Sweriduk, G. D., Cheng, V. H. L., Andre, A. D., and Foyle, D. C. (2007). Proceedings ofthe 26th Digital Avionics Systems Conference, Dallas, TX, Oct. 21, 2007

AUTOMATION TOOLS FOR HIGH-PRECISION TAXIING

G. D. Sweriduk, V. H. L. Cheng, Optimal Synthesis Inc., Palo Alto, CAA. D. Andre, Interface Analysis Associates, Gilroy, CA

D. C. Foyle, NASA Ames Research Center, Moffett Field, CA

AbstractAirport congestion is a growing problem, as

most air travelers can attest. Efforts are now underway to reduce congestion by increasing the trafficflow rate at airports through the use of automationtools. Which runway and taxiways an aircraftoccupies, and when, must be precisely scheduledand executed. Presented herein are results ondeveloping a system which can generate a 4-Dtrajectory and follow the trajectory accurately.

IntroductionThe rapid increase in air travel over the last

decade is leading to problems with airportcongestion which will only get worse in the future,if predictions are accurate. As a result, airports arebeing expanded to handle more flights, and newtechnologies are being developed to reduce theseparation between flights. These changes willmean more surface traffic that needs to be managed,and managed more carefully. With more runwaysthere will be more runway crossings required oftaxiing aircraft, and with reduced separation, thetime window for crossing a runway will be reduced.The greater complexity and requirement for greaterprecision could overwhelm a system that reliessolely on human inputs. In order to enable efficientand safe operations in this more dynamic andcomplex environment, new tools are needed forboth the airport control tower and the cockpit. Thispaper will present ongoing research into the surfacetraffic control problem and the development ofstate-of-the-art technologies to enable aircraft toexecute high-precision taxi operations.

Parallel research efforts are aimed at pathplanning for the movement of arrivals anddepartures to minimize transit times to and from thegate. The goal of that work is to produce taxiclearances with route and timing information thatthe control tower will issue to aircraft on the groundor on final approach. The focus of this research is

on a system for generating and following 4-Dtrajectories to achieve the throughput gains madepossible by the path planning. This system, calledFARGO (Flight-deck Automation for ReliableGround Operation), uses information about theairport geometry, the aircraft’s characteristics, andpassenger comfort, as well as the taxi clearance, togenerate position and velocity time histories. Theflight crew then has the option of following thetrajectory in manual mode via specially designedcockpit displays, or in fully automatic mode. Theautomatic mode uses a nonlinear controlleremploying the feedback linearization method totrack the desired trajectory. Another aspect of thisresearch is to develop contingencies to ensure safeoperations in the event that an aircraft deviates fromits intended trajectory. Simulations with a high-fidelity model of a Boeing 737 are used todemonstrate the system.

Trajectory ControlThe taxi clearance issued by the control tower

to an aircraft contains a trajectory as a list oftaxiways. These clearances are issued verballyunder the present system, although in the systembeing developed, clearances will be issued digitally.In this way the flight crew does not have to enterthe clearance into the trajectory-following systemmanually. It is also expected to be faster, and easierfor flight crews to understand if they can see astring of text. Currently clearances are issuedwithout any timing information, but it is anticipatedthat in the future, through the use of othercomputer-aided traffic management tools, theclearance will have timing information. In this workit is assumed that the timing information isincluded. Given the geometry of the airport,positions and speeds as a function of time can beextracted from the clearance. (Note: theterminology that has been adopted refers to thetrajectory as 4-dimensional (4-D), although the

altitude coordinate is superfluous since the aircraftis only taxiing.)

Trajectory GenerationGiven a clearance string, such as: "TAXI VIA

17C-35C/M5/EM CR 113," and a database of theairport’s runways and taxiways, one can generate anarray of nodes; i.e.: [180 181 182 183 232 174 162146 128 113], each of which corresponds to anintersection along the way. To generate an x-ytrajectory, it is necessary to know details about howtaxiways are designed in order to get the correctturn radii; high-speed exits from runways are laidout using an Euler’s spiral [1]. Figure 1 shows thecorresponding route for the given clearance for thesimulated airport, which is a reasonably accuraterepresentation of the Dallas-Fort Worth airport.Velocities are determined from standard taxiingspeeds, acceleration limits, and any time restrictionsthat might be placed on the route.

Figure 1. Clearance Route

System DynamicsThe control problem, then, is to get the aircraft

to track the 4-D trajectory. The tracking problemwas converted to a regulation problem by defining

the error dynamics, and designing a controller toregulate the error states to the origin.

The dynamics of the taxiing aircraft arenonlinear and of the general form:

( )uxfx ,=& (1)

where the state variables are north position (yI), eastposition (xI), north velocity ( Iy& ), east velocity

( Ix& ), and the control inputs are throttle, brakes, andtiller (nose wheel angle). In reality the left and rightbrakes can be used differentially for steering, but inthis work they are assumed to act symmetrically.Furthermore, to make the control problem moretractable, the throttle and brakes were gangedtogether to form a pseudo-control for longitudinalacceleration that can take on positive and negativevalues. Control effort was allocated to brakes orthrottle depending on the sign of the pseudo-control(see Figure 2).

