Aircraft Vertical Reference Trajectory Optimization With - AIAA ARC

Aircraft Vertical Reference Trajectory Optimization With a RTA Constraint Using the ABC Algorithm Alejandro Murrieta-Mendoza1, Audric Bunel2, and Ruxandra Mihalea Botez3 Université du Québec – École de Technologie Supérieure – LARCASE. Montreal, Canada, H3C1K3

The future traffic management systems will be time-based. Under this context, flight reference trajectories require a time of arrival constraint at a given waypoint to guarantee safety and an improved traffic management. Algorithms that compute the most economical trajectory taking into account the time based constraint are required. In this paper, an algorithm that fulfills the time of arrival constraint and computes the most economical vertical reference trajectory was developed. The time constraint was optimized by using the golden search section to find the most economical combinations of Mach numbers that complies with the required time of arrival. The combination of altitudes that provide the most economical vertical reference trajectory was computed by using the Artificial Bee Colony optimization algorithm. To avoid unnecessary Mach number variations, the solution provided by the developed algorithm was post-treated to obtain as many consecutive flight segments with the same Mach number as possible. Results showed that the RTA constraint was fulfilled while reducing the flight cost of the optimized trajectory with respect to the flight cost of the reference trajectory,

Nomenclature ETA EW Flight Time fitness Mach RTA WD WS Vsound

= = = = = = = = =

Estimated Time of Arrival Efficiency Weight Time required to fly between two waypoints Trajectory fitness Mach Number Required Time of Arrival Wind Direction Wind Speed Speed of Sound

I. Introduction

T

HE airspace has evolved since the beginning of aviation to guide aircraft to the destination and to ensure enough separation between aircraft to guarantee safe flights. Current airspace is being managed by the submissions and approval of flight plans. The general procedure is as follows. Once the flight plan has been accepted by the proper flight controller, the aircraft is supposed to follow that plan. If any modification required by either the authority, or by the aircraft is required, normally a voice message is transmitted to negotiate a new route. However, the increment in air traffic has pushed the current traffic control system to its operational limits. This becomes an important issue to be addressed by the proper authorities since it is suggested that air traffic will increment in the forthcoming years. Different agencies have decided together to improve the traffic control management in order to allow more aircraft to fly while guaranteeing their safety. Two major projects are the Next Generation Air Transport System (NextGEN) in the United States, and the Single European Sky ATM Research (SESARS) in Europe. These two programs are taking advantage of technologies to allow the aircraft to fly the Intended Based Operations (IBO) [1].

1

Ph.D. Student, Université du Québec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada Undergraduate student, LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada 3 Professor, Université du Québec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada. 2

