Denavit and Hartenberg based methods as Cardan, Fick and Euler angles
describe the position and orientation of an end-effector in Three Dimensional
(3D) space. However, the generation of unrealistic human posture in joint space
constitutes the weak point to these methods because they impose a well-defined
rotations order. A method to handle the transformation homogeneous performance
uses the dual quaternions. Quaternions have proven themselves in many fields as
providing a computational efficient method to represent a rotation, and yet,
they can not deal with the translations in 3D-space. The dual numbers can
extend quaternions to dual quaternions. This paper exploits dual quaternions
theory to provide a fast and accurate solution to the forward, inverse
kinematics and recursive Newton-Euler dynamics algorithm for 7 Degree of
Freedom (DOF) human lower limb in 3D-space.

本研究利用双四元数理论，针对七自由度人类下肢在三维空间内的正、逆运动学和递归牛顿 - 欧拉动力学算法提供了快速准确的解决方案。

使用双四元数代数对七自由度人体下肢进行运动学和动力学建模

Kinematics and Dynamics Modeling of 7 Degrees of Freedom Human Lower Limb Using Dual Quaternions Algebra

Time-varying, smooth trajectory estimation is of great interest to the vision
community for accurate and well behaving 3D systems. In this paper, we propose
a novel principal component local regression filter acting directly on the
Riemannian manifold of unit dual quaternions $\mathbb{D} \mathbb{H}_1$. We use
a numerically stable Lie algebra of the dual quaternions together with $\exp$
and $\log$ operators to locally linearize the 6D pose space. Unlike state of
the art path smoothing methods which either operate on $SO\left(3\right)$ of
rotation matrices or the hypersphere $\mathbb{H}_1$ of quaternions, we treat
the orientation and translation jointly on the dual quaternion quadric in the
7-dimensional real projective space $\mathbb{R}\mathbb{P}^7$. We provide an
outlier-robust IRLS algorithm for generic pose filtering exploiting this
manifold structure. Besides our theoretical analysis, our experiments on
synthetic and real data show the practical advantages of the manifold aware
filtering on pose tracking and smoothing.

本文提出了一种基于双四元数 Riemann 流形的新型主成分局部回归滤波器，并在 7 维实射影空间中，共同处理旋转和平移，利用此流形结构提供一种异常鲁棒的通用姿势过滤 IRLS 算法，从理论和实验上表明了该流形感知滤波对姿态跟踪和平滑处理的实际优势。