**4. Discussion**

may be facilitated.

**4. Discussion**  The first objective of this study was to show the considerable impact of different mathematical definitions on how patello-femoral kinematics are conveyed. A framework was thereby developed to enable researchers to choose the definitions for the patellar kin-The first objective of this study was to show the considerable impact of different mathematical definitions on how patello-femoral kinematics are conveyed. A framework was thereby developed to enable researchers to choose the definitions for the patellar kinematics which best fit their needs and to convert patellar kinematics between the

ematics which best fit their needs and to convert patellar kinematics between the different

different conventions. In this way, comparability and interchangeability between different studies may be facilitated.

The results obtained from the conversions we have presented using the described formulas are in excellent agreement with those obtained from the evaluation using a general multibody software package (RMSE < (6.13 <sup>×</sup> <sup>10</sup>−<sup>5</sup> ) ◦ for all sequences and angles). Therefore, the derived equations and their implementation were verified.

The presented results indicate clearly that the chosen definition for patello-femoral angles has an impact on the description of patello-femoral kinematics. The curves for patellar flexion, spin and tilt deviate not only in magnitude but also in their characteristics (see Figures 5 and 6). Therefore a correct interpretation of patello-femoral kinematic data seems to be impossible if the details of the convention used are not clear.

Deviations between the results with different definitions only occur if rotations around at least two standard directions differ from zero. This condition is based on the fact that the differences are caused by changes in rotation axes for subsequent rotations. For increasing tibio-femoral flexion angles the absolute patellar flexion angle reaches values close to 80◦ , while the absolute values of patellar spin and tilt stay relatively small if the rotations are considered around the patellar axes. Therefore, the rotation around the *x*-axis and its position in the Cardan sequence has a special impact on the data for high tibio-femoral and patello-femoral flexion angles. In the case of small patellar spin and tilt, it can be assumed that the rotation axes that occur before the patellar flexion in the Cardan sequence almost agree with the associated femoral coordinate system axes. The rotations placed after the flexion in the sequence can be interpreted as rotations around associated patellar body axes. For the first and last rotations of a Cardan sequence these statements are exactly true.

For the patello-femoral kinematics this means that the first rotation of a sequence is always around the associated axis of the femoral coordinate system and the last rotation is around a patella-fixed axis, as previously shown in Figure 2. The floating axis between these two is close to the associated patellar axis if the sequence starts with the patellar flexion and is close to the associated femoral axis otherwise. In accordance with the previous explanations the two Cardan sequences starting and ending with the patellar flexion (XYZ/XZY and YZX/ZYX) show very small differences (see Figure 5, and Tables 1–3).

For 90◦ of absolute patellar flexion angle patellar spin and tilt would swap its values if the patellar flexion would be changed from the first position (i.e., tilt and spin close to patellar body fixed rotations) to the third position (i.e., tilt and spin close to femoral body fixed rotation). This is caused by the inherent change from patellar to femoral coordinate system. For the squat cycle showed here, only 80◦ of absolute patellar flexion are reached. Nevertheless, the described effect can be seen, if the maximum patellar tilt for the sequences XYZ and XZY (15.0◦ ) is compared to the patellar spin for YZX and ZYX (15.7◦ ).

One of the main difficulties in using Cardan sequences to describe rotational poses is the gimbal lock, which occurs if the second rotation of the sequence is equal to 90◦ . In this case, the axis of the third rotation is parallel to the direction of the first one (or its negative) and the rotation loses one degree of freedom. The absolute values for spin and tilt are far from 90◦ for healthy knees [6,7]. Therefore, this situation only exists if the patellar flexion is the second rotation of the sequence. Even if the maximum absolute values for patellar flexion in the shown results are less than 80◦ , an increase of spin and tilt can be mentioned for the YXZ sequence and a comparably larger one for the ZXY sequence. (see Tables 2 and 3). The reason why ZXY differs more from the sequences without the x-rotation on second position than YXZ is that the patello-femoral rotation around the femoral *z*-axis is bigger than around the femoral *y*-axis. These effects are making the interpretation of the patellar kinematics in the ZXY and YXZ sequence quite laborious and this description differs a lot from a clinical understanding for spin and tilt. Therefore these sequences are not recommended.

The projected angles also show big differences dependent on the choice of which axis is projected onto which coordinate system. Figure 6 indicates clearly that this method is very sensitive to the choice of projection axis and that this effect increases with growing angles in the other two dimensions. The flexion angles are robust against switching from *y*- to *z*-axis for projection, while comparable changes for spin and tilt cause completely different curves due to the relatively higher patellar flexion angles.

For small angles of spin and tilt the projected angles for flexion on the patellar (or femoral) planes can be interpreted as flexion with respect to the patellar (or femoral) system. Due to the high flexion angles, the same interpretation cannot be applied to spin and tilt. For spin and tilt the values are also highly depending on the choice of axis for projection.

It was shown that the deviations between the curves for patellar flexion, spin and tilt for the different kinematic definitions can completely change in both, magnitude and characteristics of the curves. Nevertheless, the differences are not big enough in every case that inadvertently comparing results based on different definitions would be obvious at first glance. Therefore the problem explored by this study should always be considered when dealing with patellar kinematics.

Patellar maltracking is known as possible cause of anterior knee pain [4,5,52–55], but it still remains unknown, how critical patellar kinematics can be distinguished from others [6]. To increase the biomechanical knowledge about how anterior knee pain can be prevented or treated, it will help to bring the available data from the literature together. Therefore, it is essential to properly understand the underlying definitions of every single study and to carefully transform the data into one representation, that can be compared across the available literature. For most of the studies available in literature the given transformations from this study can be utilized.

If researchers are free to choose a convention for their own study, the use of Cardan sequences is recommended due to the available straight forward methods for calculation, conversion and interpretation. Even if all Cardan sequences are theoretically suitable to represent the patello-femoral kinematics in a correct way, some of them are better for intuitive interpretation from a biomechanical side of view. The definitions which are closest to the clinical understanding of patellar flexion, tilt and spin are the two Cardan sequences beginning with patellar flexion (XYZ and XZY), where patellar flexion is given around the *femoral* flexion axis. The last rotation of the sequence is given exactly around the associated *patellar* axis while the second rotation of the sequence is executed around the floating axis perpendicular to the other two. Therefore, the rotation (out of spin and tilt) with the most importance for a certain study should be chosen as last rotation of the sequence. The higher clinical relevance of tilt compared to patellar spin will qualify the Cardan sequence XYZ as recommendation for most studies.
