AEAT Longitudinal Dynamic Paper Final

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Longitudinal Flight Dynamic Analysis of an Agile U AV Abstract Purpose  The paper describes the longitudinal dynamics of a hover-capable rigid-winged UAV about various equilibrium flight conditions. The effects of the variable-incidence wing in comparison with the fixed in-incidence wing on the dynamics of UAV are also discussed. Design/methodology/approach  The aerodynamic modeling of the vehicle covers both pre-stall and post-stall regimes using a three-dimensional vortex lattice method incorporating viscous corrections. The trim states across a velocity spectrum are evaluated using a nonlinear constrained optimization scheme based on sequential quadratic programming. Then linearized dynamic analysis around trim states is carried out in order to compare the characteristics of the conventional platform with the modified platform incorporating variable-incidence wing. Findings  It is found that with the variable-incidence wing, the longitudinal equilibrium flights can be achieved with reduced thrust-to-weight ratio demands and lower elevator deflection. However, the use of the variable-incidence wing changes the dynamic characteristics of the vehicle considerably as indicated through the linear dynamic analysis. Research limitations/implications  The results presented in this paper are based on linear dynamic analysis about static trim point data. Further analysis taking into account nonlinearity, the unsteady aerodynamic effects and associated cross-coupling because of asymmetric forces may be needed to reveal the true dynamics of the vehicle under unsteady maneuvers. Practical implications  The variable-incidence wing is a useful design feature to reduce the thrust-to-weight ratio requirements and to increase elevator control authority, however its effect on the dynamics warrants further investigation. Originality/value  This is the first study highlighting the effects of variable-incidence wing on an agile hover-capable UAV. Keywords Unmanned Aerial Vehicle, longitudinal flight dynamics, variable-incidence wing, thrust-to-weight ratio, stability. Paper type Research paper.

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Longitudinal Flight Dynamic Analysis of an Agile UAV

Abstract

Purpose  – The paper describes the longitudinal dynamics of a hover-capable rigid-winged UAV about various equilibrium

flight conditions. The effects of the variable-incidence wing in comparison with the fixed in-incidence wing on the dynamics

of UAV are also discussed.

Design/methodology/approach  – The aerodynamic modeling of the vehicle covers both pre-stall and post-stall regimes

using a three-dimensional vortex lattice method incorporating viscous corrections. The trim states across a velocity

spectrum are evaluated using a nonlinear constrained optimization scheme based on sequential quadratic programming.

Then linearized dynamic analysis around trim states is carried out in order to compare the characteristics of the

conventional platform with the modified platform incorporating variable-incidence wing.

Findings  – It is found that with the variable-incidence wing, the longitudinal equilibrium flights can be achieved with

reduced thrust-to-weight ratio demands and lower elevator deflection. However, the use of the variable-incidence wing

changes the dynamic characteristics of the vehicle considerably as indicated through the linear dynamic analysis.

Research limitations/implications  – The results presented in this paper are based on linear dynamic analysis about

static trim point data. Further analysis taking into account nonlinearity, the unsteady aerodynamic effects and associated

cross-coupling because of asymmetric forces may be needed to reveal the true dynamics of the vehicle under unsteady

maneuvers.

Practical implications  – The variable-incidence wing is a useful design feature to reduce the thrust-to-weight ratio

requirements and to increase elevator control authority, however its effect on the dynamics warrants further investigation.

Originality/value  – This is the first study highlighting the effects of variable-incidence wing on an agile hover-capable

UAV.

Keywords Unmanned Aerial Vehicle, longitudinal flight dynamics, variable-incidence wing, thrust-to-weight ratio, stability.

Paper type Research paper.

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Introduction

There have been consistent efforts to enhance the flight envelope of the unmanned aerial vehicles (UAV) for versatility in

operation in confined, cluttered and urban terrains. Typical mission attributes for such vehicles include vertical take-of

and landing (VTOL), hover and cruise flight. Such efforts lead to a type of agile aircraft that can perform hover coupled

with efficient forward flight (Pines and Bohorquez, 2006). Recently (Green and Oh, 2005; Green and Oh, 2006) has

shown such versatility on fixed-wing platform by doing prop-hanging with excessively high thrust-to-weight ratios. (Green

2007) has investigated the flight either at cruise or at hover and a quick transition between these two flight modalities.

During the earlier studies by (Maqsood and Go, 2009; Maqsood and Go, 2010), such convertible platforms have been

discussed and emphasis is made on introducing variable-incidence wing to increase platform versatilities in executing

such mission profiles.

