Approximating cylinders with bundle-folding plane-symmetric Bricard linkages

This paper proposes a geometric method via isosceles trapezoids or isosceles trapezoidal prisms (ITP) to approximate a smooth planar curve or a cylinder. The method is then extended for designing double-layer cylinder-approximating deployable mechanisms (CDMs) via ITP mechanisms. A bundle-folding plane-symmetric Bricard linkage, which has an isosceles trapezoidal deployed configuration, is developed. The mappings between geometric parameters and the working configurations are discussed. Kinematics of the isosceles trapezoidal Bricard linkage are analyzed. By linking two sets of isosceles trapezoidal Bricard mechanisms and rectangular Bricard mechanisms under certain geometric conditions, the novel ITP mechanism is synthesized. Two types of ITP mechanisms, which have the same folded and deployed configurations but different intermediate configurations, are obtained. The geometric constraints for constructing general CDMs via ITP mechanisms are presented. The construction method is then validated by three cases of approximating a cylindrical cylinder, a parabolic cylinder, and a sinusoidal cylinder, respectively. The proposed method, which can be applied to all cylinders, provides the geometric guide to construct double-layer CDMs and facilitates engineering applications of the plane-symmetric Bricard linkages.