The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity, respectively, as they attest the cusp configuration of the mechanism a third-order rigidity, which conflicts with its continuous flexion. Some attempts were done to resolve the dilemma but they could not settle the problem. As cusp mechanisms demonstrate the basic shortcoming of any local mobility analysis using higher-order constraints, we present a global approach inspired by Sabitov's finite algorithm for testing the bendability of a polyhedron, which allows us (a) to compute iteratively configurations with a higher-order flexion and (b) to come up with a proper redefinition of higher-order flexibility and rigidity. We also give algorithms for computing the flexion orders as well as the associated flexes. The presented approach is demonstrated on several examples (double-Watt mechanisms and Tarnai's Leonardo structure). Moreover, we determine all configurations of a given 3-RPR manipulator with a third-order flexion and present a corresponding joint-bar framework of flexion order 23.

翻译：Connelly与Servatius提出的双瓦特机构著名实例揭示了高阶柔性与刚性经典定义存在的若干问题，因其判定该机构的尖点构型具有三阶刚性，这与其连续屈曲特性相矛盾。已有研究尝试解决该矛盾但未能彻底阐明问题。鉴于尖点机构暴露了采用高阶约束进行局部运动学分析的根本缺陷，本文受Sabitov多面体可弯性有限算法的启发，提出一种全局分析方法，使我们能够：（a）迭代计算具有高阶屈曲的构型；（b）提出高阶柔性与刚性的恰当重新定义。我们还给出了计算屈曲阶次及相关屈曲模式的算法。通过多个案例（双瓦特机构与Tarnai的达芬奇结构）验证了所提方法。此外，我们确定了给定3-RPR机械臂所有具有三阶屈曲的构型，并提出了屈曲阶次为23的对应铰接杆系结构。