By Andreas Nüchter
The monograph written via Andreas Nüchter is concentrated on buying spatial versions of actual environments via cellular robots. The robot mapping challenge is often often called SLAM (simultaneous localization and mapping). 3D maps are essential to steer clear of collisions with advanced hindrances and to self-localize in six levels of freedom
(x-, y-, z-position, roll, yaw and pitch angle). New recommendations to the 6D SLAM challenge for 3D laser scans are proposed and a large choice of purposes are presented.
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Extra info for 3D Robotic Mapping: The Simultaneous Localization and Mapping Problem with Six Degrees of Freedom
1) ⎛ cos θy cos θz − cos θy sin θz ⎜ R = ⎝ cos θz sin θx sin θy + cos θx sin θz cos θx cos θz − sin θx sin θy sin θz sin θx sin θz − cos θx cos θz sin θy cos θz sin θx + cos θx sin θy sin θz ⎞ sin θy ⎟ − cos θy sin θx ⎠ . 34). The resulting approximativ rotation is ⎛ ⎞ θy 1 −θz ⎜ ⎟ R ≈ ⎝ θx θy + θz 1 − θx θy θz −θx ⎠ .
Assume that the singular value decomposition of H is H = U ΛV T , where U and V are orthonormal 3 × 3 matrices and Λ is a 3 × 3 diagonal matrix without negative elements. Suppose R = V UT . 40 4 3D Range Image Registration R is orthonormal and RH = V U T U ΛV T = V ΛV T is a symmetric, positive deﬁnite matrix. Using the lemma above we have tr (RH) ≥ tr (BRH) for any orthonormal matrix B. Therefore the matrix R is optimal. Finally, the optimal translation is calculated as (see Eq. 7)) t = cm − Rcd .
The third part has its minimum for ˜t = 0 or t = cm − Rcd . 7) Therefore the algorithm has to minimize only the ﬁrst term, and the error function is expressed in terms of the rotation only: N 2 E(R, t) ∝ ||mi − Rdi || . 8) i=1 Computing the Transformation using the Singular Value Decomposition of a Matrix The following method was developed 1987 by Arun, Huang und Blostein . The rotation R is represented as an orthonormal 3 × 3 matrix. Theorem 1. The optimal rotation is calculated by R = V U T .