*2.1. Coordinates and Definitions*

In the following description, two reference frames will mainly be used: one attached to the femur and another to the patella, with the directions x, y and z pointing laterally, anteriorly and proximally, respectively, in full extension. If the femur or patella rotate and translate during flexion, the attached coordinate systems will follow the bones' movements. Figure 1 shows a possible system of axes for the patellar kinematics. Since the exact orientation depends on the chosen definition, it is neither described nor shown in more detail. The question of how these directions can be identified, i.e., which landmarks can be used, is highly dependent on the data and the measurement methods of every single study and will therefore not be deeply discussed. A possibility to define a reference frame for the patella is presented and validated by Innocenti et al. [19].

The most clinically relevant kinematic parameters are patellar shift, patellar flexion (α), spin (β) and tilt (γ) [5,17,20–22]. The patella's resulting movements are relative to the femur as follows: patellar flexion is defined by the relative rotation of the patella around the medio-lateral flexion axis (*x*-axis). In most cases, the *femoral* flexion axis is used as the rotation axis for patellar flexion. In a clinical context, patellar spin and tilt are represented by the rotations around the local *patellar* y- and z-axes, respectively. Alternatively, it is also common to use a floating axis for patellar tilt [23,24] or patellar spin [25–27]. The patellar shift is the translation of the patella in the medio-lateral direction with respect to the femur or the trochlear groove. Usually, the patellar shift is given as the patellar movement in the direction of the *femoral x*-axis, which can be calculated by projecting the vector of the relative patellar translations on the normalized direction of the femoral flexion axis. The values of shift, flexion, spin and tilt for a given instance of patellar motion can differ considerably from this intuitive understanding if different definitions are used.

Figure 1 shows the described components of the patellar kinematics schematically without giving precise axes and signs because these are dependent on the chosen definition.

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**Figure 1.** Overview on patellar kinematics and underlying coordinate systems. **Left**: patellar flexion in a sagittal view, **center**: medio-lateral movement (shift) and patellar spin in an anterior view, **right**: patellar tilt in a proximal view. The exact location and orientation of the rotation axes depend on the chosen definition and is therefore not shown in detail. The patellar axes are labelled with xp, yp, zp and the femoral axes with xf, yf, zf. **Figure 1.** Overview on patellar kinematics and underlying coordinate systems. (**Left**): patellar flexion in a sagittal view, (**center**): medio-lateral movement (shift) and patellar spin in an anterior view, (**right**): patellar tilt in a proximal view. The exact location and orientation of the rotation axes depend on the chosen definition and is therefore not shown in detail. The patellar axes are labelled with xp, yp, z<sup>p</sup> and the femoral axes with x<sup>f</sup> , y<sup>f</sup> , z<sup>f</sup> .

The most clinically relevant kinematic parameters are patellar shift, patellar flexion (α), spin (β) and tilt (γ) [5,17,20–22]. The patella's resulting movements are relative to the femur as follows: patellar flexion is defined by the relative rotation of the patella around the medio-lateral flexion axis (*x*-axis). In most cases, the *femoral* flexion axis is used as the rotation axis for patellar flexion. In a clinical context, patellar spin and tilt are represented by the rotations around the local *patellar* y- and z-axes, respectively. Alternatively, it is The movements of the patella in the sagittal plane are highly dependent on the definition of the origin of the femoral coordinate system. If individual anatomical landmarks are used to define this origin, the comparison between different subjects can be problematic. However, since these movements are of secondary clinical interest [17] and are controlled mainly by joint geometry [28], they are not discussed here.

### also common to use a floating axis for patellar tilt [23,24] or patellar spin [25–27]. The patellar shift is the translation of the patella in the medio-lateral direction with respect to *2.2. How Rotations Can Be Described*

the femur or the trochlear groove. Usually, the patellar shift is given as the patellar movement in the direction of the *femoral x*-axis, which can be calculated by projecting the vector of the relative patellar translations on the normalized direction of the femoral flexion axis. The values of shift, flexion, spin and tilt for a given instance of patellar motion can differ considerably from this intuitive understanding if different definitions are used. Figure 1 shows the described components of the patellar kinematics schematically without giving There are several ways to describe the rotational state of an object in a coordinate system. For example, well-known methods include rotational matrices, quaternions, projected angles, helical axes, and extrinsic and intrinsic rotation sequences. In biomechanics, representations allowing the comparison of rotations around the individual joint axes are needed. Therefore, rotation matrices, quaternions and helical axes are only suitable to process data, but not to interpret the results of studies according to patellar kinematics.