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This type of operation is a type of flight control where the aircraft proposes its trajectory and negotiates it with the core traffic control system whether it can be flown as it is, or if a modification is required. Allowing the aircraft to propose its own trajectory leads to flights which require less fuel consumption, thus reducing polluting emissions. This is very important since the aeronautical industry is responsible of 2% of the total Carbon Dioxide (CO2) released to the atmosphere. A long term environmental goal proposed by the aeronautical industry itself aims to reduce the total CO2 emissions to 50% of those recorded in 2005 [2-4]. CO2 is pointed out to affect the average winds incrementing the aircraft flight time, thus requiring more fuel [5]. The IBO trajectories are time based. In other words, it is not enough to specify the trajectory to be flown by the aircraft, but a Required Time of Arrival (RTA) to given waypoints is imposed. This is done with the goal to ensure safety, to manage more easily operations in the congested areas such as the terminal areas, and to optimize the traffic flow in the beginning of the descent phase. An important quantity of fuel can be saved during the descent phase if the procedure called Continuous Descent Approach (CDA) could be executed [6-8]. The CDA allows the aircraft to descend setting the engine to IDLE following a constant slope trajectory. However, the Top of Descent (ToD) needs to be correctly estimated as pointed out in [9, 10] to avoid the missed approach procedure which is expensive in fuel cost terms [11]. Different studies have suggested that aircraft do not fly at their optimal trajectories [12, 13]. Conventional 3D (best combination of altitude, speed, and positions) vertical reference trajectory optimization algorithms have proved to improve the flight cost by proposing optimal reference trajectories. Different optimal control techniques have been implemented to find the optimal trajectories [14-17]. These algorithms normally use the Point-Mass Equations of Motion. Nevertheless, although these algorithms provide very good results, normally they are time consuming. An alternative to the Equations of motions is the use of a numerical performance model. Algorithms using this type of models are normally used for airborne low power computation devices such as the Flight Management System. Numerical performance models are normally obtained from flight data, and might be expensive to obtain due to their availability. An alternate procedure to create a numerical performance model was discussed in [18] by using a certified Citation X Level D flight simulator. Different algorithms have been implemented to find the most economical vertical trajectory using a numerical performance model. Fuel burn estimators [19] have been used to reduce the computation time, and to find the optimal flight altitude [20]. Search space reduction techniques have also been used to quickly converge to a solution, and to allow the step climb implementations [21, 22]. The golden section search has been used to find the optimal altitude for short flights [23], a variation of the branch and cut algorithm was implemented [24], and an improvement of the B&B was later presented in [25]. Genetic algorithms were studied to find the most economical combinations of speeds and altitudes for a given flight [26] and were later used to find the most promising location to perform step climbs and step descents [27]. Using a numerical performance model, the lateral reference trajectory optimization (selection of points to follow taking advantage of winds) was explored using genetic algorithms in [28], the Dijsktra’s algorithm [29], and the Artificial Bees Colony [30]. A combination of the Vertical and the Lateral reference trajectory has been optimized using a numerical performance model in [31-33] using different techniques such as search space reduction and genetic algorithms. Dynamic programming was as well used to optimize both reference trajectories for cargo aircraft as shown in [34]. Algorithms considering the RTA have been explored. Lidén was one of the first researches to point out its important and developed an algorithm to optimize the trajectory considering the RTA [35]. Optimal Control have been implemented [36-38] as well as dynamic programming [39, 40]. Algorithms based on numerical performance models have also been explored using the ant algorithms to fulfill the RTA constrain, such as in the case shown in [41]. However, this algorithm focuses only on a fixed altitude. The objective of this paper is to present an algorithm that optimizes the vertical reference trajectory respecting the RTA constraint. This optimization was carried out by implementing the Artificial Bees Colony (ABC) to make this algorithm fit for the next generation of air transport systems using a numerical performance model. The golden search method was as well implemented to define the initial Mach number to fulfill the RTA constraint. The trajectories to optimize consist in the cruise phase, taking place normally the Top of Climb (ToC) to the ToD. Initial climb and descent are not considered. The article is organized as follows. In the first section, the numerical performance model used to compute the flight cost is explained along with the flight cost function. The Mach number needed to fulfill the RTA constraint to a given trajectory is then illustrated. In the next sub-section, the ABC algorithm to select the most economical trajectory is explained. Results and conclusions are further presented.

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II. Methodology A. Aircraft Fuel Burn: the numerical performance model The aircraft fuel information was computed with experimental flight test data. This data is arranged in the form of a database. The numerical performance model is composed of many different sub-databases for each flight phase such as climb in Indicated Air Speed (IAS), acceleration, climb in Mach number, cruise in IAS, cruise in Mach number, descent in Mach number, deceleration, and descent in IAS. Because the trajectory optimized in this paper takes place in the cruise phase, the Cruise Mach, the Climb in Mach number, and the Descent in Mach number are the sub-databases here used. The inputs and outputs for the three sub-databases are shown in Fig.1 where Altitude is provided in feet, Weight in kg, and the International Standard Temperature Standard Deviation (ISA Dev. Temp) is given in ºC. The Horizontal Traveled Distance is given in nm, the Fuel consumption in kg, and the Fuel Flow in kg/h. The main difference between the different databases is that for “Cruise Mach” only the fuel flow is provided. For “Climb Mach”, and “Descent Mach”, the required fuel to reach the required altitude as well as the required horizontal traveled distance is provided.

Figure 1. Numerical Performance Model as a Black Box While computing the flight cost, the aircraft is at constant cruise or performing a step climb / step descent. The algorithm developed in this paper decides the selection of the sub-database. Fig. 2 is a flowchart that defines the algorithms decision to select a certain database. Using these databases, the flight cost is computed as described in Section B. START

Cruise PDB

Altitude Change? No

Yes

Higher Altitude?

