AIAA 2012-4565

AIAA Modeling and Simulation Technologies Conference 13 - 16 August 2012, Minneapolis, Minnesota

Aerobatic Aircraft Modeling Based on Aerodynamic Quaternions

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F. Fisch1, J. Lenz2, and F. Holzapfel3 Institute of Flight System Dynamics, Technische Universität München, Garching, Germany

A simulation model for aerobatic aircraft is derived that accounts for a proper inclusion of static, convective and time-dependent wind fields. Two different depths of modeling are regarded, namely a point-mass simulation model and a full, nonlinear 6-DoF simulation model. Both simulation models make use of the aerodynamic flight path angles as translational states to describe the respective aircraft trajectories since the aerodynamic quantities physically determine the motion of the aircraft. The kinematic quantities are then a function of the aerodynamic quantities and the wind influence. The 6-DoF simulation model is based on a sequential structure, where the aircraft’s attitude and rotational dynamics are given with respect to its aerodynamic trajectory. Thus, the aerodynamic angle of attack and the aerodynamic sideslip angle are used to describe the attitude of the aircraft with respect to the Aerodynamic Frame instead of Euler Angles that would describe the aircraft’s attitude with respect to the North-East-Down Frame. In order to avoid the singularity that occurs for aerodynamic flight-path inclination angles of , quaternions are utilized to replace the aerodynamic flight-path course angle, the aerodynamic flight-path inclination angle and the aerodynamic bank angle instead of replacing the Euler Angles or the kinematic flight-path angles as it is commonly the case.

Nomenclature ̅

̅

⃑ ⃑⃑⃑ ⃑⃑⃑ 1

2

3

= = = = = = = = = = = = = =

Aerodynamic Frame / Aerodynamic Motion / Aerodynamic Force ) Aerodynamic Frame rotated around its -axis by ( Body-Fixed Frame Earth-Centered Earth-Fixed Frame (ECEF) Earth-Centered Inertial Frame Kinematic Flight Path Frame / Kinematic Motion Kinematic Flight Path Frame rotated around its -axis by ( ) Navigation Frame North-East-Down Frame (NED) Center of Gravity / Gravitational Force Velocity vector Force vector Moment vector Rotation vector

Dr.-Ing. Florian Fisch, Post-Doctoral Researcher, Institute of Flight System Dynamics, Technische Universität München, Boltzmannstr. 15, D-85748 Garching, Germany. [email protected], Member AIAA. Dipl.-Ing. Jakob Lenz, Research Assistant, Institute of Flight System Dynamics, Technische Universität München, Boltzmannstr. 15, D-85748 Garching, Germany. [email protected], Senior Member AIAA. Prof. Dr.-Ing. Florian Holzapfel, Director, Institute of Flight System Dynamics, Technische Universität München, Boltzmannstr. 15, D-85748 Garching, Germany. [email protected], Senior Member AIAA. 1 American Institute of Aeronautics and Astronautics

Copyright © 2012 by Institute of Flight System Dynamics. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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M m I

= = = = = = = = = = = = = = =

Velocity Northward position Eastward position Downward position Flight-path course angle Flight-path inclination angle Flight-path bank angle Angle of attack Angle of sideslip Transformation matrix Aircraft mass Inertia tensor Roll rate Pitch rate Yaw rate

Declaration:

V

Reference Frame Reference Point Type of Motion/Source of Force Notation Frame

I. Introduction

W

IND has a fundamental influence on real aircraft trajectories and a lot of research has been carried out with respect to wind effects on aircraft flight paths and optimal flight under wind influence (e.g. Refs. 1, 2, 3, 4, 5). In some flight situations, e.g. landing or take-off at severe cross-winds, wind is very safety-critical and too strong winds can lead to extremely dangerous situations. On the other hand, situations exist where wind may be beneficial for the respective flight mission. For example, by utilizing the Jetstream on transatlantic flights from the US to Europe, the required flight time respectively fuel consumption can be reduced. Because of the huge effect that wind has with regard to flight trajectories, a proper inclusion of various types of wind within the modeling and the simulation of aircraft missions is essential in order to achieve simulated flight trajectories that are as realistic as possible. At this, not only static wind fields should be taken into account but also convective and time-dependent wind fields. If aerobatic flight paths or the trajectories of military fighter aircraft are considered, special care has to be taken in order to avoid singularities within the aircraft modeling and simulation. If the attitude of an aircraft is described by Euler Angles, i.e. heading angle , inclination angle and bank angle , a singularity occurs within the simulation model for inclination angles of . To avoid these singularities, usually the set of four Quaternions and is used to replace the Euler angles (see e.g. Refs. 6, 7). Alternatively, within point-mass simulation models, the Quaternions can also be utilized to replace the kinematic flight-path course angle , the kinematic flight-path inclination angle and the kinematic flight-path bank angle in order to avoid the singularity that now occurs at flight-path inclination angles of (Refs. 8, 9). In the latter case, if wind shall be taken account, the aerodynamic flight path angles have to be restored from the aerodynamic velocity vector where again a singularity occurs for aerodynamic flight-path inclination angles of . In the paper at hand, the task of including wind effects into aircraft simulation models that allow for a singularity-free representation of aerobatic or fighter aircraft is treated. At this, two different depths of modeling are regarded that are a point-mass simulation model and a full, non-linear 6-DoF simulation model. The 6-DoF simulation model is based on a sequential structure: it is an extension of the point-mass simulation model by the attitude and rotational dynamics aligned in series to the point-mass simulation model. This means that the attitude dynamics are not given with respect to the North-East-Down Frame what would imply making use of the wellknown set of Euler Angles. Moreover, the attitude dynamics are derived with respect to the Aerodynamic Frame . At this, the aerodynamic angle of attack and the aerodynamic sideslip angle are utilized to describe the attitude of the aircraft with respect to the aerodynamic trajectory. The difference between both modeling approaches, i.e. the parallel approach utilizing Euler Angles and the sequential approach utilizing the aerodynamic angle of attack and the aerodynamic sideslip angle, is depicted in Fig. 1. Quaternions are then utilized to replace the aerodynamic course angle , the aerodynamic inclination angle and the aerodynamic bank angle instead of the Euler Angles or the kinematic flight-path angles in order to avoid the respective singularities. Because of the fact 2 American Institute of Aeronautics and Astronautics

that the Quaternions replace the aerodynamic flight-path angles, the Quaternions are termed aerodynamic Quaternions. To the authors’ knowledge, to date the Quaternions have not been utilized to replace the aerodynamic flight-path angles. The usage of aerodynamic quantities to describe the motion of an aircraft corresponds directly to the physical cause of aircraft motion: the forces and moments acting on the aircraft are caused by aerodynamic quantities while the respective kinematic quantities are a function of the aerodynamic quantities and the wind influence. If no wind influence is present at all, the kinematic quantities become identical to the aerodynamic quantities.

