Considerations in the Lateral Stability … in the Lateral Stability Characteristics of Airship...

Considerations in the Lateral Stability Characteristics of Airship Dynamics

Brian P. Danowsky

Senior Research Engineer

[email protected]

Thomas T. Myers

Vice President and Technical Director

[email protected]

Systems Technology, Inc.

Hawthorne, CA

SAE Aerospace Control and Guidance Systems Committee Meeting #102Grand Island, NY, Oct. 15-17, 2008

17 Oct. 2008 SAE ACGSC Meeting #102 2Property of Systems Technology, Inc.

Introduction• Interest has increased recently in the use of lighter-than-air craft for high

altitude, long endurance surveillance and communication missions in both the military and private sectors.

• Airship dynamics are distinct from those of heavier-than-air craft due to the fact that the mass of the airship is on the same order of magnitude as the displaced air.

• Certain unsteady aerodynamic effects, referred to as added mass and inertia, usually ignored in heavier-than-air craft, become significant and produce noticeable differences in dynamic behavior.

• Fundamental parameters that characterize the vehicles dynamic stability are directly related to these unsteady effects and differ from traditional parameters for heavier-than-air craft.

image courtesy of:http://www.aeroventure.org.uk/mainnews2007.php

Skyship 500


Purpose• Due to increased interest in airships, there is a need for further

study and determination of fundamental stability parameters.

• Fundamental parameters that characterize static stability are derived. These are for a general class of airship vehicles.

• An analytical literal approximate factor approach is utilized todetermine a yaw stability criterion.

• An alternative approach, which decouples the roll and yaw dynamics in the linear lateral equations of motion produces an identical yaw stability criterion.

• These intrinsic stability parameters for airship vehicles are shown to be fundamentally different than those obtained for traditional heavier-than-air vehicles.


Outline• Brief review of high altitude platforms (HAPs) and stratospheric

airships (SA).• Brief Overview of the Station Keeping Problem• Considerations in the equations of motion for an airship body.

0 Kirchhoff’s Equations

0 Linearized Longitudinal Dynamics

0 Linearized Lateral dynamics

• Lateral Stability Criterion for a general airship body.0 Derived using an Approximate Factor approach

0 Comparison to criterion derived by a separate approach

• Conclusions


High Altitude Platforms• There is currently considerable interest in HAP, operating in the

lower stratosphere, for both military and commercial applications.0 Intelligence, Surveillance and Reconnaissance (ISR) applications.

0 Communications with minimal terrestrial infrastructure (i.e. floating cell phone tower).

0 International Telecommunication Union

(ITU) has recently licensed several frequency

bands for HAP-based communications.

0 Operated above 65,000 feet, HAP are far

above commercial air traffic.

• HAPs promise many of the capabilities of space satellites, but at much lower cost and much easier recovery for maintenance.

NAL JAXA

Image courtesy of: http://blimp-n2a.com/projets_haa.htm


HAP Examples

Courtesy of: http://www.sti.nasa.gov/tto/spinoff2002/oa.html

Galileo

Courtesy of: http://blimp-n2a.com/projets_haa.htmBoeing: HALE

Scaled Composites: Proteus

Berkurt

Courtesy of: http://www.boeing.com

Courtesy of: http://blimp-n2a.com/projets_haa.htm


High Altitude Stratospheric Airship

• Very large super-pressure airship (operational) • Solar powered propeller thrusters • Maintain station at 70 Kft over fixed point 0Broad area coverage 0Minimum wind

• ”Atmospheric geosynchronous satellite”0Missile defense0Telecommunication relay

• Better cost and serviceability than space satellites


Stratospheric Airship

• High Altitude Station Keeping: Divided into 2 problems:0Outer Loop Station Keeping

Control: Emphasize point mass flight mechanics.

