Patent Publication Number: US-2019193845-A1

Title: Aircraft autonomous pushback

Description:
RELATED APPLICATIONS 
     The present application is a continuation of U.S. application Ser. No. 16/116,705, filed Aug. 29, 2018, now allowed; which is a continuation of U.S. application Ser. No. 15/819,922, filed Nov. 21, 2017, now U.S. Pat. No. 10,086,929, issued Oct. 2, 2018; which is a continuation of U.S. application Ser. No. 14/912,692, filed Feb. 18, 2016, now U.S. Pat. No. 9,849,972; which is the National Phase of International Application Number PCT/GB2014/052217, filed Jul. 21, 2014 and claims priority from Great Britain (GB) Application Number 1315012.3, filed Aug. 22, 2013; the entire contents of each of which are hereby incorporated by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to reversing of an aircraft on the ground using an autonomous taxiing system, and in particular the invention relates to decelerating the aircraft when reversing. 
     BACKGROUND OF THE INVENTION 
     It is sometimes necessary to move an aircraft in reverse whilst the aircraft is on the ground. For example aircraft are frequently pushed backwards away from an airport gate, a so called “pushback” manoeuvre, by a vehicle known as a pushback tractor or tug. The same tractor or tug can also move the aircraft forward if desired. Whilst some aircraft have the capability to reverse under their own power using the main aircraft engines, e.g. by using reverse thrust in a manoeuvre known as “power back”, this is not permitted for civil aircraft as the jet or prop wash from the engines can cause damage to nearby terminal buildings or facilities. 
     Recently it has been proposed to equip aircraft with an autonomous wheel drive taxi system that can drive one or more of the aircraft landing gear wheels in rotation so that the aircraft can taxi under its own power. The ability to drive the aircraft landing gear wheel in reverse using the wheel drive system enables an autonomous pushback operation without the main engines running, as well as a forward taxiing operation with or without one or more of the main engines running. The autonomous pushback operation makes the autonomous wheel drive taxi system particularly suitable for civil aircraft but the system has wide applicability to a wide variety of aircraft including: civil and military; fixed wing, rotary wing and powered lift; manned and unmanned, etc. 
     For conventional pushback operations, braking is performed by the tractor. Use of the brake pedals is generally prohibited as there is a risk of damaging the nose landing gear and the tractor. With an autonomous wheel drive taxi system, i.e. without a tractor unit, braking will need to be performed autonomously on the aircraft. Conventional aircraft braking systems are typically not designed to perform this function, being more suited to high energy dissipation in the landing phase and low energy dissipation working against the thrust of the main aircraft engines in the taxi phase. Conventional aircraft braking systems are therefore generally unsuited to braking during an autonomous pushback operation. The autonomous pushback introduces the risk of aircraft tip over (where the aircraft tends to rotate nose up about its pitch axis, possibly causing a tail strike) and aircraft runaway (where if the autonomous pushback manoeuvre is performed on a slope the longitudinal component of the gravity acceleration can become higher than the rolling resistance of the tyres on the ground such that the aircraft accelerates above the desired pushback speed). 
     SUMMARY OF THE INVENTION 
     A first aspect of the invention provides a method of controlling speed of an aircraft during backwards motion of the aircraft when in contact with the ground, the method comprising: applying a torque to at least one landing gear wheel of the aircraft, the torque being in a direction opposite to the backwards rolling direction of rotation of the landing gear wheel, wherein the torque applied does not exceed a limit for ensuring aircraft longitudinal stability. 
     A further aspect of the invention provides an autonomous pushback braking system for an aircraft having a wheel drive system for driving one or more of the aircraft&#39;s landing gear wheels in rotation, wherein the wheel drive system is operable to drive the wheel in rotation to effect backwards motion of the aircraft when in contact with the ground, and a means for applying a torque to at least one landing gear wheel of the aircraft, the torque being in a direction opposite to the backwards rolling direction of rotation of the landing gear wheel, and wherein the torque applied does not exceed a limit for ensuring aircraft longitudinal stability. 
     For longitudinal stability the torque applied to decelerate the aircraft during a pushback manoeuvre should not cause the aircraft to risk a tip-over event. Accordingly the torque limit may be selected such that a) the aircraft cannot tip back onto its tail, and/or b) a nose landing gear of the aircraft does not part contact with the ground, and/or c) a substantially vertical load through a nose landing gear wheel does not fall below a threshold at which a steering centring device maintains the wheel of the nose landing gear straight. 
     The torque limit may be based upon one or more of the following instantaneous aircraft parameters when the torque is applied: a slope angle of the ground over which the aircraft is moving; the centre of gravity of the aircraft; the mass of the aircraft; an aircraft inertia moment around the aircraft lateral (y) axis; the backwards speed of the aircraft. 
