*3.1. Cardan Angles*

Figure 5 shows the rotation angles for all possible Cardan sequences. The choice of definition has a minor effect on the patellar flexion values. All Cardan sequences except for ZXY have very similar flexion angles. The maximum RMSE within this group is 0.79◦ (see also Table 1). The patellar flexion angle of the ZXY-sequence differs up to 7.83◦ from the others (RMSE ≥ 3.71◦ ). *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 16

**Figure 5.** Patellar kinematics for the Cardan sequences. Patellar flexion (top), spin (center) and tilt **Figure 5.** Patellar kinematics for the Cardan sequences. Patellar flexion (**top**), spin (**center**) and tilt (**below**) during a squat motion for all Cardan sequences.

(below) during a squat motion for all Cardan sequences.


**Table 1.** RMSE of patellar flexion curves for Cardan sequences and projected angles in ◦ .

> In terms of patellar spin and patellar tilt the Cardan XYZ- and XZY-sequences are very close (spin: RMSE = 0.09◦ and tilt: RMSE = 0.03◦ ) as Tables 2 and 3 show. The same is true for YZX and ZYX (spin: RMSE = 0.004◦ and tilt: RMSE = 0.03◦ ). The sequence ZXY leads to curves which differ considerably from those of other sequences (RMSE up to 27.72◦ ). These deviations are particularly large and grow increasingly for higher tibio-femoral flexion angles, and reach a maximum of 49.22◦ for the patellar spin and 59.91◦ for the patellar tilt.

> > .

**Table 2.** RMSE of patellar spin curves for Cardan sequences and projected angles in ◦


**Table 3.** RMSE of patellar tilt curves for Cardan sequences and projected angles in ◦ .


*3.2. Projected Angles*

The projected angles can be divided into two groups. Projecting the patellar reference frame vectors onto the femoral planes gives patello-femoral rotations with respect to the femoral system. Therefore, the angles for patellar flexion are close to the negative of the

angles obtained by projecting onto the patellar planes (see also Figure 6, top). The flexion angles calculated by projecting the *y*- (or *z*-) axis on the femoral standard plane are equal to the patellar flexion angles from the Cardan XZY- (or XYZ- for the *z*-axis) sequence as the RMSE of 0◦ for these combinations shows in Table 1. The same pattern is visible for the patellar spin and tilt if the projections on femoral planes and Cardan sequences starting with y- and z-axes are analyzed (see also Tables 2 and 3). *Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 16

**Figure 6.** Patellar kinematics using projected angles. Patellar flexion (**top**), spin (**center**) and tilt (**below**) during a squat motion for all possible ways to calculate the angles by projection. **Figure 6.** Patellar kinematics using projected angles. Patellar flexion (**top**), spin (**center**) and tilt (**below**) during a squat motion for all possible ways to calculate the angles by projection.