Parallel Approach Rotation Dynamics

Attitude Propagation

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w.r.t. NED-Frame

Translation Dynamics

Position Propagation w.r.t. NED-Frame

Sequential Approach Rotation Dynamics

Attitude Propagation

Translation Dynamics

w.r.t. ̅ -Frame

Position Propagation w.r.t. NED-Frame

Figure 1. Difference between parallel and sequential approach The subsequent part of the paper is structured as follows: first, the wind modeling taking into account static, convective and time-dependent wind fields is depicted in paragraph II. In paragraph III, the equations for modeling the subsystems of both the point-mass simulation model and the full, non-linear 6-DoF simulation model are presented and Quaternions are applied to replace the aerodynamic flight-path angles. Finally, some conclusions are drawn in paragraph IV.

II. Wind Model Basically, the influence of wind for the simulation of aircraft flight is taken into account by the so-called wind triangle (Refs. 10, 11). At this, the true aerodynamic velocity vector ⃑ at the aircraft’s center of gravity is the difference between the kinematic velocity vector ⃑ and the wind velocity vector ⃑ :

V V V G

G

E

A O

E

G

K O

E

(1)

W O

For all the velocity vectors, the NED-Frame is used as notation frame. Differentiating the above relationship with respect to time, the following relationship for the first order time derivative of the aerodynamic velocity vector with respect to the NED-Frame results:

V V V EO G A O

EO G K O

EO G W O

(2)

Assuming flight over round, rotating Earth, the first order time derivative of the wind velocity vector with the wind measured in the Navigation Frame N then equals:

V

EO G W O

G M ON V W

EE N

ω EO

G M V ON W O

E N

(3)

where ⃑⃑⃑ denotes the transport rate necessary for keeping the North-East alignment of the moved North-EastDown Frame . is the transformation matrix between the Navigation Frame N and the NED-Frame . Now, the first order time derivative of the wind velocity vector with respect to the ECEF-Frame E and its components given in the Navigation Frame has to be evaluated: 3 American Institute of Aeronautics and Astronautics

V

G M NE V W

EE G W N

G V W

EE E

E d G VW dt

E G VW t

E E

E E

EE

(4)

E

rG

E

VWG

T

E

d G r E dt

E

(5)

E

Here, denotes the total derivative of the wind velocity vector with respect to time t while partial derivative. Inserting Eq. (5) into Eq. (4) and letting E

V G

d G r dt it holds that

V

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respectively

EE N

E G VW t

E

E VWG

E N

V

M NE E VWG

E

(6)

G V W

E

K E

E

E M NE VWG t

EE G W N

denotes the

E N

M

G

E

EN

E

(7)

K E

E

VKG

E

(8)

N

Here, denotes the gradient matrix of the wind velocity vector with respect to the position vector ECEF-Frame , i.e.

G r

VWG E

E E

E

V G

T

E

W E

E

E uWG E z E E vWG E z E E wWG E z E

E

uWG E y E

(9)

E

E

vWG E y E

(10)

E

wWG E y E

of the wind velocity vector with respect to the position vector

VWG E

E VWG

uG E W E x E vG E W E x E wG E W E x E

The gradient matrix Frame evaluates to:

E N

given in the

E E VWG N VWG N rG N N VWG T T T rG E rG N rG E

E N

given in the Navigation

M

(11)

NE

Utilizing Eq. (11), Eq. (8) for the first order time derivative of the wind velocity vector becomes:

V

EE G W N

Finally,

G V W

EO

O

E VWG t

E G M ON VW t

E N

E N

V

N VWG

G

E

E

(12)

K N

N

V ω

M ON N VWG

E

N

G

E

OE

K N

G M V ON W O

E N

(13)

Given the case that the -axis of the fixed Navigation Frame is pointing northward (i.e. no rotation between the NED-Frame and the Navigation Frame so that the respective transformation matrix becomes the identity matrix), Eq. (13) can be rewritten as:

4 American Institute of Aeronautics and Astronautics

G V W

EO

O

E E E E G E VW O ω OE O VWG O O VWG O VKG O t O E E G E VW O O VWG O VKG O t

Otherwise, if flight over flat, non-rotating Earth is considered (i.e. ⃑⃑⃑

V

EO G W O

E M ON VWG t

E N

(14)

⃑ ), Eq. (13) reduces to:

V

M ON N VWG

E

G

N

E

(15)

K N

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III. Simulation Model In the following paragraphs, the simulation model will be outlined in detail. First, the position propagation equations and the translation equations of motion will be presented that are part of both the point-mass simulation model and the full, non-linear 6-DoF simulation model. Then, the attitude propagation equations and the rotational equations of motion are given that extend the point-mass simulation model to a full 6-DoF simulation model. The principle relationship between the point-mass simulation model and the 6-DoF simulation model is illustrated in Fig. 2, where it can be seen that the attitude and rotation dynamics extend the point-mass model in series to the translation and position dynamics. The interface to the point-mass simulation model is made up of the aerodynamic angle of attack , the aerodynamic sideslip angle and the first order time derivative of the aerodynamic bank angle ̇ respectively the -component of the rotational rate ⃑⃑⃑ between the Aerodynamic Frame and the NEDFrame if Quaternions are utilized. In the case that the simulation is accomplished using solely the point-mass simulation model, those quantities become the controls of the point-mass simulation model. Furthermore, the rotation and attitude dynamics may be substituted by alternative modeling approaches that preserve the dynamic order while being less complex, like e.g. linear transfer functions or linearized state-space models that represent the rotation and attitude dynamics. In order to allow for a seamless substitution of the rotation and attitude dynamics, the alternative modeling approaches have to feature the same interface variables as the point-mass simulation model so that there is no necessity to modify the point-mass simulation model respectively its interface. First, the equations of motion are presented utilizing the aerodynamic flight-path angles. Then, the substitution of the aerodynamic flight-path angles by the Quaternions as well as the additional required calculations will be introduced.

Rotation Dynamics

Attitude Propagation

w.r.t. ̅ -Frame

,

̇ /

Translation Dynamics

Position Propagation w.r.t. NED-Frame

Point-Mass Simulation Model 6-DoF Simulation Model

Figure 2. Relationship between Point-Mass Simulation Model and 6-DoF Simulation Model A. Position Propagation Equations The position propagation equations are formulated in the NED-Frame , where the relationship between the kinematic velocity vector ⃑ , the aerodynamic velocity vector ⃑ and the wind velocity vector ⃑ results directly from Eq. (1):

5 American Institute of Aeronautics and Astronautics

E

x G y VK z O

V V G

E

G

E

A O

O

E

W O

VAG cos AG cos VAG sin AG cos V G sin G A A

G A G A

E

VWG O

E

(16)

O

Thus, the states of the position propagation equations are given by the northward position , the eastward position and the downward position . The position propagation equations are a function of the aerodynamic velocity , the aerodynamic course angle and the aerodynamic flight-path inclination angle that are in turn states of the translation equations of motion given in the next subsection.