Yes

Climb PDB

No

Descent PDB Cruise PDB

END

Figure 2. Sub-database change B. The Flight Cost Computation The flight cost is composed by the required fuel and the flight time required to perform a flight. The flight cost is not computed in currency terms, but in fuel terms. For this reason, a conversion to change flight time to fuel burn is required. This conversion is performed with a variable called Cost Index (kg/hr). This variable is normally set by airlines and it takes into account time related costs such as maintenance, cost, crew, etc. The flight cost is expressed under the form given in Eq (1).

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(

)

(

)

(

)

(1)

The fuel burn is computed using the numerical performance model described in Section II.A. A given database contains a limited number of inputs, which brings corresponds to a limited number of outputs. To compute the required values in between these inputs, Lagrange interpolations are executed. For this paper, interpolations for weight and ISA temperature deviation are considered since altitudes and Mach numbers to be considered are only those available in the numerical performance model. Altitudes are 1,000 ft multiples, and Mach numbers are only those available in the numerical performance model. The complete methodology to compute the flight cost using a numerical performance model was described in [42, 43]. Flight Time is affected by winds as tailwinds “push” the aircraft and headwinds “delay” the aircraft. Temperature has an influence on fuel burn as low temperatures make the engines require less fuel. Weather conditions are obtained from the previsions available at Weather Canada. C. Vertical Plan and Speed Management Before describing in detail the steps followed to determine the optimal trajectory using the ABC algorithm, the search space and the speed management will be explained in this sub-section. 1. The search space and its constraints The search space for this algorithm is the set of altitudes from where the aircraft can fly. There exist some constraints which are the following:  Maximal Cruise Altitude  Minimal Cruise Altitude  The step climbs occur only for 2,000 ft. The vertical plan (or space search) is a graph created from equidistant waypoints at different altitudes. Fig 3 is a description of a given search space delimited by the maximal cruise altitude of 37,000 ft and minimal cruise altitude of 29,000 ft for an initial cruise (blue line) of 33,000 ft. The red line represents a given optimized trajectory, and the green line represents different candidate trajectories that were used to wander around the search space.

Figure 3. Vertical Search Space with limits 2. RTA and the speed management To explain the speed management to fulfill the RTA constraint, a fixed altitude flight will be used as reference to compute the required Mach number. Since weather is taken into account, the trajectory is decomposed in multiple segments, along which the wind speed and the wind angle are computed. Using the aircraft True Air Speed (TAS), and the wind information, the Ground Speed (GS) can be computed. The Estimated Time of Arrival (ETA) at the end of the flight is the sum of the times required by the aircraft to fly all the flight segments. In order to fulfill the RTA constraint, the ETA at the last waypoint should be equal to the RTA. Note that the ETA can also be identified as the total flight time. The methodology followed by the algorithm to fulfill the RTA constraint requires computing local reference RTA constraints at each waypoint. Local RTAs are computed by dividing the flight time by the number of segments. For example, if a RTA is set so a given flight should take five hours to reach the objective waypoint passing over five different waypoints (being the fifth the destination waypoint), then the aircraft should take exactly one hour to 4 American Institute of Aeronautics and Astronautics

fly between each waypoint. This way, a reference GS for each segment can be computed. However, this simple computation does not take into account wind and temperature information. To compute the Mach number to attain the required GS to fulfill the RTA, local wind information should be taken into account. Providing the required GS per segment, the Mach number per segment required to fulfill the RTA constrain can be computed with the relationship given by Eq (2) and Eq (3) where GS is the ground speed, WS is the Wind Speed, Vsound is the speed of sound at a given altitude, AZ is the aircraft azimuth, and WA is the direction of wind. (

√

)

(2)

where (

)

(3)

The resulting Mach number variation with the distance can be seen in Fig 4.

Figure 4. Non restricted Mach Number Variation With the Distance Fulfilling the RTA Constraint In Fig. 4, it can be seen that the Mach number variation along the route are of importance. It is of interest to find less Mach number variations. For this reason an improvement to compute a constant Mach number for all segments to fulfill the RTA was developed. To achieve this goal, the Golden Section Search (GSS) algorithm was used. The GSS is used to find an extreme (in our case, the minimum difference between the RTA and the ETA) in an unimodal function. This function contains one minimum (or maximum) on the domain (a,b). The GSS is simple and robust, and it tends to be quick to find the optimal solution, as observed in [23] Before defining the unimodal function to optimize, the equation used to compute the flight time for a given segment is provided in Eq. (4). (

)

(

√

(

(

)

(

)

) )

(4)

Notice that Mach number is the one computed in Eq. (2). The flight time sum of all segments gives the ETA as in Eq. (5).

