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We describe a new approach to the problem of constructing gluing parameterizations for open neighborhoods of boundary points of moduli spaces arising in geometric analysis. Our approach employs general results from differential topology for smooth maps of smooth Banach manifolds with corners. We illustrate the method in the context of gluing anti-self-dual connections over closed four-dimensional manifolds, but it should apply to other problems in geometric analysis involving the gluing construction of solutions to nonlinear partial differential equations.