*Article* **The Influence of Mathematical Definitions on Patellar Kinematics Representations**

**Adrian Sauer 1,2,\* , Maeruan Kebbach <sup>3</sup> , Allan Maas 1,2, William M. Mihalko <sup>4</sup> and Thomas M. Grupp 1,2**


**Abstract:** A correlation between patellar kinematics and anterior knee pain is widely accepted. However, there is no consensus on how they are connected or what profile of patellar kinematics would minimize anterior knee pain. Nevertheless, answering this question by merging existing studies is further complicated by the variety of ways to describe patellar kinematics. Therefore, this study describes the most frequently used conventions for defining patellar kinematics, focusing on the rotations. The similarities and differences between the Cardan sequences and angles calculated by projecting axes are analyzed. Additionally, a tool is provided to enable the conversion of kinematic data between definitions in different studies. The choice of convention has a considerable impact on the absolute values and the clinical characteristics of the patello-femoral angles. In fact, the angles that result from using different mathematical conventions to describe a given patello-femoral rotation from our analyses differ up to a Root Mean Squared Error of 111.49◦ for patellar flexion, 55.72◦ for patellar spin and 35.39◦ for patellar tilt. To compare clinical kinematic patello-femoral results, every dataset must follow the same convention. Furthermore, researchers should be aware of the used convention's implications to ensure reproducibility when interpreting and comparing such data.

**Keywords:** knee joint; patello-femoral joint; kinematics; cardan sequence; euler angles; conversion