Several authors (Johnson et al., 2006; Frank et al., 2007) have worked on the techniques to broaden the fligh

envelope but mostly focus on the transition flight between hover and forward cruise. (Stone and Clarke, 2001; Stone et al.

2008) have done the experimental testing on such versatile platforms but investigations provide little insight over the flight

dynamic characteristics of such class of vehicle. (Kubo and Suzuki, 2008) has done the studies for utilization of slats and

flaps for efficient transitions. Stability of such mission segments is an important issue which has not been widely covered

in literature. During the Intelligence, Surveillance and Reconnaissance (ISR) missions in confined spaces, sometimes the

vehicles have to fly across various trimmed flight states to provide sustained intelligence information over time. This paper

emphasizes on the stability issues of such platforms across a broad velocity spectrum.

In this work a comparative analysis is presented between conventional wing configurations with the proposed variable-

incidence wing across a wide velocity spectrum. The variable-incidence wing feature has been shown to offe

performance advantages to assist in the transition maneuvers (Maqsood and Go, 2010) . In this paper, the focus is on the

type of maneuver where the aircraft flies and maintains equilibrium at specific airspeed. The overall configuration of the

vehicle for the study is depicted in Figure 2, the details of which are described in the next section. The aerodynamic forces

and moments database, which is needed for the dynamics evaluation, is developed both in pre-stall and post-stall regime

using a 3-dimensional vortex lattice code.

For the estimation of dynamic stability across the trim conditions, the conventional longitudinal dynamic equations are

modified with the angle of incidence of wing as an additional control variable. Trim analysis is done for both fixed and

variable incidence wing cases. Stability derivatives are evaluated based on the aerodynamic forces computed. The

dynamic characteristics are then analyzed based on the linearization about the trim points for both configurations and the

results are discussed in details.

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Aerodynamic Modelling

For this study, aerodynamic forces and moments are computed using a commercially available code, MultiSurface

Aerodynamics (MSA). MSA (Hanley) is based on 3 Dimensional Vortex Lattice Method (VLM3D) and can predict the

profile drag as well over the arbitrary configurations. Vortex Lattice Method (VLM) is based on panel methods techniques

derived from potential flow theory and provides inviscid aerodynamic analysis over arbitrary configurations. For the

computation of profile drag, the results from the VLM are used to compute the effective angle of attack across any span

location (for a given geometric angle of attack). Then a two dimensional vortex panel method is used on that cross-

sectional shape to compute the pressure coefficient. Subsequently, from the pressure coefficient at the cross section of

the wing for the effective angle of attack, boundary layer equations are used to compute the profile drag. The local profile

drags at various span locations are then integrated over the whole wing planform to compute the total profile drag. For the

evaluation of aerodynamic data in the post-stall regime, the numerical code works on the same boundary layer

approximation technique as transition and separation points are calculated. VLM is a well-established technique and

ample amount of literature is available on its mathematical formulation and limitations therefore readers are referred to

(Katz, 2001) for further details.

Description of the Model

The vehicle is based on a conventional aircraft model as shown in Figure 1. It has a standard wing-tail configuration with a

tractor-type propulsion system. Its airframe consists of extended polypropylene particle (EPP) foam construction with

composite landing gears. The model has a fuselage length as well as a wing span of 1 meter. The aspect ratio of the

wings is 4.31. The recommended all up weight (AUW) for enhanced performance is about 0.4 kg but the vehicle can fly

with an AUW of approximately 0.7-0.8 kg. Typical dimensional attributes of the model include span of 1 m, fuselage length

of 1 m, taper ratio of 1, propeller diameter of 0.25 m, wing and tail airfoils of NACA 0012. The centre of gravity is at 0.2 m

from the nose of the fuselage and the mean aerodynamic chord length is 0.24 m. The wing is divided into two sections

inboard and outboard sections. The inboard section (0.25 m span) is fixed with the fuselage as it will be submerged in the

slipstream of the propeller. The outboard section has an additional degree of freedom of rotation about its quarter-chord

axis (hence it is called variable-incidence wing). The outboard section of the wing will remain mostly under free-stream

flow.

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angle of attack under cruise conditions is about 22o

and the pre-stall data predicts a fairly linear lift-curve slope. The

variation of drag with the angle of attack is also plotted for several velocities in Figure 2. The conventional behavior of the

increase in drag due to the increase in velocity and angle of attack is observed. Since the aerodynamic forces of the

outboard wing and the rest of the aircraft are computed separately, the forces are added across a particular angle of

attack orientation and velocity for the variable incidence case. It is observed that the software may be inaccurate to predic

lift in post-stall regime near 90o at high velocities but these are unlikely flying conditions and the trim points of the aircraf

are not close to these conditions.