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B. Translation Equations of Motion Denoted in the Intermediate Aerodynamic Frame ̅, the translation equations of motion are a function of the total sum of forces acting at the aircraft’s center of gravity (denoted in the Intermediated Aerodynamic Frame) minus the kinematic acceleration due to the transport rate ⃑⃑⃑ , the Coriolis acceleration due to Earth rotation ⃑⃑⃑ and the centrifugal acceleration due to Earth rotation ⃑⃑⃑ (Ref. 10):

V

EO G K A

u KG vKG w G K

EO

1 m

F ω V G

EO K A

A

A

G

E

K

A

(17)

V ω ω r

2 ω IE K

G E K A

A

IE K A

The Intermediate Aerodynamic Frame ̅ results from the Aerodynamic Frame ⃑ and ⃑⃑⃑ ) around its -axis. For a flat, non-rotating Earth (i.e. ⃑⃑⃑ angle (

IE K A

G

A

rotated by the aerodynamic bank ⃑ ) the equation reduces to

EO

V

EO G K A

u KG 1 vKG m w G K A

F G

(18)

A

With the relationship between the first order time derivatives of the kinematic velocity vector ⃑ , the aerodynamic velocity vector ⃑ and the wind velocity vector ⃑ (Eq. (2)), one gets for the translation equations of motion:

V V V EO G A A

EO G K A

EO G W A

1 m

G

F

A

G V W

EO

(19)

A

So far, no aerodynamic flight-path angles are involved in the translation equations of motion. Utilizing the respective kinematic relationship resulting from the Euler differentiation rule, the first order time derivative of the aerodynamic velocity vector ⃑ can be rewritten as:

V V ω V EO G A A

V

EO G A A

EA G A A

EA VAG AG sin AG 0 0 G A A cos

G A

G A

G E A A

OA K A

E

VAG V G G GA 0 A VA cos 0 G G A A VA A

(20) EO

G A A

(21)

Solving for the first order time derivatives of the aerodynamic velocity , the aerodynamic course angle and the aerodynamic flight-path inclination angle , the translation equations of motion utilizing aerodynamic flight-path angles read:

6 American Institute of Aeronautics and Astronautics

VAG G A G A

EO

A

0 1 1 0 G V cos A 0 0

1 0 M AA m 1 G V A 0

G A

The total sum of external forces comprises the aerodynamic forces force :

F G

A

G M AO V W

, the propulsive force

EO

O

G A

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A

G P

A

G

A

G

(22)

and the gravitational

F F F F G

(23)

A

While the computation of the respective forces is not subject of this paper and thus not treated in detail, it is just mentioned that the aerodynamic forces are functions of the aerodynamic angular rates roll rate , pitch rate and yaw rate . The aerodynamic angular rates are the elements of the aerodynamic rotation vector ⃑⃑⃑ between the aircraft and the surrounding air denoted in the Body-Fixed Frame . Analogous to the aerodynamic velocity vector ⃑ , the aerodynamic rotation vector ⃑⃑⃑ is the difference between the kinematic rotation vector ⃑⃑⃑ and the wind rotation vector ⃑⃑⃑ of the surrounding air (Ref. 11):

ω ω ω AB A B

At this, the rotation vector ⃑⃑⃑

ω

OA W O

OB K B

OA W B

(24)

of the surrounding air relative to the NED-Frame O is given by:

1 rot O VWG 2

E

O

1 O G VW 2

based on the non-diagonal elements of the gradient matrix ⃑

E

O

wG E v G E W O W O y z O O E G G E 1 uW O wW O 2 z O x O vG E u G E W O W O x O y O

of the wind velocity ⃑

(25)

(Ref. 11).

C. Attitude Propagation Equations As mentioned above, the attitude of the aircraft shall not be described by the set of Euler Angles as usual, but by the aerodynamic angle of attack , the aerodynamic sideslip angle and the aerodynamic bank angle . Those angles represent the states of the attitude propagation equations of motion that can be derived utilizing the so-called strap-down equation (Refs. 6, 10) between the Intermediate Aerodynamic Frame ̅ and the Body-Fixed Frame :

Ω

AB K AA

G A G A G A

A

M M AB BA

(26)

1 KA,By cos AG KA,Bz sin AG G cos A KA,By sin AG KA,Bz cos AG AB G AB G AB G K , x tan A K , y cos A K , z sin A

A

(27)

For the computation of the attitude propagation equations, the rotational rate ⃑⃑⃑ ̅ between the Intermediate Aerodynamic Frame ̅ and the Body-Fixed Frame B is required. The rotational rate evaluates to:

7 American Institute of Aeronautics and Astronautics

ω

AB K A

ω ω OB K B

ω

M AB ωOB K

IB K B

OA

M BO ω IE

O

(28)

A

B

M BO ω EO

(29)

O

where the rotational rate ⃑⃑⃑ ̅ between the NED-Frame and the Intermediate Aerodynamic Frame ̅ has to be restored utilizing the aerodynamic flight-path angles and and their first order time derivatives. The rotational rate ⃑⃑⃑ is an outcome of the rotation equations of motions that are presented in the next subsection. D. Rotation Equations of Motion

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To describe the rotational motion of the aircraft, standard rotation equations of motion are utilized. At this, the states of the rotation equations of motion are the elements of the rotational rate ⃑⃑⃑ between the Body-Fixed Frame and the Earth-Centered Inertial Frame . The first order time derivatives of the rotational states are a function of the total sum of external moments acting at the aircraft’s center of gravity:

ω I M ω I ω IB B K B

1

G

G

BB

IB K B

B

G

BB

IB K B

(30)

where represents the inertia tensor of the regarded flight system. Here again it has to be mentioned that for the computation of the aerodynamic moments the aerodynamic rotational rates as computed by Eq. (24) have to be used. E. Translation Equations of Motion utilizing Aerodynamic Quaternions As can be seen from Eq. (22), singularities occur within the translation equations of motion at aerodynamic flight-path inclination angles of due to the division by the cosine of aerodynamic flight-path inclination angle. To avoid these singularities, the quaternions shall be implemented to substitute the aerodynamic course angle , the aerodynamic flight-path inclination angle and the aerodynamic bank angle . The corresponding equations of motion utilizing Quaternions then read (Ref. 7):

q 0 q1 q1 1 q0 q 2 q 3 2 q 3 q2

q2

q3

q3

q2

q0 q1

q1 q0

q0 q1 q2 q3

OA K ,y A OA K ,z A 2k OA K ,x A

(31)

Since the quaternions introduce a fourth state, the system is now over-determined and thus the square sum constraint ( ) is enforced. This is accomplished by accounting for the square sum error computed by equation (32) in the quaternion update equation (31) in the fourth column multiplied by a small factor .

