Derivation of the Dynamics Equations of Receiver Aircraft in Aerial Refueling

TLDR: In this article, the authors describe the derivation of a new set of nonlinear, 6DOF equations of motion of a receiver aircraft undergoing an aerial refueling, including the efiect of timevarying mass and inertia properties associated with the fuel transfer and the tanker's vortex induced wind effect.

Abstract: This paper describes the derivation of a new set of nonlinear, 6{DOF equations of motion of a receiver aircraft undergoing an aerial refueling, including the efiect of timevarying mass and inertia properties associated with the fuel transfer and the tanker’s vortex induced wind efiect. Since the Center of Mass (CM) of the receiver is time{varying during the fuel transfer, the equations are written in a reference frame whose origin is at the CM of the receiver before fuel transfer begins and stays flxed at that position even though the CM is moving during the refueling. Due to the fact that aerial refueling simulation and control deal with the position and orientation of the receiver relative to the tanker, the equations of motion are derived in terms of the translational and rotational position and velocity with respect to the tanker. Further, the derivation of the equations takes into account the momentum transfer into the receiver due to the fuel transfer. The receiver aircraft before fuel transfer is treated as a rigid body made up of ‘n’ particles. The dynamic efiects due to fuel transfer are modeled by considering the mass change to be conflned to a flnite number of lumped masses, which would normally represent the fuel tanks on the receiver aircraft. Once the refueling begins, by using the design parameters such as the shape, size and location of the individual fuel tanks and the rate of fuel ∞owing into each of them, the mass and location of the individual lumped masses are calculated and fed into the equations of motion as exogenous inputs. The new receiver equations of motion are implemented in an integrated simulation environment with a feedback controller for receiver station-keeping as well as the full set of nonlinear, 6{DOF equations of motion of the tanker aircraft and a feedback controller to ∞y the tanker on a U-turn maneuver.

Figures

Figure 8.15. Time history of receiver control surface deflections during turn.

Figure 8.16. Time history of receiver throttle setting during turn.

Figure 8.2. Initial condition for lumped mass position vector.

Figure 2.1. Intermediate reference frames: Tanker and Receiver’s body frame.

Figure 8.3. Two-view diagram of fuel tank 1.

Figure 8.6. Receiver deviation in relative orientation in straight level flight.

Full text

DERIVATI ON OF THE DYNAMICS EQUATIONS
FOR RECEIVER AIRCRAFT
IN AERIAL REFUELING
by
JAYME WAISHEK
Presented to the Faculty of the Graduate School of
The University of Texas at Arlington in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE IN AEROSPACE ENGINEERING
THE UNIVERSITY O F TEXAS AT ARLINGTON
December 2007

Copyright
c
 by Jayme Waishek 2007
All Rights Reserved

For my Fida

ACKNOWLEDGEMENTS
I would like to thank the University of Texas at Arlington as well as its Mechan-
ical and Aerospace Engineering Department, Honors College, Alumni Association, and
many other affiliates, both personal and corporate, which have financially supported
me throughout my college career. I would also like to thank the Texas Space Grant
Consortium and Bell Helicopter Textron, Inc., for their particular financial support and
recognition throughout my graduate studies.
I am highly grateful for my sup ervising professor, Dr. Atilla Dogan, and the relent-
less guidance he provided me throughout this and other projects, which enabled me to
publish a technical paper and submit a journal article during my MS studies. I greatly
appreciated the research assistantship he was able to offer during the first half of our
work together. I am further appreciative of the time spent and interest shown by Dr.
Donald Wilson and Dr. Kent Lawrence as they reviewed this thesis and participated in
my defense committee.
Finally, I would like to acknowledge all of the UTA faculty and teachers from years
before, as well as family and friends, who encouraged me, through praise and criticism, to
always learn more than the minimum required. Their advice will stay with me through
the rest of my endeavors.
November 21, 2007
iv

ABSTRACT
DERIVATION OF THE DYNAMICS EQUATIONS
FOR RECEIVER AIRCRAFT
IN AERIAL REFUELING
Publication No.
Jayme Waishek, M.S.
The University of Texas at Arlington, 2007
Supervising Professor: Atilla Dogan
This thesis describes the derivation of a new set of nonlinear, 6–DOF equations of
motion of a receiver aircraft undergoing an aerial refueling, including the effect o f time-
varying mass and inertia properties associated with the fuel transfer and the ta nker’s
vortex induced wind effect. Since the center of mass of the receiver is time–varying during
the fuel transfer, the equations ar e written in a reference frame that is geometrically
fixed in the aircraft. Due to the fact that a erial refueling simulation and control deal
with the position and orientation of the receiver relative to the tanker, the equations of
motion are derived in terms of the translational and rotational position and velocity with
respect to the tanker. Further, for the derivation, Newton’s law is applied to the system,
which consists of the receiver aircraft and the fuel before and after being transferred into
the receiver. The new receiver equations of motion are implemented in an integrated
simulation environment with a feedback controller for receiver station-keeping as well as
v

Frequently asked questions

Q1. What are the contributions in "Derivation of the dynamics equations for receiver aircraft in aerial refueling by jayme waishek presented to the faculty of the graduate school of the university of texas at arlington in partial fulfillment of the requirements for the degree of master of science in aerospace engineering the university of texas at arlington" ?

