An Application of Yaglom's Geometric Algebra to Kinematic Synthesis of Linkages for Prescribed Planar Motion of Oriented Lines

Planar motion coordination of an unoriented line passing through a point or tangent to a conic is a well-known problem in kinematics. In Yaglom's algebraic geometry, oriented lines in a plane are represented with dual numbers. In the present paper, such algebraic geometry is applied in the kinematic synthesis of an inverted slider-crank for prescribed three and four finitely separated positions of the coupler. No previous application of Yaglom's algebraic geometry in the area of linkage kinematic synthesis is recorded. To describe the planar finite displacement of an oriented line about a given rotation pole, new dual operators are initially obtained. Then, the loci of moving oriented lines whose three and four homologous planar positions are tangent to a circle are deduced. The paper proposes the application of findings to the mentioned kinematic synthesis of the inverted slider-crank. Numerical examples show the reliability of the proposed approach. Finally, it is also demonstrated that, for a general planar motion, there is not any line whose five finitely separated positions share the same concurrency point. For the case of planar infinitesimal displacements, the same property was established in a paper authored by Soni et al. (1978, "Higher Order, Planar Tangent-Line Envelope Curvature Theory," ASME J. Mech. Des., 101(4), pp. 563-568.)