## Abstract

Accurate estimation of the position and orientation (pose) of a bone from a cluster of skin markers is limited mostly by the relative motion between the bone and the markers, which is known as the soft tissue artifact (STA). This work presents a method, based on continuum mechanics, to describe the kinematics of a cluster affected by STA. The cluster is characterized by triangular cosserat point elements (TCPEs) defined by all combinations of three markers. The effects of the STA on the TCPEs are quantified using three parameters describing the strain in each TCPE and the relative rotation and translation between TCPEs. The method was evaluated using previously collected ex vivo kinematic data. Femur pose was estimated from 12 skin markers on the thigh, while its reference pose was measured using bone pins. Analysis revealed that instantaneous subsets of TCPEs exist which estimate bone position and orientation more accurately than the Procrustes Superimposition applied to the cluster of all markers. It has been shown that some of these parameters correlate well with femur pose errors, which suggests that they can be used to select, at each instant, subsets of TCPEs leading an improved estimation of the underlying bone pose.

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Associate Editor Michael R. Torry oversaw the review of this article.

## Appendices

### Appendix 1

The objective of this appendix is to give more details regarding the computation of the kinematics of the TCPEs. The symmetric right Cauchy-Green deformation tensor \(\varvec{C}\) associated with the deformation gradient tensor \(\varvec{F}\) defined in Eq. (3) is used to calculate the stretch tensor \(\varvec{M}\), as follows:

### Appendix 2

The objective of this appendix is to compare the rotation matrix of a cluster containing three markers obtained by the TCPE method with that obtained by the PS approach. Let \(\left\{ {\varvec{X}_{0} ,\varvec{X}_{1} ,\varvec{X}_{2} } \right\}\) and \(\left\{ {\varvec{x}_{0} ,\varvec{x}_{1} ,\varvec{x}_{2} } \right\}\) be the position vectors of three markers in the reference frame and in the present frame, respectively. In order to determine a \(3 \times 3\) rotation matrix \(\varvec{Q}\) and a translation \(3 \times 1\) vector \(\varvec{t}\) that map the points \(\varvec{X}_{i}\) to \(\varvec{x}_{i}\), the following least-squares problem is solved:

where \(\varvec{Q}\) is constrained to be an orthogonal rotation matrix that possess the properties

The centroids \(\left\{ {\bar{\varvec{X}} , { }\bar{\varvec{x}}} \right\}\) introduced in Eq. (5) are used to define the vectors \(\left\{ {\Delta \varvec{X}_{i} ,\,\Delta \varvec{x}_{i} } \right\}\) and the tensor \(\bar{\varvec{F}}\) by

Then, the positive polar decomposition theorem can be used to decompose \(\bar{\varvec{F}}\) into its rotation tensor \(\bar{\varvec{R}}\) and stretch tensor \(\bar{\varvec{M}}\) which is a positive-definite symmetric tensor

Different PS methods used for pose estimation32,34,37 demonstrated that the rotation matrix \(\varvec{Q}\) which satisfies Eq. (15) under the constraint condition in Eq. (16) is

The PS methods mentioned above predict the same rotation tensor \(\bar{\varvec{R}}\) and differ only by their algorithms for deriving it, as demonstrated on Appendix 3. On the other hand, the deformation gradient tensor \(\varvec{F}\) defined by the TCPE method in Eq. (3) predicts the rotation tensor \(\varvec{R}\). In this regard, it is noted that \(\varvec{F}\) transforms any material line in the analyzed triangular body from the reference to the present configuration so that

Then, with the help of Eqs. (4), (17), (18), and (20) it follows that

In general, \(\varvec{M}\) does not commute with \(\bar{\varvec{H}}\), which proves that \(\bar{\varvec{R}}\) does not equal to \(\varvec{R}\) whenever \(\varvec{M}\bar{\varvec{H}}\) is not a symmetric tensor. Moreover, it is noted that since the rotation obtained by the PS approach is used in several other methods5,9,23 for defining different metrics for defining the STA components, these definitions are also affected by the differ rotation tensor obtained by the PS method and the TCPE method.

### Appendix 3

The objective of this appendix is to demonstrate that the rotation matrices obtained by eigenvalue decomposition,34 polar decomposition37 and singular value decomposition32 are identical. The polar decomposition of the tensor \(\bar{\varvec{F}}\) defined in Eq. (17) can be used to obtain \(\bar{\varvec{R}}\) by

Similarly, the singular value decomposition of \(\bar{\varvec{F}}\) can be used to obtain the same \(\bar{\varvec{R}}\):

and the eigenvalue decomposition of \(\bar{\varvec{F}}^{T} \bar{\varvec{F}}\) yields the same \(\bar{\varvec{R}}\):

It should be noted that even though the different methods provide identical rotation matrices analytically, they rely on different numerical algorithms and their sensitivity to errors are different. Therefore, they might result in different estimates if the markers are badly configured.32

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Solav, D., Rubin, M.B., Cereatti, A. *et al.* Bone Pose Estimation in the Presence of Soft Tissue Artifact Using Triangular Cosserat Point Elements.
*Ann Biomed Eng* **44**, 1181–1190 (2016). https://doi.org/10.1007/s10439-015-1384-6

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DOI: https://doi.org/10.1007/s10439-015-1384-6

### Keywords

- Bone pose estimation
- Cosserat point theory
- Marker cluster
- Motion analysis
- Soft tissue artifact
- Soft tissue deformation
- Stereophotogrammetry
- Triangular cosserat point element