     The backwards motion of the aircraft may be effected by a wheel actuator carried by the aircraft for driving one or more of the aircraft&#39;s landing gear wheels in rotation and/or by gravity due to a slope angle of the ground over which the aircraft is moving. 
     In an autonomous pushback manoeuvre the backwards motion of the aircraft is effected autonomously in the absence of an external tractor unit. However, pushback runaway may also be caused by an inadvertent release of the park brake even when a pushback manoeuvre has not been commanded. 
     The step of applying torque to the landing gear wheel may comprise applying a braking torque to the wheel using a friction brake assembly. 
     The aircraft speed may be measured and if this speed exceeds a predetermined limit at which the aircraft longitudinal stability cannot be ensured then an indication is displayed in the aircraft cockpit. 
     The braking torque may be effected by a braking control system or by a park brake system. The braking torque may be initiated by a pilot input, e.g. using cockpit brake pedals or a park brake lever. 
     The braking control system may send a braking command to only a limited number of braking wheels. For example where the aircraft has N braking wheels the braking control system may send a braking command to a number n of the braking wheels where n&lt;N. 
     The braking control system may limit the maximum braking clamping pressure applicable to the friction brake assemblies to no more than a limit at which the aircraft longitudinal stability is ensured. 
     The maximum braking clamping pressure may be variable depending on the mass and longitudinal centre of gravity position of the aircraft. 
     The braking control system may implement a braking law that commands initially a low brake pressure which rises with increasing time. 
     The braking torque may be applied whilst the wheel actuator is driving the aircraft backwards. 
     The wheel actuator torque and the braking torque may be controlled by a common controller. The controller may receive input of the aircraft speed and control the wheel actuator torque and the braking torque towards a target speed. 
     The step of applying torque to the landing gear wheel may alternatively comprise applying a braking torque to the wheel using a generator. 
     The generator may be coupled either to an electrical network of the aircraft or to a resistor for dissipating the electrical energy generated by the generator. 
     The generator is preferably a motor/generator used to drive one or more of the aircraft&#39;s landing gear wheels in rotation to effect the backwards motion of the aircraft. 
     The motor/generator may be selectively coupled to the landing gear wheel(s) by a drive path. Preferably the drive path includes a gear mounted to the wheel rim and a pinion, wherein the pinion is moveable between an engaged position in which the pinion is in driving engagement with the wheel gear and a disengaged position in which the pinion is physically separated from the wheel gear. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments of the invention will now be described with reference to the accompanying drawings, in which: 
         FIG. 1  illustrates an aircraft performing an autonomous pushback operation; 
         FIG. 2  illustrates the aircraft on sloping ground; 
         FIG. 3  illustrates a wheel drive system of the aircraft; 
         FIG. 4  illustrates a plot of the main contributors to the longitudinal stability and performance of the aircraft during an autonomous pushback operation; 
         FIG. 5  illustrates a scheme for braking the aircraft using the aircraft&#39;s braking control system; 
         FIG. 6  illustrates a scheme for braking the aircraft using the aircraft&#39;s park brake; 
         FIG. 7  illustrates a ramp braking law of the aircraft braking control system; 
         FIG. 8  illustrates a control scheme for controlling the wheel drive system and the braking control system of the aircraft; 
         FIG. 9  illustrates a scheme for dissipating electrical power generated by a generator to the aircraft electrical power network; and 
         FIG. 10  illustrates a scheme for dissipating electrical power generated by a generator to a resistor. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENT(S) 
       FIG. 1  shows an aircraft  1  having a fuselage  2  including a nose  2   a  and a tail  2   b , wings  3 , main engines  4 , nose landing gear  5  and main landing gear  6 . The aircraft has two main landing gears  6 , one on either side of the aircraft centreline, and a single nose landing gear  5  forming a tripod. Each landing gear  5 ,  6  has a diablo configuration with two wheels. 
     The aircraft  1  is typical of a short range single aisle passenger jet aircraft, although it will be appreciated that the invention has applicability to a wide variety of aircraft types as mentioned above. In particular the aircraft may have a greater or fewer number of landing gears; and each landing gear may have any number of wheels, including one. 