1 q02 q12 q22 q32

(32)

The first order time derivatives of the Quaternions are a function of the rotational rate ⃑⃑⃑ between the Aerodynamic Frame and the NED-Frame that is in turn a function of the aerodynamic flight-path angles and their first order time derivatives. Thus, the Quaternions are termed aerodynamic Quaternions. For the computation of the rotational rate ⃑⃑⃑ , the following relationship for the first order time derivative of the aerodynamic velocity vector ⃑ ̇ is utilized (see Eq. (19)):

V V EO G A A

EA G A A

V

ωOA K

A

G

E

A

A

1 m

F G

G M V AO W A

8 American Institute of Aeronautics and Astronautics

EO

O

(33)

V

EO G A A

EA E VAG KOA, x VAG OA 1 0 K , y 0 m 0 OA A K ,z A 0 A

F F F

G x G y G z

EO

uWG G vW w G A W A

(34)

The relationship results from the translation equations of motion (Eq. (18)) notated in the Aerodynamic Frame and the Euler differentiation rule. Solving for the -component and the -component of the rotational rate ⃑⃑⃑ gives

OA K,y A

EO 1 1 FzG A w WG A G VA m EO 1 1 G G F y vWG A A VA m

OA K ,z A

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together with the differential equations for the aerodynamic velocity

1 VAG m The -component of the rotational rate ⃑⃑⃑ following relationship: OB K ,x A

(36)

:

F u

G EO W A

G

x A

(37)

cannot be restored from Eq. (34) and thus has to be computed by the

OA K ,x A

(35)

AB K ,x A

OB K ,x A

tan AG KOA, y

A

OB K,y A

(38)

where the -component of the rotational rate ⃑⃑⃑ has to be extracted from slightly modified attitude propagation equations given in the next subsection. If only a point-mass simulation model was to be considered, the -component of the rotational rate ⃑⃑⃑ would be a directly commanded input, the commanded aerodynamic roll rate :

OA K ,x A

p A,CMD

(39)

F. Attitude Propagation Equations adapted for Aerodynamic Quaternions For the use with aerodynamic Quaternions, the attitude propagation equations have to be slightly modified in comparison to the equations that would be utilized with the aerodynamic flight-path angles. The rotational rate between the Aerodynamic Frame and the Body-Fixed Frame is given by: E

ω

AB K A

AG sin AG AG cos AG G A A

(40)

Solving for the first order time derivatives of the aerodynamic angle of attack one gets as differential equations for the attitude states:

AG

1 KAB, y G cos A

A

and the aerodynamic sideslip angle

1 KOA, y G cos A

A

OB K ,y A

AG KAB,z A KOA,z A KOB, z A

(41)

(42)

At this, only the angle of attack and the sideslip angle are regarded as attitude states, while the aerodynamic bank angle is involved with the Quaternions and thus regarded as a translational state. Finally, the -component of the rotational rate ⃑⃑⃑ required in Eq. (38) evaluates to:

9 American Institute of Aeronautics and Astronautics

AB K ,x A

AG sin AG tan AG KOA, y

A

OB K,y A

(43)

G. Position Propagation Equations adapted for Aerodynamic Quaternions Finally, the position equations of motion (see Eq. (16)) also have to be reformulated so that the usage of flightpath angles is avoided at the computation of the first order time derivatives of the position coordinates:

VAG x y M OA 0 0 z O E

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where the transformation matrix between the Aerodynamic Frame quaternions is given by (Ref. 7):

M OA

(44) A

and the NED-Frame

utilizing aerodynamic

q02 q12 q22 q32 2 q1 q2 q0 q3 2 q1 q3 q0 q2 2 q1 q2 q0 q3 q02 q12 q22 q32 2 q2 q3 q0 q1 2 q1 q3 q0 q2 2 q2 q3 q0 q1 q02 q12 q22 q32

(45)

IV. Conclusion and Outlook For the simulation of aircraft flight, a wind model has been set up that takes into account static, convective and time-variable wind fields. The wind model is included into a simulation model where the aerodynamic flight-path course angle , the aerodynamic flight-path inclination angle and the aerodynamic flight-path bank angle are replaced by Quaternions. The Quaternions are therefore named aerodynamic Quaternions. Furthermore, at no instance within the simulation model there is the necessity to restore any of the aerodynamic flight-path variables if Quaternions are used. By the simulation model with aerodynamic Quaternions, the influence of wind fields can directly be taken into account for the simulation of aerobatic or fighter aircraft trajectories where singularities would occur at flight-path inclination angles of if the flight-path angles were used instead of the Quaternions. It is mentioned that if there is no wind, the aerodynamic Quaternions become the kinematic Quaternions utilized to replace the kinematic flight-path angles just as the aerodynamic flight-path angles would become the kinematic flight-path angles. Instead of the Quaternions, the so-called Modified Rodriguez Parameters (Ref. 12) might be utilized to replace the aerodynamic flight-path angles and to avoid the singularities that occur at the specific values for the flight-path inclination angles. Accordingly, the Modified Rodriguez Parameters might then be called aerodynamic Modified Rodriguez Parameters. The simulation model presented here shall be utilized for the investigation of the wind influence on various aerobatic flight maneuvers, like e.g. the Half-Cuban Eight.

References 1

Franco, A., and Rivas, D., “Minimum-Cost Cruise at Constant Altitude of Commercial Aircraft Including Wind Effects”, Journal of Guidance, Control, and Dynamics, Vol. 34, No. 4, Aug. 2011, pp. 1253-1260. doi: 10.2514/1.53255 2 Guo, W., Zhao, Y. J., and Capozzi, B., “Optimal Unmanned Aerial Vehicle Flights Seeability and Endurance in Winds”, Journal of Aircraft, Vol. 48, No. 1, Feb. 2011, pp. 305-314. doi: 10.2514/1.C031114 3 Jardin, M. R., and Bryson, A. E., “Neighboring Optimal Aircraft Guidance in Winds”, Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, 2001, pp. 710-715. 4 Bulirsch, R., Montrone, F., and Pesch, H. J., “Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem - Part 1: Necessary Conditions”, Journal of Optimization Theory and Applications, Vol. 70, No. 1, 1991. 5 Bulirsch, R., Montrone, F., and Pesch, H. J., “Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem - Part 2: Multiple Shooting and Homotopy”, Journal of Optimization Theory and Applications, Vol. 70, No. 2, 1991. 6 Stevens, B. L., and Lewis, F. L., Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.

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7

Phillips, W. F., Mechanics of Flight, John Wiley & Sons, New York, 2004. Plas, H. v. d., and Visser, H. G., “Trajectory Optimization of an Aerobatic Air Race”, 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2009, AIAA 2009-53. 9 Fisch, F., Weingartner, M., Pfifer, H., Holzapfel, F., Sachs, G., and Myschik, S., “Airframe and Trajectory Pursuit Modeling for Simulation Assisted Air Race Planning”, AIAA Modeling and Simulation Technologies Conference and Exhibit, Honolulu, Hawaii, Aug. 2008, AIAA Paper 2008-6532. 10 Holzapfel, F., Flugsystemdynamik I & II, Lecture Notes, Lehrstuhl für Flugsystemdynamik, Technische Universität München, München, 2011. 11 Brockhaus, R., Flugregelung, 2nd ed., Springer-Verlag, Berlin, 2001. 12 Schaub, H., and Junkins, J. L., “Stereographic Orientation Parameters for Attitude Dynamics: A Generalization of the Rodrigues Parameters”, Journal of the Astronautical Sciences, Vol. 44, No. 1, Jan. 1996, pp. 1-19.

