A kinematic approach to segmented-trajectory generation for the total loss of thrust emergency

AbstractContemporary twin-engine airliners are more vulnerable to total loss of thrust than yesterday's three and four engine airliners, due to reduced engine redundancy. In the event of a total loss of thrust, flight crews have only one chance for landing, because the aircraft cannot gain altitude. Therefore, there is a pressing need to explore the idea of an engines-out landing trajectory optimization for commercial jets. A few past studies addressed this safety issue for general aviation aircraft and fighter jets but not commercial jets, primarily because the essential aircraft-specific aero-dynamic data are not publicly available. To fill in this gap, this study adopts a kinematic approach to aircraft trajectory optimization. Unlike conventional trajectory optimization methods, the kinematic algorithm requires minimal amount of aircraft-specific aerodynamic data that can be effortlessly collected in a full flight simulator. The paper describes the kinematic algorithm and applies it to a realistic bird...


Notations
Compressibility correction to convert calibrated airspeed into equivalent airspeed. The transition segment connecting linear segment j to circular segment j.
, 1 j j T + The transition segment connecting linear segment j+1 to circular segment j.
v Calibrated airspeed. Maximum allowable bank angle (°) for the engines-out landing maneuver.

1.1.The total loss of thrust emergency
While 87% of the U.S. commercial fleet consisted of three and four engine aircraft in 1965 (Dolbeer et al. 2013), today more than 96% of air travelers are transported by twin-engine jets (RITA 2014). Contemporary twin-engine jets offer superior fuel efficiency and lower noise levels than three or four engine jets (Peeters et al. 2005). However, they have cut down engine redundancy due to the reduced number of engines. A statistical analysis of the FAA Wildlife Strike Database shows that contemporary twin-engine jets are 15 times more likely to undergo a total loss of thrust in the event of a bird strike when compared to three and four engine jets (Avrenli, Dempsey 2015). In the near future, a total loss of thrust due to a bird strike is expected to occur more frequently due to the substantial increase in North American large bird populations (Dolbeer et al. 2013;Dolbeer 2009;Nicholson, Reed 2011), and the fact that modern-day turbofan engines are not tested for large birds (Transport Canada 2004). Unlike single-engine failure or partial loss of thrust, the aircraft cannot gain altitude in the occurrence of a total loss of thrust. Thus, the flight crew has only one chance for landing, and there is no room for human error. Otherwise, a total loss of thrust may lead to severe consequences such as the Dana Air Flight 992 that resulted in 163 fatalities in 2012 (Holland 2012).
To address the total-loss-of-thrust emergency, quick reference handbooks are designed to enable speedy and successful recovery of at least one engine. Airliner type-rating programs assume that a total loss of thrust culminates in at least one engine recovery, and do not require simulator training for an engines-out emergency landing (FAA 2008;CFR 2014). If an engine restart cannot be achieved in a real-life emergency, airline pilots are left with virtually no guidance on how to manage the emergency situation. Therefore, there is a pressing need to develop the idea of an engines-out landing trajectory optimization for commercial aircraft.

Definition of the problem
A number of studies (Rogers 1995;Hoffren, Raivio 2000;Hyde 2005;Shapira, Ben-Asher 2005;Atkins et al. 2006;Brinkman, Visser 2007;Adler et al. 2012) addressed the engines-out landing trajectory optimization. These studies targeted single-engine general aviation aircraft such as the Beech Bonanza Model 33A 285 (Rogers 1995), Van's Aircraft RV-4 (Hyde 2005) and Cessna 172 Skyhawk (Adler et al. 2012), and single-engine fighter jets such as the Bae Hawk Mk.51 (Hoffren, Raivio 2000) and Lockheed-Martin F-16 (Shapira, Ben-Asher 2005;Brinkman, Visser 2007). However, none of these studies addressed commercial jets, primarily because conventional trajectory optimization algorithms require aircraft-specific aerodynamic-coefficient data 1 , which is not released by commercial aircraft manufacturers. The unreleased data may be collected in full flight simulators, but it would require several hours of flight simulation tests, which would render it costly and impractical. Due to the inaccessibility of the aerodynamic-coefficient data, the idea of an engines-out landing trajectory optimization considering commercial jets has not been addressed in detail.

