This paper shows that the canonical system and the instantaneous invariants for a moving plane, which is connected to the fixed plane by a revolute-revolute crank, are functions of the derivatives of the crank angle. Then closed-form expressions are derived for the curvature ratios of the path generated by an arbitrary point fixed in the moving plane, in terms of the coordinates of the point and the instantaneous invariants of the plane. For illustrative purposes, numerical results are presented for the instantaneous invariants (up to the fourth-order) of the coupler of a specified crank-rocker mechanism, as a function of the input angle. In addition, the paper shows the variation in the first and second curvature ratios of an arbitrary coupler curve during the complete operating cycle of the mechanism. The authors hope that, based on the results presented here, a variety of useful tools for the kinematic design of planar mechanisms, with a rotary input, will be developed for plane rigid body guidance as well as curve generation.

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