[Connelly and Servatius, 1994] shows the difficulty of properly defining n-th order rigidity and flexiblity of a bar-and-joint framework for higher order (n >= 3) through the introduction of a cusp mechanism. The author proposes a "proper" definition of the order of rigidity by the order of elongation of the bars with respect to the arclength along the path in the configuration space. We show that the classic definition using formal n-th derivative of the length constraint is a sufficient condition for the n-th flexiblity in the proposed definition and also a necessary condition only for n = 1, 2.

翻译：[Connelly and Servatius, 1994] 通过引入尖点机制，揭示了为高阶（n ≥ 3）杆件-铰接框架正确定义 n 阶刚度和柔性的困难。作者提出一种“恰当”的刚度阶次定义，其依据是杆件长度沿构型空间中路径弧长的伸长阶次。我们证明，使用长度约束的形式 n 阶导数的经典定义是所提定义中 n 阶柔性的充分条件，并且仅当 n = 1, 2 时才是必要条件。