By Andreas Nüchter
The monograph written through Andreas Nüchter is targeted on buying spatial types of actual environments via cellular robots. The robot mapping challenge is usually often called SLAM (simultaneous localization and mapping). 3D maps are essential to steer clear of collisions with advanced stumbling blocks 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 functions are presented.
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Extra resources for 3D Robotic Mapping: The Simultaneous Localization and Mapping Problem with Six Degrees of Freedom
To determine the trace tr (RH) one uses the above method and yields tr (RH) = = tr (RP S) √ √ λ1 tr RP u1 uT1 + λ2 tr RP u2 uT2 √ + λ3 tr RP u3 uT3 . 13) The following lemma is true for all matrices: Lemma 5. For all matrices X and Y , for which the products XY and Y X yield a square matrix, it holds tr (XY ) = tr (Y X). 13) holds: tr RP ui uTi = tr uTi RT P ui = tr RT ui · P ui = RT ui · P ui . 42 4 3D Range Image Registration Since the vectors ui are unit vectors by construction and P and R are orthonormal matrices, we have RT ui · P ui ≤ 1.
It is: E T t0 = (T R)T t0 = −RT T × t0 = −RT t0 × t0 = RT 0 = 0 · T Lemma 2. The singular value decomposition of E yields E = U SV with the singular values σ0 = 1, σ1 = 1 and σ2 = 0. Proof. See . The singular value decomposition E = U SV yields orthonormal matrices U and V . S is a diagonal matrix that contains the singular values σi : ⎞ ⎛ σ0 0 0 ⎟ ⎜ S = ⎝ 0 σ1 0 ⎠ 0 0 σ3 Lemma 3. It holds : E 2 = tr E T E = 2 Proof. See . Ê Ê Unfortunately the 3D coordinates pl , pr ∈ 3 are usually unknown.
The depth resolution of a stereo camera is the smallest distinguishable depth. It depends on the smallest distinguishable disparity d and the distance between the cameras b and the focal length f . , decrease the size of the pixel. , objects that are only visible from one camera or that are between the two cameras and close to the stereo rig cannot be seen. To reduce the blind area it is often necessary to place the cameras close to each other, which in turn reduces depth resolution. 16 does not occur, since it is not possible to adjust the cameras such that the image sensors lie in one plane.