Ehrig, R.M. and Heller, M.O.
(2019)
On intrinsic equivalences of the finite helical axis, the instantaneous helical axis, and the SARA approach. A mathematical perspective.
*Journal of Biomechanics*.
(doi:10.1016/j.jbiomech.2018.12.034).

## Abstract

Accurate determination of joint axes is essential for understanding musculoskeletal function. Whilst numerous algorithms to compute such axes exist, the conditions under which each of the methods performs best remain largely unknown. Typically, algorithms are evaluated for specific conditions only limiting the external validity of conclusions regarding their performance. We derive exact mathematical relationships between three commonly used algorithms for computing joint axes from motion data: finite helical axes (FHA), instantaneous helical axes (IHA) and SARA (symmetrical axis of rotation approach), including relationships for an extension to the mean helical axes methods that facilitate determining joint centres and axes. Through the derivation of a sound mathematical framework to objectively compare the algorithms we demonstrate that the FHA and SARA approach are equivalent for the analysis of two time frames. Moreover, we show that the position of a helical axis derived from the IHA using positional data is affected by a systematic error perpendicular to the true axis direction, whereas the axis direction is identical to those computed with either the FHA or SARA approach (true direction). Finally, with an appropriate choice of weighting factors the mean FHA (MFHA) method is equivalent to the Symmetrical Centre of Rotation Estimation (SCoRE) algorithm for determination of a Centre of Rotation (CoR), and similarly, equivalent to the SARA algorithm for determination of an Axis of Rotation (AoR). The deep understanding of the equivalences between methods presented here enables readers to choose numerically efficient, robust methods for determining AoRs and CoRs with confidence.

Keywords

Functional gait analysis

Joint axes

Joint centres

Helical axis

SARA

SCoRE

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1. Introduction

Accurate determination of joint kinematics is of great importance for understanding musculoskeletal function, where the former are often expressed based on the identification of centres and axes of rotation (Akbari Shandiz et al., 2016, Boeth et al., 2013, Bonny et al., 2017, Ellingson and Nuckley, 2015, Heller et al., 2011, Hooke et al., 2015, Meireles et al., 2017, Michaud et al., 2016, Monnet et al., 2007, Most et al., 2004, Piazza and Cavanagh, 2000, Sauret et al., 2016, Trepczynski et al., 2014, Trepczynski et al., 2012). The general concept of describing joint kinematics using screw theory is well established (Corke, 2017, Hunt, 1990, Reuleaux, 1875) and a number of approaches for determining an axis of rotation have been described in the literature (Cerveri et al., 2005, Ehrig et al., 2007, Gamage and Lasenby, 2002, Halvorsen et al., 1999, Schwartz and Rozumalski, 2005). However, the helical axes methods which include e.g. the Finite Helical Axis (FHA) and the Instantaneous Helical Axis (IHA) appear to be among the most often utilised approaches and the importance of the helical axes methods more generally is reflected in a wide spectrum of application areas including analyses of the movement of the hip (Besier et al., 2003, Camomilla et al., 2006), the intact and replaced knee (Akbari Shandiz et al., 2016, Besier et al., 2003, Colle et al., 2012, Colle et al., 2016, De Rosario et al., 2017), the talocrural and subtalar joints of the foot (Beimers et al., 2008, Ferraresi et al., 2017, Lewis et al., 2006, Sheehan, 2010), the shoulder (Amabile et al., 2016, De Rosario et al., 2014, Lempereur et al., 2010, Monnet et al., 2007, Nikooyan et al., 2011, Wu et al., 2005), the elbow (Stokdijk et al., 1999, Veeger and Yu, 1996), the joints of the wrist and hand (Goislard de Monsabert et al., 2014, Hooke et al., 2015, Pfaeffle et al., 2005, Tay et al., 2010), as well as the analysis of the motion of the spine (Aiyangar et al., 2017, Anderst et al., 2013, Beyer et al., 2015), the assessment of the temporomandibular joint (Hayashi et al., 2009), and the movement of teeth (Hayashi et al., 2007). Whilst the IHA has a physical meaning and the FHA is considered to be a theoretical axis, they are related yet computationally different approaches based on the helical axis concept, where the motion of a segment relative to another at a specific time frame is described by a rotation around a line in space and a translation along this line. The use of this concept in biomechanics dates back to the nineteen seventies, see e.g. (Soudan et al. (1979), but has acquired considerable popularity through a series of seminal papers by Woltring and co-workers (Woltring and Huiskes, 1985; Woltring et al., 1983; Woltring, 1990; Woltring et al., 1987; Woltring et al., 1985, Woltring et al., 1994), and Spoor and Veldpaus (1980). Since then, both types of helical axes were studied in many publications and are now routinely used in motion analysis. Both helical axis methods were furthermore used to estimate joint centre and joint axis positions. Here, the centre is often defined as the point which is nearest to all elements of a set of helical axes while the axis is defined as the line which is most parallel to all helical axes, although they may also be derived from the optimisation of a cost function. More recently, an alternative methodology for determining an axis of rotation from motion data - SARA, Symmetrical Axis of Rotation Approach - was developed by Ehrig and co-workers (2007) through a different mathematical approach to identify the axis based on singular value decomposition of a set of rotation matrices.