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8

11 American Institute of Aeronautics and Astronautics

AIAA Modeling and Simulation Technologies Conference 13 - 16 August 2012, Minneapolis, Minnesota

Aerobatic Aircraft Modeling Based on Aerodynamic Quaternions

Downloaded by Matthias Bittner on May 2, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2012-4565

F. Fisch1, J. Lenz2, and F. Holzapfel3 Institute of Flight System Dynamics, Technische Universität München, Garching, Germany

A simulation model for aerobatic aircraft is derived that accounts for a proper inclusion of static, convective and time-dependent wind fields. Two different depths of modeling are regarded, namely a point-mass simulation model and a full, nonlinear 6-DoF simulation model. Both simulation models make use of the aerodynamic flight path angles as translational states to describe the respective aircraft trajectories since the aerodynamic quantities physically determine the motion of the aircraft. The kinematic quantities are then a function of the aerodynamic quantities and the wind influence. The 6-DoF simulation model is based on a sequential structure, where the aircraft’s attitude and rotational dynamics are given with respect to its aerodynamic trajectory. Thus, the aerodynamic angle of attack and the aerodynamic sideslip angle are used to describe the attitude of the aircraft with respect to the Aerodynamic Frame instead of Euler Angles that would describe the aircraft’s attitude with respect to the North-East-Down Frame. In order to avoid the singularity that occurs for aerodynamic flight-path inclination angles of , quaternions are utilized to replace the aerodynamic flight-path course angle, the aerodynamic flight-path inclination angle and the aerodynamic bank angle instead of replacing the Euler Angles or the kinematic flight-path angles as it is commonly the case.

Nomenclature ̅

̅

⃑ ⃑⃑⃑ ⃑⃑⃑ 1

2

3

= = = = = = = = = = = = = =

Aerodynamic Frame / Aerodynamic Motion / Aerodynamic Force ) Aerodynamic Frame rotated around its -axis by ( Body-Fixed Frame Earth-Centered Earth-Fixed Frame (ECEF) Earth-Centered Inertial Frame Kinematic Flight Path Frame / Kinematic Motion Kinematic Flight Path Frame rotated around its -axis by ( ) Navigation Frame North-East-Down Frame (NED) Center of Gravity / Gravitational Force Velocity vector Force vector Moment vector Rotation vector

Dr.-Ing. Florian Fisch, Post-Doctoral Researcher, Institute of Flight System Dynamics, Technische Universität München, Boltzmannstr. 15, D-85748 Garching, Germany. [email protected], Member AIAA. Dipl.-Ing. Jakob Lenz, Research Assistant, Institute of Flight System Dynamics, Technische Universität München, Boltzmannstr. 15, D-85748 Garching, Germany. [email protected], Senior Member AIAA. Prof. Dr.-Ing. Florian Holzapfel, Director, Institute of Flight System Dynamics, Technische Universität München, Boltzmannstr. 15, D-85748 Garching, Germany. [email protected], Senior Member AIAA. 1 American Institute of Aeronautics and Astronautics

Copyright © 2012 by Institute of Flight System Dynamics. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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M m I

= = = = = = = = = = = = = = =

Velocity Northward position Eastward position Downward position Flight-path course angle Flight-path inclination angle Flight-path bank angle Angle of attack Angle of sideslip Transformation matrix Aircraft mass Inertia tensor Roll rate Pitch rate Yaw rate

Declaration:

V

Reference Frame Reference Point Type of Motion/Source of Force Notation Frame

I. Introduction

W

IND has a fundamental influence on real aircraft trajectories and a lot of research has been carried out with respect to wind effects on aircraft flight paths and optimal flight under wind influence (e.g. Refs. 1, 2, 3, 4, 5). In some flight situations, e.g. landing or take-off at severe cross-winds, wind is very safety-critical and too strong winds can lead to extremely dangerous situations. On the other hand, situations exist where wind may be beneficial for the respective flight mission. For example, by utilizing the Jetstream on transatlantic flights from the US to Europe, the required flight time respectively fuel consumption can be reduced. Because of the huge effect that wind has with regard to flight trajectories, a proper inclusion of various types of wind within the modeling and the simulation of aircraft missions is essential in order to achieve simulated flight trajectories that are as realistic as possible. At this, not only static wind fields should be taken into account but also convective and time-dependent wind fields. If aerobatic flight paths or the trajectories of military fighter aircraft are considered, special care has to be taken in order to avoid singularities within the aircraft modeling and simulation. If the attitude of an aircraft is described by Euler Angles, i.e. heading angle , inclination angle and bank angle , a singularity occurs within the simulation model for inclination angles of . To avoid these singularities, usually the set of four Quaternions and is used to replace the Euler angles (see e.g. Refs. 6, 7). Alternatively, within point-mass simulation models, the Quaternions can also be utilized to replace the kinematic flight-path course angle , the kinematic flight-path inclination angle and the kinematic flight-path bank angle in order to avoid the singularity that now occurs at flight-path inclination angles of (Refs. 8, 9). In the latter case, if wind shall be taken account, the aerodynamic flight path angles have to be restored from the aerodynamic velocity vector where again a singularity occurs for aerodynamic flight-path inclination angles of . In the paper at hand, the task of including wind effects into aircraft simulation models that allow for a singularity-free representation of aerobatic or fighter aircraft is treated. At this, two different depths of modeling are regarded that are a point-mass simulation model and a full, non-linear 6-DoF simulation model. The 6-DoF simulation model is based on a sequential structure: it is an extension of the point-mass simulation model by the attitude and rotational dynamics aligned in series to the point-mass simulation model. This means that the attitude dynamics are not given with respect to the North-East-Down Frame what would imply making use of the wellknown set of Euler Angles. Moreover, the attitude dynamics are derived with respect to the Aerodynamic Frame . At this, the aerodynamic angle of attack and the aerodynamic sideslip angle are utilized to describe the attitude of the aircraft with respect to the aerodynamic trajectory. The difference between both modeling approaches, i.e. the parallel approach utilizing Euler Angles and the sequential approach utilizing the aerodynamic angle of attack and the aerodynamic sideslip angle, is depicted in Fig. 1. Quaternions are then utilized to replace the aerodynamic course angle , the aerodynamic inclination angle and the aerodynamic bank angle instead of the Euler Angles or the kinematic flight-path angles in order to avoid the respective singularities. Because of the fact 2 American Institute of Aeronautics and Astronautics

that the Quaternions replace the aerodynamic flight-path angles, the Quaternions are termed aerodynamic Quaternions. To the authors’ knowledge, to date the Quaternions have not been utilized to replace the aerodynamic flight-path angles. The usage of aerodynamic quantities to describe the motion of an aircraft corresponds directly to the physical cause of aircraft motion: the forces and moments acting on the aircraft are caused by aerodynamic quantities while the respective kinematic quantities are a function of the aerodynamic quantities and the wind influence. If no wind influence is present at all, the kinematic quantities become identical to the aerodynamic quantities.

