Abstract
In this paper it is demonstrated that the solution space of the inverse kinematic problem of an anthropomorphic, redundant 7R chain for a given pose does consist of eight different coinciding circles instead of a single circle that has been reported as of today. By modeling the structure using the convention by Sheth and Uicker, the displacements within the kinematics of the chain are partitioned in time-invariant displacements along rigid links and time-variant displacements along the seven rotative joints. In particular, the subchains of shoulder, elbow, and wrist are preserved. By respecting the ‘flips’ of these three substructures the eight-fold occupancy of the redundancy circle is obtained. The result corresponds to the eight IK solutions for regional-spherical arms and provides a prerequisite for using all capabilities of respective robots in practical applications.
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Notes
- 1.
In terminology of [12], the complete solution reported here incorporates the inversion of the three cosine-type into the solution procedure. In comparison to the recent work [10] which employs a parametrization with respect to the second joint, the parametrization via the redundancy angle permits the direct interpretation as eight coinciding elbow circles.
- 2.
Given the triangle \(\triangle \) in Fig. 3a with interior angles \(\varsigma , \upsilon , \varphi \in [0,\pi ]\), the oriented angles of the positive-elbow triangle \(\triangle _{+} \) on the right hand side of Fig. 3b are given as the supplementary angles
(with 
). The angles of the negative-elbow triangle \(\triangle _{-} \) on the left hand side of Fig. 3b are given as the \(\pi \)-shifted angles 
(with 
). - 3.
In [12] only one solution is reported. Note, that the two solutions are not covered by an elbow rotation of \(\pi \): while rotating along the circle, the elbow angle remains constant. However, the elbow configuration \(q_{45-} \) is the negative of the configuration \(q_{45+}\): apart from any ‘stretched-out’ posture, where the two values coincide \(q_{45+} = q_{45-} = 0\), they differ in general postures. In Fig. 3b, this distinction is reflected by the counter-clockwise orientations of all six angles.
- 4.
For determining the shoulder angles and wrist angles from the matrices \(\pmb {S}_{14}\) and \(\pmb {S}_{58}\), the method in [12] is based on orthogonal decomposition of rotation matrices and solving for \(( q_{12}, q_{23}, q_{34})\) and \((q_{56}, q_{67}, q_{78})\), by coefficient comparison with respect to \(\psi \). For the shoulder, the equation
with 
, and the coefficients of \(\pmb {S}\) are analyzed. For the wrist, a similar approach is chosen. In both cases, the coefficient analysis only reports one of the two feasible solutions.
(with 
). The angles of the negative-elbow triangle 
(with 
).
with 
, and the coefficients of References
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Acknowledgements
The work was performed within the projects Capio and Recupera, funded with federal funds from the German Federal Ministry of Education and Research (BMBF) (Grant 01-IW-10001 and 01-IM-14006A). The author would like to thank Sankaranarayanan Natarajan, Wiebke Drop, and Arnaud Sengers for their contributions.
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Bongardt, B. (2018). Inverse Kinematics of Anthropomorphic Arms Yielding Eight Coinciding Circles. 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_60
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