∑

()

(5)

The difference between the required RTA and the ETA computed in Eq. (5) should theoretically be 0. However, as stated before, there exists a high Mach number variation. The function f to optimize using the GSS is expressed in Eq. (6). [∑ (

√

(

(

) )

)

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]

(6)

Notice that the first term of Eq. (6) is almost identical as Eq. (4), the difference is that the GSS is searching for a unique Mach number instead of a set of Mach numbers. The WS is still being considered as a set of values; this is also true for all values that compose B. Equation (6) is an unimodal function as shown in Fig. 5 where the y axis is the flight time in seconds, and the x axis is the Mach number. The GGS should find a solution where f approaches to zero.

Figure 5. Unimodal behavior of function f The algorithm uses the Mach number computed in Eq (6) as the Mach number of reference. This Mach number computed in Eq. (6) also allows the algorithm to find the reference ETA for each waypoint. These ETAs will later serve as reference for the algorithm to find the optimal Mach number when changing altitudes (thus changing the speed of sound and the weather parameters) as the ETA should always be respected. 3. RTA and Altitude management For a fixed altitude, the speed is managed to fulfill the RTA as discussed in the previous section. However, changing the flight altitude the aircraft is inevitably exposed to different weather, and thus a different speed of sound. As a consequence, for the same Mach number, the ground speed changes, and also the ETA. For this reason, the Mach number needs to be recalculated with respect the next waypoint reference ETA. Thus the Mach number will no longer be constant and it will deviate from the previously computed Mach number of reference. The reference ETA provided as a consequence of the GSS, it is possible to compute the required Mach number using exclusively Eq (2) and Eq (3). Figure 6 illustrates the Mach number behavior due to altitudes changes for a given trajectory.

(a)

(b)

Figure 6. Mach Number Variation with Altitude Change

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In Fig. 6 the black lines represent the altitude and Mach number limits, the blue lines represent the reference Mach number and altitude, and the red lines represent the computed Mach number and altitudes. As seen on Fig. 6 (a), right after the first change of altitude, the optimal trajectory speed on Fig. 6 (b) does not follow the reference speed. After the second change of altitude, the difference between the optimal speed and the reference speed is generally even more important. At the 17th waypoint, it can be observed important speed changes, which is due to a new change of altitude and weather changes. Although the speed variation in Fig. 6 (b) is lower than the one in Fig. 4, it is desirable to have the same Mach number at as many consecutive segments as possible. It is also desirable to have Mach number values available in the numerical performance model; this is Mach numbers with a step difference of 0.005. To accomplish this objective, the GGS technique was used once again to find a Mach number for different consecutive segments as shown in Fig 7.

Figure 7. Vertical Trajectory, With Constant Mach number for Different Segments The blue line represents the speed of the reference trajectory; the red line represents the Mach number selection of the optimal trajectory, and the black lines are the Mach number limits. The red line represents the selection of Mach number per waypoints. The methodology explained in this section is executed for every trajectory generated by the ABC algorithm described next.

D. The Artificial Bees Colony and the Vertical Reference Optimization Algorithm The collaboration between bees is the key of food finding and recollection to guarantee the hive survival and development. Over time, bees have developed an interesting way of accomplishing their tasks. Different “types” of bees perform different tasks such as exploring the search space, exploiting the known food sources, reproduction, among others. A general algorithmic description of the ABC algorithm can be found in [44] The Artificial Bees Colony (ABC) algorithm for the vertical reference trajectory optimization mimics the behavior of honey bees in their quest and exploitation of food sources. The ABC algorithm implementation phases are shown next. 1. Inputs The inputs required for the algorithm are the initial and final coordinates - normally the Top of Climb (ToC) and the Top of Descent (ToD), the aircraft weight, the waypoints to follow, the weather conditions along the route, the size of the swarm, the maximal number of successive failed mutations, the maximal number of iterations (stop condition), and zones where the user requires constant Mach number, and constant altitude. 2. Initialization The reference trajectory should be created and evaluated. In this paper, the reference trajectory is the geodesic route. Its reference trajectory cost is evaluated using Eq. (1), and the weather information. The altitude and speed boundaries explained in Section II.C.1 are also created. The swarm is divided in two halves; one half corresponds to the employed bees, and the other half correspond to onlooker bees. A different semi random trajectory is generated and assigned to every “employed bee”.

