Proportional Navigation with Adaptive Terminal Guidance for Aircraft Rendezvous

Austin L. Smith1

Air Force Research Laboratory, Wright-Patterson AFB, OH, 45433

Future unmanned aerial vehicle missions will include aircraft rendezvous maneuvers that are currently performed by today’s manned aircraft when rejoining for formation flight or aerial refueling operations. A means to compute a trajectory or generate guidance commands is necessary for an autonomous rendezvous execution. One major issue in aircraft rendezvous is the necessity to meet space and time constraints while satisfying operational and vehicle limitations. Failure to realize these constraints increases the risk of midair collisions between an unmanned receiver aircraft and the tanker during refueling operations, and introduces the possibility of a receiver aircraft running out of fuel. Proportional navigation guidance has been used extensively and is well established in two-body intercept problems, and has been recognized as an alternative to route planning for aircraft rendezvous. It can easily be implemented into a vehicle flight control system and is not computationally burdensome on the host flight control computer. Thus, a proportional navigation system, augmented with adaptive terminal guidance and a coupled velocity controller, is proposed and developed for a receiver to tanker rendezvous. The proportional navigation algorithm will steer a receiver to a tanker aircraft and the velocity control loop will accelerate the receiver to execute a successful rendezvous.

Nomenclature Af = acceleration command at rendezvous RTR = distance from receiver to tanker s = distance from receiver to target Vc = closing velocity Vf = final velocity constraint Vmax = maximum receiver velocity Vmin = minimum receiver velocity VR = receiver velocity, airspeed VRg = receiver velocity, groundspeed VT = tanker velocity, airspeed VTg = tanker velocity, groundspeed xR = receiver East position xRZ = rendezvous East position xT = tanker East position yR = receiver North position yRZ = rendezvous North position yT = tanker North position ∆yRFE = distance from receiver to tanker in down-track direction (rotated flat-earth frame) Фf = final line-of-sight constraint Ψf = final heading constraint ψR = receiver heading ψRZ = tanker heading ψT = desired heading at rendezvous ( ˙ ) = derivative with respect to time

1 Associate Aerospace Engineer, Air Vehicles Directorate, 2180 8th St, Member AIAA.

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AIAA Modeling and Simulation Technologies Conference and Exhibit20 - 23 August 2007, Hilton Head, South Carolina

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This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

AFRL-WS 07-161

I. Introduction oday’s manned aircraft rendezvous are facilitated by systems such as the Airborne Tactical Air Navigation system (TACAN) or air-to-air radar systems. However, for unmanned air vehicles (UAVs), a system that can

provide the capability for both rendezvous and formation flight is desired to increase rendezvous precision, and reduce the total number of onboard systems for all aircraft involved in the rendezvous. A near-term solution to unmanned aircraft rendezvous and formation flight is precision differential GPS (PGPS)1. Use of PGPS is being investigated in the Air Force’s Automated Aerial Refueling (AAR) system in order to enable UAVs to fly in close formation flight with an aerial refueling tanker aircraft. This same system can be used to provide spatial information to the receiver and tanker aircraft to achieve a successful rendezvous during refueling operations. Using PGPS data exchanged over an airborne communication network, unmanned aircraft can autonomously generate a trajectory or guidance commands to the rendezvous location. The difficulty in autonomous trajectory generation and guidance schemes is the ability to accurately satisfy space and time constraints and alleviate the risks of airborne collisions and a fuel shortage. The receiver aircraft must arrive behind the tanker at a specified trail location at the same heading and speed as the tanker.

T

The Air Force Research Laboratory is utilizing an Automated Aerial Refueling simulation2 to identify and evaluate different techniques for aircraft rendezvous that will minimize the associated risks while satisfying tactical, operational, and vehicle constraints. Optimal trajectory generation, with dynamic optimization techniques, is one candidate method currently being studied for application to UAV route planning and rendezvous with a tanker. Dynamic Optimization techniques have been used to solve for suitable trajectories in 2D or 3D space, given temporal and spatial requirements, while avoiding threats and obstacles3,4. When applied to rendezvous during aerial refueling operations, aircraft constraints such as turn rate or acceleration limits can be applied. Optimal trajectory generation has been applied to the refueling rendezvous problem in 2D space in a non-realtime simulation environment, and successful rejoins have been performed. Unfortunately, a major obstacle to implementing this and other optimization algorithms is the fact that they are computationally intensive and lack convergence guarantees, which precludes the use of such algorithms in flight critical UAV flight control systems. Another potential rendezvous algorithm is based on a Dubins’ path5, and forms a non-optimal trajectory solution, given multiple geometric combinations of constant-radius turns and straight lines. This limited 2D trajectory generator is less computationally demanding and can be embedded successfully into a flight control system computer and run in real-time, but still must be iterated to meet the given time constraint. Furthermore, it limits UAV maneuvering to only constant radius turns and straight line flight. An algorithm is desired that will compute guidance commands to enable a UAV to arrive at a specified rendezvous location at the same time as the tanker aircraft. It must obey tactical, operational, and performance limitations of the host-aircraft and it must be implemented efficiently into the UAV flight control system for real-time operation. The algorithm should also be robust enough to compensate for tanker maneuvering and arbitrary winds.

