sin sin sin tan cos cos cos 2 2 2 ... - Utah State...

Reentry Air Data System for a Sub-orbital

Spacecraft Based on X-34 DesignJoel Ellsworth

Utah State University

4130 Old Main Hill

Logan, Utah 84322

Joel.Ellsworth@gmail.com

Stephen A. Whitmore

Utah State University

4130 Old Main Hill

Logan, Utah 84322

swhitmore@engineering.usu.edu

Background

Several companies have been formulated during the past decade

with the intention of developing a sup-orbital space tourism

market. These companies include Okalahoma City based

Rocketplane-Kistler, Roswell New Mexico based Virgin Galactic,

and San Diego based SpaceDev among others.

Most designs are intended to glide back to earth after using a

rocket engine to boost themselves out of the atmosphere.

Knowledge of the wind-relative vehicle state parameters -

dynamic pressure, Mach number, angle-of-attack and -sideslip,

and surface winds - are especially critical for the landing phase

for energy management and runway alignment.

Why use the X-34 as a Baseline Vehicle?

The X-34 was chosen for this study because it incorporates many of the features needed by a commercial suborbital tourist spaceflight system.

The X-34 aerodynamic database was developed using public dollars and is available in the public domain.

For this design study a typical X-34trajectory was used to generate simulated nosecap surface pressure values based on wind-tunnel derived calibration models.

1heat

le

qR

∝&

High temperatures and dynamic

pressures associated with

attached shockwaves (reentry

conditions) will destroy

traditional pitot probes and

directional vanes.

Heating is related to leading

edge curvature

Why Develop a New System?

Solution:

Flow Model:

The FADS algorithm uses the

pressures measured by a matrix

(at least five) of flush mounted

pressure ports to produce a full

air data state.

Flow model is a blend of

Modified-Newtonian Flow and

Potential Flow over a sphere.Potential flow for a sphere

(M <<1)

Modified Newtonian flow

(M>>1)

( )( )

( )( )

2 2

2

5cos sin

4

cos

p pCp

qc

p pCp

qc

θθ θ θ

θθ θ

∞

∞

−= = −

−= =

( ) 2 2

2 cos sinp qc pθ θ ε θ ∞ = + +

FADS flow model for a spherical nosecap

X-34 Trajectory and Simulation Results

The X-34 trajectory was used to generate surface pressures on the

nosecap of the vehicle, which were then corrupted with random

noise, pneumatic lag, sensor resolution, and sensor latency before

being passed to the FADS algorithm Results are shown with and

without the inertial filtering algorithm active.

Simulation results without inertial

filtering active.

(s)

(s) (s)

(s)

Simulation results with inertial

filtering active.

(s)

(s)

(s)

(s)

FADS Solution Algorithm:

The parameter αind can be de-coupled from βind

by using only pressures triples aligned along a

vertical meridian. The result is a quadratic

expression in terms of tan αind.

2 2 2 2 2 2 2sin sin sin tan cos cos cos

2 cos sin cos cos sin cos cos sin cos tan 0

ik j ji k kj i ind ik j ji k kj i

ik j j j ji k k k kj i i i ind

φ φ φ α φ φ φ

φ φ λ φ φ λ φ φ λ α

Γ + Γ + Γ + Γ + Γ + Γ +

Γ + Γ + Γ =

Once we know Angle of Attack, we can

calculate Angle of Sideslip using any

combination of ports exclusive of sets with all

ports on the vertical meridian.

By taking strategic differences of three surface sensor readings

("triples") the parameters qc2, p

∞, and ε, are eliminated.

Γ ik cos2 θ j + Γ ji cos2 θk + Γkj cos2 θi = 0 Γ ik = pi − pk , Γ ji = p j − pi , Γ kj = pk − p jwhere

2' tan 2 ' tan ' 0ind ind

A B Cβ β+ + =

( )

( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

2 2 2

2 2 2

'

'

'

cos cos sin sin cos

sin sin

ik j ji k kj i

ik j j ji k k kj i i

ik j ji k kj i

ind indijk ijk ijk ijk

ijk ijk ijk

A b b b

B a b a b a b

C a a a

a

b

α λ α λ φ

λ φ

= Γ + Γ + Γ

= Γ + Γ + Γ

= Γ + Γ + Γ

= +

=

where

Altitude is found by table look-up of freestream pressure, p∞ .

Mach number is found through the ratio of dynamic compressible

pressure, qc2, and free stream pressure.

2

1

1 1 1 1

2 2 2 21 2 1 2

0 0 1 0 0

0 0 1 0 0

0 0 0 01 1 1 1 1 1

0 0 1 0 0

c n n

n n n n

q q p

q q q p

p

q q p

−

∞

Θ ΘΘ Θ Θ Θ Θ Θ =

Θ

L L

M ML L

M O M M M O ML L

L L

2 2cos sini i iθ ε θΘ = +where

2

1 1 1

2

1 1 1

2

2

1 1 1

n n n

i i i i i i

i i i

n n n

i i i i i ic i i i

n n n

i i i i i

i i i

q q p q

q q p qq

pq q q

= = =

= = =

∞

= = =

− Θ Θ

− Θ Θ =

Θ − Θ

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

This can be simplified to

Subsonic Mach number

2

1

21 1

1

cq

Mp

γ

γ

γ

−

∞

∞

= + − −

Supersonic Mach number

2

12

1

12

1

21

2 1

1 1

c

Mq

p

M

γγ

γ

γ

γ γ

γ γ

−

∞

∞ −

∞

+ = −

−− + +

Noise reduction is accomplished through the use of an inertial

filtering algorithm that combines the unbiased FADS data with

the biased INS data to give a high fidelity airdata state with a

minimum of system noise. Although all states are filtered, the

sideslip filter is presented here.

After adding an altitude scheduled weighting factor, Awght, to

further reduce noise at high altitudes, and applying a bi-linear

transform to convert to the time domain we have

( ) ( )( )

( )1 1 11

1 tan tan 1 tan12 2 2

1 tan 1 tan 1 tan 1 tan2 2 2 2

k k k k k k

wght wght

k k inert inert FADS FADS inert inert

t t tA A

t t t t

τ τ τβ β β β β β β β

τ τ τ τ

− − −−

∆ ∆ ∆ − −

= + − + − + −

∆ ∆ ∆ ∆ + + + +

1

inertial FADSs

s

τ β ββ

τ

+=

+In the Laplace domain, sideslip is given by

Place matrix of pressure sensors

flush with surface behind a

detached shockwave.

Matrix of ports allows for

determination of the full airdata

state, as well as the potential for

fail-operational capability and

full redundancy.

This requires a flow model

simple enough to invert in real-

time, yet accurate enough to

still provide useful data.

The effect of the filter can be seen in the figures at the far left.