Objectives
In order to fill in the gap in the literature sources, the objective of this study is to develop a trajectory optimization algorithm for commercial jets in a total loss of thrust emergency. Contrary to conventional trajectory generation algorithms, the proposed algorithm will adopt a kinematic approach, and will not require the input of aerodynamic-coefficient data for the aircraft in 1 Such as the lift coefficient versus angle-of-attack and the drag polar question. To achieve the objective, the study aims to carry out the following tasks: 1) develop a trajectory optimization algorithm for the total-loss-of-thrust emergency using a kinematic approach; 2) demonstrate the application of the algorithm through a realistic total-loss-of-thrust scenario. 3) assess the accuracy of the proposed algorithm through flight simulation tests; 4) assess the sensitivity of the proposed algorithm for modeling uncertainties; The findings can enable aviation practitioners to identify safe landing maneuvers in possible total-loss-ofthrust emergencies for commercial jets. Consequently, the findings can be utilized in airliner type-rating programs, design of airport environments and evaluation of aircraft-airport compatibility. The findings can also be utilized to develop adaptive flight planners that can achieve real-time trajectory optimization in the totalloss-of-thrust emergency.

Background
A number of studies have been focused on the engines-out trajectory optimization for general aviation aircraft and fighter jets, but not for commercial jets. Rogers (1995) utilized a simplified point-mass model to compute the optimal steady-state turn-back maneuver to the departure airport for a Beech Bonanza aircraft undergoing a total loss of thrust during the initial climb. Hoffren and Raivio (2000) used the point mass dynamics to compute the maximum-glide-range trajectory for a BAe Hawk Mk.51 aircraft. Hyde (2005) utilized the "Optimal Trajectories by Implicit Simulation" (OTIS) software package to compute the minimum-altitude-loss trajectory to the departure airport for a Van's Aircraft RV-4 undergoing a total loss of thrust during the initial climb. Shapira and Ben-Asher (2005) formulated an analytical algorithm that converts airspeed excess into altitude and range in the occurrence of a total loss of thrust, and demonstrated the application of the algorithm on a Lockheed-Martin F-16 aircraft. Atkins, Portillo and Strube (2006) employed the point-mass dynamics to develop a real-time segmented-trajectory generation algorithm for the totalloss-of-thrust emergency, and demonstrated the application of the algorithm for a general aviation aircraft. Brinkman and Visser (2007) utilized the point-mass equations of motion to formulate the conditions under which a return to the departure runway is a safe option for a Lockheed-Martin F-16 aircraft undergoing a total loss of thrust during the initial climb. Adler, Bar-Gill and Shimkin (2012) formulated a six-dimensional optimal control problem to develop a 3-D trajectory planning algorithm, and applied the algorithm to a Cessna 172 Skyhawk undergoing a total loss of thrust. connected by segments of constant-trim states 2 as follows: In the occurrence of a total loss of thrust, a rapid computation of a flyable landing trajectory is required. Thus, an overly complicated formulation of the optimum trajectory should be avoided. A segmented trajectory can be quickly computed. Atkins, Portillo, and Strube (2006) showed that the execution time for a segmented trajectory generation can be achieved in less than 1.0 second in an emergency situation. A segmented trajectory can also be reduced to basic pilot and ATC commands and more easily interpreted by pilots and air traffic controllers (Atkins et al. 2006).

Aerodynamic input data
In the occurrence of a total loss of thrust, the post-failure performance characteristics of the distressed aircraft depend on the specific aerodynamic design. Therefore, a kinematic approach still requires some aircraft-specific aerodynamic data. In the proposed algorithm, the required input data is the steady-speed 3 engines-out glide ratio of the aircraft at a minimum of four bank angles as follows: i) 0°; ii) max ϕ ; iii) two intermediate ϕ values between 0° and max ϕ . A simulation methodology in (Avrenli, Dempsey 2014b) can be followed to effortlessly collect the input data in a full flight simulator in less than an hour. Once the data is collected, the least squares estimation method (Weisberg 2014) is applied to the data to build a third-degree, piecewise continuous polynomial function given in Equation (1): Equation (1) is used as the aerodynamic input data in the formulation of the optimization problem.

