Abstract
When a mechanism moves, the twist system \(S\) of the end-effector generally varies. In significant special cases, \(S\) is a subalgebra of the Lie algebra of the special Euclidean group, and it remains constant. In more general cases, \(S\) remains invariant up to a proper isometry, thus preserving its class. A mechanism of this kind is said to generate a persistent screw system (PSS) of the end-effector. PSSs play an important role in mobility analysis and mechanism design. This paper presents the serial generators of \(4\)-dimensional PSSs with a constant class of the general type.
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Notes
- 1.
Indeed, \(\mathbf{S}_1\) may be reciprocal, for arbitrary values of \(\theta _2\), to a screw of pitch \(-h_{p1}\) intersecting \(\ell _3\) at right-angle, even if \(p_{32}=0\), \(\alpha _{32}=\pi /2\) and \(h_2=h_{p1}\), with \(\mathbf{S}_{r1}\) being, in this case, aligned with \(\mathbf{S}_2\). However, requiring \(\mathbf{S}_1\) to be also reciprocal to \(\mathbf{S}_{r2}\), i.e. a screw both perpendicular to \(\mathbf{S}_{r1}\) and intersecting it, straightforwardly leads to the condition that \(\mathbf{S}_1\) and \(\mathbf{S}_2\) must be collinear. Explicit calculations are not reported for the sake of brevity.
References
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Carricato, M. (2014). Four-Dimensional Persistent Screw Systems of the General Type. In: Thomas, F., Perez Gracia, A. (eds) Computational Kinematics. Mechanisms and Machine Science, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7214-4_33
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DOI: https://doi.org/10.1007/978-94-007-7214-4_33
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