     Each main landing gear  6  has a wheel drive system  10 , shown in detail in  FIG. 3 . The wheel drive system  10  is for driving one wheel of the main landing gear  6  (typically the outboard wheel but may alternatively be the inboard wheel) in rotation to taxi the aircraft on the ground. The wheel drive system  10  is operated without the main engines  4  running for reversing the aircraft. The wheel drive system  10  is operated either with or without one or more of the main engines running for taxiing the aircraft forward. For the avoidance of doubt the forward direction is along the aircraft longitudinal axis nose first and the reverse direction is along the aircraft longitudinal axis tail first. 
     In  FIG. 1  the aircraft is shown reversing, indicated by the direction arrow R, whilst the landing gear wheels are in contact with the ground G which is substantially level, i.e. zero slope with the horizontal. In  FIG. 2  the aircraft  1  is shown reversing, indicated by the direction arrow R, whilst the landing gear wheels are in contact with the ground G which has a slope angle alpha (a) to the horizontal, h. 
     Wheel Drive System 
       FIG. 3  shows a partial view of the wheel drive system  10 . The main landing gear  6  includes a telescopic shock-absorbing main leg  12 , including an upper telescopic part (main fitting) and a lower telescopic part (slider)  13 . The upper telescopic part is attached to the aircraft fuselage or wing (not shown) by its upper end (not shown). The lower telescopic part supports an axle  14  carrying a pair of wheels  16 , one on either side of the main leg. Each wheel comprises a tyre supported by a hub  18  (only the hub  18  of one wheel  16  is shown in  FIG. 3 , for clarity). The wheels  16  are arranged to rotate about the axle  14  to enable ground movement of the aircraft, such as taxiing or landing. 
     Each wheel hub  18  has a rim  19  for holding the tyre (not shown). The wheel drive system  10  includes a driven gear  20  attached to the hub  18  so as to be rotatable with the wheel  16 , the driven gear  20  comprising a roller gear  34  formed by two rigid annular rings  35  connected together by a series of rollers  36  extending around the rings to form a continuous track. The rollers  36  are each rotatable about a pin (not shown) which extends between the annular rings  35  to form a rigid connection between the annular rings  35 . One of the annular rings  35  comprises a plurality of connection extension tabs  37  which provide a rigid connection to the hub  18 . 
     The wheel drive system  10  further comprises a wheel actuator  50  comprising a motor  52  which rotates an output sprocket  60  (drive pinion) via a gearbox  70 . The sprocket  60  is a wheel-type sprocket with radially-extending teeth which can interlock with the rollers  32  of the roller gear  34 . 
     The wheel actuator  50  is supported by a bracket which is rigidly connected to the axle  14  of the landing gear and pivotally connected to the motor  52  about a pivot axis. The wheel actuator  50  may alternatively be mounted on the upper telescopic part (main fitting) or lower telescopic part  13  (slider). A linear actuator  58 , such as a direct-drive roller-screw electro-mechanical linear actuator, extends between the bracket  56  (at an end nearest the axle  14 ) and the motor  52 . Thus, linear movement of the actuator  58  is translated to rotational movement of the wheel actuator  50  causing the sprocket  60  to move between an engaged position in which the sprocket teeth interlock with the rollers  32  of the roller gear  34 , and a disengaged position in which the sprocket teeth are physically separated from the rollers  32  of the roller gear  34 . The sprocket  60  is therefore moved in a substantially radial direction with respect to the roller gear  34  axis of rotation between the engaged and disengaged positions. 
     It will be appreciated that the wheel drive system  10  may take a variety of forms. The wheel drive system  10  illustrated is an example of an open-geared arrangement where the engagement/disengagement of the wheel drive system is by moving the drive pinion in a substantially radial direction into/out of positive driving engagement with the driven gear. The drive pinion and driven gear respectively may be formed as a sprocket and roller gear (as illustrated); a sprocket and roller chain; a roller gear and sprocket; a roller chain and sprocket; or toothed gears, e.g. spur gears. Alternatively the drive pinion may move in a substantially axial direction (along the axis of rotation of the drive pinion) into and out of driving engagement with the driven gear. 
     Yet further alternatively a clutch device may be provided between the motor and the driven wheel. The motor may be disposed within the wheel hub or mounted adjacent the wheel. The driven wheel is permanently engaged with a portion of a drive path between the motor and the driven wheel, and the clutch device may make or break the drive path between the motor and the driven wheel. The engagement between the drive path and the driven wheel may be a geared drive or may be a friction drive. 
     Although the figures only show features of the wheel drive system  10  for driving one of the wheels  16 , these features be mirrored for the other wheel  16 . That is, one wheel drive system  10  may be provided for each wheel  16 . For a landing gear  10  with four or more wheels  16 , a wheel drive system  10  may be provided for each of the wheels  16 , or for only two of them. In other embodiments it may be possible to have one motor  52  shared between two wheel drive systems  10 . That is, the motor  52  may be arranged to rotate the output sprockets of each drive system. Additionally or alternatively a wheel drive system may drive one or more wheels of the nose landing gear  5 . 
