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：本文详细推导了刚体逆/正动力学二阶偏导数。基于空间向量代数给出了若干性质与恒等式，并提供了详细的推导过程。相关表达式建立在作者先前关于逆动力学一阶偏导数的工作基础之上。同时，本文将一阶/二阶导数推广至含外力系统。最后，推导了含外力的KKT正动力学及碰撞动力学的导数。