By Andreas Nüchter

The monograph written by way of Andreas Nüchter is targeted 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 keep away from collisions with advanced hindrances and to self-localize in six levels of freedom
(x-, y-, z-position, roll, yaw and pitch angle). New strategies to the 6D SLAM challenge for 3D laser scans are proposed and a large choice of functions are presented.

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Extra info for 3D Robotic Mapping: The Simultaneous Localization and Mapping Problem with Six Degrees of Freedom

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We conclude, that ˙ = α21 λ1 + α22 λ2 + α23 λ3 + α24 λ4 q˙ T N q˙ = q˙ · (N q) holds. 1 The ICP Algorithm 45 q˙ T N q˙ ≤ α21 λ1 + α22 λ1 + α23 λ1 + α24 λ1 = λ1 . This shows that the quadratic form is never larger than the largest eigenvalue. Choosing α1 = 1 and α2 = α3 = α4 = 0 the maximal value is reached and the unit quaternion is q˙ = e˙ 1 [61]. After the computation of the rotation matrix R the translation is t = cm − Rcd . Computing the Tranformation using Dual Quaterions The transformation that minimizes Eq.

The analogous algorithm is derived directly from this theorem. Proof. 1 The ICP Algorithm 39 N N N 2 ||mi || − 2 E(R, t) = i=1 2 mi · Rdi + i=1 ||di || . i=1 The rotation affects only the middle term, thus it is sufficient to maximize N N mi · Rdi = i=1 mi T Rdi . 9). Now we have to find the matrix R that maximizes tr (RH). Lemma 4. For all positive definite matrices AAT and every orthonormal matrix B it holds: tr AAT ≥ tr BAAT . Proof. Assume ai to be the i-th column of A. With that we can calculate tr BAAT = tr AT BA N aTi (BaTi ).

There is no influence to following computations; the amount of memory is usually reduced from O(n2 ) to O(n). Closed Form Solution Four algorithms are currently known that solve the error function of the ICP algorithm in closed form [76]. The difficulty of this minimization is to enforce the orthonormality constraint for the rotation matrix R. In the following the four algorithms are are presented. The first three algorithms separate the computation of the rotation R from the computation of the translation t.

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