0 Inner Loop Vehicle Dynamics:

– Dynamic Stability and Control.– Inner Loop Control: i.e.,

Stability Augmentation.– Rigid Body Vehicle Dynamics,

Emphasized in the talk here.NOAA GAINS Balloon


Overview: Evolved Conception of Station-keeping Problem

• Partition station-keeping problem: 0 “Sub thrust limit” (STL) regime: winds below thruster control power

limit

0 “Post thrust limit” (PTL) regime: winds above thruster control power limit

– MDA extreme wind event (90 kt peak over 7 days)• PTL control problem distinct from STL:0 Focus shifts from vehicle characteristics to wind field

0 Emphasize flight mechanics over vehicle dynamics

0 PTL control strategy intrinsically nonlinear

0 Use wind gradient as a function of altitude for maneuverability.

• Initially used simplest adequate vehicle and environment models


Rigid Body Equations of Motion for an Airship

• Airships are designed to neutralize gravity with buoyancy forces. These vehicles are the closest approximation to "antigravity machines" available.

• Due to the fact that the mass of the aircraft is comparable to the displaced air mass, there are distinct differences in the conventional formulation of the equations of motion.

• EOM for airships are inherently similar to EOM for submarines.• Notably, the process of formulating the aircraft aerodynamic

forces and moments is basically uncoupled from formulating the rigid body equations (Kirchhoff).

• Namely, certain aerodynamic forces and moments, specifically “added mass” and “added inertia,” which are negligible for heavier-than-air craft, become important.


Rigid Body Equations of Motion for an Airship

• These “added mass” and “added inertia” terms take the form of aerodynamic stability derivatives taken with respect to accelerations (e.g. ).

• These forces and moments have been traditionally estimated from ideal fluid aerodynamics under the assumption of acyclic (zero circulation) flow.

• It can be shown that potential flow can be used to derive analytical solutions to these forces for a prolate spheroid. Prolatespheroids approximate traditional airship hull shapes well.

• Problem with potential flow approximations:0 Viscous effects not accounted for, no vorticity.

0 Difficult to apply Kutta-Joukowski condition to a bluff body.

0 D’Alembert’s paradox – zero forces in the steady state

wM


Kirchhoff’s Non-linear Equations of Motion

• The fluid and the solid are considered as a single dynamic system.

• Kinetic energy of the system is defined consisting of the fluid (Tf) and the solid (Ts).

• The derivative of the scalar kinetic energy with respect to the translational and angular velocity vectors gives the translational and rotational impulse vectors respectively. Impulse “wrench” is the combination of a given translational and rotational impulse

• The impulse wrench can be differentiated with respect to time and set equal to the corresponding force and moment wrench accordingto Newton's second law.

• EOM derived about the hull center of volume rather than c.g.0 Complicates rigid body dynamics but simplifies unsteady potential flow

aerodynamics


Kirchhoff’s Non-linear Equations of Motion

• Similar to Lagrange’s Equations.

• s - solid, f - fluid

• F, L – external forces and moments (gravity, buoyancy, viscous aerodynamics).

• RHS derivatives give rise to “virtual mass” and “virtual inertia.”

• Combined with actual mass and inertia produce “apparent mass”and “apparent inertia.”

B B f fs s

o o o o

T TT Td ddt dt

∂ ∂⎛ ⎞⎛ ⎞∂ ∂+ = − −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ω F ω

V V V V× ×

B B f f fs s so o

o o

T T TT T Td ddt dt

∂ ∂ ∂⎛ ⎞∂ ∂ ∂⎛ ⎞ + + = − − −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ω V L ω V

ω ω V ω ω V× × × ×

Translational:

Rotational:


Linearized Equations of Motion• Linearization is done about a basic reference condition: steady

level flight at zero angle of attack and zero sideslip.

• Coupling exists between longitudinal-directional equations and lateral-directional equations.0 Due to lateral offset of c.g., which is negligible as in traditional heavy aircraft.

0 Due to aerodynamic cross coupling – also negligible.

• Longitudinal and Lateral linear equations can be decoupled.

• Resulting linearized equations of motion take on the following form:

Mx = Ax + BuM – Mass matrix (includes apparent mass terms)A – state matrix (contains stability derivatives)B – Input effectiveness matrix (contains input

effectiveness derivatives)


Linearized Equations of Motion

• One set for Longitudinal EOM and one set for Lateral EOM.

• This form is consistent with that of traditional heavier-than-air craft.

