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4.4. QUALITY OF TRANSFORMATIONS 33
• x_frame → y_frame:
⎡
⎤
0.7071 −0.7071 0 1 ⎡
⎤
M b = ⎢0.7071 0.7071 0 2
0.001 0 0
⎥
⎣ 0 0 1 0⎦ , Σ b = ⎣ 0 0.5 0⎦
0 0 0
0 0 0 1
The uncertainties are given in their local frame and look like shown in Figure 4.13. There it can be
seen that the uncertainty for the x_frame is in x-direction and for the y_frame it is in y-direction.
To merge these uncertainties, to get an overall quality for the chain, the single variances have to
be converted into a single reference frame. This frame is the root of the chain and would be the
base_frame in the example.
x_frame
y_frame
(a) 3σ ellipse of the x_frame.
(b) 3σ ellipse of the y_frame.
Figure 4.13: Single uncertainties for the frames.
To change the coordinate frame of a covariance matrix Σ the equation
Σ + = RΣR T
can be used, where R is the Rotation of the transformation. The rotation for the x_frame would be
R(M a ) whereas for the y_frame it would be R(M a M b ).
After mapping the variances to a single reference frame, the next and last step is to merge them.
The result of this step is the position uncertainty for the last frame of the chain in relation to the root
frame of the chain. Thus the uncertainty Σ ab describes the quality of the whole transformation chain.
It can simply be calculated by taking the sum of the single covariance matrices. For the example this
would be
Σ ab = Σ a + Σ b .
The resulting tf chain with the single uncertainties Σ a , Σ b and the total uncertainty Σ ab is shown in
Figure 4.14. In this figure it gets obvious, that the single uncertainties add up to a bigger uncertainty.
This of course makes sense, because each joint introduces some additional uncertainty into the chain,
which must increase the overall uncertainty.
If a hinge joint is included in the chain it is not enough to determine the position uncertainty
of the Child frame and to add it to the overall position uncertainty. This is due to the fact that the