Figure 2 Lift and drag plots for α = 0o

to 90o

at several velocities

The aircraft pitching moments are plotted in Figure 3 as a function of the aircraft angle of attack ( α , from 0o

to 90o)a

several velocities. The center of gravity of the aircraft is assumed fixed at 20 cm aft of the nose. The pitching moment is

greatly governed by the center of pressure over the lifting surfaces. The center of pressure moves backward with the

increase in angle of attack resulting the trends shown in Figure 3.

Figure 3 Pitching moment plot for α = 0o

to 90o

at several velocities

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Longitudinal Dynamics

Since the aircraft has a vertically symmetric configuration and is assumed to perform only symmetric flight, it is reasonable

to assess its longitudinal dynamics separately. With these assumptions the dynamics can be represented by the nonlinea

equations of motion as shown below

qw m X u   

qu m 

Z w  (2)

yy I 

M q   

where w  u , are horizontal and vertical velocities respectively; X  and Z are the horizontal and vertical force vectors; M  is

the pitching moment; g is the acceleration due to gravity; q is the pitch rate; m is the mass of the aircraft and yy I  is the

moment of inertia in the longitudinal mode. The forces and moment involved may be represented in the following manner:

Other Grav Thrust Aero  X X X X X  (3)

Other Grav Thrust Aero  Z Z Z Z Z  (4)

Other Thrust Aero  M M M M  (5)

where Grav Thrust Aero  ,, and Other represents aerodynamic, propulsive, gravitational and miscellaneous

contributions respectively.

The longitudinal equations mentioned above are non-linear in nature. A common practice is to linearize them around a

specific trim point using small disturbance theory. In applying the small-disturbance theory, we assume that the motion of

the airplane consists of small deviations about a trimmed flight condition. All the variables in the equations of motion are

replaced by a reference value plus a perturbation or disturbance as shown below

M M M Z Z Z X X X 

q q q w w w u u u 

o o o 

o o o 

(6)

where o (.) is the trim state and (.) is the perturbation or disturbance. (Nelson, 1998) has presented these linearized

equations in state-space form by neglecting several stability derivatives like w w  Z M  , and q Z  . For the dynamics at hove

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and slow forward flight at high angles of attack, a more detailed longitudinal dynamic model is considered such that the

stability derivatives w w  Z M  , and q Z  are not neglected, as follows.

w w w 

q w u 

o o q w u 

o w u 

yy w 

i i T T e e 

i T e 

i T e 

Z M M Z M M Z M M 

Z Z Z 

X X X 

M M M 

mg mU Z Z Z 

mg X X 

I M 

Z m 

000

ˆˆˆˆˆˆˆˆˆ

ˆˆˆ

ˆˆˆ

0100

0ˆˆˆ

sinˆˆˆ

cos0ˆˆ

1000

0ˆ0

00ˆ0

000

(7)

where c  is the control parameter. Equation (7) can be written in the following state-space form

E A B x x u

(8)

uxuxx B AB AE  )(1(9)

where T q w u  )(x , T i T e  )(u and the matrices E, A, and B are obvious through comparison of Equation

(9) with Equation (7).

By defining m Z Z m X X  u u u u  / ˆ, / ˆ and yy q q  I M M  / ˆ

,the comprehensive form of matrix A can be presented as

0100

sin)(

1

sin

111

cos0

00

00

0

g U Z M Z M Z M 

U Z 

g X X 

A

q q w w u u 

w w 

w u 

(10)

wherew 

1(11)

The stability derivatives in Equation (10) can be evaluated from the numerical aerodynamic data/empirical methods by

assuming linear aerodynamic variation about the trimmed flight conditions. The technique for finding the stability

derivatives is relatively mature and available in literature (Nelson, 1998; Phillips, 2004) and will not be elaborated here.