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3. The Employed bee phase The employed bee memorizes its assigned trajectory and evaluates its cost. The employed bee then mutates its trajectory and evaluates its cost again with this new mutation. The mutation performed by the employed bee is a randomized mutation influenced by a neighbor trajectory. A random waypoint of the trajectory is selected, and its altitude is modified. By changing the altitude, the speed of sounds also changes. For this reason, the Mach number is adjusted to respect the RTA. The valid mutated trajectory is evaluated. If the mutated trajectory is more efficient (lowest cost) than the original assigned trajectory, the mutated trajectory is saved to the employed bee memory, and the original trajectory is discarded. If the mutated trajectory fails to improve the flight cost, the original trajectory is kept. This process is repeated for every single trajectory. 4. The Onlooker bee phase The “onlooker bees” behave in the same way as the “employed bees”. The only difference is that onlooker bees select their trajectories instead of having trajectories assigned to them. These bees are elitist and tend to select the most promising trajectories. A test is performed to decide the trajectory potential and influence the onlooker bees to select or not a given trajectory. For this test, a parameter called “Efficiency Weight” (EW) is computed to give the most economical trajectories more chances to be selected by the onlooker bee. The EW is computed for each trajectory using Eq. (7): (

)

(7)

where EW is the efficiency weight, and fitness is a parameter that depends on the trajectory cost. The more economical the trajectory is, the more fitness it is. Maxfitness is the fitness of the most economical trajectory, and Minfitness is the fitness of the most expensive trajectory. Once the EW for a given trajectory is known, a random value between 0 and 1 is generated. If the random number is less than the current trajectory EW, the trajectory is selected by the onlooker bee, otherwise the current on-looker bee performs the test for the next trajectory. Note that the rejected trajectory can still be selected by another onlooker bee. This phase ends when every single onlooker bee has mutated a trajectory. At this point the algorithm searches for the most economical trajectory. If the trajectory found is more economical than the current most economical trajectory, it is then selected as the global optimal so far. 5. The Scout bee phase The scout bee phase mimics the search of new food sources. Every trajectory has an associated counter which keeps track of the successive number of times that a mutation fails to improve the trajectory cost. Once the failure counter reaches its pre-defined limit value, the associated trajectory is discarded, and a new semi-random trajectory is generated. For the new generated trajectory, the failure counter is set to zero. This trajectory rejecting process avoids spending too many resources in trying to improve an unfavorable trajectory and to explore the search space for more promising trajectories. As the global optimal is independently saved in memory, if the rejected trajectory is the current global optimal it can be safely rejected as the solution is safely stored. 6. Stop criteria The algorithm stops when the pre-defined number of iterations is reached. If necessary, time-based stop criteria can be used, as well as a saved fuel objective criteria. 7. Mach number processing At this point, the Mach number to fulfill the RTA constrains used was “too accurate” for an aircraft. For this reason, after the determination of the optimal trajectory by the ABC algorithm, a Mach number processing was executed using a pre-defined Mach grid. The first segment Mach number was adjusted to the closest Mach value available in the user defined grid. For the other values, the Mach number was selected, not by the closest Mach number, but in a time based frame. In other words, if the aircraft was delayed the closest faster Mach number is selected, if the aircraft is in advance the closest slowest Mach number was selected. This process is repeated until the last segment is reached. 8 American Institute of Aeronautics and Astronautics

If the ETA does not fulfill the RTA tolerance (+/- user requirement) the method changes for the last segment. The two Mach numbers circling the exact Mach number value are memorized. Knowing the flight time on the last segment with these two values, a new waypoint is created where the Mach number will have to change in order to respect the RTA with accuracy less than one second.

III. Results To explore the algorithm saving potentials, long haul trajectories from Montreal to Paris were explored using a two turbofans engine aircraft with a Maximal Take-Off weight of 160 tones, and a Mach number range from 0.6 to 0.82. The flight was conducted for three different dates. During the tests, the fuel burned and the time difference against the imposed RTA were evaluated. The flights cost for all flights was compared against the same flight at constant flight level The output trajectory provided by the algorithm for flight A, the first flight from Montreal to Paris is shown in Fig. 8.