A number of studies have attempted to compare the different algorithms, however, there is no general agreement about the conditions under which each of the methods would perform best or when a method is likely to produce less accurate results compared to its competitors. Whilst some theoretical analyses exist in the literature (De Rosario et al., 2017, De Rosario et al., 2014), we are not aware of any work presenting in detail a sound mathematical analysis of the respective algorithms and their intrinsic relationships. Because comparisons are typically performed for specific, selected data sets only, the external validity of any conclusion regarding the performance of the methods based on experimental data remains inherently limited (Colle et al., 2016, Gastaldi et al., 2015, Monnet et al., 2007).

Therefore, we derive the exact mathematical relationships between three commonly used algorithms for computing joint axes from motion data to provide a sound mathematical framework to objectively evaluate the relative strengths and weaknesses of the respective algorithms.

2. Mathematical Preliminaries

For the ease of presentation we assume that one segment of the joint is stationary. Under these conditions it is possible to describe the movement of the non-stationary segment by time

dependent rigid body transformations, i.e. rotations and translations that transform local coordinates defined on the moving segment into global coordinates. In practice, only discrete values and

will be known.

Rotation matrices are typically interpreted as orthogonal 3×3 matrices with a determinant equal to one. For clarity of the derivations here it is advantageous to express rotations directly in terms of the rotation axis unit vector

, defined in the stationary system, and the angle of rotation around that axis using the Rodrigues formula (Shabana, 2005) in the form:

(1)

where

is the identity matrix and is the cross-product matrix, i.e.:

(2)

where

holds for any vector With these definitions one may easily check that the following relationships, which will be repeatedly used in the following, also hold true:

(3)

The angle

and the vector can be determined from for using the relations:

(4)

where

is the trace of the matrix R.

2.1. Helical Axes

The determination of a helical axis requires information about the moving segment at two discrete instances of the motion (time frames), usually given by two rotations

, which, together with the associated translation vectors set up the transformations from local to global coordinates. In the following we will construct a continuous interpolation between these two time frames. For this purpose we apply the Rodrigues formula from equation (1) to the transformation which describes the rotational part of the transformation from the segment position in time frame 0 to those in time frame 1 and obtain:

(5)

with the axis unit vector

and the angle of rotation , i.e. , holds, which defines as the rotation axis of

.

We further define a parameterized interpolation of

betweenand over a unit time step by:

(6)

In this notation, we have

and1. It is of note that in (6) is not an arbitrary parametrization but is in fact the parameterization of the shortest geodesic path in, the Lie-group of orthogonal transformations in from to. We use a time parameterization for the translation between and . We further consider that each point on the local axis is transformed from local coordinates of the moving segment to a point in the global representation of the helical axis by the equation:

(7)

(Ehrig et al., 2006). To obtain a similar parameterization for

as for in equation (6) we decompose in components perpendicular to the axis vector and parallel to:

(8)

Using equation (7) the first term in the right hand side (r.h.s) of this equation can be rewritten as:

(9)

This term describes the pure rotational part of the helical motion. For the second term, which captures the translation parallel to

, we assume a constant speed along the axis, i.e.:

(10)

Taken together, (9), (10) yield:

(11)

The interpolation schemes for the rotational part

of the motion given in (6) and for the translational part

given in (11) are the formalization of the implicit assumption behind all methods for the determination of helical axes, namely that the motion between two time frames is continuous, and can in fact be described as a rotation around an axis together with a translation parallel to this axis. If these assumptions do not hold, the computed helical axis will not reflect the true kinematics of the joint.

