The details of second-order partial derivatives of rigid-body Inverse/Forward dynamics are provided. Several properties and identities using Spatial Vector Algebra are listed, along with their detailed derivations. The expressions build upon previous work by the author on first-order partial derivatives of inverse dynamics. The first/second-order derivatives are also extended for systems with external forces. Finally, the KKT Forward dynamics and Impact dynamics derivatives are derived.

翻译：提供了刚体逆/正动力学中二阶偏导数的细节。列出了空间向量代数中的若干性质与恒等式，并给出了其详细推导。这些表达式建立在作者先前关于逆动力学一阶偏导数工作的基础之上。一阶/二阶导数结果也扩展至包含外力的系统。最后，推导了KKT正动力学与冲击动力学的导数。