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Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. Instead of using local methods, we conduct global design searches by direct computation of all critical points. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, which allows for a full exploration of the design space, whereas local solvers can only find the minimum nearest to an initial guess. We form equality-constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. We consider the general case where all mechanism dimensions may vary and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. The discretization points are contained within the coefficients of the stationarity polynomials, so the algebraic structure of the synthesis equations remains invariant to the number of points. The results of our computational work were used to design a mechanism that coordinates the folding wings. We also use this method to parameterize mechanism dimensions for a family of hyperbolic curves. 

Abstract Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. In this work, we conduct global design searches by direct computation of all critical points through stationarity conditions. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, serving as a powerful design exploration technique. We form equality constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. For each problem considered, Lagrangian stationarity conditions set up a square system of polynomials. We consider the most general case where all mechanism dimensions may vary, and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit an estimated maximum of 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. Approximate motion is specified as discretized points from a desired input-output angle function. The coefficients of the stationarity polynomials can be expressed as summations of symmetric matrices indexed by the discretization points. We take the sums themselves to parameterize these polynomials rather than constituent terms (the discrete data). In this way, the algebraic structure of the synthesis equations remains invariant to the number of discretization points chosen. The results of our computational work were used to design a mechanism that coordinates the unfolding of wings for a deployable aircraft.