Abstract Hyper-Dual Quaternions, HDQ, are first introduced in this paper. Utilizing the Hyper-Dual Numbers, HDN, Dual Quaternions, DQ, and hyper-dual angle, the general expression of HDQ is obtained. Rigid single- and multi-body kinematics such as in a serial robot kinematics, are then expressed in HDQ form. Taking advantage of the dual numbers’ “automatic differentiation” feature, HDQ encompasses both body pose (translational and rotational) and body velocities in a compact form with no need for further pose differentiation.

中文翻译：

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刚体运动学的超双四元数表示

摘要 本文首先介绍了超双四元数 HDQ。利用超对偶数、HDN、对偶四元数、DQ、超对偶角，得到HDQ的一般表达式。刚性单体和多体运动学（例如在串行机器人运动学中）然后以 HDQ 形式表示。利用双数的“自动微分”功能，HDQ 以紧凑的形式包含身体姿势（平移和旋转）和身体速度，无需进一步的姿势微分。