In the following, we will examine whether three commonly used algorithms for determining an axis of rotation, namely the Symmetrical Axis of Rotation Approach - SARA (Ehrig et al., 2007), the Finite Helical Axis approach - FHA (Spoor and Veldpaus, 1980; Woltring et al., 1983; Woltring et al., 1985), and the Instantaneous Helical Axis – IHA (Woltring and Huiskes, 1985, Woltring, 1990; Woltring et al., 1987; Woltring et al., 1994) can correctly identify the direction of the axis vector

and its position from two measurements

.

2.2. The Symmetrical Axis of Rotation Approach - SARA

The Symmetrical Axis of Rotation Approach - SARA (Ehrig et al., 2007) determines axis parameters by solving a linear least squares system. The problem that is solved using SARA for the case in which the motion between two time frames is considered and only one segment is moving can be derived from the mathematical definition of a axis joint, equation (7), as:

(12)

This system may be solved directly w.r.t.

by elimination of which yields

(13)

The matrix

has rank 2, therefore the 3x3 system is underdetermined, and the minimum norm solution for can be computed through a SVD decomposition as follows: , where and are the matrices of the left and right singular vectors, and is the diagonal matrix of the singular values. The smallest of the 3 singular values, , is equal to 0 (a consequence of the matrix having rank 2), a condition than can be described by: . Thus, one obtains. The right singular vector associated to the singular value is thus identified as the axis unit vector of. Thus the SARA algorithm correctly reproduces the axis vector and its position; the accuracy of the direction of the axis vector depends only on the accuracy of the rotations ,

, but not on the accuracy of the translation vectors.

2.3. The Finite Helical Axis - FHA

The Finite Helical Axis (FHA) approach was developed by Spoor and Veldpaus (1980) and Woltring and co-workers (Woltring et al., 1983; Woltring et al., 1985) for computing the helical axis from two time frame using geometrical considerations. As we will show, the FHA and SARA are closely related and therefore we base the derivation of the FHA on the SARA approach. As a first step, we combine two equations of type (13), one each for

and ,

(14)

as follows:

(15)

After replacing the matrix

by the Rodrigues representation (5) the matrix on the left hand side of this term can be re-written as:

(16)

For the norm-minimal point

on the axis the following conditions holds:, and equation (15) now reads:

(17)

Using relation (5) again for the r.h.s. of (17) we obtain the following closed solution for

:

(18)

This latter expression is exactly the original definition of the FHA axis position as it appears in the papers of Woltring, e.g. (Woltring et al., 1985). Moreover, the definition of

and

used in the current manuscript (equations (4)) is identical to the definition of these parameters by Woltring and co-workers (1985). We have therefore proven the following statement:

Statement 1: The FHA algorithm and the SARA approach for two time frames are equivalent.

Since both algorithms are mathematically equivalent they are affected by any errors in the input data in exactly the same way.

2.4. The Instantaneous Helical Axis – IHA

The time dependent representation of any point

defined in a local coordinate system, with respect to a global coordinate system, is given by. Following the original definition in (Woltring and Huiskes, 1985, Woltring, 1990; Woltring et al., 1987; Woltring et al., 1994) the IHA at time point is defined as the set of points with minimum velocity which yields the following time dependent representation for the position of the IHA:

(19)

with the direction vector of the angular velocity

. Note that

is not necessarily the point on the helical axis with minimum norm. We refrain from presenting the details of the derivation of equation (19) and refer the interested reader to the detailed revision of Woltring’s original derivation provided by Reichl and Auzinger (2012).

The derivation shows that the cross-product matrix

associated to the vector can be written as and the computation of the IHA requires the time derivatives of and. These derivatives are usually estimated by discretization, a process for which different suggestions exist in the literature (Reichl and Auzinger, 2012, Woltring et al., 1994). Here, using the representations (6), (11) for and, we obtain exact expressions for the derivatives w.r.t. the time i.e.

(20)

(21)

Using the relationships between

and, equations (3), yields the simple equation, see derivation S1 in the supplementary material,

(22)

Thus:

(23)

As expected, the unit direction vector of the instantaneous helical axis is identical to the direction

of the rotation axis of the matrix

, and, of course, also the finite helical axis.

In order to determine the position of the axis, we insert the terms for

, equation (11), equation (23), , equation (21), , equation(20), into equation (19) and obtain after some calculations (for details see derivation S2 in the supplementary material) the following relationship: . Moreover, for the minimum norm point on the axis we obtain

. Thus, also the axis position is correctly reproduced by the analytical IHA approach.

