Abstract
A lot of different kinematic chains have been investigated focusing on constraint equations, singularities, assembly modes and motion capabilities. The approach to obtain constraint equations via inverted chains however is rarely considered. We provide a detailed look on the constraint varieties of inverted chains, beginning with the basics of quaternion conjugation. The transformation of the Denavit- Hartenberg parameters needed for the quaternion conjugation is discussed in the paper. The quaternion conjugation is a fast way to obtain the variety corresponding to the inverted kinematic chain. Geometrically the conjugation is a reflection in the kinematic image space \(\mathbb {P}^7\) with respect to a line and a five- dimensional subspace. Some examples of constraint equations of kinematic chains and their inverted chains complete the paper.
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Notes
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The kinematic images of planar and spherical displacements subordinate completely to this description because both cases are obtained by three dimensional sub-spaces of \(\mathbb {P}^7\). The corresponding geometry of their image spaces and the algorithms to derive these geometries can be found in [1] p. 393ff. resp. [5] p. 60ff.
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Acknowledgements
The authors acknowledge the support of the FWF project I 1750-N26 “Kinematic Analysis of Lower-Mobility Parallel Manipulators Using Efficient Algebraic Tools”.
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Stigger, T., Husty, M.L. (2018). Constraint Equations of Inverted Kinematic Chains. In: Zeghloul, S., Romdhane, L., Laribi, M. (eds) Computational Kinematics. Mechanisms and Machine Science, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-60867-9_56
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