precise axes and signs because these are dependent on the chosen definition. The movements of the patella in the sagittal plane are highly dependent on the definition of the origin of the femoral coordinate system. If individual anatomical landmarks are used to define this origin, the comparison between different subjects can be problematic. However, since these movements are of secondary clinical interest [17] and are con-First, a fundamental property of rotations should be noted: rotations are not commutative. This means that—regardless of the mathematical description—the resulting pose for a sequence of rotations about multiple axes is generally not equal to the same rotations in another sequence. This statement remains valid for both multiple rotations in time and rotation sequences to describe a pose at one point in time.

trolled mainly by joint geometry [28], they are not discussed here. *2.2. How Rotations Can Be Described*  There are several ways to describe the rotational state of an object in a coordinate system. For example, well-known methods include rotational matrices, quaternions, projected angles, helical axes, and extrinsic and intrinsic rotation sequences. In biomechanics, representations allowing the comparison of rotations around the individual joint axes are To describe any rotation in three-dimensional (3D) space, a rotation sequence consisting of three consecutive rotations around three axes can be used. If the orientation of the axes is changed by each elemental rotation, the sequence is referred to as *intrinsic*. For initial coordinate axes ξ, η, ζ, the axes after the first rotation are written as ξ', η', ζ' and after the second rotation as ξ", η", ζ". If all rotations occur around the axes as they are in their initial orientation, independent of previous rotation steps, the rotation sequence is called an *extrinsic* rotation.

needed. Therefore, rotation matrices, quaternions and helical axes are only suitable to process data, but not to interpret the results of studies according to patellar kinematics. First, a fundamental property of rotations should be noted: rotations are not commutative. This means that—regardless of the mathematical description—the resulting pose for a sequence of rotations about multiple axes is generally not equal to the same rotations In biomechanical literature, sometimes an intrinsic sequence of rotations around *three different axes* is called an Euler rotation, and the associated angles are termed *Euler angles* [29,30]. The correct term for these angles is in fact *Cardan angles* or *Tait-Bryan-angles*. Actual Euler angles are an intrinsic rotation sequence, which uses the same axis for the first and the third rotation (e.g., rotations around the axes ξ, η', ξ") [31]. While Cardan angles

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rotation sequences to describe a pose at one point in time.

an *extrinsic* rotation.

can be used for a biomechanical description of patello-femoral rotations, Euler angles are difficult to relate to the clinical terms of patellar flexion, spin and tilt. Figure 2 shows three different rotation sequences which lead to a particular patellar orientation. of the sequence is ±90° [33]. In this case, the axes of the first and third rotation are collinear, and one degree of freedom is lost. Since the patellar spin and tilt do not reach absolute values of 90° [6,7] sequences with either of these in the middle position are free from this problem for the description of patellar kinematics.

Two disadvantages are occasionally associated in connection with Cardan angles. The first, sequence dependency, can simply be solved by appropriate standardization of the sequence [32]. The second is termed gimbal lock, which occurs when the second angle

different rotation sequences which lead to a particular patellar orientation.

in another sequence. This statement remains valid for both multiple rotations in time and

To describe any rotation in three-dimensional (3D) space, a rotation sequence consisting of three consecutive rotations around three axes can be used. If the orientation of the axes is changed by each elemental rotation, the sequence is referred to as *intrinsic*. For initial coordinate axes ξ, η, ζ, the axes after the first rotation are written as ξ', η', ζ' and after the second rotation as ξ'', η'', ζ''. If all rotations occur around the axes as they are in their initial orientation, independent of previous rotation steps, the rotation sequence is called

In biomechanical literature, sometimes an intrinsic sequence of rotations around *three different axes* is called an Euler rotation, and the associated angles are termed *Euler angles* [29,30]. The correct term for these angles is in fact *Cardan angles* or *Tait-Bryan-angles*. Actual Euler angles are an intrinsic rotation sequence, which uses the same axis for the first and the third rotation (e.g., rotations around the axes ξ, η', ξ'') [31]. While Cardan angles can be used for a biomechanical description of patello-femoral rotations, Euler angles are difficult to relate to the clinical terms of patellar flexion, spin and tilt. Figure 2 shows three