Adjustables
The adjustables for the optimization problem are defined as follows: - Hence, the optimization problem involves a total of 10 adjustables. 2 A constant-trim state cannot occur during the transition segments due to the changing roll state of the aircraft 3 Steady-speed refers to the constant calibrated airspeed All aforementioned studies involved methodologies that require the input of aircraft-specific aerodynamic-coefficient data, which is not publically available for commercial jets. Without the essential aerodynamic input data, these methodologies are not applicable to commercial aircraft. This study is the first of its type to explore the idea of an engines-out trajectory optimization for commercial jets, and will bring this underdeveloped idea to maturity by adopting a kinematic approach.

Practical feasibility
To be of practical value, the optimum landing trajectory should: -not require non-trivial and complex changes in the flight path angle and bank (roll) angle (Hoffren, Raivio 2000); -be simple enough to follow in an emergency situation (Kelly et al. 1982); -be compressible to simple and general pilot and air traffic control (ATC) commands (Hoffren, Raivio 2000). Therefore, the optimization algorithm will be formulated allowing for the following: 1) the optimum landing trajectory will be defined assuming a constant calibrated airspeed, because the airspeed is directly related to the flight path angle (Anderson 2007), and it would be significantly more practical and intuitive for flight crews to maintain a given airspeed rather than to follow a set of complex pitch attitude directives; 2) the optimum landing trajectory will not require more than three banked turns. Otherwise, too many changes in the bank (roll) angle may not be practical to follow in an emergency situation.

Segmented trajectory
To adopt a kinematic approach, the engines-out landing trajectory is divided into three types of segments based on the bank (roll) angle state: 3) linear segments, where the bank angle equals 0° and a wings-level flight is performed; 4) transition segments, where the bank angle changes linearly from 0° up to max ϕ and vice versa; 5) circular segments, where the bank angle retains a constant positive value of less than or equal to max ϕ . The presumed landing trajectory consists of four linear segments (L) interconnected with three circular segments (C). There is also one transition segment (T) directly before and after each circular segment to simulate a continuous change in the bank angle. Hence, the presumed landing trajectory is a sequence of waypoints

Objective function
The goal of trajectory optimization is to achieve efficient energy management. In the occurrence of a total loss of thrust, the energy of the distressed aircraft stems from its airspeed and altitude. To achieve efficient energy management, the objective function aims to minimize the altitude loss required for the distressed aircraft to glide to an intended landing site within the degraded flight envelope. The objective function is formulated in Equation (2) as follows: where: - loss over all linear, circular and transition segments, respectively. Using , cot( ) v ϕ γ found from Equation (1), the altitude loss over different segments is computed based on Equations (3a), (3b) and (3c): where: - equals the radius of the banked turn for circular segment j (Anderson 2007); equals the length of each transition segment connected to circular segment j; is the true airspeed for subsonic flights below 10,000 ft above mean sea level (AMSL) (Blake 2009a). If the total-loss-of-thrust emergency occurs above 10,000 ft AMSL, the compressibility correction (i.e. V C ∆ ⊥ ) must be applied to the calibrated airspeed to predict the required altitude loss accurately (Blake 2009a).

Constraints
The landing trajectory starts at the initial aircraft position, and ends at the intended touchdown point. This is formulated as a set of geometric constraints as given in Equations (4a) through (4c) as follows: Sign v X g A clothoid (i.e. Euler's spiral) can accurately represent a transition segment because a linear change in the bank angle results in an approximately linear change in the curvature (Wilde 2009).
The ( ) S u and ( ) C u functions in Equations (8a) through (9b) are the Sine and Cosine Fresnel integrals that can be accurately approximated using the series expansions of the Sine and Cosine functions (Knopp 1990).

