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Kresling origami has recently been widely used to design mechanical metamaterials, soft robots and smart devices, benefiting from its bistability and compression-twist coupling deformation. However, previous studies mostly focus on the traditional parallelogram Kresling patterns which can only be folded to cylindrical configurations. In this paper, we generalize the Kresling patterns by introducing free-form quadrilateral unit cells, leading to diverse conical folded configurations. The conical Kresling origami is modelled with a truss system, by which the stable states and energy landscapes are derived analytically. We find that the generalization preserves the bistable nature of parallelogram Kresling patterns, while enabling an enlarged design space of geometric parameters for structural and mechanical applications. To demonstrate this, we develop inverse design frameworks to employ conical Kresling origami to approximate arbitrary target surfaces of revolution and achieve prescribed energy landscapes. Various numerical examples obtained from our framework are presented, which agree well with the paper models and the finite-element simulations. We envision that the proposed conical Kresling pattern and inverse design framework can provide a new perspective for applications in deployable structures, shape-morphing devices, multi-modal robots and multistable metamaterials.

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