Proportional navigation (PN) guidance has been used extensively and is well established in two-body intercept problems. It can easily be implemented into a vehicle guidance, navigation, and flight control system and is not computationally burdensome on the host flight control computer. Terminal guidance has been introduced into PN by adding a bias term to achieve an angular constraint at impact of a moving target6. However, this guidance system assumed a constant interceptor velocity. Conversely, PN has been used to intercept stationary targets by an interceptor moving at a varying velocity profile7. Combining terminal guidance and time-varying velocity, an adaptive proportional navigation guidance approach has also been shown, in three-degree-of-freedom simulations with full-nonlinear point-mass dynamics, to be effective and accurate in guiding a lifting vehicle during its terminal phase8. Thus, a proportional navigation system, augmented with adaptive terminal guidance, is proposed. The PN algorithm will steer a UAV to a rendezvous, or “intercept” with a tanker aircraft, and the velocity control loop will accelerate the UAV to meet aerial refueling rendezvous constraints while obeying aircraft saturation limits. This algorithm is not limited to intercepts in 2D space, and could be expanded to more difficult 3D rendezvous problems.

This paper presents a proportional navigation system with adaptive terminal guidance and a coupled velocity controller developed for a constant-altitude UAV (receiver)-to-tanker rendezvous for the purpose of aerial refueling. Rendezvous is assumed to occur at a one nautical mile trail position behind the moving tanker, so small spatial and temporal errors are acceptable. Rendezvous at this trail position ensures that the tanker crew has the ability to assess UAV behavior before authorizing closure to a formation position close to the tanker. The UAV receiver is commanded to fly to an estimated rendezvous location chosen to prevent prolonged “tail-chases”, and should arrive at that rendezvous location at the same time as the desired one nautical mile tanker trail position. The PN control law aboard the UAV is defined so that the turn rate command to the UAV is proportional to the rate of change of the line-of-site between the UAV receiver and the estimated rendezvous location. Furthermore, the UAV receiver is

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subject to turn rate, velocity, and acceleration limits embedded into the PN algorithm. The PN algorithm is evaluated with six degree-of-freedom (6DOF) vehicle dynamics for both the receiver and tanker and with external disturbances9.

II. Proportional Navigation Figure 1 defines the aerial refueling geometry, including the axis system and the angles necessary for the

guidance system formulation. The line of sight angle, Φ, from the target location to the receiver is defined positive counter-clockwise from east and –π ≤ Φ ≤ π. Aircraft heading, ψ, is defined positive clockwise from north and –π ≤ ψ ≤ π. All positions and velocities are defined in North and East directions in the flat earth plane. This corresponds to PGPS data. Aircraft heading is defined as the direction in which the nose of the aircraft is pointing, not the actual ground-track direction. This becomes important when winds are acting on the aircraft.

Proportional navigation has been shown to execute an intercept of a stationary target by a vehicle moving with an arbitrary time-varying velocity4. Reference 4 describes the course with which the interceptor will approach the target based on the proportional gain value in the PN guidance law, shown in Eq. (1).

Figure 1. Aerial Refueling Rendezvous Geometry in Horizontal Flat-Earth Plane.

φλψ && 0−=com (1)

For λ0 > 2, the intercept trajectory will end in a direct collision course with the target and . Therefore, by commanding a turn-rate proportional to the rate of change of the line-of-sight angle between the interceptor and a target, the interceptor can be steered onto a direct collision course with the target given a proper proportional gain.

0→φ&

In the aerial refueling rendezvous problem addressed in this paper, the interceptor will be referred to as the receiver aircraft. The derivation of the time derivative of the line of sight angle in Eq. (1) is different from that shown in Ref. 5 and 6 in order to make the guidance data compatible with PGPS data. The line of sight angle between the receiver and the target location is defined in Eq. (2), where tan-1 is a four quadrant inverse tangent.

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

= −

RZR

RZR

xxyy1tanφ (2)

The derivative of the line of sight angle with respect to time is found by first defining an intermediate variable, X.

RZR

RZR

xxyy

−−

=Χ (3)

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Then, taking the tangent of both sides of Eq. (2) and taking the derivative with respect to X, one obtains:

( )

Χ=

Χ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−

d

d

dd RZxRx

RZyRy

φtan (4)

1sec2 =Χd

dφφ (5)

Solving for Χddφ in Eq. (5) and multiplying by dtdΧ , one obtains:

( )( ) ( )( )

( ) ⎥⎦

⎤⎢⎣

⎡

−−−−−−

=Χ

Χ 22sec1

RZR

RZRRZRRZRRZR

xxyyxxxxyy

dtd

dd &&&&

φφ

(6)

Using the trigonometric identity in Eq. (7), Eq. (6) can be re-written and simplified to obtain Eq. (8).

222 11

tan11

sec1

Χ+=

+=

φφ (7)

( )( ) ( )( )( ) ( )22

RZRRZR

RZRRZRRZRRZR

yyxx −+−yyxxxxyy −−−−− &&&&

=&φ (8)

In order for the intercept conditions of a direct impact specified in Ref. 5 to occur, the target location must be stationary. Therefore, an assumption is made in the formulation of the proportional guidance law that the velocity of the target location is zero. As such, Eq. (9) is used in the guidance law instead of Eq. (8). Any errors caused by this assumption due to tanker uncertainties will be addressed in Section III.

