In the realm of nonlinear structural mechanics, tracing load-displacement curves becomes complex when approaching the limit points caused by instability. Multiple ideas have addressed this challenge, dating back to the late 1960s. Despite the integration of various algorithms into commercial finite element software, no single algorithm generally solves all nonlinear structural problems. Moreover, many existing methods are sensitive to initial parameters, e.g. step-length and might have problem with large values. This paper introduces a family of minimum residual displacement methods for tracing equilibrium paths. Our approach can be seen as a generalized framework that provides new methods and encompasses some existing ones as specific instances. We develop and apply straightforward techniques to control residual displacement in nodes, elements, or specific deformation components. We present a comprehensive library of methods implemented in OpenSees, enabling a comparison between our presented methods and the minimum residual displacement method, as well as four well-established techniques. We apply these methods to solve complex geometrically nonlinear problems involving truss, frame, and shell structures. The results highlight the enhanced ability to converge with larger step-length even in highly complex behavioral scenarios and the capacity to reduce the required iterations to converge

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