     Inside the wheel hub  18  is a friction brake arrangement indicated generally at  40 . The friction brake arrangement  40  may be of conventional type and so will not be described in detail here. Generally, however, the aircraft friction brake arrangement includes a stator part and a rotor part comprising a stack of carbon disks. A brake actuator, which may be hydraulically or electrically operated, builds the pressure on the carbon stack to convert rotational torque of the wheel into heat thereby decelerating the aircraft. The brake actuator(s) are controlled by a braking control system (BCS) of the aircraft. The BCS responds to inputs, e.g. pilot and autopilot inputs, and commands a brake pressure accordingly. 
     Aircraft Longitudinal Stability 
     Whilst the tripod arrangement of landing gears  5 ,  6  is generally stable in the aircraft longitudinal direction when the aircraft is travelling forwards on the ground and the aircraft is decelerated by the friction brakes of the main landing gear, studies have shown that when the aircraft is performing an autonomous pushback operation application only a low level of braking torque using the friction brakes may be sufficient to disturb the aircraft longitudinal stability and risk a tip over event. 
     Pushback Tip Over 
     For longitudinal stability three different tip over cases have been identified: 
     Tail strike—The aircraft nose lifts and the tail impacts on the ground. 
     Nose lift up—The wheels of the nose landing gear momentarily lose contact with the ground but the aircraft does not sit on the tail. 
     Steering cam engagement—The load on the nose landing gear steering centring cam goes below a predetermined level at which the centring cam engages to maintain the nose landing gear wheels inline (zero degree position facing forwards along the aircraft centreline). 
       FIG. 4  illustrates the effect of particular aircraft parameters on the aircraft longitudinal stability (x-axis) and aircraft performance (y-axis). The parameters considered are: 
     Aircraft Mass 
     Aircraft longitudinal centre of gravity (CG X) position 
     Apron (ground) slope 
     Inertia moment about pitch (y) axis, Iyy 
     Autonomous Pushback Speed 
     Brake Gain 
     Braking Rise time (time to build brake pressure to commanded level) 
     It has been identified that of these parameters a combination of an aft CG position, with a high pushback speed and a high apron slope angle may pose the greatest risk of a tip over event during an autonomous pushback manoeuvre. Although not shown in  FIG. 4 , it should also be noted that with an aft CG a higher mass is also more detrimental to the aircraft longitudinal stability. 
     Pushback Runaway 
     Simulations have also been performed for the pushback runaway condition if the autonomous pushback is performed on a slope. The longitudinal component of the gravity acceleration adds a component of the gravity force in the direction of the motion. The runaway condition is encountered when the gravity component on the aircraft longitudinal axis is higher than the sum of the aerodynamic drag and the friction forces on the ground. The runaway phenomenon is not affected by the aircraft mass. The highest slope angle achievable at the gate is assumed to be 1.15 degrees, or 2%. The study has shown if the autonomous pushback was to be performed without restrictions on the taxiway slope, the aircraft would need to be provided with a means of applying a relatively low retardation force to protect from pushback runaway. 
     Design Solution to Address Pushback Tip Over and/or Pushback Runaway 
     Various design solutions have been identified and these will be described detail. 
     1. Pushback Speed Limitation with Pilot Braking 
     One solution is to limit the pushback speed. By selecting the worst case combination of parameters (Brake Gain, Slope, Mass, C G, . . . ) a speed limit can be identified for a particular aircraft type up to which the aircraft BCS can apply full brake pressure without risks of tip over. Whilst this solution would negate risks of tip over the speed limit is likely to be very low, e.g. around 1 knot. This may be considered too low a pushback speed to be commercially viable. 
     As shown in  FIG. 5  the aircraft speed is detected, e.g. by a wheel speed sensor  100  (or alternatively from the aircraft inertial reference sensing system, a GPS system or a resolver in the wheel drive system  10 ), and if this is judged to be greater than or equal to the speed limit  102  then a cockpit indication  104  alerts the pilot that the reversing speed limit has been reached. Upon receiving the indication, e.g. as a visual, audible or tactile warning, the pilot would be required to react by using the conventional foot brake pedals  106 . The BCS  108  would interpret the braking input in the usual manner and command a brake pressure to the wheel brake assemblies  110  depending on the degree of brake pedal deflection. Since the autonomous taxi speed limit is selected such that any brake pressure up to the full brake pressure will not risk a tip over event the pilot can make any brake input to stop the aircraft. 
     The components of conventional aircraft BCS generally have a significant level of inaccuracy, particularly at low brake pressures. For example, in a conventional hydraulic aircraft braking system the zero torque pressure (ZTP), i.e. the braking system pressure that will apply zero braking torque, may have a nominal value of around 15-20 bar. However, the actual ZTP may be significantly lower than this nominal value, e.g. around 10 bar. Furthermore, a valve regulating the hydraulic pressure at the brake actuators may have a tolerance of ±5 bar or more at low brake pressures. There are yet further sources of inaccuracy. 