( )( ) ( )( )

( ) ( )22RZRRZR

RZRRRZRR

yyxxyyxxxy

−+−−−−

=&&&φ (9)

The proportional gain in Eq. (1) must ensure a direct collision course with the target, i.e. λ0 > 2, and it must satisfy rendezvous constraints which impose a terminal constraint. At rendezvous, the receiver heading and velocity must equal the heading and velocity of the tanker. The heading constraint is given by:

RZf ψ=Ψ (10)

where ψRZ is the target heading at the predicted point of rendezvous. Reference 6 shows that when perfect tracking is assumed and the guidance law is integrated, the proportional gain that ensures Eq. (10) also satisfies Eq. (11). Hence, there is a unique proportional gain that will satisfy the rendezvous heading constraint.

( )

0

00 φ

ψλ

−Φ

−Ψ−=

f

f (11)

Reference 5 establishes, assuming perfect tracking of the guidance command, that the guidance law will result in the condition shown in Eq. (12). The offset term in Eq. (12) exists because Φ is defined 90 degrees clockwise of ψ. This condition reveals that the final line of sight angle will be exactly opposite in direction to the final heading as the

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range approaches zero. The only exception is if the PN is engaged when the receiver is flying away from the target along the line of sight, which would result in no turn being commanded. This exception would be a rare instance in reality, and is easily avoided by slightly maneuvering the receiver before PN engagement.

ff Ψ−−=Φ 2π (12)

Although aircraft heading is defined as –π ≤ ψ ≤ π, ψ0 in Eq. (11) has a range of –2π ≤ ψ0 ≤ 2π to calculate λ0. This is done to make certain there is some λ0 > 2 for a given receiver location relative to the initial target location, even though the receiver may not be at that heading. For example, if the terminal constraint is Ψf = 0, then Фf = -π/2. The proportional gain becomes λ0 = ψ0/(-π/2 – Φ0). By setting λ0 > 2, it follows that ψ0 > –π - 2Φ0. If Φ0 = π/4, then ψ0 < -3π/2, which is not in the range of ψ.

The proportional navigation is not engaged until the proportional gain is greater than two in order to ensure a direct collision course with the target given perfect tracking and a stationary target. There exists some heading or range of headings at any given line of sight angle that will result in the proportional gain being greater than two. However, if the receiver heading is not in that range, it must change position and orientation with respect to the target location in order to achieve the necessary gain condition. Reference 6 suggests commanding a bank angle with the same sign as the line-of-sight error that will guarantee an increase in the proportional gain. Here, an analogous command will be generated to increase the gain.

( )φψψ −Φ= fcom sgnmax&& (13)

Equation (13) will turn the receiver until Eq. (11) returns a gain greater than two. At that instant, the proportional navigation will be engaged to command a rendezvous. A guidance law of the form shown in Eq. (1) that satisfies Eq. (11), will result in a receiver trajectory that approaches the tanker from behind and ends with the condition in Eq. (10). This is the desired result for an aerial refueling rendezvous.

III. Adaptation of Proportional Gain The guidance law developed in Section II will achieve the final heading constraint with the assumption of perfect

tracking and no external disturbances. However, when subjected to tracking errors, winds, modeling uncertainties, etc., the guidance law with the gain from Eq. (11) may not result in a trajectory satisfying Eq. (10) at the point of rendezvous. Consequently, the gain must be updated to account for any guidance errors. The update is considered adaptive in the sense that it is a function of time. As with the guidance law derivation, the adaptation derivation is similar to that shown in Ref. 6, but is tailored to be compatible with PGPS guidance information. From Ref. 6, the closed-loop adaptation law of the proportional gain with range as the independent variable is

⎟⎟⎠

⎞⎜⎜⎝

⎛∆∆

+=φψλκλ

sdsd

(14)

where,

Rf ψψ −Ψ=∆ (15)

φφ −Φ=∆ f (16)

Reference 6 suggests representing the closed-loop adaptation law with time as the independent variable. Because of the application in this paper of PN for rendezvous instead of intercept, the range to the target may not monotonically decrease due to tanker maneuvers. Therefore, it is preferred to represent the adaptation law with time as the independent variable.

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dtds

sdtd

⎟⎟⎠

⎞⎜⎜⎝

⎛∆∆

+=φψλκλ

(17)

( )( ) ( )(( )RZRRZRRZRRZR yyyyxxxxs

&&&&& −−+−−⎟⎟⎠

⎞⎜⎜⎝

⎛∆∆

+=φψλκλ 2 ) (18)

κ is an arbitrary tunable gain that scales the rate of adaptation. The assumption made for Eq. (9) is made again for Eq. (18). A direct collision course is guaranteed for a stationary target and λ > 2.

