Rigid Body Points Approximating Concentric Circles in given Sets of Its Planar Displacements

The problem of determining the points in a plane undergoing rigid body co-planar motion which deviate least from concentric circles in a given sequence of its ordered position sets is considered. The sought for approximation is the one which minimizes the sum of squared algebraic deviations (distances) of such points from the concentric circles approximating their paths in the given alternating sets of positions. Algebraic deviation functions introduced to substitute the geometric (orthogonal) deviations define the powers of the points of interest with respect to the approximating circles and are expressed by linear functions in circle parameters. The theory and methods developed here can be applied to the synthesis of multi degree of freedom adjustable planar mechanisms for approximate generation of the given multi-phase motions and multiple point-paths.