     Due to these BCS inaccuracies when the pilot makes a brake input to maintain the speed below the autonomous taxi speed limit it may be difficult for the pilot to bring the aircraft to a smooth stop. This would result in an unpleasant and discontinuous pushback manoeuvre. 
     A partial solution would be to limit the autonomous taxi speed to a higher speed limit than that mentioned above, and reducing the allowable Mass-CG combination for the aircraft, excluding the areas of the CG envelope where the tip over risk is higher, e.g. high mass and aft CG. These extremities of the mass-CG envelope of the aircraft would need to be restricted accordingly. However a limitation in the CG envelope is a limitation in the flexibility of the operability of the aircraft. 
     2. Use of Park Brake 
     Another solution is to use a modified park brake to stop the aircraft. In particular it would be possible to limit the rise rate of the braking application through one or more restrictors limiting the flow rate in the hydraulic system. Simulations have been carried out to evaluate for each speed the acceptable rise rate that would stop the aircraft, within acceptable passenger comfort level (assumed longitudinal deceleration lower than 0.2 G). The study indicates that the brake rise time would increase rapidly with increasing pushback speed. 
     In order to implement this solution the maximum pushback speed would still need to be limited to a low speed if the brake rise time is to be kept within acceptable limits. Since the park brake rise time is generally applicable for all aircraft operations it is not considered feasible to increase the brake rise time to the extent that its performance is limited, e.g. for taxiing with engines or hybrid (combined wheel taxi and engine taxi) operations. Accordingly, a low pushback speed limit would need to be imposed which may be similar to solution 1 above. Indeed solution 1 may be preferred. 
       FIG. 6  illustrates an implementation of the park brake solution, which is reliant on a cockpit indication  104  of when the pushback speed is greater than or equal to the pushback speed limit. In response to the cockpit indication the pilot would need to activate the park brake lever  112  in the cockpit which controls the park brake  116  having a brake rise limiter  114  to ensure than the deceleration is within the acceptable passenger comfort level (assumed longitudinal deceleration lower than 0.2 G) at the speed limit. 
     3. Braking with Limited Number of Brakes 
     A further solution involves commanding braking through a limited number of brakes. In order to achieve the same aircraft level retardation force a reduced number of brakes need to be commanded with a higher braking pressure. 
     For alternate braking the braking command is sent to a wheel pair (e.g. one main landing gear wheel on each side of the aircraft), hence the minimum number of braking wheels would be two. In this case the clamping braking pressure commanded could be raised to approximately double that of the baseline case where all four wheels of the main landing gears  6  are braked. 
     For normal braking a single braking command can be sent to one of the main landing gear wheels. In this case the clamping braking pressure commanded could be raised further to approximately quadruple that of the baseline case where all four wheels of the main landing gears  6  are braked. 
     This solution alone may not be able to accommodate full braking pressure without any risk of tip over, but may be used in combination with one or more other solutions presented here. 
     4. Single or Variable Maximum Brake Pressure Limitation 
     In this solution an overall brake pressure limitation is set to ensure the braking clamping pressure is never so high as to cause aircraft tip over. Preferably such pressure limitation is customised for each operational point in the CG-Mass diagram to optimise the maximum pressure applicable by the system depending on the aircraft load condition. By accurately measuring the vertical load through each landing gear leg (the sprung mass), e.g. using strain gauges on the main strut, and with prior knowledge of the un-sprung mass of each landing gear, the CG X position and mass of the aircraft can be determined. By plotting these values on the CG-Mass diagram the maximum brake clamping pressure can be determined. 
     5. Ramp Braking Law 
     Analysing the dynamic response of the aircraft in the limit case for nose lift-up we observe that there is a nose lift-up ‘peak’ immediately after the brake application, where the nose landing gear shock absorbers extend and the pitch angle reaches a maximum. Subsequently the aircraft tends to put the nose back down due to its weight. 
     This solution aims to prevent nose lift-up and obtain an optimised performance during braking by implementing a braking law that commands initially the maximum pressure that ensures no nose lift-up, and after the nose lift-up ‘peak’ increases gradually the braking command, up to the maximum system pressure. 
     The increase in pressure is initially very gentle and subsequently very rapid once the nose lift-up peak is safely overcome. Such behaviour can be modelled with an exponential function, e.g.: 
     