In practice, however, equation (21) for the time derivative

cannot be used since is not previously known. When positional rather than velocity data serves as input, is thus usually estimated by time discretisation of using finite differences or by analytic derivatives of spline interpolants for (Woltring and Huiskes, 1985, Woltring, 1990, Woltring et al., 1994). To further explore this case we analyse the result for the simplest approach, a linear interpolation between and, even though we would certainly recommend using such methodology in practice only if the angle is very small. With a linear interpolation , we obtain . For a particular point on the helical axis we then obtain, using equations (19), (23), and for the mean of and, i.e. for, (c.f. Woltring et al., 1985):

(24)

The unique point

on the helical axis with minimum norm is then given by:

(25)

Equation (25) is very similar to equation (18) obtained for the FHA, only the denominators in the second terms differ: whilst this denominator is here

, it is for the FHA. However, the numerical differences between and are small only for very small angles. Thus, compared to the FHA, the position of the IHA is affected by an error perpendicular to the axis direction when using positional data as input. The reason for this error is the interpolation for , which approximates the circular arc around the helical axis, here using a straight line. The error may be substantially reduced if more accurate approximations for were used and if this particular source of error was completely eliminated, e.g. when using accurate measurement data of velocities as input, the results would be identical to those of the FHA approach. We should note that additional errors which may be introduced by numerical approximations of

are not considered here. We summarize these considerations by the following statement:

Statement 2: The analytical solution of the IHA approach for two time frames is equivalent with the FHA or SARA approach. If the derivatives of t are calculated by numerical discretization because positional data are used as input the IHA algorithm yields a helical axis position which may be affected by an error perpendicular to the true axis direction, i.e. an error in the axis position. The axis direction

is identical to those computed with the FHA or SARA approach, i.e. the true axis direction, as long as the direction vector of the angular velocity is known exactly, but additional errors in both axis position and direction may be introduced by numerical approximation of

.

If the velocity

and the angular velocity

can be measured accurately, e.g. by inertial measurement units (IMUs), the determination of the axis parameters will not suffer from discretization errors. Using equation (19) one will therefore obtain the same result as with the FHA or SARA methodologies.

3. Mean Finite and Mean Instantaneous Helical axes and their Relation to SCoRE and SARA

A further extension and important application of the helical axis methodology is the computation of a joint centre as the point with minimum root mean square distance from a set of helical axes, c.f. (Camomilla et al., 2006, De Rosario et al., 2014, Woltring, 1990), an approach that has therefore been referred to as mean helical axis (MHA) approach. As a next step, we will therefore analyze the relationship between the centre computed using either a helical axes approach or the Symmetric Centre of Rotation Estimation (SCoRE) method (Ehrig et al., 2006). We firstly consider a set of finite helical axes, and thus denote the resulting algorithm as mean finite helical axis approach, MHFA; this approach for locating centres of rotation is sometimes also called the helical pivot method (Camomilla et al., 2006, De Rosario et al., 2017, De Rosario et al., 2014, Woltring, 1990).

Given a set of n time frames, the first step for determining the MFHA, is to compute the FHA positions

with minimal norm and the direction vectors for all possible combinations. Since the determination of the FHA is sensitive to measurement errors in particular if the FHA is determined from small ranges of motion (c.f. equation (18)), empirical weighting terms are typically used which depend on the angle in a manner that for and thereby minimize the influence of less optimally defined FHAs. The minimization problem defines the joint centre and is then given by:

(26)∊

see e.g. (Woltring, 1990, Stokdijk et al., 1999, Camomilla et al., 2006). Using

it can be shown that the minimizer of the functional in (26) is the least squares solution of the linear system:

(27)

As shown in supplementary material S3

is identical to the closed form solution of the SCoRE approach for the determination of the centre of rotation of a ball joint if one defines the weighting factors as

. We have thus proven the following generalization of the statement 1:

Statement 3: Using weighting factors

the mean finite helical axis (MFHA) method as an approach for the joint centre determination is equivalent to the SCoRE algorithm.

Yet, any definition of the weighting terms differing from the setting

chosen above leads to a system of equations which is very different from the SCoRE approach, and therefore also results in a different solution. Woltring and co-workers (1985) proposed an empirical weighting factor. Whilst their algorithm seems to be widely used in the literature (Camomilla et al., 2006, Tay et al., 2010), only the weighting leads to a solution which optimally fits to the set of equations defining a joint centre, i.e. for all possible combinations

. Moreover, detailed tests in simulations (Ehrig et al., 2006) and experimental validation (Monnet et al., 2007) strongly suggest that SCoRE yields noticeably more precise results especially for small ranges of motion.