Figure 2. Pseudo-control Allocation

The clearance trajectory gives the desiredvalues of the state variables. However, the statevariables are in an inertial frame, and from a controlpoint of view, it makes more sense to deal withquantities in the aircraft’s frame of reference. Also,it is more practical to control the trajectory of theaircraft’s nose wheel rather than its center of mass,which is what the equations of motion beingintegrated in the simulation produce.

The desired trajectory is used to define amoving reference frame. This reference frame canbe thought of as a body-fixed frame of a phantomaircraft, whose origin is at its center of mass. Thenose wheel is located at a known distance along thelongitudinal axis of this frame. The error betweenthe actual nose wheel states and the reference nosewheel states can then be expressed as longitudinal

and lateral errors in the actual aircraft’s body-fixedframe.

The frames of interest are shown in Figure 3.The inertial frame, or runway frame, is shown in thelower left corner of the figure. The dashed lineshows the desired trajectory that is traced by thepoint Rn , which is the reference nose wheel. The

body frame ( Bx , By ) is the standard frame used inflight mechanics where the origin is at the center ofgravity, the x axis is aligned with the centerline ofthe fuselage, and the y axis is positive on the rightside of the aircraft. The point n is the nose wheelposition in the body frame. The objective is toregulate the error vector between the points n and

Rn and its derivative.

Figure 3. Error Between Aircraft and ReferenceTrajectory

The error vector is:

nn rre R −= (2)

where Rnr is the position of Rn in the body frameand rn is the position of n in the body frame.Taking the derivative in the body frame (indicatedby the superscript B) produces:

0−=−= RR nBnB

nBB

vrdtd

rdtd

edtd

(3)

The velocity of the nose wheel reference point Rnrelative to the body frame can be obtained from theexpression for the velocity of the nose wheelreference point relative to the inertial frame:

RRR nBnInI vvv += (4)

orRRR nInInB vvv −= (5)

where RnI v is the inertial velocity of a point Rnfixed in the body frame which is coincident with thepoint nR in the reference frame at a given instant intime. The inertial velocity of Rn is:

RR OnRIOInI rvv ×+= ω (6)

where OI v is the inertial velocity of the origin of

the reference frame, RIω is the angular velocity ofthe reference frame relative to the inertial frame,which can be determined from the reference

trajectory, and ROnr is the position vector of thenose wheel reference point in the reference frame,which is known from the aircraft geometry. Itshould be noted that the first term on the right handside is given in inertial coordinates while the secondterm is computed in reference frame coordinates, soa transformation is necessary to perform theaddition correctly. The inertial velocity of the point

Rn is:

RR ncgBIcgInI rvv ×+= ω (7)

where cgI v is the inertial velocity of the c.g. (center

of gravity) of the aircraft, BIω is the angularvelocity of the aircraft relative to the inertial frame,and Rncgr is the position vector of Rn relative to theaircraft’s c.g. Again, the first term on the righthand side is given in inertial coordinates while thesecond term is computed in body frame coordinates,so a transformation is necessary to perform the

addition correctly. The position vector Rncgr can beexpressed in the body frame as:

( ) RRR OnBR

cgOBI

nOcgOncg rCrrCrrr +−=+= (8)

where BIC and BRC are transformation matrices tothe body frame from the inertial and trajectoryreference frames, respectively, Or is the position ofthe origin of the reference frame in the inertial

frame, and cgr is the position of the c.g. in theinertial frame.

Differentiating the expression for velocity, theacceleration is:

RRRR nBBInInInB vaaa ×−−= ω2 (9)

where

( ) RRR OnRIOnRIRIOInI rraa ×+××+= αωω (10)

( ) RRR ncgBIncgBIBIcgInI rraa ×+××+= αωω (11)

OI a and cgI a are the inertial accelerations of theorigins of the reference and body frames,

respectively, and RIα and BIα are the angularaccelerations of the reference and body frames,respectively, relative to the inertial frame. In thereference trajectory, turns are made at constantspeed, so the angular acceleration RIα is zeroexcept instantaneously at the beginning and end of aturn.

It is assumed that the trajectory generatorgenerates the inertial position, velocity, andacceleration of the nose wheel, so it is not necessaryto compute them using the equations above.

For reasons explained in the following, first-order linear dynamics were added at the inputs, sothat the augmented system is linear with respect tothe inputs. The additional dynamics could beconsidered as filters or actuator dynamics.

( ) ( )suas

asu c+= (12)

where uc is the output of the controller, and the poleis at -a, and a is chosen to be sufficiently large thatthe dynamics of the filters do not interact with thedynamics of the aircraft. In this case a = 20.

