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of the camera intrinsic parameters and the knowledge of the distance among two
points in the 3D space, in order to estimate the norm of the translation). The screen
size is 659
×
493 pixels, and the estimate of the intrinsic parameters matrix is
⎛
⎞
624
.
21
.
035 318
.
1
⎝
⎠
.
A
=
0
609
.
7 246
.
3
0
0
1
) and
s
t
(
Table 9.3 shows the upper bounds
s
r
(
δ
δ
) in (9.24)-(9.25) and the lower
bounds
s
r
(
) and
s
t
(
δ
δ
) in (9.27) for some values of
δ
.
Ta b l e 9 . 3
Example 3: upper and lower bounds of the worst-case location errors for the object
points in Figure 9.3
δ
s
r
(δ)
s
r
(δ)
s
t
(δ)
s
t
(δ)
[pixels] [deg]
[deg]
[mm] [mm]
0
.
5
0
.
164 1
.
102 1
.
091 1
.
701
.
.
.
.
.
1
0
0
329 2
342 2
339 3
399
1
.
5
0
.
494 3
.
496 3
.
376 5
.
108
.
.
.
.
.
2
0
0
662 4
733 4
593 6
828
9.5
Conclusion
This chapter has addressed the estimation of the worst-case robot positioning error
introduced by image measurement errors, which is a fundamental problem in visual
servoing. Specifically, it has been shown how upper and lower bounds of this worst-
case error can be computed by solving optimization problems over polynomials with
LMIs and barrier functions. Future work will investigate the possibility of improving
the technique presented in this chapter in order to establish and guarantee exactness
of the derived bounds.
Acknowledgements.
This work was supported in part by the Research Grants Council of
Hong Kong Special Administrative Region under Grants HKU711208E and HKU712808E.
References
[1] Allotta, B., Fioravanti, D.: 3D motion planning for image-based visual servoing tasks.
In: Proc. IEEE Int. Conf. on Robotics and Automation, Barcelona, Spain (2005)
[2] Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Sys-
tem and Control Theory. SIAM, Philadelphia (1994)
[3] Chaumette, F.: Potential problems of stability and convergence in image-based and
position-based visual servoing. In: Kriegman, D., Hager, G., Morse, A. (eds.) The con-
fluence of vision and control, pp. 66-78. Springer, Heidelberg (1998)
[4] Chaumette, F., Hutchinson, S.: Visual servo control, part I: Basic approaches. IEEE
Robotics and Automation Magazine 13(4), 82-90 (2006)
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