An algebraic method is presented to synthesize quick-returning R-S-S-R mechanisms which satisfy the given time-ratio and follower oscillation angle requirements. In these designs, the three parameters, which define the follower spheric joint, satisfy a quadratic condition. When the shaft angle between the input and output shafts is zero, this quadratic condition reduces to the equation of a circle which is a familiar classical result for the planar case. The solution space for the quick-returning R-S-S-R linkage is such that, for each set of choices for crank length a2, follower axial distance S4, and initial follower angle φ0, there are four sets of follower length a4, initial crank angle θ0, crank axial distance S2, and coupler length a3. These designs are screened so that they do not have branch defect, have transmission angle values in a given range, and have reasonable link length proportions.

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