Parallel Approach Rotation Dynamics

Attitude Propagation

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w.r.t. NED-Frame

Translation Dynamics

Position Propagation w.r.t. NED-Frame

Sequential Approach Rotation Dynamics

Attitude Propagation

Translation Dynamics

w.r.t. ̅ -Frame

Position Propagation w.r.t. NED-Frame

Figure 1. Difference between parallel and sequential approach The subsequent part of the paper is structured as follows: first, the wind modeling taking into account static, convective and time-dependent wind fields is depicted in paragraph II. In paragraph III, the equations for modeling the subsystems of both the point-mass simulation model and the full, non-linear 6-DoF simulation model are presented and Quaternions are applied to replace the aerodynamic flight-path angles. Finally, some conclusions are drawn in paragraph IV.

II. Wind Model Basically, the influence of wind for the simulation of aircraft flight is taken into account by the so-called wind triangle (Refs. 10, 11). At this, the true aerodynamic velocity vector ⃑ at the aircraft’s center of gravity is the difference between the kinematic velocity vector ⃑ and the wind velocity vector ⃑ :

V V V G

G

E

A O

E

G

K O

E

(1)

W O

For all the velocity vectors, the NED-Frame is used as notation frame. Differentiating the above relationship with respect to time, the following relationship for the first order time derivative of the aerodynamic velocity vector with respect to the NED-Frame results:

V V V EO G A O

EO G K O

EO G W O

(2)

Assuming flight over round, rotating Earth, the first order time derivative of the wind velocity vector with the wind measured in the Navigation Frame N then equals:

V

EO G W O

G M ON V W

EE N

ω EO

G M V ON W O

E N

(3)

where ⃑⃑⃑ denotes the transport rate necessary for keeping the North-East alignment of the moved North-EastDown Frame . is the transformation matrix between the Navigation Frame N and the NED-Frame . Now, the first order time derivative of the wind velocity vector with respect to the ECEF-Frame E and its components given in the Navigation Frame has to be evaluated: 3 American Institute of Aeronautics and Astronautics

V

G M NE V W

EE G W N

G V W

EE E

E d G VW dt

E G VW t

E E

E E

EE

(4)

E

rG

E

VWG

T

E

d G r E dt

E

(5)

E

Here, denotes the total derivative of the wind velocity vector with respect to time t while partial derivative. Inserting Eq. (5) into Eq. (4) and letting E

V G

d G r dt it holds that

V

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respectively

EE N

E G VW t

E

E VWG

E N

V

M NE E VWG

E

(6)

G V W

E

K E

E

E M NE VWG t

EE G W N

denotes the

E N

M

G

E

EN

E

(7)

K E

E

VKG

E

(8)

N

Here, denotes the gradient matrix of the wind velocity vector with respect to the position vector ECEF-Frame , i.e.

G r

VWG E

E E

E

V G

T

E

W E

E

E uWG E z E E vWG E z E E wWG E z E

E

uWG E y E

(9)

E

E

vWG E y E

(10)

E

wWG E y E

of the wind velocity vector with respect to the position vector

VWG E

E VWG

uG E W E x E vG E W E x E wG E W E x E

The gradient matrix Frame evaluates to:

E N

given in the

E E VWG N VWG N rG N N VWG T T T rG E rG N rG E

E N

given in the Navigation

M

(11)

NE

Utilizing Eq. (11), Eq. (8) for the first order time derivative of the wind velocity vector becomes:

V

EE G W N

Finally,

G V W

EO

O

E VWG t

E G M ON VW t

E N

E N

V

N VWG

G

E

E

(12)

K N

N

V ω

M ON N VWG

E

N

G

E

OE

K N

G M V ON W O

E N

(13)

Given the case that the -axis of the fixed Navigation Frame is pointing northward (i.e. no rotation between the NED-Frame and the Navigation Frame so that the respective transformation matrix becomes the identity matrix), Eq. (13) can be rewritten as:

4 American Institute of Aeronautics and Astronautics

G V W

EO

O

E E E E G E VW O ω OE O VWG O O VWG O VKG O t O E E G E VW O O VWG O VKG O t

Otherwise, if flight over flat, non-rotating Earth is considered (i.e. ⃑⃑⃑

V

EO G W O

E M ON VWG t

E N

(14)

⃑ ), Eq. (13) reduces to:

V

M ON N VWG

E

G

N

E

(15)

K N

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III. Simulation Model In the following paragraphs, the simulation model will be outlined in detail. First, the position propagation equations and the translation equations of motion will be presented that are part of both the point-mass simulation model and the full, non-linear 6-DoF simulation model. Then, the attitude propagation equations and the rotational equations of motion are given that extend the point-mass simulation model to a full 6-DoF simulation model. The principle relationship between the point-mass simulation model and the 6-DoF simulation model is illustrated in Fig. 2, where it can be seen that the attitude and rotation dynamics extend the point-mass model in series to the translation and position dynamics. The interface to the point-mass simulation model is made up of the aerodynamic angle of attack , the aerodynamic sideslip angle and the first order time derivative of the aerodynamic bank angle ̇ respectively the -component of the rotational rate ⃑⃑⃑ between the Aerodynamic Frame and the NEDFrame if Quaternions are utilized. In the case that the simulation is accomplished using solely the point-mass simulation model, those quantities become the controls of the point-mass simulation model. Furthermore, the rotation and attitude dynamics may be substituted by alternative modeling approaches that preserve the dynamic order while being less complex, like e.g. linear transfer functions or linearized state-space models that represent the rotation and attitude dynamics. In order to allow for a seamless substitution of the rotation and attitude dynamics, the alternative modeling approaches have to feature the same interface variables as the point-mass simulation model so that there is no necessity to modify the point-mass simulation model respectively its interface. First, the equations of motion are presented utilizing the aerodynamic flight-path angles. Then, the substitution of the aerodynamic flight-path angles by the Quaternions as well as the additional required calculations will be introduced.

