By Andreas Nüchter
The monograph written by means of Andreas Nüchter is concentrated on buying spatial versions of actual environments via cellular robots. The robot mapping challenge is often known as SLAM (simultaneous localization and mapping). 3D maps are essential to stay 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 options to the 6D SLAM challenge for 3D laser scans are proposed and a large choice of purposes are presented.
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Extra resources for 3D Robotic Mapping: The Simultaneous Localization and Mapping Problem with Six Degrees of Freedom
3, Lemma 7). It follows, that tr (RP S) ≤ λ1 + λ2 + λ3 = tr (S) holds. According to this the maximum of tr (RP S) is reached, if RT P = ½ and accordingly R = P (cf. 3, Lemma 9 and Corollary 1). 12), is the rotation matrix and it holds: R = P = HS −1 = H(H T H)−1/2 . Again, the optimal translation is calculated as (cf. Eq. 7)) t = cm − Rcd . Computing the Tranformation using Unit Quaternions The transformation for the ICP algorithm can be calculated by a method that uses unit quaternions. This method was invented 1987 by Horn .
10. All rays go through the focal point o. Objects of same size that are at the same distance to o yield projections of the same size. A point p in the three-dimensional space has the coordinates (cx, c y, cz) relative to the camera center c. The point p is projected to the image point p in the image plane with the coordinates (u, v). The latter coordinates are relative to the image center c that is the intersection between the optical axis and the image plane (cf. 11). The dependence of the 3D coordinates in the camera coordinate system and the image coordinates with a focal length f is given by: u v f −c = c = c .
The optimal displacement calculated by Eq. 31) corresponds to an aﬃne motion. Therefore in a post processing step a rigid transformation (R, t) is calculated from (¯ c, c). 2 presents the displacement of a point using the aﬃne transformation and rigid transformation. G xi xi + v(xi ) p·ϕ ϕ xi Fig. 2 The aﬃne position of a 3D point xi + v(xi ) is diﬀerent from the rigid transformation that results in point xi .. Based on . The rigid transformation is calculated as follows: If c = 0 only a translation ¯.