Solution of the optimization problem
the altitude loss required for flying an optimal trajectory depends on the true airspeed, which depends on air density Blake 2009a). This introduces a circular reference in the optimization problem, because air density depends on altitude (Blake 2009b), and the flight altitude can only be predicted once the optimum trajectory is computed. To overcome the circular reference, an iterative procedure is proposed as follows. 1) Replace all T v (i.e. true airspeed) terms in Equations (3), (6), (7), (8) and (9) with v (i.e. calibrated airspeed). 2) Find the preliminary solution to the optimization problem using the differential evolution algorithm (Price et al. 2005). The differential solution algorithm is a fast and robust optimization algorithm for continuous domains that can handle non-differentiable and nonlinear objective functions, as in this problem (Price et al. 2005;Simon 2013 Equation (11) is formulated based on the fact that the true airspeed ( T v ) corresponding to a given calibrated airspeed (v) increases by around 1.5 percent for every 1,000-ft increase in altitude up to 10,000 ft above sea level (Blake 2009b).
1) Replace each T v in Equations (3), (6), (7), (8) and (9) with , T j ν , and compute the solution to the optimization problem using the differential evolution algorithm (Price et al. 2005). 2) Raised on the latter solution, re-compute the average true airspeed ( , T j ν ) over each circular segment using Equations (10) and (11). 3) Compute the average relative difference ( d ) between the , T j ν values from the latter solution and prior solution as shown in Equation (12): 4) If the average relative difference ( d ) is greater than 1.0 percent, repeat steps 5) through 7). Otherwise, the procedure has converged, and the latter solution is accurate enough for practical purposes.

Total-loss-of-thrust scenario
The application of the trajectory optimization algorithm is demonstrated through a realistic bird strike scenario for the Airbus A320 aircraft, which is the best-selling commercial jet as of May 2015 (Airbus Industrie 2013; The Boeing… 2013). The data in the FAA Wildlife Strike Database shows that a total loss of thrust due to a bird strike is, statistically, most likely to occur during the initial climb below 5000 ft above ground level (AGL) (Avrenli, Dempsey 2014a). Hence, the following bird strike scenario is assumed. 1) An Airbus A320 performs a northbound takeoff from Runway 36, shown in Figure 1, and starts the initial climb. The runway is situated at mean sea level. The take-off wing configuration of the A320 aircraft is 1+F, in which the slats are extended to 18° and the flaps are extended to 10°. Wing configuration 1+F induces less drag than other takeoff configurations, and, hence, it gives a superior climb gradient and fuel efficiency (Airbus Industrie 2002).
2) Although flap retraction on the A320 aircraft may occur as early as 400 ft (Airbus Industrie 2002), it is assumed that the flap retraction is scheduled to occur at 3,000 ft AGL at the time of the departure, in accordance with the International Civil Aviation Organization's (ICAO) noise abatement departure procedure for closein noise monitors (Roberson, Johns 2008). 3) Before the flaps are retracted during the initial climb, the A320 encounters a flock of birds, and multiple birds are ingested into both engines, as a result both engines undergo a total loss of thrust. It is assumed that the airport environment is surrounded by an inhospitable terrain and large bodies of water, so the pilots have to attempt a turn-back maneuver to the departure airport to increase the odds of survivability. 4) While the aircraft is heading north at a distance of 2.0 nautical miles (nm) north of Runway 18, the pilots initiate a turn-back maneuver to Runway 18 (i.e. 0°; see Fig. 1). At this point, the aircraft weighs 70.0 tons. 5) It is assumed that the A320 does not have sufficient hydraulic power to change the wing configuration due to the dual-engine failure. Thus, the wing configuration remains at 1+F throughout the landing maneuver. 6) Since the flaps cannot be extended, the landing speed has to be greater than typical landing speeds. It is assumed that the pilots maintain 160 KCAS (knots in calibrated airspeed) throughout the landing maneuver, which approximately equals the lowest selectable airspeed (i.e. LS V ) in this particular configuration. The LS V provides an appropriate margin to the stall speed. For fly-bywire aircraft, like the A320, LS V equals 1.23 times 1 S g V (Airbus 2002). By minimizing the airspeed, the pilots maximize the time aloft, and minimize the runway length requirements for landing roll.