**Figure 2.** Illustration of three exemplary rotation sequences representing one particular patellar orientation. All sequences start in a patellar orientation where the patellar axes are parallel to the femoral axes. The upper and middle paths show the Cardan sequences XYZ and XZY, respectively, and the lower path represents the Euler sequence XYX. **Figure 2.** Illustration of three exemplary rotation sequences representing one particular patellar orientation. All sequences start in a patellar orientation where the patellar axes are parallel to the femoral axes. The upper and middle paths show the Cardan sequences XYZ and XZY, respectively, and the lower path represents the Euler sequence XYX.

If femoral axes are used to describe the patello-femoral rotations [34], the interpretation of the rotations around the femoral anterior-posterior and proximo-distal axes as patellar spin and patellar tilt, respectively, are in general no longer correct. For a patellar Two disadvantages are occasionally associated in connection with Cardan angles. The first, sequence dependency, can simply be solved by appropriate standardization of the sequence [32]. The second is termed gimbal lock, which occurs when the second angle of the sequence is ±90◦ [33]. In this case, the axes of the first and third rotation are collinear, and one degree of freedom is lost. Since the patellar spin and tilt do not reach absolute values of 90◦ [6,7] sequences with either of these in the middle position are free from this problem for the description of patellar kinematics.

If femoral axes are used to describe the patello-femoral rotations [34], the interpretation of the rotations around the femoral anterior-posterior and proximo-distal axes as patellar spin and patellar tilt, respectively, are in general no longer correct. For a patellar flexion near 0◦ the deviation remains small, but for one of 90◦ the femoral tilt axis is parallel to the patellar spin axis in a sagittal view. The same applies to the femoral rotation axis and the patellar tilt axis. In this case, the calculated values for tilt and spin of the patella are switched, in addition to a possible inversion of their sign, relative to the common clinical interpretation [17] (see also Figure 3). Researchers should be aware that Cardan sequences ending with patellar flexion (ZYX and YZX) show the same pattern (e.g., [35]).

For knee kinematics, it seems that there exists another valuable way to describe the rotations. It is called the *three-cylinder open-chain representation*. For the tibio-femoral joint, it was first mentioned by Grood and Suntay in 1983 [18]. It was suggested in 2009 by Merican and Amis [36] and later on proven in 2014 by MacWilliams and Davis [37] that the Grood and Suntay definition is equivalent to a Cardan XYZ-sequence. This sequence was regardless found suitable to describe the tibio-femoral kinematics [38,39].

[35]).

regardless found suitable to describe the tibio-femoral kinematics [38,39].

**Figure 3.** Illustration of the switch of meaning of patellar spin and tilt for an absolute patellar flexion angle of 90°, if described with respect to the femoral coordinate system. Rotating the patella around the anatomical patellar proximo-distal axis (zp) equals a rotation around the femoral anterior-posterior axis (yf) for 90° of patello-femoral flexion. The zf- and yp-axes show a similar pattern. **Figure 3.** Illustration of the switch of meaning of patellar spin and tilt for an absolute patellar flexion angle of 90◦ , if described with respect to the femoral coordinate system. Rotating the patella around the anatomical patellar proximo-distal axis (zp) equals a rotation around the femoral anteriorposterior axis (y<sup>f</sup> ) for 90◦ of patello-femoral flexion. The z<sup>f</sup> - and yp-axes show a similar pattern.

flexion near 0° the deviation remains small, but for one of 90° the femoral tilt axis is parallel to the patellar spin axis in a sagittal view. The same applies to the femoral rotation axis and the patellar tilt axis. In this case, the calculated values for tilt and spin of the patella are switched, in addition to a possible inversion of their sign, relative to the common clinical interpretation [17] (see also Figure 3). Researchers should be aware that Cardan sequences ending with patellar flexion (ZYX and YZX) show the same pattern (e.g.,

