Abstract—Accurately representing spatial transformations in robotics is crucial for reliable system performance. Traditional methods often fail to account for internal inaccuracies and environmental factors, leading to significant errors. This work introduces a framework that incorporates uncertainty into transformation trees using Lie Algebra, offering a consistent and realistic computation of spatial transformations. Our approach models inaccuracies from sensor decalibration, joint position errors, mechanical stress, and gravitational influences, as well as environmental uncertainties from perception limitations. By integrating probabilistic models into transformation calculations, we provide a robust and adaptable solution for various robotic applications. The framework is implemented using a C++ library with a Python wrapper, leveraging hierarchical transformation trees to simplify kinematic chains and apply uncertainty propagation. Real-world examples demonstrate the framework’s effectiveness: compensating for gravitational bending in a robotic arm and handling uncertainties in a mapping task with an uncertain kinematic. These applications highlight the framework’s ability to enhance the accuracy and reliability of tasks such as manipulation, navigation, and interaction with environments. This contribution aims to advance robotic systems’ performance by providing a comprehensive method for managing spatial transformation uncertainties.

Keywords—robotics, transformation tree, uncertainty modeling, Lie Algebra

Cite: Marco Sewtz, Lukas Burkhard, Xiaozhou Luo, Leon Dorscht, and Rudolph Triebel, "Representing Uncertain Spatial Transformations in Robotic Applications in a Structured Framework Leveraging Lie Algebra," International Journal of Mechanical Engineering and Robotics Research, Vol. 14, No. 1, pp. 1-9, 2025. doi: 10.18178/ijmerr.14.1.1-9

Copyright © 2025 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).