Abstract
In many applications, it is interesting or necessary to extract a rotation component in a plane from a general 3D rotation. Examples are the knee flexion/extension in the sagittal plane or the torsion about the normal direction of a parallel platform. Due to the non-Abelian structure of spatial rotations, such component extractions cannot be accomplished by projections as in vector spaces. Classical methods are thus to use Euler angles concatenations or projections of rotated coordinate axes to the target plane. This paper proposes an alternative, novel method based on Quaternion vector projection. It is shown that the equations become very simple and that for applications such as biomechanics it yields better results than the other classical rotation extraction methods.
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Abbreviations
- \({\mathcal K}_{W}\) :
-
Space-fixed reference frame
- \({\mathcal K}_{B}\) :
-
Body-fixed frame
- \(\underline{Q}\) :
-
Quaternion, describing the orientation of \({\mathcal K}_{B}\) relative to \({\mathcal K}_{W}\)
- \(\underline{Q}^{\text {z}}\) :
-
Decomposed component of \(\underline{Q}\) in the z-direction of \({\mathcal K}_{W}\)
- \(\underline{Q}^{\text {xy}}\) :
-
Decomposed component of \(\underline{Q}\) normal to \(\underline{Q}^{\text {z}}\)
- \(\overline{\underline{Q}}^{\text {xy}}\) :
-
Conjugate of \(\underline{Q}^{\text {xy}}\)
- \(q_{0}, q_{0}^{\text {z}}, q_{0}^{\text {xy}}\) :
-
Real part of \(\underline{Q}\), \(\underline{Q}^{\text {z}}\), \(\underline{Q}^{\text {xy}}\), respectively
- \(\underline{q},\underline{q}^{\text {z}} ,\underline{q}^{\text {xy}}\) :
-
Vector part of \(\underline{Q}\), \(\underline{Q}^{\text {z}}\), \(\underline{Q}^{\text {xy}}\), respectively
- \(q_{x}, q_{y}, q_{z}\) :
-
Elements of \(\underline{q}\)
- \(q_{z}^{\text {z}}\) :
-
Element of \(\underline{q}^{\text {z}}\)
- \(q_{x}^{\text {xy}}, q_{y}^{\text {xy}}\) :
-
Elements of \(\underline{q}^{\text {xy}}\)
- \(\text{ Rot }\,[\,i,\psi \,]\) :
-
Rotation about the i-axis of \({\mathcal K}_{B}\) with an angle \(\psi \)
- \({\mathbf {\mathsf{{R}}}}\) :
-
General rotation matrix
- \({}^W{\mathbf {\mathsf{{R}}}}_B\) :
-
Rotation matrix, describing the orientation of \({\mathcal K}_{B}\) relative to \({\mathcal K}_{W}\)
- \({}^W{\mathbf {\mathsf{{R}}}}^\text {Eyx}_B,{}^W{\mathbf {\mathsf{{R}}}}^\text {Exy}_B\) :
-
\({}^W{\mathbf {\mathsf{{R}}}}_B\) formed by Z-Y-X and Z-X-Y order of Euler-angles, respectively
- \(\alpha \) :
-
Single-axis rotation angle from \({\mathcal K}_{W}\) to \({\mathcal K}_{B}\)
- \(\alpha ^{\text {z}}\) :
-
Extracted in-plane rotation from \({\mathcal K}_{W}\) to \({\mathcal K}_{B}\) about the z-axis of \({\mathcal K}_{W}\)
- \(\alpha ^{\text {z}}_{\text {Eyx}},\alpha ^{\text {z}}_{\text {Exy}}\) :
-
\(\alpha ^{\text {z}}\) determined via Euler Z-Y-X and Z-X-Y decomposition, respectively
- \(\alpha ^{\text {z}}_{\text {Ax}}, \alpha ^{\text {z}}_{\text {Ay}}\) :
-
\(\alpha ^{\text {z}}\) determined via Axis projection of x and y-axis of \({\mathcal K}_{B}\), respectively
- \(\alpha ^{\text {z}}_{\text {Q}}\) :
-
\(\alpha ^{\text {z}}\) determined via Quaternion decomposition
- \(\alpha ^{\text {xy}}\) :
-
Rotation angle of \(\underline{Q}^{\text {xy}}\)
- \(\varphi ,\Theta \) :
-
Euler-angle about body-fixed x and y-axis, respectively
- \(\overline{\underline{x}}, \overline{\underline{y}}\) :
-
Planar projection of x and y-axis of \({\mathcal K}_{B}\), respectively, on xy-plane of \({\mathcal K}_{W}\)
- \(\overline{\underline{x}}_{x},\overline{\underline{x}}_{y},\overline{\underline{y}}_{x},\overline{\underline{y}}_{y}\) :
-
x and y components of \(\overline{\underline{x}}\), \(\overline{\underline{y}}\), respectively
- \(\overline{\underline{e}}_{x}, \overline{\underline{e}}_{y}\) :
-
planar projection of x and y-axis of \({\mathcal K}_{W}\), respectively, onxy-plane of \({\mathcal K}_{W}\)
- \(\omega _x,\omega _y,\omega _z\) :
-
Angular velocity components about x,y and z-axis of \({\mathcal K}_{B}\), respectively
References
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Acknowledgment
The funding of this work by the ZIM (Central Innovation Program for small and medium-sized enterprises) is gratefully acknowledged. The authors also wish to thank the reviewers for their valuable comments which have contributed to integrating important complementary thoughts in this paper.
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Ghiassi, M., Maibaum, J., Kecskeméthy, A. (2022). Methods Comparison and Proposition of New Quaternion-Based Approach for Extraction of In-plane Rotations Out of 3D Rotations. In: Kecskeméthy, A., Parenti-Castelli, V. (eds) ROMANSY 24 - Robot Design, Dynamics and Control. ROMANSY 2022. CISM International Centre for Mechanical Sciences, vol 606. Springer, Cham. https://doi.org/10.1007/978-3-031-06409-8_14
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DOI: https://doi.org/10.1007/978-3-031-06409-8_14
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