Numerical Nonlinear ControlA controller was developed using the feedback

linearization methodology [2,3]. The feedbacklinearization of the dynamics is achieved byredefining the system dynamics in terms ofderivatives of the error states. It may be observedfrom the preceding formulation that the controlvariables will appear in the third derivative of theposition errors. Hence, feedback linearization can

be achieved by redefining the dynamics in terms ofthe position errors and three of its successive timederivatives as the state variables.

Starting from Equation (2), it can be seen thatthe error states are functions of position, say

( )nrfe 1= (13)

Differentiating this expression three times willproduce a complex expression of the form:

() () cue ⋅+⋅= ψφ&&& (14)

where φ(⋅) and ψ(⋅) are complex nonlinear functionsthat depend on the state variables. Defining new

state variables as z =[ ]Teee &&& , the dynamics canbe recast in the form:

BvAzz +=& (15)

where

,A

=

000100010

=100

B (16)

and the transformed control variable v is given by:

() () cuv ⋅+⋅= ψφ (17)

In this work, Nonlinear Synthesis Tools,software for numerically computing the feedbackcontrol signals, was used [4]. The nonlinear controlsystem design software automatically constructs thenonlinear functions φ(⋅) and ψ(⋅) from an embeddedmodel of the aircraft. An assumption of thenumerical method used is that the control appearslinearly in the dynamics in order to obtain the formof Equation 17. To address this problem, thedynamics shown in Equation 12 were added.

Since the transformed system is in a linear,time-invariant form with respect to the transformedcontrol variable, linear control techniques can beused to design a stable feedback control law; i.e.:

v = [k1 k2 k3] z (18)

In this study, pole placement was used to computethe gains using standard techniques. The actualcontrol variables can be recovered from thetransformed control variables using the inverserelationship from (17):

() ()[ ]⋅−⋅= − φψ vuc1 (19)

If the system nonlinearities φ(⋅) and ψ(⋅) are knownreasonably well, the resulting closed-loop systemwill have dynamic properties close to thetransformed system.

Due to the assumptions of the numericalmethod, the embedded model needed to bemodified. Both the throttle and brake responses arenearly constant for inputs between 0 and about10%. This is practically the same situation as asaturated actuator, and leaves the system effectivelyuncontrollable. To avoid this problem, curves werefitted to the throttle and brake response, but weregiven more slope near zero (see Figure 4 and Figure5). These curves were then substituted in theembedded model for the actual dynamics.

In the design model, the angular acceleration isalways assumed to be zero.

Figure 4. Throttle Function Approximation

Figure 5. Brake Function Approximation

The poles selected for the lateral channel were[-2, -3, -20]. The poles selected for the longitudinalchannel were [-1, -2, -20] when the aircraft wastaxiing, and during rollout the poles were a functionof total velocity: [σ, 2σ, -20] where:

−−−−=σ

201

vvexp TD (20)

and vTD = 205.0 is the touchdown velocity.

Figure 6 shows the desired and actualtrajectories in the horizontal plane. The totalvelocity is shown in Figure 7. It can be seen that theaircraft comes to a stop midway because it crossesanother runway. Figure 8 shows the x and y axiserrors in nose wheel position, in the body frame. Itcan be seen that the error never exceeds about 2.5feet for the given choice of poles. From Figure 9 itcan be seen that the velocity errors never exceedabout 2 feet/second. The largest errors occuraround the time that the aircraft starts moving fromrest. The body-frame accelerations are shown inFigure 10 and show that the peak longitudinalacceleration is approximately 0.2 g, and the peaklateral acceleration is approximately 0.1 g. Thetime histories of the control inputs are shown inFigure 11. The control allocation scheme createdsome undesirable switching behavior. In the futurea better blending scheme will be required.

ConclusionsInitial development of automated taxiing tools

was presented. Taxi clearances from the tower were

converted into time-dependent ground trajectories.A nonlinear controller was developed for a taxiingaircraft that can accurately follow a trajectory as afunction of time.

References[1] Federal Aviation Administration, 1989,Airport Design, Advisory Circular 150/5300-13,Washington, D. C., U.S. Dept. of Transportation.

[2] Isidori, A., 1989, Nonlinear Control Systems -An Introduction, New York, NY, Springer-Verlag,pp. 145-172.

[3] Marino, R., and P. Tomei, 1995, NonlinearControl Design, New York, NY, Prentice Hall, pp.41-60.

[4] Menon, P. K., et al., 2000, Nonlinear SynthesisTools™ for Use with MATLAB®, Palo Alto, CA,Optimal Synthesis Inc.

Figure 6. North Position vs. East Position

Figure 7. Total Velocity vs. Time

Figure 8. Longitudinal and Lateral PositionError vs. Time

Figure 9. Longitudinal and Lateral VelocityError vs. Time

Figure 10. Longitudinal and LateralAcceleration vs. Time

Figure 11. Control Inputs vs. Time

AcknowledgementsThis work was supported by NASA under the SmallBusiness Innovations Research program, contractno. NNA06AA06C

26th Digital Avionics Systems Conference,October 21, 2007