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46 CHAPTER 4. METHOD
• slide_frame → pan_frame:
• pan_frame → tilt_frame:
⎡
⎤
0.9947 −0.1019 0 9 ⎡ ⎤
M b = ⎢0.1019 0.9947 0 0
0 0 0
⎥
⎣ 0 0 1 3⎦ , Σ b = ⎣0 0 0⎦
0 0 0
0 0 0 1
⎡
⎤
0.9921 0 0.1253 1 ⎡
⎤
M c = ⎢ 0 1 0 0
0 −0.0003 0
⎥
⎣−0.1253 0 0.9921 2⎦ , Σ c = ⎣−0.0003 0.0027 0⎦
0 0 0
0 0 0 1
• tilt_frame → sensor_frame:
⎡ ⎤
1 0 0 2 ⎡
⎤
M d
⎢0 1 0 0
0.0002 0 −0.0015
⎥
⎣0 0 1 3⎦ , Σ d = ⎣ 0 0 0 ⎦
−0.0015 0 0.0115
0 0 0 1
The above uncertainties describe the position uncertainty of the Child frame relative to the Parent
frame. To get the overall uncertainty Σ f of the sensor_frame relative to the base_frame all
single variances in the chain Σ − must be summed up. The variances Σ a and Σ b can directly be used for
the variance of the chain. The variances introduced by the hinge joints must be calculated as describe
in Section 4.4.2. In particular this means that the uncertainties Σ − c and Σ − d
for the sensor_frame
must be calculate relative to the pan_frame and tilt_frame. After doing this, the variances
Σ − a. . . d
must be converted into a single frame. The most suitable frame therefore is the base_frame.
The conversion is done with the rules described in Section 4.4.3. The final variance Σ f which consists
off all summed up single variances after conversion Σ + a. . . d
can be calculated as
Σ f = Σ + a + Σ + b + Σ+ c + Σ + d
⎡
4.0100 −5.647 · 10 −4 −5.093 · 10 −5 ⎤
= ⎣−5.647 · 10 −4 2.611 · 10 −3 3.258 · 10 −5 ⎦
−5.093 · 10 −5 3.258 · 10 −5 1.169 · 10 −2
The final transformation M f from the base_frame to the sensor_frame can be computed by
multiplying the single transformation matrices M a...d . Thus M f is calculated as
M f = M a M b M c M d
⎡
⎤
0.9869 −0.1019 0.1247 37.3428
= ⎢ 0.1011 0.9947 0.0127 2.3424
⎥
⎣−0.1253 0 0.9921 15.7256⎦
0 0 0 1