Completely Rational $\text{SO}(n)$ Orthonormalization

The rotation orthonormalization on the special orthogonal group
$\text{SO}(n)$
, also known as the high dimensional nearest rotation problem, has been revisited. A new generalized simple iterative formula has been proposed that solves this problem in a completely rational manner. Rational operations allow for efficient implementation on various platforms and also significantly simplify the synthesis of large-scale circuitization. The developed scheme is also capable of designing efficient fundamental rational algorithms, for example, quaternion normalization, which outperforms long-exisiting solvers. Furthermore, an
$\text{SO}(n)$
neural network has been developed for further learning purpose on the rotation group. Simulation results verify the effectiveness of the proposed scheme and show the superiority against existing representatives. Applications show that the proposed orthonormalizer is of potential in robotic pose estimation problems, e.g., hand-eye calibration.