• Notable difference is the presence of the added mass and inertiaterms which are present in both M and A.

• These differences substantially affect fundamental stability criteria. These criteria differ from traditional heavier-than-air stability criteria

Mx = Ax + Bu


Stability Criterion for an Airship

• As noted, the added mass and inertia have profound effects on classical stability criteria.

• The requirement for static stability defines requirements on the pitching moment slope (Mα) and the yawing moment slope (Nβ).

• Traditional Requirement for heavier-than-air craft:

• It can be shown that this is a sufficient, but not necessary condition for airships.

• For most airships, the c.g. is below the hull center of volume so the condition on longitudinal stability is met due to a restoring moment from gravity and buoyancy.

• It will be shown that there can be cases where the yawing momentslope is negative and the vehicle is still statically stable.

, 0 , 0w vM M N Nα β< >


Lateral Stability Criterion Derived using Literal Approximate Factors

• An approximate expression for lateral stability has been derivedusing a literal approximate factor approach.

• Literal approximate factors are symbolic (as distinct from numerical) approximations of the poles and zeros of transfer functions.

• Literal approximate factors are only usually practical to develop for relatively low order systems like rigid vehicles.

• When available, they provide an explicit connection between the dynamics and the system parameters for an entire class of vehicles.

• The formulation is somewhat artful in that numerical evaluationsare routinely used to identify negligible terms, but the final results are symbolic forms of significant generality.


Lateral System Characteristic Polynomial

• The poles of the system, which define the stability, are found using the determinant below.

• Roots of the resulting polynomial equation are the poles and thereal parts must all be negative valued for stability to be guaranteed.

• Looks relatively simple, however:0 Lateral system is 5th order.

0 It has been proven that polynomials of degree 5 and higher cannot be factored explicitly (Abel-Ruffini theorem).

0s − =M A


Characteristic Polynomial Approximations

• The lateral characteristic polynomial takes on the following form in general:

• Coefficients (c1, c2, ..., c5) are functions of the elements of M and A.

• Using typical ballpark values for an airship, the roots of the characteristic polynomial take on the following form (an integrator, 2 1st

order poles and 1 set of 2nd order poles).

• This form holds up for a wide range of vehicles in this class.

( )5 4 3 2

5 4 3 2 14 3 2

5 4 3 2 1

0

0

c s c s c s c s c s

s c s c s c s c s c

+ + + + =

+ + + + =

( )3 3 3

2 4

1 12 0as s s s sT T

ξ ω ω⎛ ⎞ ⎛ ⎞

+ + + + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


Approximating Roots• Knowledge of the relative magnitude of the coefficients (c1, c2, ...,

c5) can be used to factor out approximate root estimates from the polynomial.

• This approximation is valid as long as the following holds:

• The developed algorithm only checks one of these approximations to retain simplicity.

• In cases where roots are close together, the check will fail and a 2nd order root is factored out, followed by a 3rd, etc.

( ) ( )4 3 2 3 2 15 4 3 2 1 5 4 3 2 2

4 3 25 1 3 14 15 4 3 2 12 2 2

cs c s c s c s c s c s c s c s c s c s c

c c c cc cs c s c s c s c s cc c c

⎛ ⎞+ + + + ≅ + + + +⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞≅ + + + + + + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

5 1 3 14 14 4 3 3 2 22 2 2 c c c cc cc c c c c cc c c+ ≅ + ≅ + ≅


Approximate Characteristic Polynomial• Using ballpark values for a typical airship, this process produced

the approximate characteristic polynomial.

• where:

( )3 3 3

2 4

1 12 0as s s s sT T

ξ ω ω⎛ ⎞ ⎛ ⎞

+ + + + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

25 3 0

4 1

3 5 2 1 43 3 2

4 2 2 4 54

1 0

1 12

ca c c

c T

c c c c cc T c T cc

ω

ξ ω

= = = =

⎛ ⎞= − = =⎜ ⎟⎜ ⎟

⎝ ⎠

This real root is the closest to the right half plane, making it the most vulnerable pole for instability. Also happens to be a function of the yawing moment slope.


