Vertical Darboux motion and its parallel mechanical generators

Abstract

Vertical Darboux motion termed VDM is a special kind of general Darboux motion, in which all the trajectories of the points belonging to the moving body are planar ellipses. The self-conjugation of a VDM in a cylindrical displacement is introduced. The properties and metric constraint of the amplitude of VDM are derived in an intrinsic frame-free vector calculation. For utilizations, single-loop one-degree-of-freedom (1-DoF) primitive VDM generators including isoconstrained and overconstrained realizations are briefly recalled. The main purpose of our article is to synthesize new two-, three- or multi-loop parallel mechanical generators of a VDM. Interestingly, the removal of the fixed cylindrical pair leads to an additional new family of VDM generators with a trivial, exceptional, or paradoxical mobility. The detection of the possible failure actuation of a fully parallel manipulator via the VDM parallel generators is revealed too.

Appendix: Terminology

Technical words or expressions that are used in the literatures [10–20, 23, 24] are summarized below.

1.1 Axode

An extension of the concept of planar centrodes for spatial motion of a rigid body in terms of two ruled surfaces which are known as fixed and moving axodes (sometimes called axoids) [17, 24].

1.2 Mechanical generator

It is a physical realization of mechanical bond, which is the mathematical model of a coupling between two rigid bodies. In practice, a kinematic chain generating a given bond is named a mechanical generator of the kinematic bond [13, 14, 16, 20], which is the set of allowed relative displacements between two rigid bodies.

1.3 Overconstrained chain

The mobility of the chain disobeys the Chebyshev–Gruebler–Kutzbach formula [13–15, 17, 18]. It can move only when geometric conditions are achieved.

1.4 Isoconstrained chain

The chain is not overconstrained and there are no excess constraints. The word “isoconstrained” is equivalent to “non-overconstrained” [13–15, 17, 23].

1.5 Trivial chain

When all relative motions belong to one displacement Lie subgroup, a generalized Chebyshev–Gruebler–Kutzbach formula including the dimension of the subgroup can be used to determine its mobility and the corresponding chain is said to be trivial (or banal) [18].

1.6 Exceptional chain

When the mobility can be established by using the product closure in two or several displacement subgroups, a formula with the dimension of one group cannot be implemented and the mobility is qualified as exceptional [13–15, 18, 24].

1.7 Ordinary chain

The mobility type in mechanism is qualified as ordinary when the mobility results from the product closure in one or several displacement Lie subgroups. This kind of chain includes trivial and exceptional ones [13–15, 18].

1.8 Paradoxical chain

The chain mobility is subject to geometric conditions that require the use of the Euclidean metric, i.e., the Euclidean (or Pythagorean) definition of the distance between two points. Its mobility cannot be deduced only from the product closure in displacement Lie subgroups. The conditions of ordinary (trivial and exceptional) mobility belong to affine geometry whereas the paradoxical chain is movable only when the geometric conditions are based on the Euclidean metric [13–15, 17].

1.9 Schoenflies (or Schönflies) motion

The motion contains the spatial translations and all the rotations around any axis parallel to a given direction. Schönflies is spelt Schoenflies in the original relevant publications [19, 20].

1.10 Self-conjugation

Let N be a subset of a group G. We say that N is normal (or self-conjugate) [10–12, 24] in G, if g N g ⁻¹ = N for all elements g ∈ G. In an Abelian group, such as the group of cylindrical displacements, any subset is self-conjugate.