Analysis and Evaluation of Fiber Orientation Reconstruction Methods

The calculation of the fiber orientation of short fiber-reinforced plastics with the Fokker–Planck equation requires a considerable numerical effort, which is practically not feasible for injection molding simulations. Therefore, only the fiber orientation tensors are determined, i.e., by the Folgar–Tucker equation, which requires much less computational effort. However, spatial fiber orientation must be reconstructed from the fiber orientation tensors in advance for structural simulations. In this contribution, two reconstruction methods were investigated and evaluated using generated test scenarios and experimentally measured fiber orientation. The reconstruction methods include spherical harmonics up to the 8th order and the method of maximum entropy, with which a Bingham distribution is reconstructed. It is shown that the quality of the reconstruction depends massively on the original fiber orientation to be reconstructed. If the original distribution can be regarded as a Bingham distribution in good approximation, the method of maximum entropy is superior to spherical harmonics. If there is no Bingham distribution, spherical harmonics is more suitable due to its greater flexibility, but only if sufficiently high orders of the fiber orientation tensor can be determined exactly.

Probabilistic pose estimation using a Bingham distribution-based linear filter

Pose estimation is central to several robotics applications such as registration, hand–eye calibration, and simultaneous localization and mapping (SLAM). Online pose estimation methods typically use Gaussian distributions to describe the uncertainty in the pose parameters. Such a description can be inadequate when using parameters such as unit quaternions that are not unimodally distributed. A Bingham distribution can effectively model the uncertainty in unit quaternions, as it has antipodal symmetry, and is defined on a unit hypersphere. A combination of Gaussian and Bingham distributions is used to develop a truly linear filter that accurately estimates the distribution of the pose parameters. The linear filter, however, comes at the cost of state-dependent measurement uncertainty. Using results from stochastic theory, we show that the state-dependent measurement uncertainty can be evaluated exactly. To show the broad applicability of this approach, we derive linear measurement models for applications that use position, surface-normal, and pose measurements. Experiments assert that this approach is robust to initial estimation errors as well as sensor noise. Compared with state-of-the-art methods, our approach takes fewer iterations to converge onto the correct pose estimate. The efficacy of the formulation is illustrated with a number of examples on standard datasets as well as real-world experiments.