Figure 8. Flight A: Output trajectory from Montreal to Paris In Fig 8, it can be seen that the algorithm suggested two step climbs to improve the fuel consumption. In the right hand side of the figure, the Mach number variation in a 0.005 step grid to fulfill the RTA constraint can was observed. In the left hand side, the altitude variation was noticed. For this particular flight, the saved amount of fuel was of about 402 kg of fuel compared to a direct fixed altitude flight. The trajectory with the Mach number grid was able to respect the RTA with a time difference of as low as 0.16 seconds. The second flight evaluated was the Flight B from Paris to Montreal that used the same weather parameters as Flight A. The trajectory output is shown on Fig 9.

Figure 9. Flight B: Output Trajectory from Paris to Montreal

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As expected, the output trajectory was different as the weather changed. It can be observed that the altitude changed as the ToC was reached; so the flight should begin at 35,000 ft instead of 33,000 ft as the initial reference flight plan suggested. It can be as well observed that towards the end of flight, the algorithm preformed a step descent; which was due to better weather parameters in that direction. The aircraft’s speed was less stable, which was due to turbulent winds along the trajectory. It can be observed that the speed is higher; this can be explained by the fact that the prevailing wind going west is a headwind. Fuel saving results was higher as around 820 kg were saved and the RTA was respected within a time difference of around 0.60 seconds. The last evaluated flight was Flight C from Montreal to Paris using different weather information. The output is shown in Fig 10.

Figure 10. Altitude and Speed of the Optimized Trajectory A similar behavior for this flight (Fig. 10) is found from the first flight test. For this particular flight, a total of around 632 kg of fuel were saved, and the RTA was respected within a time difference of 0.3s. During the executed simulations, it was observed that the maximum saved amount of fuel was influenced by both the initial weight, and the average Mach number. If the aircraft weight was low, it was able to immediately fly at higher altitudes where the air density diminishes and temperatures are normally lower. If the aircraft average Mach number was high, the aircraft consumed more fuel, and more optimization was performed. An example of the results obtained was that with an initial weight of 115 tons, an average speed of 0.778 Mach for a RTA of 6h30min, the algorithm was able to save 1.3 tons of fuel. The RTA was respected within a time difference of 0.15seconds. The algorithm execution time is very short as it takes less than 12 seconds for every trajectory to be optimized. The algorithm showed to be stable as it provided the same solution consistently.

IV. Conclusion In this paper, a 4D trajectory was optimized with the Artificial Bees Colony (ABC), the Golden Search Section (GSS) was used to find the best combination of Mach numbers to fulfill the RTA constraint. This algorithm can greatly decrease the cost of a given flight by consuming less fuel. Results have shown that the algorithm can efficiently reduce the fuel consumption due to its ability to find adequate waypoints to perform the step climb by taking into account the aircraft performance, and weather. It was as well observed that the algorithm was able to find the most economical trajectory for a space search with many waypoints in a short calculation time. This is a powerful advantage against most algorithms available in the literature. The time efficiency can bring as consequence longest and most accurate trajectories and for future addons on other dimensions or constraints such as adding the lateral reference trajectory and no flight zones. The algorithm was able to adjust the Mach number according to the weather in order to respect the fixed RTA, an important constraint for future traffic control systems. This algorithm was able to follow constant speed and/or altitude segments as required by the user to make this algorithm able to conform to current traffic control regulations. The ABC algorithm is very modular, for this reason, future research to be focused in adding lateral and speed optimization in order to improve even more the flight optimization.

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Acknowledgments This research was conducted in The Research Laboratory in Active Controls, Avionics and Aeroservoelasticity (LARCASE) for the global project “Optimized Descent and Cruise” with funds from the Business-led Network of Centers of Excellence Green Aviation Research & Development Network (GARDN). For more information please visit http://larcase.etsmtl.ca. The authors would like to thank Rex Haygate, Dominique Labour and Yvan Blondeau from CMC-Electronics – Esterline, and Oscar Carranza from LARCASE. The authors would like to thank CONACYT in Mexico and the FQRNT in Quebec, Canada.

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