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30 CHAPTER 4. METHOD
Hinge Joint
A hinge joint does not have any influence on the position accuracy of the joint frame, because it represents
a rotation and not a translation. Thus it only effects the position accuracy of Child frames. The
angular uncertainty of the model must therefor be applied when calculating the position uncertainty
of the Child frames.
To perform the mapping for a model of a hinge joint to Cartesian space, additionally the distance
d must be known. This value describes the distance from the origin of the Child frame to the origin of
the joint frame. This distance is measured in the plane perpendicular to the axis of rotation. Together
the distance d and the angle s from the model describe a point in polar coordinates. In the following
equations, which describe the mapping steps the value s and σ 2 s are named α and σ 2 α. The velocity in
the model is only necessary for the propagation of the position and thus not used for the mapping.
The mapping of polar coordinates to Cartesian coordinates can be done by simply using trigonometrical
functions. Figure 4.10 visualizes the relationship of polar and Cartesian coordinates, where
Y
1
d
U
α
1 2
X
Figure 4.10: Mapping of polar to Cartesian coordinates.
the point U with polar coordinates d, α is mapped into Cartesian coordinates with x, y. The equations
for calculating the values of x and y simply are
x = d · cos α
y = d · sin α
This relationship can also be seen as a function F which transforms the polar coordinates [ d
into Cartesian coordinates [ x
y ] T . This relationship can be written as
([ d
F
α])
=
[ x
=
y]
[ ] d · cos α
d · sin α
α ] T
The normal distributed uncertainty Σ dα can be mapped from polar to Cartesian space with the
same function F . Since the new uncertainty should also be normal distributed, the mean and variance
must satisfy the equations
with
[ x
y]
[
µx
µ y
]
∼
N
= F
([
µx
µ y
]
, Σ xy
)
([
µd
µ α
])
=
[ ]
µd · cos α
µ d · sin α
The Cartesian covariance matrix Σ xy can be calculated out of the polar covariance matrix Σ dα by