3.1. Mean Finite Helical Axes and their Relation to SARA

Given a set of helical axes, following again Woltring (1990), one may further define an optimal direction vector

through , which is as “parallel as possible” to all helical axes. Together with the specification of the position of a joint centre this approach then also provides a representation of an axis of rotation. As a next step, we will therefore analyze the relationship between such an optimal helical axis and an axis computed using the SARA method. In order to derive an axis of rotation satisfying such criteria, we consider the minimization of the following expression over all vectors with :

(28)

Expression (28) provides an implementation of the idea to define an optimal direction vector as described above; the more parallel

is to all the smaller the whole term will be. A detailed discussion of the expression (28) may also be found in Woltring (1990). Defining the weighting factors in (28) again as and further proceeding in the same way as described in the supplementary material (S3) it follows that the determination of the optimal direction vector requires to minimize:

(29)

This expression is minimal if

is chosen as the eigenvector associated to the smallest eigenvalue of the symmetric matrix given in brackets or, correspondingly, the eigenvector associated to the largest eigenvalue of Together with this

then defines the unit rotation vector of the mean finite helical axis. This approach is exactly the SARA algorithm as described by Ehrig and co-workers (2007, supplemental material). We summarize these findings in the following statement:

Statement 4: With the weighting factors chosen to be

the mean finite helical axis method as an approach for determination of an axis of rotation is equivalent to the SARA algorithm.

For any other choice of values for the weighting factors one may also compute estimates for the helical axis direction. The results, however, will be significantly different, and the error increases with a decreasing range of motion.

3.2. Mean Instantaneous Helical Axes and their Relation to SARA

Mean instantaneous helical (MIHA) axes were also discussed in the literature; see e.g. (De Rosario et al., 2017, De Rosario et al., 2014, Monnet et al., 2007, Stokdijk et al., 1999, Veeger and Yu, 1996, Woltring, 1990). Whilst the IHA approach reproduces the exact direction

of the helical axis for each pair of frames, the positions of the axes derived in this manner are error-prone (statement 2), especially for larger rotation angles. It is therefore difficult to estimate the error in the computational results of the position of a joint centre or axis by an approach similar to equation (26) or (28). However, within the derivation of the equivalence between SARA and mean helical axes the specifics of how these axes were determined, i.e. whether they are FHA, IHA axes based on measurements of positions or velocities, or any other method, was not used in any way. Thus, also for computation of the mean IHA the weighting is optimal since this is the only weighting which is consistent with the definition of a joint centre, equation (26), or a mean helical axis, equation (28). Since the inexact positions are not involved in the computation of the mean helical axis direction, see equation (27), the computed mean axis direction is identical to the corresponding result of the MFHA or SARA as long as errors arise from discretizations of

only.

4. Discussion

This study sought to derive exact mathematical relationships between three commonly used algorithms for computing joint axes from motion data. Here we have shown that the SARA/SCoRE methods are numerically equivalent to FHA approaches when computing an axis for two frames. This equivalence holds also more generally if the axes are computed from more frames i.e. the MFHA is used, as long as the weighting factors are appropriately chosen. If such weighting factors are sub-optimally chosen, e.g. using those originally proposed by Woltring and co-workers (1985), a small yet noticeable difference in the Axis of Rotation (AoR) parameters can be observed (Ehrig et al. 2007, Figures 3 and 4). The same equivalence of helical axes and SARA does not generally hold for the IHA approach. Whilst an IHA method based on velocities is mathematically equivalent to SARA and therefore also to FHA, care needs to be taken if position data serves as input to the method. Fed with such position data the IHA requires the calculation of time derivatives of the rotations and translations which can only be approximated numerically. Under these conditions IHA derived axes may suffer from reduced accuracy. If the IHA could be computed without any errors that would typically result from the time discretization of

and identical results to those from the FHA approach could be obtained. Since it is next to impossible to completely eliminate any such errors when using positional data as input, IHA axes derived under such conditions may suffer from errors in the axis position due to the discretisation of and from errors in both the axis position and direction due to the discretisation of

. Using highly accurate direct measurement of velocities which may be possible using e.g. inertial measurement units (IMUs) such problems will likely be substantially reduced and may result in an increased appreciation of the IHA approach in future.

