Abstract
A hexapod is a parallel manipulator where the platform is linked with the base by six legs, which are anchored via spherical joints. In general, such a mechanical device is rigid for fixed leg lengths, but, under particular conditions, it can perform a so-called self-motion. In this paper, we determine all hexapods possessing self-motions of a special type. The motions under consideration are so-called plane-symmetric ones, which are the straight forward spatial counterpart of planar/spherical symmetric rollings. The full classification of hexapods with plane-symmetric self-motions is achieved by formulating the problem in terms of algebraic geometry by means of Study parameters. It turns out that besides the planar/spherical symmetric rollings with circular paths and two trivial cases (butterfly self-motion and two-dimensional spherical self-motion), only one further solution exists, which is the so-called Duporcq hexapod. This manipulator, which is studied in detail in the last part of the paper, may be of interest for the design of deployable structures due to its kinematotropic behavior and total flat branching singularities.
Highlights
In planar kinematics, the instantaneous pole P traces the so-called fixed/moving polode in the fixed/moving system during the constrained motion of a given mechanism
Krames reasoned this by the fact that the path x of a point X under a line-symmetric motion can be generated by the reflexion of a point X0 on each generator g of Γ; for example, x can be obtained by a central doubling of X0 ’s pedal-curve f with respect to Γ’s rulings
One of the author’s main research interests are hexapods with self-motions, i.e., overconstrained parallel manipulators where the platform is linked with the base by six legs, which are anchored via spherical red joints
Summary
The instantaneous pole P traces the so-called fixed/moving polode in the fixed/moving system during the constrained motion of a given mechanism. The pedal-point of the fixed point X0 with respect to the pol-tangent t is denoted by F From another perspective, a planar/spherical symmetric rolling can be generated by reflecting the fixed system in a 1-parametric continuous set of lines/great circles. These motions are obtained by reflecting the moving system in a 1-parametric continuous set of lines, which form the so-called basic surface Γ (cf Figure 2, left) Krames reasoned this by the fact that the path x of a point X under a line-symmetric motion can be generated by the reflexion of a point X0 on each generator g of Γ; for example, x can be obtained by a central doubling of X0 ’s pedal-curve f with respect to Γ’s rulings. In order to avoid confusions, we point out that we do not mean this superset by using this wording
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
