Appunti Space missions and systems

V1 2The required attitude matrix follows: T (22) ŷ ŷ ŷ x̂ x̂ x̂ x̂ x̂ x̂ ŷ ŷ ŷA = =⇒ A =1 2 3 1 2 3 1 2 3 1 2 32.3 The Wahba problem

The TRIAD algorithm requires that the observation vectors are produced at the same time. In order to take advantage of multiple measurements from many sensors, attitude determination is generally tackled as least squares problem.

Let and be the unit vectors of known celestial objects of the OBF and ICRF at the same time. The Wahba problem can be stated as: find the proper orthogonal matrix (attitude matrix) that minimises the cost function: 1 X 2|W − |a AVJ(A) = i i i2 where is the appropriate measurement weights and n is the number of simultaneous observations.

a iSolutions to the Wahba problem:

If the weights are chosen like the cost function may be reduced to:P −a = 1 1 g(A).i The solution requires to find the maximum of the function we can manipulate and obtain:g(A),24X T Tg(A) = a W AV = tr(AB )1 iiX TB = a W Vi i

The matrix is called (external scalar product of W and V).B attitude profile matrix

So the Wahba problem is now maximisation of Tg(A) = tr(AB ).

The QUEST algorithm

Very efficient method to solve the Wahba problem. The maximisation of may be transformed into ang(A)eigenvalue problem for a 4x4 matrix, built of the attitude quaternions.

Figure 30: QUEST algorithm

B becomes: X X XT T T T−→B = a W V 2q tr(q̃B ) = q ( aṼ W )q + q ( a(Ṽ W ) )qi i 4 i i 4 4 i ii

With these manipulations becomes:g(A) Figure 31: transformedg(A)

• quaternions must have unitary norm
• we solve we have to find out if there’s a maximum or minimum, by solving the eigenvalue−KQ λQ = 0,problem
• is body ref. is ICRF frame, are coordinate vectors through a star recognition methodW Vwe obtain: T Tg(A) = Q KQ = λQ Q = λand its maximum is the largest eigenvalue, that is very close to unity, so that the characteristic equationcan be solved iteratively with

Newton-Raphson method starting from as initial value. λ = 1252.4

Attitude Kinematics

The attitude of a rigid body can be described with:

• Euler angles
• direction cosine
• axis-angle
• quaternions (much more common): do not have geometric singularities, do not require evaluation of trigonometric functions, satisfy a set of linear differential equations.

2.5 Attitude determination sensors

• star trackers
• gyroscopes
• earth sensors
• sun sensors

2.5.1 Star sensors

Why stars are the best option to determine attitude:

• stars are used to realise inertial frames
• RA and DEC known with great precision
• always available to optical imaging
• star trackers technology is very reliable (using CCD)

however:

• cost is high
• ST require significant computing power
• every mission requires tailoring

Figure 32: Magnitude and type

The process is to find the identity curve of each star through their coordinates using magnitude and spectrum (visual mag).

Stars are classified in spectral type + 9

sub-groups in each type. The spectral character-OBAFGKMLTistics are important in the selection of the optical filters used to reduce background noise.

Main categories of star sensors: 26

1. star scanners (in spin-stabilised sats)
2. fixed-head star trackers (most accurate)
3. gimbaled star trackers (complex, less accurate)

Main components of a star sensor:

• optical system
• light detector (CCD)
• readout electronics

We want to determine (latitude) and (azimuth), measured in pixels in the CCD.

λ ϕ 27

Usually in the inside the star sensors have a corrugated shield that protects from straight light, reflecting back (Moon and Earth albedo are also a problem).

Appropriate corrections to the star position must be applied, to account for: proper motion, aberration, parallax. ∗ −Ŝ = (1 + Ŝ k)Ŝ kA T T|v| ≈β = sinθ 20arcsecc v v− ∗Ŝ = (1

• accounted•
• number of tracked stars•
• smaller star catalog/FoV can be used in some missions (GEO)

The stars are identified by direct match or angular separation match:

−11 1∗ |d(i, −d(O, S) < ϵ d 2 = cos (S S ) j) d 2| < ϵ1 2m m2.5.

2 Laser Gyroscopes

Modern gyro are based either upon a general relativistic effect known as Sagnac effect (ring laser gyros, RLG,and fiber optical gyros, FOG) or upon mechanical resonance effect (hemispheric resonating gyros, HRG).

In space applications are always used together with star sensors, they provide attitude between ST frameacquisitions. ST and gyros data are generally combined in an attitude estimator based upon a Kalman filter.

Advantages:

• no moving parts
• simple design, few components (<20)
• no maintenance costs
• wide dynamic range (>10 , can detect small oscillations)9
• fast update rate
• long and reliable lifetime

The operating principle is the Sagnac effect:

Figure 36: Sagnac effect

After the beamsplitter,

one beam moves CW and one CCW, then they're reflected back on it.

Fundamental equations:

The important outcome is the phase differences accumulated with time:

Figure 37: Eqs for laser gyros

ΔL = ωi = 2πtν

Δφ = tΔω = ωt L Lc L

When the gyro is rotated by an angle at constant angular speed fringes are produced by theθ = Ωt Ω, combiner. By computing the fringes passing by, we can evaluate the rotation angle:

Δφ = 4A 4A (23)

N = Δν = ν = θ2π lλ Lλ

Δφ = 2ΩR 2RΩt L (24)

N = Δν = tν = θ2π λ λ πλ2.

5.3 Fiber optic gyro FOG

The path length is increased by injecting the laser signal into a fiber optic coil, we gain which is the numbernof coils. Figure 38: FOG

5.4 Hemispherical resonating gyro

Exploits the principle for which when the hemispherical resonator vibrates and rotates, the wave pattern is not fixed wrt the resonator, but drifts about 0.3 times the rotation rate.

due to the Coriolis acc.2.5.5 Farrenkopf ’s gyro noise model

Separates gyro noise into three noise types:

Figure 39: HRG• electronic noise : colored noise but can be treated as white noise if the gyro time constant is muchk ismaller that the gyro readout time interval.

• float torque noise : white Gaussian noise.η i

• float torque derivative noise : is the casue of gyro drift rate bias (integration is rate random walk).

bi g = (1 + k )ω + b + b + ηi i i i0 i i1gyro measurement, true angular speed, initial bias, white noise, is scale factor. The idealg ω b η ki i i0 i1 imeasurement would be .g = ωi iGenerally not reliable for long term measurements, because the scale factor multiplies and makes itωidiverge.

2.5.6 Earth horizon sensors

Attainable altitude errors: 0.02-0.5 deg.

Appearance of Earth is much more uniform in the infrared than in the visible range (no albedo, no ill-definedterminator). So the IR profile of Earth is