Conformal Geometric Algebra in Stochastic Optimization Problems ...

7.2. ESTIMATING A RIGID BODY MOTION 195
7.2 Estimating a Rigid Body Motion
This section is preparatory for the coming sections dealing with pose estimation in
that not a geometric object but a geometric operator, namely a motor, is estimated.
Such an element is also the objective of the estimation presented in chapter 4, where
an approach completely different from the GH-method is chosen. A comprehensive
discussion of motors can be found in section 3.4.5.
Fig. 7.4: The estimation of an RBM with points forming a cube.
Now it is being focused on the transformation interrelating two (almost) congruent
sets of 3D-points. Let {a 1...N } and {b 1...N } be the two sets, where it is assumed
that each point ai ∈ � 4,1 is a constant, whereas the corresponding point bi ∈ � 4,1
represents an observation with associated uncertainty Σbibi ∈ �5×5 , i ∈ [1,N] � . An
example scenario is depicted in figure 7.4. Recall that in CGA, transformations are
expressed in the form of
Ma � M = b. (7.2)
Unfortunately, it is no particular algebraic operation known by means of which the
motor M that best transforms the points {a 1...N } into {b 1...N } could be computed
at once. However, switching to the tensor representation of CGA the above equation
can be reformulated [98]. The first step consists in exploiting that a motor is a
unitary versor, i.e. M � M = 1. On multiplying with M from the right, equation
(7.2) can be rewritten as
Ma � M = b ⇐⇒ Ma − bM = 0,
whence the tensor representation follows as
Φ :
M ai − bi M = 0
↓ ↓ ↓ ↓ ↓
p k G t kl ai l − bi l G t lk p k = 0 t
,
1 ≤ t ≤ 5
1 ≤ i ≤ N.
(7.3)
It is Φ(a) = a ∈ � 5 , Φ(b) = b ∈ � 5 and Φ(M) = p ∈ � 8 , which includes the scalar
component of a motor, see below. Accordingly, the tensor for the geometric product
is [(G t kl)] ∈ � 5×8×5 and [(G t lk)] ∈ � 5×5×8 , respectively. The Φ-mask associated with