https://orcid.org/0000-0001-9906-8037
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Abstract
This paper presents a screw theory approach for the computation of the instantaneous rotation axes of indeterminate spherical linkages. Since the last part of the XIX century, the determination of the instantaneous rotation, or velocity, centers of planar mechanisms has been studied using the Aronhold-Kennedy theorem, extended to the spatial case until the second part of last century. In the beginning of the XX century, it was found that there were planar mechanisms for which the application of the Aronhold-Kennedy theorem was unable to find all the instantaneous rotation centers; these mechanisms where denominated complex or indeterminate. The beginning of this century saw a renewed interest on the complex or indeterminate mechanisms, both planar and spherical; and very soon, taken advantage of the results obtained for planar mechanisms, there were several methods for the determination of the indeterminate instantaneous rotation axes. The new screw theory approach, presented here, provides a more straightforward method for setting up the equations; furthermore, the algebraic equations to be solved are simpler than the ones published up to date. In addition, the method is part of a comprehensive method for the determination of the instantaneous screw axes of spatial mechanisms or their many special cases. The method is based on a systematic application of screw theory, which is isomorphic to the Lie algebra, se(3), of the Euclidean group, SE(3), and the invariant symmetric bilinear forms defined on se(3).
Acknowledgments
The first author thanks the Conacyt, the Mexican National Council of Science and Technology, for the support through an scholarship, grant 458523, to pursue a M. Sc. degree at the Universidad of Guanajuato. The authors thank Conacyt and also to the Universidad of Guanajuato for their continuous support.
Notes
1 The detailed process is shown in Appendix A.
2 The notation Eqijk means that the term, equated to 0 to form an equation, allows to find the rotation axis associated to the movement between the links i and j and employs link k to form the equation.
3 In the graphical methods employed by kinematicians until the 1950s, it was customary to prefer the primary rotation axes compared with those already obtained by using the Aronhold-Kennedy theorem, since the drawing errors made the secondary rotation axes not as palatable than the primary rotation axes. However, when doing these computation algebraically, with the help of a mathematical software, such as Maple©, the disadvantage is completely eliminated.
4 Only the required instantaneous rotation axes required for the comparison are indicated; however, all the secondary rotation axes that can be located are marked accordingly in the successive upper triangular arrays.
5 Only the required instantaneous axes are indicated; however the upper triangular array indicate all the secondary rotation axes that can be found.
6 It should be noted that these vectors are not unitary. However, to avoid the introduction of additional notation, this error will be left unnoticed.
7 Only those required in the procedure are indicated. However, the upper triangular arrays show all that can be obtained.
8 Only the required secondary rotation axes are indicated in the text. However, the upper diagonal array shows all the newly found secondary rotation axes.
9 It should be noted that these vectors are not, in general, unitary.
10 It should be noted that all the rotation axes have its z-component greater or equal to zero. Therefore, all the unit vectors point towards the upper semi-space, if the computation provides a unit vector with z-component negative, the whole unit vector is multiplied by –1. However, in this particular case, this multiplication was not required. Not all the rotation axes are shown due to space limitations.
11 Only the required rotation axes for the process are indicated. However, the upper triangular array shows all the instantaneous rotation axes that can be found.
12 Only the required secondary rotation axes are indicated. However, the upper diagonal array indicates all newly found secondary rotation axes.
13 Only the required secondary rotation axes are indicated in the text. However, the upper diagonal array shows all those newly found secondary rotation axes.
14 The arrangement also shows additional secondary rotation axes that can be found at this stage but they are not used in the process.
15 Rational expressions for the rotation axes are too cumbersome and do not add anything important to the problem solution. Not all the secondary rotation axes are provided due to space limitations.