( )( ) ( )(( RRZRRRZR yyyxxxs

&&& −+−⎟⎟⎠

⎞⎜⎜⎝

⎛∆∆

+=φψλκλ 2 )) (19)

It is essential for both ∆ψ and ∆Φ → 0 as s → 0 for a successful rendezvous to occur. However, a singularity exists as s → 0 because ∆Φ is in the denominator of Eq. (14). Accordingly, the adaptation is stopped when Φ is within some range of Фf, and λ becomes a constant gain. This could cause maximum position errors of a few hundred feet in the terminal region of the rendezvous. Nevertheless, current operational concepts call for the rendezvous to occur at trail position of over one nautical mile, making position errors of this magnitude insignificant. It is possible for ∆Φ → 0 when a rendezvous is not imminent. This could occur when the receiver is turning at its maximum allowable turn rate and cannot merge onto the line-of-sight directly behind the tanker. In this case, the adaptation is stopped and the receiver is commanded to continue turning at its maximum turn rate. When the receiver crosses Фf, but ∆ψ does not go to zero, the PN is re-initialized and must again acquire λ > 2 at its new position and orientation. This prevents the receiver from turning in the wrong direction after leaving the region where adaptation is halted. If re-initialization of the PN occurs too close to the rendezvous location, over-steering of the receiver may occur resulting in a bang-bang type control. This is unacceptable when operating aircraft in close proximity. As such, rendezvous tolerances and adaptation regions are defined conservatively to prevent over-steering when conservative turn rate limits are imposed.

IV. Rendezvous Point Estimator Current concepts of operation of automated aerial refueling require the receiver to maintain a trail position

approximately one nautical mile behind the tanker until it receives clearance to approach the refueling positions. However, in this paper, the actual tanker location will be used to define the target for the proportional navigation in order to simplify the algorithm. Hence, during algorithm development and demonstration in simulation, the receiver will fly to the actual tanker location, but would not do so in full mission simulations or in reality.

Rendezvous geometry for the rendezvous point estimator is shown in Fig. 2. In order to prevent prolonged tail chases by the receiver aircraft, the PN guidance system is commanded to fly the receiver to a point where the tanker is expected to be in the future using a rendezvous point estimator. This estimator predicts a rendezvous point based on

America

Figure 2. Rendezvous geometry and axis systems for rendezvous point estimator.

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current measurements each time the guidance system updates. This position estimate is referred to as the target location. It is desired for the receiver to approach the terminal region of the rendezvous directly behind the tanker as

0)( →− TR ψψ , and fly straight to the tanker. Accordingly, the target location must eventually coincide with the tanker. The projection of the target is devised using a worst-case scenario where the receiver has no knowledge of what the tanker is going to do next. The target location is predicted based on the tanker’s current state and the amount of time to project out into the future. The target location is only an estimate, but as the receiver approaches the tanker, the estimate will become more accurate and will eventually coincide with the tanker location. A new axis system is defined so that the flat-earth axes are rotated by the tanker’s heading. It is referred to as the rotated flat-earth (RFE) system. Measurements in this axis system are used to determine if the receiver is in line with the tanker’s longitudinal axis. Simple kinematics based on the current speed and turn rate of the tanker are used to calculate the target location, assuming constant speed and constant turn rate throughout the projection time. The instantaneous estimate of the time-to-rendezvous is given by:

( ) ( )

( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ −∆−∆−

∆+= 3

2

321ˆ

π

πψψ

Rg

RFE

Rg

TR

Vy

VR

t (20)

where,

( ) ( )22TRTRTR yyxxR −+−= (21)

TR ψψψ −=∆ (22)

The first term on the right hand side of Eq. (20) computes an estimate of the time-to-rendezvous based on the distance between the tanker and receiver and the difference in their headings. The second term on the right hand side of Eq. (20) is a factor that ensures, as the receiver approaches a position directly along the negative yRFE direction and the receiver heading is approaching the tanker heading, the second term will approach the first term. The t will go to zero, and the target rendezvous location will be exactly at the tanker’s location. This guarantees that the receiver will eventually be commanded to fly to the tanker. ∆y

ˆ

RFE will always be less than or equal to RTR, and will only equal RTR when the receiver is directly in front of or behind the tanker.

The cubed function in Eq. (20) is a multiplier that allows the second term on the right side to approach the first term only when the receiver and tanker headings are equal. This multiplier was chosen instead of others, such as cos(∆ψ), because of the manner in which it approaches zero, ±π/2, and ±π. Its purpose can be grasped with the following scenario. If the multiplier is one and ∆yRFE is equal to RTR, then will equal zero, and the target location will coincide with the tanker location.

t̂

If knowledge is available about what the tanker is going to do in the future, such as a scheduled turn or a complete racetrack pattern, the rendezvous time estimate could be used with that information to predict the rendezvous location with greater accuracy. The estimator could then be based on the geometry of the known tanker trajectory and the kinematics of its current states. This hybrid estimator scheme would be accurate and could account for tanker perturbations and external disturbances such as winds.

V. Velocity Controller The PN guidance algorithm steers the aircraft at a varying velocity until a rendezvous occurs subject to a

terminal heading constraint. The receiver must also arrive at the rendezvous location with the same velocity as the tanker. This is achieved by accelerating the vehicle throughout its trajectory to meet the tanker at the rendezvous location at the same time and with zero acceleration.

