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16.346 Astrodynamics
Fall 2008
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Lecture 21
Space Navigation —The Position Fix
Chapter 13
Fig. 13.1 from An Introduction to the
Mathematics and Methods of
Astrodynamics. Courtesy of AIAA.
Used with permission.
The Line of Position
ir · i 1 = − cos A1
ir · i 2 = − cos A2
where
=⇒
ir = α i 1 + β i 2 + γ i 1 × i 2
α sin2 ϕ = cos A2 cos ϕ − cos A1
β sin2 ϕ = cos A1 cos ϕ − cos A2
γ 2 sin2 ϕ = 1 + α cos A1 + β cos A2
and cos ϕ = i 1 · i 2
The Position Fix
ir · i 1 = − cos A1
ir · i 2 = − cos A2
ir · rp = r − |rp − r| cos A3
where rp is the position vector of a planet or other near object.
Henceforth, we will linearize the measurements so that we can deal with a set of redundant
measurements using Gauss’s Method of Least Squares.
16.346 Astrodynamics
Lecture 21
Determining the Measurement Geometry Vector
#13.2
For an arbitrary angle A(r) , we calculate the measurement geometry vector h from the
Taylor Series expansion about a reference position r0 and discarding all terms of higher
order in δr :
�
∂A ��
δr + · · · = A0 + h T δr + · · ·
A(r) = A(r0 ) +
∂r
�
r=r0
Hence
�
∂A ��
T
T
δA = h δr where h =
∂r �r=r0
Measuring the Angle between a Near Object and a Star
The angle between the line-of-sight to a near object, e.g., the sun or a planet, and the
line-of-sight to a distant star is defined by
r cos A = −iT r
from which
∂r
∂A
cos A − r sin A
= −iT
∂r
∂r
The derivative of the scalar r with respect to the vector r is obtained from
r2 = r · r = r T r
=⇒
2r
∂r
∂r
= 2r T
= 2r T I
∂r
∂r
=⇒
∂r
1
= r T = irT
∂r
r
Therefore,
h=
1
(cos A ir + i )
r sin A
or
h=
1
i
r n
The vector in is a unit vector in the plane of the measurement and perpendicular to the
line-of-sight to the near object.
Measuring the Angle between Two Near Objects
The angle between the two position vectors r and d produces the measurement equation
d T r = dr cos A
Since
r − d = constant , then δd = δr .
2d
∂d
= 2dT I
∂r
Again, from
or
d2 = dT d ,
we have
∂d
= idT
∂r
Hence:
h=−
1
1
(id − cos A ir ) −
(i − cos A id )
r sin A
d sin A r
or
h=
1
1
in + im
r
d
Both in and im are unit vectors in the plane determined by the spacecraft and the two
near bodies.
16.346 Astrodynamics
Lecture 21
The Measurement Geometry Matrix
For several measurements we define
H = [h1
where
h2 · · · hn ]
so that
⎡
δq ⎤
1
δq
⎢ 2⎥
⎥
δ q
=
⎢
⎣
.. ⎦
.
δq = H T δr
⎡
⎤
δx ∂r
=
⎣
δy
⎦
δz
and
δqn
Gauss’ Method of Least Squares
Given mij and ci : To determine xi so that
n
�
mij xj = ci
where
#13.5
i = 1, 2, . . . , N > n
j=1
is “as nearly satisfied as possible.”
n
�
• Define: Residuals ei =
mij xj − ci
j=1
• Choose: Weighting factor wi > 0 for i th residual
• Determine: x1 , x2 , . . . , xn so that w1 e21 + w2 e22 + · · · + wN e2N is a minimum.
Solution of Least Squares Problem
•
Vector of residuals:
• Weighting matrix:
e
= Mx − c
⎡w
0
1
0
w
⎢
2
W =
⎢
..
⎣
..
.
.
0
0
• Weighted squares:
···
···
..
.
0
0
.
..
⎤
⎥
⎥ = WT
⎦
· · · wN
e T We = (x T M T − c T )W(Mx − c) = x T M T WMx − c T WMx − x T M T Wc + c T Wc
• Least value of the weighted squares:
∂ T
(e We) = 2x T M T WM − 2c T WM = 0 T
∂x
or M T WMx = M T Wc
x = (M T WM)−1 M T Wc
Note: If y = x T Bx = (x T Bx) T = x T B T x then
16.346 Astrodynamics
∂y
= x T BI + x T B T I
∂x
Lecture 21
Application of Gauss’ Method of Least Squares to Space Navigation
x = (M T WM)−1 M T Wc
In our notation
x = δ�
r
W = A−1
MT = H
wi =
1
σi2
�
c = δq
so that
�
δ�
r = F δq
F is called the Estimator Matrix where
F = PHA−1
P = (HA−1 H T )−1
and
The deviation from the reference position is an estimate, denoted by the “hat” over the
position vector
δ�
r = δr + and is the sum of the actual deviation and the error in the estimate. Similarly,
� = δq + α
δq
where α is the vector error in the determination of the quantities measured. (The vector
δq is the actual deviation in those quantities from their reference values.)
The Information Matrix
The matrix P−1 is called the Information Matrix because of the property
HA−1 H T =
h1 h1T
h2 h2T
+
+ ···
σ22
σ12
Each new measurement adds a new term to the series and each term contains all the
information about the new measurement.
−1
P
= HA
−1
H
T
N
�
hi hiT
=
σi2
i=1
The factor σi2
is called the variance. The larger the i th variance the less weight is given
to the i th measurement.
16.346 Astrodynamics
Lecture 21