Rotation Dynamics

Attitude Propagation

w.r.t. ̅ -Frame

,

̇ /

Translation Dynamics

Position Propagation w.r.t. NED-Frame

Point-Mass Simulation Model 6-DoF Simulation Model

Figure 2. Relationship between Point-Mass Simulation Model and 6-DoF Simulation Model A. Position Propagation Equations The position propagation equations are formulated in the NED-Frame , where the relationship between the kinematic velocity vector ⃑ , the aerodynamic velocity vector ⃑ and the wind velocity vector ⃑ results directly from Eq. (1):

5 American Institute of Aeronautics and Astronautics

E

x G y VK z O

V V G

E

G

E

A O

O

E

W O

VAG cos AG cos VAG sin AG cos V G sin G A A

G A G A

E

VWG O

E

(16)

O

Thus, the states of the position propagation equations are given by the northward position , the eastward position and the downward position . The position propagation equations are a function of the aerodynamic velocity , the aerodynamic course angle and the aerodynamic flight-path inclination angle that are in turn states of the translation equations of motion given in the next subsection.

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B. Translation Equations of Motion Denoted in the Intermediate Aerodynamic Frame ̅, the translation equations of motion are a function of the total sum of forces acting at the aircraft’s center of gravity (denoted in the Intermediated Aerodynamic Frame) minus the kinematic acceleration due to the transport rate ⃑⃑⃑ , the Coriolis acceleration due to Earth rotation ⃑⃑⃑ and the centrifugal acceleration due to Earth rotation ⃑⃑⃑ (Ref. 10):

V

EO G K A

u KG vKG w G K

EO

1 m

F ω V G

EO K A

A

A

G

E

K

A

(17)

V ω ω r

2 ω IE K

G E K A

A

IE K A

The Intermediate Aerodynamic Frame ̅ results from the Aerodynamic Frame ⃑ and ⃑⃑⃑ ) around its -axis. For a flat, non-rotating Earth (i.e. ⃑⃑⃑ angle (

IE K A

G

A

rotated by the aerodynamic bank ⃑ ) the equation reduces to

EO

V

EO G K A

u KG 1 vKG m w G K A

F G

(18)

A

With the relationship between the first order time derivatives of the kinematic velocity vector ⃑ , the aerodynamic velocity vector ⃑ and the wind velocity vector ⃑ (Eq. (2)), one gets for the translation equations of motion:

V V V EO G A A

EO G K A

EO G W A

1 m

G

F

A

G V W

EO

(19)

A

So far, no aerodynamic flight-path angles are involved in the translation equations of motion. Utilizing the respective kinematic relationship resulting from the Euler differentiation rule, the first order time derivative of the aerodynamic velocity vector ⃑ can be rewritten as:

V V ω V EO G A A

V

EO G A A

EA G A A

EA VAG AG sin AG 0 0 G A A cos

G A

G A

G E A A

OA K A

E

VAG V G G GA 0 A VA cos 0 G G A A VA A

(20) EO

G A A

(21)

Solving for the first order time derivatives of the aerodynamic velocity , the aerodynamic course angle and the aerodynamic flight-path inclination angle , the translation equations of motion utilizing aerodynamic flight-path angles read:

6 American Institute of Aeronautics and Astronautics

VAG G A G A

EO

A

0 1 1 0 G V cos A 0 0

1 0 M AA m 1 G V A 0

G A

The total sum of external forces comprises the aerodynamic forces force :

F G

A

G M AO V W

, the propulsive force

EO

O

G A

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A

G P

A

G

A

G

(22)

and the gravitational

F F F F G

(23)

A

While the computation of the respective forces is not subject of this paper and thus not treated in detail, it is just mentioned that the aerodynamic forces are functions of the aerodynamic angular rates roll rate , pitch rate and yaw rate . The aerodynamic angular rates are the elements of the aerodynamic rotation vector ⃑⃑⃑ between the aircraft and the surrounding air denoted in the Body-Fixed Frame . Analogous to the aerodynamic velocity vector ⃑ , the aerodynamic rotation vector ⃑⃑⃑ is the difference between the kinematic rotation vector ⃑⃑⃑ and the wind rotation vector ⃑⃑⃑ of the surrounding air (Ref. 11):

ω ω ω AB A B

At this, the rotation vector ⃑⃑⃑

ω

OA W O

OB K B

OA W B

(24)

of the surrounding air relative to the NED-Frame O is given by:

1 rot O VWG 2

E

O

1 O G VW 2

based on the non-diagonal elements of the gradient matrix ⃑

E

O

wG E v G E W O W O y z O O E G G E 1 uW O wW O 2 z O x O vG E u G E W O W O x O y O

of the wind velocity ⃑

(25)

(Ref. 11).

C. Attitude Propagation Equations As mentioned above, the attitude of the aircraft shall not be described by the set of Euler Angles as usual, but by the aerodynamic angle of attack , the aerodynamic sideslip angle and the aerodynamic bank angle . Those angles represent the states of the attitude propagation equations of motion that can be derived utilizing the so-called strap-down equation (Refs. 6, 10) between the Intermediate Aerodynamic Frame ̅ and the Body-Fixed Frame :

Ω

AB K AA

G A G A G A

A

M M AB BA

(26)

1 KA,By cos AG KA,Bz sin AG G cos A KA,By sin AG KA,Bz cos AG AB G AB G AB G K , x tan A K , y cos A K , z sin A

A

(27)

For the computation of the attitude propagation equations, the rotational rate ⃑⃑⃑ ̅ between the Intermediate Aerodynamic Frame ̅ and the Body-Fixed Frame B is required. The rotational rate evaluates to:

7 American Institute of Aeronautics and Astronautics

ω

AB K A

ω ω OB K B

ω

M AB ωOB K

IB K B

OA

M BO ω IE

O

(28)

A

B

M BO ω EO

(29)

O

where the rotational rate ⃑⃑⃑ ̅ between the NED-Frame and the Intermediate Aerodynamic Frame ̅ has to be restored utilizing the aerodynamic flight-path angles and and their first order time derivatives. The rotational rate ⃑⃑⃑ is an outcome of the rotation equations of motions that are presented in the next subsection. D. Rotation Equations of Motion

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To describe the rotational motion of the aircraft, standard rotation equations of motion are utilized. At this, the states of the rotation equations of motion are the elements of the rotational rate ⃑⃑⃑ between the Body-Fixed Frame and the Earth-Centered Inertial Frame . The first order time derivatives of the rotational states are a function of the total sum of external moments acting at the aircraft’s center of gravity:

ω I M ω I ω IB B K B

1

G

G

BB

IB K B

B

G

BB

IB K B

(30)

where represents the inertia tensor of the regarded flight system. Here again it has to be mentioned that for the computation of the aerodynamic moments the aerodynamic rotational rates as computed by Eq. (24) have to be used. E. Translation Equations of Motion utilizing Aerodynamic Quaternions As can be seen from Eq. (22), singularities occur within the translation equations of motion at aerodynamic flight-path inclination angles of due to the division by the cosine of aerodynamic flight-path inclination angle. To avoid these singularities, the quaternions shall be implemented to substitute the aerodynamic course angle , the aerodynamic flight-path inclination angle and the aerodynamic bank angle . The corresponding equations of motion utilizing Quaternions then read (Ref. 7):

q 0 q1 q1 1 q0 q 2 q 3 2 q 3 q2

q2

q3

q3

q2

q0 q1

q1 q0

q0 q1 q2 q3

OA K ,y A OA K ,z A 2k OA K ,x A

(31)

Since the quaternions introduce a fourth state, the system is now over-determined and thus the square sum constraint ( ) is enforced. This is accomplished by accounting for the square sum error computed by equation (32) in the quaternion update equation (31) in the fourth column multiplied by a small factor .