Tf VV = (23)

0=fA (24)

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The following formulation is a novel approach at coupling a velocity controller to the PN guidance and rendezvous point estimator to achieve the terminal velocity constraint. Reference 10 proposes a velocity controller of the form:

( )

TR

RT

RVVka

5.1

2−−= (25)

Eq. (25) is based on the kinematics equation for constant deceleration, ( )sVa c2

21−= , that results in relative

speed, Vc, and range, s, going to zero at the same time. This controller, however, only provides deceleration commands and would not be robust to tanker maneuvers such as turns or accelerations. Using Eq. (25) as a baseline, a controller is developed that is valid for any relative velocity or position.

Acceleration commands must be generated based on the distance and relative orientation to the rendezvous location for both aircraft. Both distance and orientation can be combined to produce a time-to-rendezvous estimate for both the receiver and tanker. These two terms can then be compared to get a relative sense of how far ahead or behind the tanker the receiver is from reaching the rendezvous location. Because they are being used to get only a relative comparison, the time estimates do not have to be actual estimates of when the rendezvous will occur. The form of the time-to-rendezvous estimate is similar to the first term in Eq. (20). Equation (26) is the time to rendezvous for the receiver.

( ψ∆+= 1Rg

RRZR V

Rt ) (26)

where,

RZR ψψψ −=∆ (27)

( ) ( )22RZRRZRRRZ yyxxR −+−= (28)

Similarly, the time to rendezvous for the tanker is given by:

( ψ∆+= 1Tg

TRZT V

Rt ) (29)

where,

RZT ψψψ −=∆ (30)

( ) ( )22RZTRZTTRZ yyxxR −+−= (31)

The receiver’s desired velocity at rendezvous is the tanker’s velocity, but throughout the trajectory, the receiver must be commanded to fly whatever velocity is necessary to ensure its arrival to the tanker with equal velocity and its acceleration approaching zero. Hence, the time-to-rendezvous estimates in Eq. (26) and Eq. (29) are used to scale the tanker velocity to create a target velocity. The signum function is used to determine whether the command should be an acceleration or deceleration command. K is an arbitrary tunable gain that scales the rate of acceleration. Equation (32) shows the proposed velocity controller.

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⎟⎟⎠

⎞⎜⎜⎝

⎛

++

−⎟⎟⎠

⎞⎜⎜⎝

⎛

++

−

−= TT

RR

RRZ

TT

RR

Vtt

VR

Vtt

Vka

11

sgn11

2

(32)

The scaling ratio for VT has the following properties: 1) if tR = tT = 0, then the ratio is equal to one, and the receiver is commanded to fly at the tanker’s velocity, 2) if tR = tT, then the ratio is equal to one, and the receiver is commanded to fly at the tanker’s velocity, 3) if tR > tT, then the ratio is greater than one, and the receiver is commanded to fly faster than the tanker because it is estimated to arrive at the current rendezvous location after the tanker, 4) if tR < tT, then the ratio is less than one, and the receiver is commanded to fly slower than the tanker because it is estimated to arrive at the current rendezvous location before the tanker. Acceleration commands are greater in magnitude closer to the rendezvous location because the ability to maintain a precise position becomes crucial when in close proximity to other aircraft. The target velocity and acceleration command are subject to limits set in the algorithm.

In order to show Eq. (32) results in the closing velocity and range-to-target going to zero at the same time, it must revert back to the constant deceleration kinematics equation form. First, assume the receiver and tanker are currently estimated to arrive at the rendezvous location at the same time, i.e. tR = tT. Also assume VR > VT which will make the command a deceleration command. RRRZ is defined the same as s, so Eq. (32) becomes:

( )

sVVka TR

2−−= (33)

As fΦ→φ , the receiver will be behind the tanker, and assuming the receiver is close to the tanker relative to any

existing tanker turn radius and any winds act equally on both aircraft, ( ) cTR VVV =− . Defining gain k as ½, Eq. (33) becomes:

s

Va c

2

2

−= (34)

Eq. (34) is of the constant deceleration kinematics equation form. The same result would be found for acceleration, but would be opposite in sign. The instantaneous acceleration commanded to the receiver is the acceleration necessary to cause VR to approach a target velocity as range goes to zero, and is scaled by gain k.

A check is performed in the algorithm to determine if the receiver has enough time to slow down to the tanker’s velocity depending on its acceleration limitations. These limitations can be vehicle limitations or operational limitations defined in the algorithm. If, based on its current velocity, it is too close to the rendezvous location to match the tanker’s velocity, the acceleration command is scaled to command the maximum deceleration possible in an attempt to prevent an overrun situation where the receiver overtakes the tanker. Procedures exist that deal with this situation in the event it occurs. Additionally, operational concepts call for a receiver altitude offset of at least 1000 feet from the tanker’s altitude during a rendezvous. Only after a safe rendezvous is the receiver given permission to approach the tanker and proceed to pre-contact position.

VI. Tanker Tracking Controller The PN guidance and velocity controller may not achieve their respective rendezvous constraints at the same

time. The PN guidance may reach the desired line of sight and heading before the distance to the tanker has been closed. In this case, the PN guidance is disengaged and a simple proportional controller is used to maintain the correct line of sight and heading while the receiver closes the distance to the tanker along the tanker’s track. This is important in the terminal region of the rendezvous to prevent large control commands being generated by the PN guidance because of a small range-to-target squared term in the denominator of Eq. (8).