The FHA and IHA approaches were already compared by Woltring and co-workers (1994) who suggested that both methods are less accurate for small angles. At the time, tools for the direct measurement of angles or velocities were not available and the IHA approach was based on positional data. Under these conditions, according to the analyses of Wachowski and co-workers (2010), IHA parameters can be reliably computed only if the frames

and are sufficiently close, i.e. “differentially close-by”. However, accurate determination of these rotations and translations is known to be difficult for small angular velocities (Stokdijk et al., 1999), requiring angular velocities and therefore also to remain above a critical lower limit which is in direct conflict with the above requirement for frames to be “differentially close-by”. This issue is further amplified if non-optimal weighting factors (e.g.

) are used for the calculation of the IHA. Our analyses suggest that the FHA, as long as the weightings are appropriately chosen, and SARA provide the best approach to compute an AoR for small angles when using positional data. When accurate measurements of linear and angular velocities are not available or when using positional data as input the use of the IHA is therefore not recommended.

Although methods for computing joint axes have been compared in a number of experimental studies, comparisons specifically between FHA, IHA or SARA are only available from a small number of studies (Colle et al., 2016, De Rosario et al., 2014). The study by De Rosario and co-workers (2014) provided evidence that an improvement in axis precision can be obtained by the specific choice of weightings when using the mean IHA, but the chosen weightings differed from the optimal ones and the authors did not evaluate the FHA. The study by Colle and co-workers (2016) compared implementations of a mean FHA and SARA approach and found differences in the angles between the anatomical transepicondylar axis (TEA) and the axes computed from either algorithm. Whilst one may fundamentally challenge the use of such models to describe the motion of the knee joint more generally, in the context of this manuscript it is most relevant that details regarding the helical axes (FHA vs. IHA, weightings) are not provided, however, and it remains unclear whether the observed differences resulted from the use of an IHA approach and the related limitations, from the use of sub-optimal weightings with the FHA, or their combination. In view of the theoretical derivation of the conditions for equivalence between helical axis methods and SARA presented here, recommendations on the use of either method for the computation of an AoR should be critically re-considered. Whilst the analyses presented here focused on deep theoretical analyses only with the aim to eliminate the need for data driven comparisons for seemingly different methods that are in fact intrinsically equivalent, this does not remove the need for more detailed analyses (De Rosario et al., 2017, De Rosario et al., 2013, De Rosario et al., 2014) of the behavior of methods that are truly distinct. Whilst it remains challenging to obtain a comprehensive understanding of the joint function, recent studies utilising the SARA method suggest that this approach is sufficiently sensitive to capture the load dependency of a functional AoR (Meireles et al., 2017). The deep understanding of the conditions of mathematical equivalence between the SARA/SCoRE approaches and the helical axis methods derived here enables users of either approach to focus on addressing such important questions on the load and activity dependency of joint function including the influence of soft tissue artefact (De Rosario et al., 2017, De Rosario et al., 2013) in the future and eliminates the need to consider the effect of the chosen analysis method when doing so. Such ability to focus on a more comprehensive understanding of the dynamic function of joints is increasingly important as emerging imaging and sensor technology are offering more detailed studies of kinematics than ever before.

5. Conclusion

Through the derivation of a sound mathematical framework to objectively compare the algorithms we could demonstrate that the FHA algorithm and the SARA approach are equivalent for the analysis of two time frames. For the IHA the same statements holds only if its implementation is based on direct measurements of linear and angular velocities. If on the other hand an IHA algorithm is based on the use of marker positions, the helical axis position may be affected by a systematic error perpendicular to the true axis direction, whereas the axis direction is identical to those computed with the FHA or SARA approach, i.e. the true axis direction. Finally, with an appropriate choice of weighting factors the MFHA and the MIHA (under the conditions described above) methods are equivalent to the SCoRE algorithm for determination of a CoR, and similarly, equivalent to the SARA algorithm for determination of an AoR.

Uncited references

Woltring et al., xxxx, Woltring et al., 1986.

Acknowledgements

We acknowledge the support of the research provided by the German Research Foundation (DFG EH-422/1-1), the Engineering and Physical Science Research Council (EP/K034847/1), and Arthritis Research UK (grant reference: 20194). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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1

Equation (6) is most often written as

, where exp and log are the exponential and logarithmic map, respectively. The alternative formulation here facilitates the following derivations.

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**Demystifying helical axes J Biomech author accepted version - Accepted Manuscript**

**De-mystifying helical axes J Biomech Supplementary material author accepted version**

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