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Trim Analysis

In this section, analysis of steady-state trimmed flight conditions at various airspeeds are described. In order to obtain the

trim flight conditions, i.e. w u  , and q equal to zero, a numerical approach is used. The problem is formulated as a nonlinear

constrained optimization problem and the MATLAB routine, fmincon , is used to find the trimmed states. Fmincon is based

on hybrid Sequential Quadratic Programming (SQP), which represents state-of-the-art in nonlinear programming

methods, and Quasi-Newton methods. The method allows to closely mimic Newton’s method for constrained optimization

  just as is done for unconstrained optimization. Fmincon  finds the constrained minimum of scalar function of severa

variables starting with an initial estimate. This is generally referred to as constrained nonlinear optimization or nonlinear

programming (Coleman and Branch, 1999). The input is the initial guess of the variable/s to be optimized. At each

iteration, the scalar objective function is evaluated subjected to the constraints posed to the dynamics of the body. The

output at each iteration is the input for the next iteration. In order to increase the convergence rate; the initial guess should

be realistic and near to the optimal output. If the initial guess is remote from optimal values, then the convergence will be

very slow. The motion is three degree of freedom and with the addition of the wing incidence as a control variable, there

are overall four parameters to be optimized.

The control parameters to be optimized for the trim states from 0 to 15 m/sec velocity range are:

T elev wing fus  W T  ]; / ;;[c

(12)

where elev is the elevator angle with respect to the fuselage.

The cost function to be minimized in the optimization is as follows:

222 M Z X J  (13)

which corresponds to the trim flight conditions, where the components of the resultant forces and moments X, Z and M as

shown in Equation (2) are in equilibrium state. The constraints posed to the state variables are shown below.

00

00

00

2020

00.1 / 0

)(

)(300

900

elev 

wing fus 

wing 

fus 

W T 

Incidence Fixed 

Incidence Variable 

(14)

From several initial guesses, single trim states across different velocities are evaluated as shown in Figures 4 and 5 for

the fixed and variable-incidence wing cases. As shown in Figure 4, the T/W gradually decreases from the perfect hover

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condition at 0 m/s to the cruise conditions at 15 m/s for both cases. The T/W  requirement for the variable-incidence

scheme is substantially lower than the fixed-incidence scheme until about 12 m/s. In this regime, the propulsive forces are

aided by the additional lift augmentation due to the variable incidence wing, which is always in pre-stall regime. The pitch

angle of the aircraft reduces from 90o

in hover to the cruise pitch angle as the velocity of the aircraft increases for both

schemes. For the variable-incidence wing case, an interesting sharp reduction in pitch is observed between 7 and 8 m/s

whereas, for the fixed-incidence case, the sharp reduction in pitch is not observed until about 10 m/s. This is due to the

shift in the flight condition from thrust borne to aerodynamic borne that takes place during these regimes. This shift occurs

at lower speeds for the variable-incidence wing case because of the improved aerodynamic efficiency due to effective

wing angle in the pre-stall regime. In Figure 5, it can be observed that the wings remain at the Cl max  state before the

transition speed of 7-8 m/sec in the variable-incidence case. Note that for the fixed-wing case, wing remains aligned with

fu s .

Figure 4 Trim thrust-to-weight ratio and pitch angle of the fuselage

Figure 5 Wing incidence and elevator deflection angles for trimmed flights

An advantage of the variable-incidence wing in the elevator control effort is also observed. The elevator deflection for the

fixed-wing case is higher than that of the variable-incidence wing one at low speeds, thereby reducing the elevator contro

authority to counter disturbances. Ideally, reduced control efforts are desirable during slow speeds to have enough

margins to counter any disturbances. From Figures 4 and 5 above, the variable-incidence wing has shown advantages in

terms of reduced T/W requirement and elevator deflection to achieve the trimmed states.

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Analysis of Dynamic Characteristics

Based on the estimation of the trim points calculated in the previous section, the stability derivatives required to compute

Equation (10) are evaluated as mentioned in previous section. The migration of eigenvalues of matrix A with the varying

trim conditions are examined to evaluate the open-loop stick-fixed stability of the aircraft in its operational velocity range.

The dynamic stability characteristics of the variable-incidence wing case are compared with the fixed-incidence wing.

In Figure 6, the eigenvalues associated with the short period mode for various trim airspeeds are plotted for the fixed and

variable-incidence wing cases. At higher speeds (beyond 8 m/s), the short-period mode of both cases is stable and its

damping increases with the increase in airspeed. This mode is also stable at the low speeds. For the variable-incidence

wing case, the short period mode becomes unstable between 7 and 8 m/s speeds. Unlike the variable-incidence wing

case, in the fixed-wing configuration, the aircraft exhibits stable short period mode over the whole airspeed range.

Figure 6 Short-period eigenvalue variation of the fixed incidence (left) and variable incidence (right) wing cases with airspeed

In comparison with the variable-incidence wing case (Figure 6), it is evident that the short-period natural frequency for the

fixed-wing case is substantially higher than for the variable-incidence wing case. This is due to the fact that the frequency

of short-period mode is influenced mainly byM  as can be seen from the approximation below (Nelson, 1998).