φκψκψ ∆+∆= 21com& (35)

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Eq. (35) will command the receiver to follow the tanker in straight or turning flight, assuming small heading and line of sight errors prior to the switch from PN guidance to this controller. The two components on the right side of Eq. (35) act jointly when the angle errors have the same sign, turning the receiver back to the line of sight. Conversely, the two components act separately to turn the receiver onto the line of sight when the angle errors are opposite in sign. κ 1 and κ 2 are tunable gains.

VII. Simulation The preceding guidance

system was developed assuming point mass dynamics and perfect tracking of turn rate and acceleration commands. It is tested using medium-fidelity 6DOF dynamics in an unmanned aerial vehicle model7, along with a high-fidelity 6DOF KC-135 tanker model, to represent an actual GPS-guided rendezvous. The UAV flight control system is designed for station-keeping and precise maneuvering around the tanker by using tanker-relative PGPS data. The tanker-relative portion of the flight control system is bypassed and commands are sent directly from the PN guidance system to the inner loop. Perfect communication is assumed between the receiver and tanker when transmitting and receiving PGPS data. The dynamics models and communication are integrated and transmitted, respectively, at 80Hz. Flight control system commands are generated from the guidance commands without altering the inner loops of the existing station keeping AAR flight control system.

Figure 3 shows simulation results of a rendezvous with the tanker starting at the origin and flying north, the UAV starting approximately 6.5 nautical miles west-north-west of the tanker and flying east, and the tanker turning after 30 seconds of simulation time. The UAV starts with an airspeed of 750 ft/s, and the tanker is commanded to fly at 670 ft/s. Triangles represent the initial position and direction of both aircraft. Differences in trajectories can clearly be seen between the point-mass representation with perfect tracking and the 6DOF representation. However, the PN guidance system responds successfully to UAV dynamics, imperfect UAV flight control system tracking, and tanker dynamics introduced with the high fidelity models, and commands a successful rendezvous. The guidance system is also constrained to commanding within theoretical operational limits that conservatively limit UAV maneuvering. These limits are shown in Table 1.

-6 -4 -2 0 2 4 6 8 10 12

0

2

4

6

8

10

12

East, NM

Nor

th, N

M

o Point-Mass Tanker

[] 6DOF Tankerx Point-Mass UAV

* 6DOF UAV

Figure 3. Overhead View of Point Mass and 6DOF Trajectories.

Turn Rate

Acceleration

Airspeed

A. Simulation with Wind A rendezvous guidance system must be ab

trajectory. Free-stream winds will be used to tethe PN guidance system’s ability to withstand th

American Insti

Table 1 UAV Limits 22 ≤≤− ψ& Deg/s

22 ≤≤− tV& Ft/s2

800600 ≤≤ tV Ft/s

le to account for external disturbances introduced throughout the st the PN guidance system. A wind model is used that will exercise ese disturbances by using steady winds whose strength is a function

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of each vehicle’s East position. The tanker and receiver will be subjected to varying and unequal wind strengths throughout their respective flights, unless they are at equivalent East positions. Figure 4 shows the wind profile used; airstreams will blow in the North and South directions. Because both the tanker and receiver are turning throughout their respective trajectories, the wind will act in a changing relative direction on both aircraft, further exercising the adaptive PN.

Multiple initial positions and orientations were used to demonstrate the PN guidance’s ability to achieve a rendezvous in winds independent of the starting location and direction, and subject to the limits in Table 1. Table 2 lists the UAV initial conditions for the rendezvous cases shown in this paper. The tanker starts at the origin flying North at 670 ft/s for all cases. Rendezvous is declared in the simulation when the closure rate is less than 1 ft/s for more than 20 seconds. A significant amount of time is spent closing the distance while approaching the tanker’s speed in the simulation. This, however, may not be the case in actual autonomous rendezvous operations because permission to approach the tanker may be given before the receiver arrives at the trail location. As such, the receiver would not decelerate to the tanker’s speed, but would continue to an observation or pre-contact position at the allowed closure rate. Therefore, the simulation represents a worst-case scenario in terms of time-to-rendezvous. Multiple runs of the simulation have shown that the final trajectory flown by the receiver is very sensitive to small changes in initial position and orientation. Sensitivities are caused by the extreme coupling between the adaptive PN, the rendezvous point estimator, and the velocity controller, and by the disengagement and reengagement of the PN for certain re-initialization contingencies in the algorithm discussed in Section III. Wind also causes some sensitivity. Consequently, the method of guidance prior to PN engagement must be able to accurately align the receiver to a desired orientation for a given position relative to the tanker, such as a waypoint following mode. Figure 5 shows three rendezvous trajectories commanded by the PN at different starting positions listed in Table 2 with the wind profile shown in Fig. 4.

