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32 CHAPTER 4. METHOD
Prismatic Joint
The mapping of a prismatic joint is more straight forward, because the model is already defined in
Cartesian space. The only thing that must be done is to set the other two axis, in which the joint is not
moving, to zero. Thus the mapping of a prismatic joint with the model x k , P k can be done as
[ [ ]
s σ
2
x k = , P
v]
k = s 0
0 σv
2
⎡ ⎤ ⎡
s
σ 2 ⎤
s 0 0
move in x-axis: µ xyz = ⎣0⎦ , Σ xyz = ⎣ 0 0 0⎦
0
0 0 0
⎡ ⎤ ⎡ ⎤
0
0 0 0
move in y-axis: µ xyz = ⎣s⎦ , Σ xyz = ⎣0 σs 2 0⎦
0
0 0 0
⎡ ⎤ ⎡ ⎤
0
0 0 0
move in z-axis: µ xyz = ⎣0⎦ , Σ xyz = ⎣0 0 0 ⎦
s
0 0 σs
2
4.4.3 Quality of Transformation Chains
Until now only the quality of a single transformation was considered. This section will show how
single position uncertainties can be merged to describe the uncertainty of a transformation chain.
To do this, the simple tf chain shown in Figure 4.12 is used. This chain consists of a root frame
Figure 4.12: Tf tree with three frames.
called base_frame and two other frames, which are linked through prismatic joints. The joint that
links the base_frame with the x_frame moves in the x-direction whereas the joint linking the
x_frame with the y_frame moves in y-direction. As an example the transformation M, as well as
the position uncertainty Σ can be defined as:
• base_frame → x_frame:
⎡ ⎤
1 0 0 5 ⎡
⎤
M a = ⎢0 1 0 1
1 0 0
⎥
⎣0 0 1 3⎦ , Σ a = ⎣0 0.001 0⎦
0 0 0
0 0 0 1