2 / 1M Z M  w q sp  (15)

The higher the magnitude ofM  , the higher the short-period natural frequency will be and vice-versa. TheM  comparison

between the fixed and variable-incidence wing cases for the complete speed envelope is given in Figure 7. It can be seen

that the magnitude of M  for the fixed-wing case is substantially higher in most velocity regime compared to that of the

variable-incidence wing case.

The damping of the short period mode can be approximated using (Nelson, 1998)

w w o q sp sp  Z M u M  2 (16)

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which is a function of w q  M M  , and w Z  . These parameters are plotted in Figures 7 and 8 as functions of airspeed. From

these figures, it can be deduced that the main difference in the short period damping is due to the difference in w Z 

between the two cases. It has a positive value for the variable-incidence wing case at 7-8 m/velocities airspeed range,

while it is negative for the fixed-wing case. The build-up of aerodynamic forces in this flow regime starts playing an

important role and the primary difference is that the net LC  slope is negative (post-stall regime) for the fixed-incidence

wing and is positive (pre-stall regime) for the variable-incidence one. This makes the total magnitude of the right hand side

of Equation (16) negative and thereby contributes to the instability for the variable-incidence wing case.

Figure 7 Comparison of  M α (left) and  M q (right) for the fixed and variable-incidence wing cases

Figure 8 Comparison of  w Z  (left) and w M  (right) for the fixed and variable-incidence wing cases

It should be noted that the current analysis is based on linearization about steady trim points. Hence, even though the

current analysis indicates short period instability in the 7-8 m/s speed regimes, the nature of the departure from the trim

point is not necessarily exponential. The crossing of the eigenvalues from the left-half of the complex plane to the right  –

half plane often indicates the presence of Hopf bifurcation in the associated nonlinear system, where limit cycle type of

oscillation appears instead of exponential instability. Indeed, that is the case here. Through numerical simulation of

Equation (2), it is observed that limit cycles appear in this speed regime. Figure 9 shows an example of the aircraf

response when the trim point associated with variable-incidence wing case at 8 m/s airspeed is perturbed.

Figure 9 Nonlinear response of velocity (left) and pitch rate (right) to numerical perturbation for trim point of 8 m/sec

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In Figure 10, the variation of the phugoid eigenvalues with airspeed is plotted for the fixed-wing and the variable-incidence

wing cases. For the variable-incidence wing case, the aircraft shows an unstable phugoid behavior below 13 m/s. In this

speed region, the fuselage angle of attack is higher than 10o. For the fixed-incidence case, the aircraft maneuver is

unstable in two velocity regions: 10-12 m/sec and 0-7 m/sec. Comparison of the phugoid mode between the fixed and

variable-incidence wing cases indicates that the variable-incidence wing aircraft has a reduced phugoid damping

compared to the fixed-wing. The phugoid damping is affected by the lift to drag ratio as can be seen from the phugoid

damping approximations below (Nelson, 1998).

D Lph 

 / 

1

2

1(17)

The higher the lift to drag ratio, the lower the damping will be as shown in Equation (17). For the variable incidence

scheme, L/D is substantially higher because the wings remain in the pre-stall regime across all trimmed states, as shown

in Figure 5, whereas for the for the fixed-wing case, the wing stays in post-stall regime for the most trimmed conditions

especially at low speeds. This causes the reduction in L/D for the fixed-incidence case and therefore, its phugoid damping

is relatively higher to that of the variable-incidence one. In general, phugoid instability is less of a concern compared to the

short period one due to its relatively low frequency. For this class of UAV, autonomous operations are generally designed

to fly with minimal pilot input therefore; the dynamic analysis gives us the baseline reference conditions to develop future

closed loop control.

Figure 10 Phugoid eigenvalue variation for the fixed (left) and variable-incidence (right) wing cases with airspeed

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Conclusion

The longitudinal dynamic analysis for a small agile UAV having fixed and variable-incidence wing configuration in various

trimmed states has been carried out. The results reveal the advantages of the variable-incidence wing feature to reduce

T/W  requirements and to lower the elevator control deflection for achieving trimmed flight, especially at low speeds

However, linear analysis indicates that the variable-incidence wing leads to short-period instability at certain airspeed

range. Nonlinear simulation reveals the existence of stable limit cycle in that region. This suggests that the implementation

of the variable-incidence wing may alter the control strategy needed to achieve the desired stable trimmed flights.

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