Aerodynamic input data
The steady-speed engines-out glide ratio of the Airbus A320 is assessed using the simulation methodology (Avrenli, Dempsey 2014b) in a Joint Aviation Requirements Flight Simulation Training Device (JAR-FSTD) A, Level D full flight simulator. The simulator is certified under the European Aviation Safety Agency (EASA) and Joint Aviation Authorities (JAA). The exact model of the simulated aircraft is the Airbus A320-232 with wingtip fences. The simulator has been tested in "normal", "alternate", and "direct" control laws of the Airbus A320 aircraft, and simulates real aircraft behavior in all normal and abnormal flight operations. All simulations are conducted in standard day conditions, in which the air pressure equals 101.3 kPa and the air temperature equals 15°C at AMSL.
The steady-speed engines-out glide ratio is assessed at 160 KCAS for the bank angles of 0°, 10°, 20°, and max ϕ = 33°. The maximum bank (roll) angle is limited to 33° because the bank angle is one of the flight dynamics parameters limited by the Airbus' fly-by-wire system. If the bank angle exceeds 33° with a side-stick input from the pilot, the aircraft automatically reduces the bank angle to 33° upon side-stick release in the "normal control law". Thus, it is not practically possible 4 to maintain a constant bank angle greater than 33° on the A320 aircraft when the aircraft is in the "normal control law" (Bugaj 2011).
The total simulation time required for assessing the steady-speed engines-out glide ratios is less than 30 minutes, which is substantially less than the time that would otherwise be required to assess the aircraft-specific aerodynamic-coefficient data. The results from the flight simulation tests are plotted in Figure 2. As shown in Figure 2, the steady-speed engines-out glide ratio at a given airspeed decreases with an increasing bank angle because 4 In the "normal law", the maximum achievable bank angle equals 45° or 67°, depending on the particular flight conditions. In the "normal control law", if the side-stick is not released, the bank angle keeps increasing up to 45° or 67° and then remains constant at this value until the side-stick is released. It is also possible to exceed the 33° bank angle if the aircraft is in "alternate law". the aircraft partially loses lift, and there is increased drag acting on the aircraft during the banked turns (Anderson 2007). A piece-wise continuous, third-degree polynomial function is fitted to the data in Figure 2 using the least squares estimation method (Weisberg 2014): Equation (13) returns a coefficient of determination, 2 1.0 R = , which is plotted in Figure 2. It is used as an aerodynamic input datum for formulating the optimization problem.

Modeling assumptions
The following assumptions are incorporated into the optimization problem.
1) The aerodynamic roll rate of the A320 aircraft equals ϕ = 10°/sec. 2) Once the flight crew configures "landing gear down", it takes no longer than 15 seconds for the landing gear to be fully extended and locked.
3) The aircraft is aligned with Runway 18 by the time it is 100 ft above ground level. This allows for the insertion of a "final approach waypoint", which precedes a stabilizing final approach segment. The "final approach waypoint" projects onto the beginning of Runway 18 threshold, and the landing trajectory is to be generated from the initial aircraft state to the "final approach waypoint". 4) The pilots configure "landing gear down" no later than 15 seconds before the aircraft reaches the "final approach waypoint". 5) Standard day conditions prevail during the engines-out landing, which corresponds to an air pressure of 101.3 kPa and an air temperature of 15°C at mean sea level. If standard day conditions do not prevail, the pressure altitude is used in calculations instead of the geographic elevation. The pressure altitude can be read from the pressure altimeter of the aircraft. The first and the second assumptions are based on the observations made in the JAR-FSTD A, Level D full flight simulator. The third assumption basically prohibits banked turns below 100 ft to prevent possible wingtip collisions with ground objects. The fourth assumption enables sufficient time for the landing gear extension prior to touchdown. The landing gear extension results in an increased drag and a reduced engines-out glide ratio (Anderson 2007). When the landing gear is fully extended in the full flight simulator, the resultant reduction in the steady-speed engines-out glide ratio is measured as being approximately 14% at 160 KCAS.

