Kinetic structures possess the unique ability to adapt their shape in response to both internal and external environmental conditions, as well as functional requirements. This study introduces a geometry-driven design and implementation of kinetic grid structures that harness straight and flat planks to weave asymptotic networks on negatively curved surfaces. The proposed approach strategically utilises the strong and weak bending axes of slender planks, capitalising on their elastic deformation capabilities to transform the overall geometry while maintaining high structural stability for external loads. The design features two interconnected families of planks, connected by scissor joints to form a doubly curved grid and a semi-compliant mechanism. A geometry-driven algorithmic workflow computes the geometries for subsequent service stages, seamlessly integrating the initial geometry design with kinetic behaviour. Three different kinetic asymptotic structures showcase the design approach and the architectural application of the novel system. Grounded in differential geometry, this method enables precise control of grid curvature and tracking of elastic strain energy throughout the transformation process. The construction of physical prototypes and corresponding finite element simulations validate the predicted transformation behaviour and demonstrate compatibility between digital and physical models. This advancement in kinetic behaviour offers new insights into transformable structures for adaptable buildings, broadening their potential applications within the field of architectural engineering.