Figures 6 and 7 show guidance control histories for the trajectory denoted with a square in Figure 5. The control command limits can be seen in each figure. Significant commands are noticeable after 30 seconds into the simulation and approximately 295 seconds into the simulation. At 30 seconds, the tanker begins its turn, causing the estimated rendezvous location inside the PN algorithm to drastically change, and therefore, change the receiver’s current course and speed. After 295 seconds, the receiver is in the terminal region near the tanker and makes the switch from PN directional control to the tanker tracking controller. The receiver is then aligned with the tanker at its correct line-of-sight and heading angles, and only needs to close the distance to the tanker.

At the point of simulation termination for all three trajectories, the receiver is within 19 feet of the tanker, is within 0.01 degrees of the tanker’s heading, and is within 0.7 ft/s of the tanker’s airspeed. The errors would

-25 -20 -15 -10 -5 0 5 10 15 20 25-50

-40

-30

-20

-10

0

10

20

30

40

50

East, NM

Win

d S

peed

, KT

Figure 4. Wind Profile.

Table 2 Simulation Initial Conditions, UAV Trajectory North, ft East, ft Heading, deg Airspeed, ft/s ■ 15000 -80000 90 750 ♦ 100000 -60000 135 750 ● 180000 180000 -110 750

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eventually go to zero if the simulation were allowed to run indefinitely. The trajectory denoted with a square took 453 seconds for a rendezvous, the diamond took 316 seconds, and the circle took 496 seconds.

B. Optimization The benefits of using

proportional navigation versus other methods for aerial rendezvous are that it has been used extensively for intercept problems, it is relatively simple in terms of the mathematics involved in its implementation, and it is easily embedded into flight control systems running in real-time. A drawback to using PN may be that the resultant trajectory is undesirable for an aerial vehicle trying to save fuel. Fuel burn, is, of course, of utmost importance for a vehicle trying to refuel when already low on fuel. Let us assume the receiver is low on fuel prior to PN engagement, and therefore, must conserve fuel throughout the trajectory. Because the trajectory is constrained to limits set in Table 1, the characteristics of a resultant trajectory can only be changed by varying initial conditions. It is suggested that, given an initial receiver position relative to the tanker, an initial heading and velocity can be found that will result in a trajectory using the least fuel. A generic optimization can be performed on the PN guidance system by only including trajectory data and control variables that relate to fuel consumption in the cost function. A point-mass model with perfect tracking will be used so that no vehicle-specific dynamics are included in the final cost and, as a result, the optimization result is vehicle independent.

A simple optimization routine that runs the point-mass rendezvous simulation, with no winds, for a range of all possible initial headings and velocities at a given initial position is used to retrieve trajectory data and control variables for the matrix of initial conditions. Trajectory data includes inertial position, airspeed, and heading.

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-15

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East, NM

Nor

th, N

M

Tanker

UAVWind Vector Field

Figure 5. Rendezvous Trajectories with Wind.

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-2

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1

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Time, s

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eler

atio

n, ft

/s2

PN Cmd

Actual

Figure 7. Acceleration Control History for “Square” Trajectory

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Figure 6. Turn Rate Control History for “Square” Trajectory

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Control variables include commanded turn rate and acceleration. Initial airspeed for the test matrix ranges from 600 ft/s to 800 ft/s. Initial heading for the test matrix ranges from -180 to 180 degrees. The range of each independent variable is decreased throughout multiple iterations to achieve the desired level of detail for the answer. A cost is calculated for each trajectory in the test matrix using Eq. (36). The cost is equivalent to the weighted total distance traveled plus the weighted and normalized total amount of acceleration commanded. Both have a significant effect on the amount of fuel used during the execution of the trajectory.

( )∫∫ −+= dtVVkVdtkJ min21&& (36)

A maximum time limit is set for the rendezvous to occur for all the simulation runs. This is to ensure the optimum trajectory does not take an excessive amount of time to fly. Although minimum time is not the objective of the optimization, time wasted is undesirable during most refueling operations. For that reason, the trajectories where a rendezvous does not occur in the allotted time are not considered when choosing the optimum trajectory. The following optimization case is carried out with k1=0.0001 and k2=0.002. The optimum initial conditions for a trajectory subject to the cost function in Eq. (36) are: an airspeed of 789 ft/s and a heading of 110.5 deg.

The penalty associated with using proportional navigation to fly a receiver to a tanker versus a receiver flying a dynamically optimized trajectory to the tanker can be gauged by comparing the actual trajectories and their respective costs flown by 6DOF models. An algorithm based on techniques shown in Ref. 4 and previous work from Ref. 3 has been used to produce a dynamically optimized trajectory during the execution of a receiver-to-tanker rendezvous. This algorithm is embedded into a simulation using the same 6DOF UAV and tanker models as the PN simulation. The optimal PN initial conditions are used as initial conditions in the dynamic optimization simulation, and the trajectories are generated by the two simulations and compared in Fig. 8. Heading and airspeed comparisons are shown in Figures 9 and 10.

The ground-track of the PN guided trajectory and the dynamically optimized trajectory are very similar, as well as the heading time-history shown in Fig. 10. The major difference lies in the velocity control near rendezvous, seen in Fig. 9. The PN velocity controller slowly closes the distance to the tanker instead of quickly approaching the tanker and using the maximum deceleration to slow to the tanker’s speed. The PN velocity control results in a much larger time-to-rendezvous than the pre-planned trajectory; however, if time is decided to be the most important factor for a rendezvous, the properties of the controller can be changed.

