coverage Minimum turn radius satisfied Third best average score Moderate efficiency CONS: None, but not the best The outward spiral is the same path as the previous spiral, with the only modification being the direction of flight. This path begins at the center and spirals outward. This is a more efficient path because the inefficient searching is done at the end, which is likely after all targets have been discovered. With the same area coverage, it is unsurprising that the picture efficiency is the same at 3.2 pictures per point. We were surprised at how different the scores between the two spirals actually is, and as a result will consider the direction of whichever path we settle on. Including the second stage of the path, we can have an average score of about 55.4. Min Average Max Variance 1 Deviation Score 15.1 32.3 120 114 10.7 Time to Sight 23 90.7 155 648 25.4 Pictures of Targets 5 9.6 16 2.1 1.4 37

Concentric Circles A: PROS: Full area coverage Minimum turn radius satisfied Tied for best average score Tied for most efficient Fastest average time to sight CONS: None This path of concentric circles was developed to have as little overlap as possible between images. The thought behind this was that such a path would take less time and have a better score due to the time to sight being minimized. The picture efficiency of this path was 2.7 pictures per point, which is tied for the best of all of the paths considered. This path is tied for the best average score as well, but it has the fastest time to sight. Because of this, we consider this our best path. Including the second stage of the path, we can have an average score of about 57.4. Min Average Max Variance 1 Deviation Score 18.6 32.8 116 94.4 9.7 Time to Sight 23 83.1 113 385 19.6 Pictures of Targets 3 8.2 15 2.0 1.4 39

Concentric Circles B: PROS: Full area coverage Minimum turn radius satisfied Tied for best average score Tied for most efficient Second fastest average time to sight CONS None, but not the best at anything Similar to Concentric Circles A, Concentric Circles B is only different in the transition between the inner and outer circle. Instead of performing a near 180 o turn, we have a straight line connecting the inner and outer waypoints. The efficiency is very similar and still 2.7, and the average score has no slight improvement on the previous path design. It satisfies all of the constraints, has the best average score, and has the second lowest average time to sight. Including the second stage of the path, we can have an average score of about 57.3. Min Average Max Variance 1 Deviation Score 18.7 32.8 110 94.5 9.7 Time to Sight 23 83.2 113 385 19.6 Pictures of Targets 3 8.2 15 2.0 1.4 40

Modified Lawnmower : PROS: Full area coverage Third most efficient CONS: Low average score Min. turn radius violated Another common search and rescue pattern is the lawnmower pattern. This pattern is particularly suited for an undefined region or a rectangular region. Because of this, we have modified it slightly to make it more suited for our circular search area. The efficiency of this pattern is pretty good at 2.88, but the average score was not very good. Including the second stage of the path, we can have an average score of about 54.4. Min Average Max Variance 1 Deviation Score 17.0 29.9 120.3 84.5 9.2 Time to Sight 23 95 125 471.3 21.7 Pictures of Targets 4 8.6 15 2.1 1.4 41

Seven Sector: PROS: Full area coverage CONS: Low average score Worst efficiency Min. turn radius violated Longest average time to sight The sector search is another common search technique for search and rescue efforts. We did not expect this to perform well because of the intense amount of overlap in the center of the search area. This is the least efficient pattern, having over 5.9 pictures per point. Including the second stage of the path, we can have an average score of about 47.9. Min Average Max Variance 1 Deviation Score 11.1 29.1 108 165 12.9 Time to Sight 23 137.3 206 2160 46.5 Pictures of Targets 4 17.7 59 47.6 6.9 Battery: The battery we used for the Bixler 2 was able to perform for a 15 minute flight of fairly vigorous maneuvering at 50-60% throttle. We will use this to derive the first model of our battery power consumption. It should be a sufficient starting point to consider a linear drain 42

of the battery and that our aircraft will have a similar power consumption to the Bixler 2. The battery for the competition is 1100mAh while that which we used for the Bixler is 1500mAh. Since the Bixler battery lasted for 15 minutes, we will expect an average draw of approximately 6A. We can therefore expect that our endurance for the competition flight will be eleven minutes. Calculating the distance of the flight, and using the assumption of constant airspeed (12 m/s) and no wind, we can determine the length of time required for each path. The worst case scenario for the second stage of the flight is that the targets are all along the circumference of the circle and equally spaced apart. This would result in a second stage flight with an approximate length of 571.6m, which would take 47.6s. For a worst case scenario of five additional pictures at each target, we would need another 15 seconds each for a total of 45 seconds. The total flight time is initial climb, traversing the path, loitering, and traveling between known targets, or 112.6s plus the time to traverse the path. Path Length Path Time Total Flight Time Total Battery Consumed Inward Spiral 1614.4 m 134.5s 4.1 minutes 37% Outward Spiral 1614.4 m 134.5s 4.1 minutes 37% Concentric Circles A 1092.6 m 91.1s 3.4 minutes 31% Concentric Circles B 1080.4 m 90.0s 3.4 minutes 31% Modified Lawnmower 1243.6 m 103.6s 3.6 minutes 33% Seven Sectors 2481.7 m 206.8s 5.3 minutes 48% We see therefore that the aircraft should be well within its endurance limits for the paths we are considering. Not surprisingly, the paths with the shortest time consume the least battery. Our consumption should also be less than this because we will not have to traverse the entire path in most cases, and we should already have multiple pictures of each target, requiring a shorter loitering time. Optimization With these results, we take the Concentric Circles A path and use that as the starting point for the optimization algorithm. Using FMINCON, we can solve this nonlinear optimization by writing a cost function and a nonlinear constraint function. The nonlinear constraint function enforces the limits on turn radius, coverage, and max climb/descent. The cost function returns the negative of the average score of many simulations. The same 43

assumptions and conditions as earlier still apply for this first optimization. Further investigation will be done in height optimization, the battery model, and the strategy for getting better than 3m accuracy after seeing all three targets. To help FMINCON reach reasonable results, we introduce several new restrictions to the problem. The waypoints are constrained to be within a square grid which circumscribes the search area. These upper and lower bounds on the design variables prevent the algorithm from testing wild paths that cannot be efficient. After running for only a couple iterations and not making much change, we introduced the limit that the minimum step in design variable change is.1m. The default is zero, and increasing this forces FMINCON to consider larger steps and potentially develop a much more different path. The other important change was to force FMINCON to only consider paths which satisfied the constraints. Without this, it frequently happened that it would reach an infeasible solution and become unable to find a direction to improve. The scoring function used is the negative of the average score after 100 simulations. The following results were for an optimization without the turn radius constraint, which was preventing the optimizer from converging. It s interesting to note that the path is not substantially changed from the original, nor is the score. The efficiency of the new route is 2.8 pictures per point. This leads us to conclude that for our limited problem, ours is a near optimal solution. Min Average Max Variance 1 Deviation Score 18.8 32.8 109 91.9 9.6 44