Series Solution for Finite Displacement of Planar Four-Bar Linkages

A time-varying instantaneous screw characterizes the motion of a rigid body. The kinematic differential equation expresses the path taken by any point on that rigid body in terms of this screw. Therefore, when a revolute joint is attached to a moving link in a planar kinematic chain, the path taken by the center of that revolute joint is the solution to such an equation. The instantaneous screw of a link in that chain is in turn determined by the action of the joints connecting that link to ground, where the contribution of each joint to that instantaneous screw is determined by its actuation rate and center point. Substituting power series expansions for joint rates into the kinematic differential equations for joint centers, and expressing loop closure as a linear constraint on the instantaneous screws of the links, a recurrence relation is established that solves for the coefficients in those power series. The resulting solution is applied to determine the equilibrium pendulum tilt of the United Aircraft TurboTrain. Comparing that power series approximation with an exact kinematic analysis shows convergence properties of the series.