       
         
           
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       FIG. 7  illustrates graphically how the system braking pressure varies with time, where P2 is the final pressure (max system pressure), P1 is the initial pressure (max initial pressure to ensure no nose lift-up), and T is the total time from brake application to full brakes applied. The parameter, τ, regulates the behaviour of the ramp law. A high value of τ makes flatter the first part of the ramp law and steeper the second part, and vice versa. 
     This solution may be preferred, as it guarantees stability without compromising on performance (in all cases full braking application within T seconds), but has to take into account the limitations introduced by the low accuracy of conventional braking systems. 
     6. Braking Against the Wheel Actuator Motors 
     The main limitations of using the conventional braking system to provide the aircraft with retardation force include both the low limit of braking pressure that can be applied without risking aircraft tip-up and the low accuracy of conventional braking systems at very low pressures. 
     A solution that could mitigate both limitations includes using the autonomous wheel drive taxi system in conjunction with the BCS. The amount of wheel drive taxi system torque in the direction of motion is additional torque that needs to be demanded to the braking system to provide some retardation, and the inaccuracies of the BSCS can be balanced by a fine tuning of the wheel drive taxi system torque, to ensure the global retardation torque is sufficiently accurate. 
     This solution is particularly beneficial for the pushback runaway problem, for which the amount of required braking torque is very low and also very dependent on certain parameters (Slope, Mass, Brake Gain, . . . ). 
       FIG. 8  shows an example of a closed loop speed controller  200  for regulating the wheel drive taxi system torque during pushback. When a binary pushback command is sent from the pilot the autonomous wheel drive taxi system (or “eTaxi”) motors  204  target the pushback speed  202 , and in absence of apron slope the speed  206  is achieved and maintained by the aircraft, and there is no need to apply braking. 
     In the presence of a downhill slope (in the direction of motion) the aircraft tends to accelerate, as described above, and when it overshoots the target speed plus a defined tolerance, a threshold speed  208 , a set amount of braking is commanded  210  to the BCS  212 . 
     The additional braking torque  214  will act as an ‘additional’ resistive force that will tend to slow down the aircraft, hence the eTaxi speed controller  200  will reduce the amount of torque required to maintain the speed, so the overall contribution of eTaxi and braking will be a retardation force that is accurately controlled by the eTaxi controller. 
     When the slope decreases and drops below the pushback runaway limit the eTaxi system will saturate the torque controller to match the amount of braking provided. When this happens the eTaxi controller  200  will command braking release, as it will have detected the end of the slope. 
     7. Braking Through the eTaxi System 
     The solution uses the dissipation of the power generated in the eTaxi electric motors (used as generators)  300  to provide the torque necessary to stop the aircraft. This solution requires the capability to sink power back into the aircraft electrical network  302  (as shown in  FIG. 9 ), or to use a resistor  304  to dissipate the energy through the Joule Effect (as shown in  FIG. 10 ). 
     Considering the maximum aircraft mass, M, and the pushback speed, V, the maximum kinetic energy, K, to be dissipated is (in the ideal case of 100% motor/generator efficiency): 
       K=½MV 2 :
 