Numerical solutions
The optimum solution is computed using the MathWorld Software by Wolfram Research, which is a free resource built with Mathematica technology (Wolfram Research 2015). Since the optimization problem is formulated over a continuous domain, the procedure enables the computation of the global optimum solution (Price et al. 2005). Table 1a lists the solutions for the adjustables. Table 1b provides the predicted altitude loss for each segment, and Table 1c shows the convergence criterion. Since 0.03% 1.0% d = < , the convergence criterion is met in the second iteration, and the second solution is the final solution to the optimization problem. Table 1b shows that the predicted altitude loss in order to fly along the optimum trajectory is 2,590 ft, but this does not include the additional altitude loss due to the landing gear extension. Based on the modeling assumptions described in section 7.3, the following procedure is used to predict the additional altitude loss due to the landing gear extension.

Additional altitude loss due to landing gear extension
1) The predicted altitude AGL vs. time is plotted over the optimum trajectory based on the solution in Table 1b. The plot is shown in Fig. 3. 2) Using Figure 3, the time corresponding to 15 seconds before the aircraft is at 100 ft AGL is computed. This is the moment when the pilots must have the "landing gear down" configuration.
3) The required altitude loss from this moment on is increased by 14%. In this example, the additional altitude loss due to the landing gear extension is predicted as 500 · 0.14 70 ft = . Hence, the total altitude loss required for flying along the optimum trajectory is predicted as 2,590 70 2,660 ft + = . Figure 4 illustrates the ground tracks of the optimum segmented trajectory. The bank angle state is also shown at the beginning and end of each segment. Figure 4 shows that the optimum trajectory is divided into four distinct flight phases as follows: 1) an initial left-turn with a 33° bank, where the absolute change in aircraft heading equals 221°; 2) a wings-level, equilibrium glide for a horizontal distance of 1.2 nm; 3) a final 41° right turn at a 33° bank that aligns the aircraft with Runway 18; 4) a landing flare and a safe touchdown on Runway 18.

ATC commands for the optimum trajectory
Since the autopilot of the A320 aircraft disengages in the total loss of thrust emergency (Baltic Aviation… 2013), the optimum landing trajectory has to be followed by the pilots manually. The pilots can be guided over the optimum landing trajectory through standard and simple ATC commands. In order to generate the ATC commands, the following procedure is applied.
1) The altitude loss is plotted versus the aircraft heading over the optimum trajectory as shown in Figure 5. 2) Using Figure 5, the altitude loss corresponding to the beginning of a heading change is computed. For instance, the heading change from 139° to 180° has to be initiated at an altitude loss of approximately 2,150 ft (Fig. 5). 3) Using the results from Figure 5, the altitude AGL corresponding to the beginning of a heading change is computed. For example, if the emergency landing starts at an altitude of 2,700 ft AGL, the heading change from 139° to 180° has to be initiated at approximately 550 ft AGL. 4) The ATC command requiring the heading change is issued to the pilot-in-command shortly before the aircraft descends to the predicted altitude at which the heading change has to be initiated. For example, if the emergency landing starts at an altitude of 2,700 ft AGL, the command "turn right heading 180° at 33° bank" is issued shortly before the aircraft descends to 550 ft AGL. The altitude AGL can be obtained from the aircraft's radar altimeter. The following section utilizes the oral ATC commands to validate the accuracy of the optimum landing trajectory in full flight simulator of an A320. All simulations are conducted in standard day conditions. Three type-rated A320 pilots participated in the flight simulation tests. A typical simulation run starts with freezing the aircraft position at 2.0 nautical miles from the runway threshold at an altitude of 2,700 ft AGL. The aircraft weight is frozen at 70.0 tons throughout the simulations, and the aircraft heading is initially frozen at 0°. The wing configuration is set to 1+F, and the landing gear is initially fully retracted. The simulator is programmed to simulate dual-engine failure on the A320. When the dual-engine failure results in a total loss of thrust, both the aircraft's position and heading are "released", and the aircraft starts an engines-out glide at 160 KCAS. From this moment on, one of the three typerated A320 pilots flies the aircraft, while one member of the research team issues the following ATC commands to the pilot-in-command: -turn left heading 139° with 33° bank, maintain 160 KCAS; -maintain present heading and speed, descend to 550'; -turn right heading 180° with 33° bank, maintain 160 KCAS; -landing gear down (at 600 ft AGL); -clear to land on Runway 18. After each run, the simulator plots the ground tracks and the descent profile of the A320 aircraft. Using the plots of the ground tracks, the research team verifies that the landing trajectory is simulated properly. Using the plots of the descent profile, the research team measures the total altitude loss rounded up to the nearest 50 ft. The average simulated altitude loss is found as 2,600 ft, which differs from the predicted altitude loss of 2,660 ft by 2.3 percent. Hence, the kinematic method can accurately estimate the altitude loss required for flying along the optimum trajectory.