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-15

-10

-5

0

5

10

East, NM

Nor

th, N

M

Tanker

PN Trajectory

Dyn. Opt. Trajectory

Figure 8. Optimized Proportional Navigation Trajectory VersusDynamically Optimized Trajectory.

The cost of the optimum trajectory using the point mass model with perfect tracking is 3.6886; using the 6DOF model with proportional navigation, it is 5.0349; and using the 6DOF model with a dynamically optimized trajectory, it is 3.4281. The maximum, and worst, cost for the point mass model was 6.2757. It is clear the PN implementation into the UAV 6DOF model causes a significant rise in cost because of imperfect tracking of the PN guidance commands. A cleaner approach to PN integration and generation of inner loop commands from the guidance commands could result in a trajectory closer to the point-mass trajectory, and as a result, its cost would approach the point-mass cost. The point-mass trajectory cost is only 7.6% more than the dynamically optimized trajectory, proving a near-optimal proportionally navigated trajectory is obtainable.

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VIII. ConA proportional navigation guidance system, augment

control loop, has been shown to execute a successful mreceiver aircraft and tanker aircraft for the purpose of shown robustness to six degree-of-freedom dynamics of of guidance commands, and varying winds. Proportional by convoluted mathematics or iterative loops used ioptimization. When combined with terminal guidance effectively flies a receiver aircraft within operational limterminal heading and speed constraints. It has also bconditions, the resultant trajectory commanded by thdynamically optimized trajectory. A possible operationalwaypoint-guidance scheme that terminates at the appresulting in the execution of a near-optimal trajectoryguidance and velocity control. This guidance method is expanded to varying altitudes and be used to execute othe

AcknowA. L. Smith thanks Dave Doman from the Air Fo

suggesting this research topic and Mark Mears and Amy dynamic optimization.

Refe

1 Nalepka, J.P. and Hinchman, J.L., “Automated Aerial RefAIAA Paper 2005-6005, Aug. 2005

2 Burns, R.S., Clark, C.S., and Ewart, R., “The AutomatedPaper 2005-6008, Aug. 2005

3 Larson, R.A., Mears, M.J., and Blue, P.A., “Path PlannEnvironments,” AIAA Paper 2005-7173, Sept. 2005

4 Bryson, A.E. Jr., Dynamic Optimization, Addison-Wesley-5 Dubins, L.E., “On Curves of Minimal Length with a C

Terminal Positions and Tangents,” American Journal of Mathem6 Kim, B.S., Lee, J.G., and Han, H.S., “Biased PNG La

Aerospace and Electronic Systems, Vo. 34, No. 1, 1998, pp. 277

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0 100 200 300 400 500640

660

680

700

720

740

760

780

800

820

Time, s

Airs

peed

, ft/s

Tanker

PN TrajectoryDyn. Opt. Trajectory

Figure 10. Optimization Airspeed Time-History

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Hea

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Figure 9. Optimization Heading Time-History

clusion ed with adaptive terminal guidance and a coupled velocity id-air constant-altitude rendezvous between an unmanned automated aerial refueling. The PN guidance system has both aircraft, unknown tanker maneuvers, saturation limits navigation is a simple guidance method that is not hindered n other methods such as route planning and dynamic and a coupled velocity controller, this guidance system its to a rendezvous with a tanker, where it is subject to

een shown that when engaged with appropriate initial e PN guidance can be near-optimal with respect to a guidance system for aerial rendezvous could begin with a ropriate initial conditions for PN guidance engagement using the proportional navigation system with terminal also not limited to two dimensional trajectories; it can be r forms of intercepts and rejoins.

ledgments rce Research Laboratory’s Air Vehicles Directorate for Linklater, both also from Air Vehicles, for their part in the

rences

ueling: Extending the Effectiveness of Unmanned Air Vehicles,”

Aerial Refueling Simulation at the AVTAS Laboratory,” AIAA

ing for Unmanned Air Vehicles to Goal States in Operational

Longman, Menlo Park, CA, 1999, pp 154-157 onstraint on Average Curvature, and with Prescribed Initial and atics, Vol. 79, No. 3, 1957, pp. 497-516

w for Impact with Angular Constraint,” IEEE Transactions on -288

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7 Lu, P., “Intercept of Nonmoving Targets at Arbitrary time-Varying Velocity,” Journal of Guidance, Control and Dynamics,

Vol. 21, No. 1, 1998, pp. 176-178 8 Lu, P., Doman, D.B., and Schierman, J.D., “Adaptive Terminal Guidance for Hypervelocity Impact in Specified Direction,”

Journal of Guidance, Control and Dynamics, Vol. 29, No. 2, 2006, pp. 269-278 9 Barfield, A.F. and Hinchman, J.L., “An Equivalent Model for Automated Aerial Refueling Research,” AIAA Paper 2005-

6006, Aug. 2005 10 Ochi, Y. and Kominami, T., “Flight Control for Automatic Aerial Refueling via PNG and LOS Angle Control,” AIAA

Paper 2005-6268, Aug. 2005

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