     The amount of heat dissipated through the resistor and the maximum reachable brake temperature, Tmax, sizes the minimum required mass of the resistor at aircraft level. Assuming three consecutive full brakes application, and an initial brake temperature T1, the mass of the resistor m res : 
     
       
         
           
             
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     where Cs is the specific heat coefficient of the material chosen for the resistor and ΔT=Tmax−T1. The mechanical power to be dissipated for each brakes application through the resistor, assuming no loss in the electro/mechanical chain is: 
     
       
         
           
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     The mechanical power is the kinetic energy over the time interval Δt for the braking, and is equal to the electric power dissipated through Joule effect, expressed as a function of the back emf ε induced in the motor: 
     
       
         
           
             
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     Assuming a resistor formed as a simple wire dissipating heat (hence not considering the insulation, and the resistor case) the resistance R can be expressed through the second Ohm&#39;s Law as a function of mass of the resistor and geometry: 
     
       
         
           
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     Where ρ is the material&#39;s resistivity, 1 the length of the wire in the resistor, A the wire cross section&#39;s area, V the wire&#39;s total volume, and δ the material&#39;s density. Thus we obtain: 
     
       
         
           
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     where C m  is constant, depending on the chosen material. 
     ε is the amount of back emf that the system has to be able to tolerate. In order to minimise this quantity the time interval (duration of the braking) has to be increased, and a material providing a low Cm has to be selected. This means to select a material with low resistivity (good conductor), a high density and a high specific heat coefficient. Suitable candidate materials include steel, copper and aluminium, with copper or steel being preferred. 
     The weight and space impact of the resistor will also need to take into account a consistent amount of insulating material around the resistor wire, adding mass and volume. It may be preferable to split the amount of power to dissipate among more resistors, e.g. at least one per landing gear, in order to optimise the dissipation within the available space. 
     For dissipation of the power generated in the electric motors the wheel drive system  10  needs to be capable of being back-driven. The wheel drive system  10  shown in  FIG. 3  is particularly suitable as the sprocket  60  is in positive meshing engagement with the roller gear  34  when the wheel drive system is in operation. Other wheel drive systems that include an over-running clutch in the drive path between the wheel and the motor would not be suitable as these only permit drive torque to be passed from the motor to the wheel and not vice versa. 
     The clutch-less wheel drive system  10  shown in  FIG. 3  is also particularly beneficial as the physical separation between the drive pinion and the driven gear when the system is moved to the disengaged position ensures that on landing there is no drive path between the wheel and the motor. 
     Although the invention has been described above with reference to one or more preferred embodiments, it will be appreciated that various changes or modifications may be made without departing from the scope of the invention as defined in the appended claims.