Advanced robotics and autonomous vehicles rely on filtering and sensor fusion techniques to a large extent. These mobile applications need to handle the computations onboard at high rates while the computing capacities are limited. Therefore, any improvement that lowers the CPU time of the filtering leads to more accurate control or longer battery operation. This article introduces a generic computational relaxation for the unscented transformation (UT) that is the key operation of the Unscented Kalman filter-based applications. The central idea behind the relaxation is to pull out the linear part of the filtering model and avoid the calculations for the kernel of the nonlinear part. The practical merit of the proposed relaxation is demonstrated through a simultaneous localization and mapping (SLAM) implementation that underpins the superior performance of the algorithm in the practically relevant cases, where the nonlinear dependencies influence only an affine subspace of the image space. The numerical examples show that the computational demand can be mitigated below 50% without decreasing the accuracy of the approximation. The method described in this article is implemented and published as an open-source C++ library RelaxedUnscentedTransformation on GitHub.