Waishek et al. this paper derived a set of nonlinear, 6DOF equations of motion of a receiver aircraft undergoing an aerial refueling, including the effect of time-varying mass and inertia properties associated with the fuel transfer and the tanker 's vortex induced wind effect. 

Q2. What are the future works mentioned in the paper "Derivation of the dynamics equations for receiver aircraft in aerial refueling by jayme waishek presented to the faculty of the graduate school of the university of texas at arlington in partial fulfillment of the requirements for the degree of master of science in aerospace engineering the university of texas at arlington" ?

In future work, the equations will be utilized in more effective control system development as well as analysis of other systems having time-varying mass properties. 

Q3. What are the techniques used to model the dynamic effect of nonuniform wind?

To model the dynamic effect of nonuniform wind, various techniques such as strip theory, averaging, and look-up tables are used [54, 55, 56].3 

Q4. What is the angular velocity of the receiver relative to the inertial frame?

The angular velocity of the receiver relative to the inertial frame can be vectoriallyexpanded asωBR = ωBRBT + ωBT (5.7)where ωBT is the angular velocity vector of the tanker relative to the inertial frame. 

Q5. What is the resultant force acting on the receiver in Eq. (3.16)?

Ẇ (3.16)The resultant force acting on the receiver in Eq. (3.16), F , is considered to be the sum of the gravity force vector MR, the aerodynamic force vector AR and the propulsive force vector PR. 

Q6. What is the advantage of writing the equations with respect to a geometrically fixed frame?

Another advantage of writing the equations with respect to a geometrically fixed frame manifests itself when the aerodynamic varibles such as airspeed angle of attack, side slip angle, and stability derivatives are used. 

Q7. What is the advantage of the matrix form?

Another advantage of the matrix form is that the couplings between rotational and translational dynamics can be easily identified through the appearance of the rotational variables in the translational equations and vice versa. 

Q8. Why is the rotation matrix not used in aerial refueling?

The reason for not using the rotation matrix from the inertial frame directly8 to BR-frame is because, in aerial refueling, the focus is on the motion of the receiver relative to the tanker. 

Q9. What is the yaw rate for the tanker in a racetrack maneuver?

The commanded yaw rate for the tanker in a racetrack maneuver is generated from a 1.7 deg/sec step response of fourth order linear filter with time constants of 10, 10, 10 and 1 seconds. 

Q10. What is the significance of the mass transfer effect on vehicle dynamics?

Variable mass systems are studied extensively in the area of space flight [29, 30, 31, 32, 33, 34, 35, 36, 37], which show the significance of the mass transfer effect on vehicle dynamics. 

Q11. What is the property of vector product introduced in Eq. (2.15)?

using the property of vector product introduced in Eq. (2.15),ξ̇ = [B̂T ] T ξ̇ − [B̂T ] TS(ωBT )ξ (5.2)where ωBT is the representation of ωBT in BT -frame. 

Q12. what is the angular momentum of the system around the origin of the inertial frame?

At time t−∆t, the angular momentum of the system around the origin of the inertial frame isH1 =n ∑i=1ri × Miṙi +k ∑j=1rmj × ṙmj + rR × ∆mV 0 (4.1)where each term represents the moment of the corresponding linear momentum term in Eq. (3.4) about the origin of the inertial frame. 

Q13. What is the derivative of with respect to BT -frame?

BT is the derivative of ξ with respect to BT -frame and ωBT is the angular velocity vector of BT -frame with respect to the inertial frame. 

Q14. what is the vectorial relation of the origins of the reference frames?

The vectorial relation of the origins of the reference frames ( Fig. 2.1) yieldsrBR = rBT + ξ (3.1)Further, the effect of the wind is incorporated into the kinematics asṙBR = U + W (3.2)where U is the velocity vector of the receiver relative to the surrounding air and W is the wind velocity vector, i.e. the velocity of the air relative to the inertial frame. 

Q15. what is the first term of the expression in eq. (5.47)?

W + Ẇ }(5.48)33Making use of Eqs. (5.47) and (5.48), the first term of the expression in Eq. (5.45) may be written ask ∑j=1ρ mj × mj(U̇ + 

Q16. What is the yaw angle of the refueling?

Figures 8.11 through 8.16 depict the results of the simulation when executed for aU-turn maneuver scenario which involves the tanker beginning to turn 25 seconds into the simulation with a specified yaw rate until the yaw angle change reaches 180 degrees. 

Q17. how to use the equations in control systems?

In future work, the equations will be utilized in more effective control system development as well as analysis of other systems having time-varying mass properties. 

Q18. What is the rotation matrix from the inertial frame to the receiver’s body frame?

Thus,[Î] = RT BTI [B̂T ] (2.6)When the rotation matrix from the inertial frame to the receiver’s body frame is needed, as Fig. 2.2 implies, in this thesis, it is written through BT -frame, i.e.RBRI = RBRBTRBTI (2.7)which means a transformation from the inertial frame to BT -frame and then from there to BR-frame.