Rigid bodies and motions

Abstract

This chapter has shown that behind the use of Lie groups for rigid body motions, there are implicit assumptions that, even if more or less “natural”, are not intrinsic and just depend on a choice of relative reference between spaces (see Remark. 1.2.2, p.14).

It is always important to pinpoint those assumptions which are not intrinsic: any hypothesis which is not intrinsic can in fact be considered a modeling hypothesis and should be made explicit as such.

Twists have been analyzed in detail and their mappings and relations have been shown. The intrinsic mappings using right translations are the ones which give the motion of a space with respect to an observer space directly and have been denoted with only one subscript and one superscript: t ^j _i or t ⁱ _j .

A complete formal relation between the Lie group approach and Screw theory has then been presented. This one-to-one relation between the Lie group approach and the screw approach is due to the existence of two biinvariant forms known as the Killing form and the Hyperbolic or Klijn form.

The space of screw is equivalent in this way to the projectivization of the Lie algebra of rigid body motions.