This paper uses screw algebra to derive relatively simple and explicit expressions for the complex motion-invariant rotational first- and sequin-order geometric influence coefficients for a planar multi-loop mechanism in terms of reciprocal products of screws and differential screws. The concept of he differential screws is presented and used as and effective tool in characterizing he second-order geometric influence coefficients. Furthermore, the geometrical interpretation of both the screw product and the reciprocal product of screws in two-dimensional space is presented and implemented to simplify the formulation. It is believed that this is the first time to be able to drive the motion-invariant geometric influence coefficients of multi closed-loop mechanisms in explicit and relatively simple forms. The sue of the geometric infuse coefficients eases the kinematics and dynamic modeling probles of mechanisms. The derived simple expressions for geometric influence coefficients are then implemented to drive he angular velocity-squared/acceleration ratio of he output link. If the single-degree of freedom (dof) planar multi-loop mechanism is running with consign input velocity this ratio becomes motion-invariant too. An illustrative example is presented. This technique has he potential of belong extended to cover other spatial and spherical mechanisms as well as multi-degree of freedom multi-loop mechanisms. It could also be expended to derive simple expressions for he translational geometric influence coefficients.
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