Uncertainty analysis
To be of practical value, the optimum trajectory should not be overly sensitive to the assumed aerodynamic parameters, such as the assumed roll rate (Hoffren, Raivio 2000). In the example, the optimum trajectory is computed assuming an aerodynamic roll rate of ϕ = 10°/sec. The goal of this section is to find out how the required altitude loss would change when different roll rates are assumed. For this purpose, the optimum trajectory is re-computed based on the roll rates of ϕ = 12.5°/sec, and ϕ = 7.5°/sec. The resulting altitude vs. time histories are plotted in Figure 6, which shows that the predicted altitude loss remains virtually the same for the assumed roll rates of ϕ = 12.5°/sec, ϕ = 10.0°/ sec and ϕ = 7.5°/sec. The resultant relative difference in the required altitude loss is in the order of ±0.6 percent, and is trivial. Therefore, the kinematic method is not sensitive to the assumed roll rate, and is promising for real-world applications.

Limitations
While the kinematic methodology is applicable to virtually all possible events of the total loss of thrust, it is primarily intended for an emergency at a low altitude, since an engine failure due to a bird strike is most likely to occur below 5,000 ft AGL (Avrenli, Dempsey 2014a). If a total loss of thrust occurs at an altitude higher than 10,000 ft AMSL, a compressibility correction (i.e. C V ∆ ) needs to be incorporated into the methodology, which would marginally increase the required computational time (Blake 2009b).
Another limitation of the methodology is that the optimum landing trajectory is computed based on a minimized altitude loss. However, cases may occur in which the aircraft's altitude is greater than that required for flying along the optimum trajectory. In such a case, the methodology does not provide guidance on the excess altitude dissipation, and the flight crew would have to dissipate the excess altitude by applying one or more of the following strategies: -early extension of the landing gear; -flaps extension 5 (if the aircraft has sufficient hydraulic power); -a side-slip maneuver, as in the case of Air Canada Flight 143 (Williams 2003); -s-turn maneuvers, as in the case of Air Transat Flight 236 (AAPID 2003).

Conclusions and recommendations
-This study adopted a kinematic approach to trajectory optimization for the total-loss-of-thrust emergency, which particularly endangers contemporary twin-engine jets with reduced engine redundancy.
-Contrary to current trajectory optimization methods, the kinematic method does not require aircraft-specific aerodynamic-coefficient data. Hence, it is readily applicable to commercial jets, for which the aerodynamic-coefficient data is not released. -In the event of a total loss of thrust, the kinematic method computes the minimum-altitude-loss trajectory to an accessible landing site through an iterative procedure that converges rapidly. -To guide the pilots over the optimum trajectory, the method generates standard ATC commands that are simple to follow in an emergency. -Flight simulation tests show that the method can compute the optimum trajectory with less than a 3.0 percent error for the Airbus A320 aircraft, which is the best-selling commercial jet as of May 2015 (Airbus Industrie 2013; The Boeing… 2013). For future research, the following are recommended: -the method can be developed to provide guidance on the excess altitude dissipation. -using the findings, an adaptive flight planner can be developed for a real-time trajectory optimization in the total-loss-of-thrust emergency. -the designed experiment can be conducted in a full flight simulator to test if the adaptive flight planner significantly increases the odds of a safe touchdown in the event of a total loss of thrust.