Stability Characterization• Focusing on the vulnerable real root, the following statement can

be made to ensure stability:

• Coefficients c1 and c2 are known and are functions of the elements of the mass (M) and state (A) matrices. Therefore, a completely symbolic expression can be determined for the lateralstability criteria.

• The expressions for c1 and c2 are extensive and contain many terms – not practical.

• Order of magnitude approximations can be applied to vastly simplify the expression.

• Still retains the general “class” of airship vehicles assumption.

2

1 0T

> 1

2 2

1 c

T c=


Approximate Lateral Stability Criteria

• Still somewhat complicated.• It is noted that the denominator is always positive for typical airship

parameters – only need to focus on the numerator (NUM > 0).

• Critical yawing moment slope solved for:

• If NV is less than the above, an instability is encountered.

( ) ( ) ( )( )( )

( ) ( )

001

2 2

1v u v r ob r u v

r v v ru v u

v r zz r v r r

N X Y Y x Y X YN Y N Y V

V X Y Xc NUMT c Y N I N Y Y Y DEN

ρ ρ

ρρ

⎛ ⎞− + − + −− + ⎜ ⎟⎜ ⎟+ − −⎝ ⎠≅ = =

− + − −

V V

V

V

( ), 0, ,

, 0

v r TOT obv TOT crit

r TOT

Y N x VN

Y V

ρ

ρ

−=

−

V

V

General for an entire class of

airship vehicles


Independent approach to Lateral stability criterion

• An expression for lateral stability has been determined independantly.

• Assumptions are made to decouple the roll dynamics from the linear equations of motion, reducing the order to two.

• Determinant is much simpler in this case.

• This independent analysis produces equivalent results as that determined using the approximate factor approach.

( ) ( ) 0yv nr ob yr nvc c m x c m c′ ′− − − <


Summary• There is current interest in Stratospheric

Airships (SA) as a High Altitude Platform (HAP) capable of autonomous long duration missions for both military and commercial applications.

• SA as a HAP can accomplish many of the same objectives as space satellites at a much lower cost and are easier to maintain.

• There is a long history of development and study of airships leading back to before the turn of the twentieth century.

• Although the buoyant vehicle has been the subject of strong analytical scrutiny there is still a need for further study due to the current interest in SA as a useful platform.

• “Added mass” and “added inertia” effect the traditional requirements for static and dynamic stability and differ from that of heavier-than-air craft.

• Analytical expressions for these stability criteria can be determined using a literal approximate factor approach and will be valid for an entire class of airship vehicles.

image courtesy of: http://blimp-n2a.com/projets_haa.htm

StratSat


Conclusions• A general analytical expression has been developed to express the

requirement of the yaw moment slope for stability for an entire class of airship vehicles.

• Starting from the linearized lateral dynamic equations of motion, a literal approximate factor approach was utilized to derive this expression.

• A separate approach, which separates the roll and yaw dynamics to determine the equivalent expression concerning the lateral stability has been carried out. This method requires more insight into the system of interest to justify the assumptions.

• The literal approximate factor approach, described herein, has benefits in the fact that it is entirely general whereas only relative magnitudes of various terms are utilized to determine a literal expression concerning the stability of the proposed dynamic system.

• Any other stability criteria can be determined using this literal approximate factor approach in exactly the same way.

• These general algorithms have been automated in a symbolic “literal toobox” constructed for use with Matlab.


Questions?

Image courtesy of: http://blimp-n2a.com/projets_haa.htm


extra slides


http://blimp-n2a.com/projets_haa.htmGalileo

http://www.aeroventure.org.uk/mainnews2007.phpSkyship 500

http://www.sti.nasa.gov/tto/spinoff2002/oa.html

http://blimp-n2a.com/projets_haa.htmX-Station


http://blimp-n2a.com/projets_haa.htmBerkurt

http://blimp-n2a.com/projets_haa.htmStratSat

http://blimp-n2a.com/projets_haa.htmTechnosphere http://blimp-n2a.com/projets_haa.htm

NAL JAXA

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Considerations in the Lateral Stability … in the Lateral Stability Characteristics of Airship...

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