Conformal Geometric Algebra in Stochastic Optimization Problems ...

5.3. GAUSS-HELMERT MODEL BASED ESTIMATION 171
Combining all conditions into a single block matrix equation then gives
⎡ ⎤ ⎡
X1 Z1
⎢ ⎥ ⎢
⎢
⎣ .
⎥
⎦∆θ
+ ⎢
⎣
. ..
0
⎤ ⎡ ⎤ ⎡
∆y1 z
⎥ ⎢ ⎥ ⎢
⎥ ⎢
⎦ ⎣ .
⎥
⎦ = ⎢
⎣
Xk
� �� � �
0
��
Zk
� �
∆y1
�� �
=: X =: Z =: ∆y
′ 1
.
z ′ ⎤
⎥
⎦
k
� �� �
=: z ′ ,
g
where X ∈ � Ng×M , Z ∈ � Ng×K , ∆y ∈ � K and correspondingly z ′ g ∈ � Ng .
Fig. 5.4: Situation after the t th iteration: estimated values should in each iteration
fulfill the G-condition. After linearization, the relation ∆y = ǫ − (ˆy [t] − y) has to
be obeyed, cf. [34].
Taking into account the situation depicted in figure 5.4 the important relation
∆y = ǫ − (ˆy [t] − y) (5.27)
can be inferred. This is, for example, shown by Förstner in [34]. It follows that
−z ′ g + Z∆y + X∆θ = g(ˆy [t] , ˆ θ [t] ) + Z(y − ˆy [t] ) + Zǫ + X∆θ
= g(y, ˆ θ [t] )
� �� �
=: −zg
+ Zǫ + X∆θ,
where the unprimed zg denotes the contradictions. Hence by means of the block
Jacobians X and Z, the pendant of the GH-model (5.21) in terms of the residuals is
The Constraint Equations
X∆θ + Zǫ = zg. (5.28)
The respective Taylor series expansion of first order for the H-constraint reads
h( ˆ θ [t] + ∆θ) ≈ h( ˆ θ [t] ) + ∂h
∂θ (ˆ θ [t] ) ∆θ
= −zh + H∆θ.