1 q02 q12 q22 q32

(32)

The first order time derivatives of the Quaternions are a function of the rotational rate ⃑⃑⃑ between the Aerodynamic Frame and the NED-Frame that is in turn a function of the aerodynamic flight-path angles and their first order time derivatives. Thus, the Quaternions are termed aerodynamic Quaternions. For the computation of the rotational rate ⃑⃑⃑ , the following relationship for the first order time derivative of the aerodynamic velocity vector ⃑ ̇ is utilized (see Eq. (19)):

V V EO G A A

EA G A A

V

ωOA K

A

G

E

A

A

1 m

F G

G M V AO W A

8 American Institute of Aeronautics and Astronautics

EO

O

(33)

V

EO G A A

EA E VAG KOA, x VAG OA 1 0 K , y 0 m 0 OA A K ,z A 0 A

F F F

G x G y G z

EO

uWG G vW w G A W A

(34)

The relationship results from the translation equations of motion (Eq. (18)) notated in the Aerodynamic Frame and the Euler differentiation rule. Solving for the -component and the -component of the rotational rate ⃑⃑⃑ gives

OA K,y A

EO 1 1 FzG A w WG A G VA m EO 1 1 G G F y vWG A A VA m

OA K ,z A

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together with the differential equations for the aerodynamic velocity

1 VAG m The -component of the rotational rate ⃑⃑⃑ following relationship: OB K ,x A

(36)

:

F u

G EO W A

G

x A

(37)

cannot be restored from Eq. (34) and thus has to be computed by the

OA K ,x A

(35)

AB K ,x A

OB K ,x A

tan AG KOA, y

A

OB K,y A

(38)

where the -component of the rotational rate ⃑⃑⃑ has to be extracted from slightly modified attitude propagation equations given in the next subsection. If only a point-mass simulation model was to be considered, the -component of the rotational rate ⃑⃑⃑ would be a directly commanded input, the commanded aerodynamic roll rate :

OA K ,x A

p A,CMD

(39)

F. Attitude Propagation Equations adapted for Aerodynamic Quaternions For the use with aerodynamic Quaternions, the attitude propagation equations have to be slightly modified in comparison to the equations that would be utilized with the aerodynamic flight-path angles. The rotational rate between the Aerodynamic Frame and the Body-Fixed Frame is given by: E

ω

AB K A

AG sin AG AG cos AG G A A

(40)

Solving for the first order time derivatives of the aerodynamic angle of attack one gets as differential equations for the attitude states:

AG

1 KAB, y G cos A

A

and the aerodynamic sideslip angle

1 KOA, y G cos A

A

OB K ,y A

AG KAB,z A KOA,z A KOB, z A

(41)

(42)

At this, only the angle of attack and the sideslip angle are regarded as attitude states, while the aerodynamic bank angle is involved with the Quaternions and thus regarded as a translational state. Finally, the -component of the rotational rate ⃑⃑⃑ required in Eq. (38) evaluates to:

9 American Institute of Aeronautics and Astronautics

AB K ,x A

AG sin AG tan AG KOA, y

A

OB K,y A

(43)

G. Position Propagation Equations adapted for Aerodynamic Quaternions Finally, the position equations of motion (see Eq. (16)) also have to be reformulated so that the usage of flightpath angles is avoided at the computation of the first order time derivatives of the position coordinates:

VAG x y M OA 0 0 z O E

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where the transformation matrix between the Aerodynamic Frame quaternions is given by (Ref. 7):

M OA

(44) A

and the NED-Frame

utilizing aerodynamic

q02 q12 q22 q32 2 q1 q2 q0 q3 2 q1 q3 q0 q2 2 q1 q2 q0 q3 q02 q12 q22 q32 2 q2 q3 q0 q1 2 q1 q3 q0 q2 2 q2 q3 q0 q1 q02 q12 q22 q32

(45)

IV. Conclusion and Outlook For the simulation of aircraft flight, a wind model has been set up that takes into account static, convective and time-variable wind fields. The wind model is included into a simulation model where the aerodynamic flight-path course angle , the aerodynamic flight-path inclination angle and the aerodynamic flight-path bank angle are replaced by Quaternions. The Quaternions are therefore named aerodynamic Quaternions. Furthermore, at no instance within the simulation model there is the necessity to restore any of the aerodynamic flight-path variables if Quaternions are used. By the simulation model with aerodynamic Quaternions, the influence of wind fields can directly be taken into account for the simulation of aerobatic or fighter aircraft trajectories where singularities would occur at flight-path inclination angles of if the flight-path angles were used instead of the Quaternions. It is mentioned that if there is no wind, the aerodynamic Quaternions become the kinematic Quaternions utilized to replace the kinematic flight-path angles just as the aerodynamic flight-path angles would become the kinematic flight-path angles. Instead of the Quaternions, the so-called Modified Rodriguez Parameters (Ref. 12) might be utilized to replace the aerodynamic flight-path angles and to avoid the singularities that occur at the specific values for the flight-path inclination angles. Accordingly, the Modified Rodriguez Parameters might then be called aerodynamic Modified Rodriguez Parameters. The simulation model presented here shall be utilized for the investigation of the wind influence on various aerobatic flight maneuvers, like e.g. the Half-Cuban Eight.

References 1

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Phillips, W. F., Mechanics of Flight, John Wiley & Sons, New York, 2004. Plas, H. v. d., and Visser, H. G., “Trajectory Optimization of an Aerobatic Air Race”, 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2009, AIAA 2009-53. 9 Fisch, F., Weingartner, M., Pfifer, H., Holzapfel, F., Sachs, G., and Myschik, S., “Airframe and Trajectory Pursuit Modeling for Simulation Assisted Air Race Planning”, AIAA Modeling and Simulation Technologies Conference and Exhibit, Honolulu, Hawaii, Aug. 2008, AIAA Paper 2008-6532. 10 Holzapfel, F., Flugsystemdynamik I & II, Lecture Notes, Lehrstuhl für Flugsystemdynamik, Technische Universität München, München, 2011. 11 Brockhaus, R., Flugregelung, 2nd ed., Springer-Verlag, Berlin, 2001. 12 Schaub, H., and Junkins, J. L., “Stereographic Orientation Parameters for Attitude Dynamics: A Generalization of the Rodrigues Parameters”, Journal of the Astronautical Sciences, Vol. 44, No. 1, Jan. 1996, pp. 1-19.

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