For knee kinematics, it seems that there exists another valuable way to describe the rotations. It is called the *three-cylinder open-chain representation*. For the tibio-femoral joint, it was first mentioned by Grood and Suntay in 1983 [18]. It was suggested in 2009 by Merican and Amis [36] and later on proven in 2014 by MacWilliams and Davis [37] that the Grood and Suntay definition is equivalent to a Cardan XYZ-sequence. This sequence was

For the patello-femoral joint, the three-cylinder open-chain representation was first applied in 1992 by Hefzy et al. [16]. Since this way to describe patellar kinematics was recommended in a method paper [17], it is frequently used. The proof of the equivalence of the Grood and Suntay rotations and a Cardan sequence for the tibio-femoral joint [37] can be carried out analogously for the patello-femoral joint. Therefore, the three-cylinder open-chain representation of the patello-femoral kinematics is—from the rotations point of view—equivalent to the use of the Cardan sequence XYZ. Since Bull et al. [17] do not describe translations in a sagittal plane, their recommendation is to measure the patellar For the patello-femoral joint, the three-cylinder open-chain representation was first applied in 1992 by Hefzy et al. [16]. Since this way to describe patellar kinematics was recommended in a method paper [17], it is frequently used. The proof of the equivalence of the Grood and Suntay rotations and a Cardan sequence for the tibio-femoral joint [37] can be carried out analogously for the patello-femoral joint. Therefore, the three-cylinder open-chain representation of the patello-femoral kinematics is—from the rotations point of view—equivalent to the use of the Cardan sequence XYZ. Since Bull et al. [17] do not describe translations in a sagittal plane, their recommendation is to measure the patellar shift relative to the femoral medio-lateral axis and describe the patellar rotations with respect to the femoral axes as a Cardan XYZ-sequence.

The methods, which use a sequence of rotations around rotated or not rotated coordinate axes, lead to the inherent possibility to receive a description of the pose of the associated body very straightforward by executing the rotations one after the other. Nevertheless, some authors [40,41] determine the rotations of the patella by projecting the patellar coordinate vectors on the planes of the femoral coordinate system and calculate the angle between these projected vectors and the associated vectors from the femoral system. For instance, the patellar flexion can be calculated by projecting the y-vector of the patella onto the sagittal plane of the femoral coordinate system and calculating the angle between this projection and the y-vector of the femoral system. If a projection is carried out onto the patellar coordinate planes, a rotation relative to the patellar axes can be calculated. This method in general does not lead to values for patellar flexion, spin and tilt which can be executed in sequence to acquire the full rotation of the patella relative to the femur. For every projection plane, there are two vectors that can be projected to calculate the rotation around the axis perpendicular to this plane. As the results of this paper will show, the calculated angles for these two projected directions differ a lot for

3D rotations. For example, the calculation of the patellar flexion as the angle between the patellar *y*-axis projected onto the sagittal plane of the femoral system differs from the same calculation with the z-axes. Therefore, the angles are highly dependent on the choice of planes and axes. around the axis perpendicular to this plane. As the results of this paper will show, the calculated angles for these two projected directions differ a lot for 3D rotations. For example, the calculation of the patellar flexion as the angle between the patellar *y*-axis projected onto the sagittal plane of the femoral system differs from the same calculation with the zaxes. Therefore, the angles are highly dependent on the choice of planes and axes.

shift relative to the femoral medio-lateral axis and describe the patellar rotations with re-

The methods, which use a sequence of rotations around rotated or not rotated coordinate axes, lead to the inherent possibility to receive a description of the pose of the associated body very straightforward by executing the rotations one after the other. Nevertheless, some authors [40,41] determine the rotations of the patella by projecting the patellar coordinate vectors on the planes of the femoral coordinate system and calculate the angle between these projected vectors and the associated vectors from the femoral system. For instance, the patellar flexion can be calculated by projecting the y-vector of the patella onto the sagittal plane of the femoral coordinate system and calculating the angle between this projection and the y-vector of the femoral system. If a projection is carried out onto the patellar coordinate planes, a rotation relative to the patellar axes can be calculated. This method in general does not lead to values for patellar flexion, spin and tilt which can be executed in sequence to acquire the full rotation of the patella relative to the femur. For every projection plane, there are two vectors that can be projected to calculate the rotation
