20230208 对偶四元数的乘法

---

前言

对偶四元数也称为螺旋算子，能够有效描述位置和姿态的耦合关系。

---

一、对偶四元数是什么？

对偶四元数是同时描述位置和姿态的一种工具，本文对其乘法进行讲解。

二、对偶四元数乘法

给定两个对偶四元数，其乘法按照四元数乘法进行处理。

$\mathbf{a}=[a_i]\in {\mathbb{R}}^4$，$\mathbf{b}=[b_i]\in {\mathbb{R}}^4$，$\mathbf{c}=[c_i]\in {\mathbb{R}}^4$，$\mathbf{d}=[d_i]\in {\mathbb{R}}^4$为四个四元数，其中 $a_1$，$b_1$，$c_1$，$d_1$为相应的四个标量，$\epsilon$ 为对偶四元数的单位。

$$\left[\begin{array}{c}{a}_1+\epsilon b_1\\ a_2+\epsilon b_2\\ a_3+\epsilon b_3\\ a_4+\epsilon b_4\end{array}\right]\circ \left[\begin{array}{c}{c}_1+\epsilon d_1\\ c_2+\epsilon d_2\\ c_3+\epsilon d_3\\ c_4+\epsilon d_4\end{array}\right]$$

参考四元数的乘法，可以对上式进行展开。
$$\left[\begin{array}{c}{a}_1+\epsilon b_1\\ a_2+\epsilon b_2\\ a_3+\epsilon b_3\\ a_4+\epsilon b_4\end{array}\right]\circ \left[\begin{array}{c}{c}_1+\epsilon d_1\\ c_2+\epsilon d_2\\ c_3+\epsilon d_3\\ c_4+\epsilon d_4\end{array}\right]=\left[\begin{array}{c}(a_1+\epsilon b_1)(c_1+\epsilon d_1)-(a_2+\epsilon b_2)(c_2+\epsilon d_2)-(a_3+\epsilon b_3)(c_3+\epsilon d_3)-(a_4+\epsilon b_4)(c_4+\epsilon d_4)\\ (c_1+\epsilon d_1)\left[\begin{array}{c}{a}_2+\epsilon b_2\\ a_3+\epsilon b_3\\ a_4+\epsilon b_4\end{array}\right]+(a_1+\epsilon b_1)\left[\begin{array}{c}{c}_2+\epsilon d_2\\ c_3+\epsilon d_3\\ c_4+\epsilon d_4\end{array}\right]+{\left[\begin{array}{c}{a}_2+\epsilon b_2\\ a_3+\epsilon b_3\\ a_4+\epsilon b_4\end{array}\right]}^{\times }\left[\begin{array}{c}{c}_2+\epsilon d_2\\ c_3+\epsilon d_3\\ c_4+\epsilon d_4\end{array}\right]\end{array}\right]$$
同时，有
$${\left[\begin{array}{c}{a}_2+\epsilon b_2\\ a_3+\epsilon b_3\\ a_4+\epsilon b_4\end{array}\right]}^{\times }=\left[\begin{array}{ccc}0& -(a_4+\epsilon b_4)& a_3+\epsilon b_3\\ a_4+\epsilon b_4& 0& -(a_2+\epsilon b_2)\\ -(a_3+\epsilon b_3)& a_2+\epsilon b_2& 0\end{array}\right]$$

进一步展开，
$$\left[\begin{array}{c}{a}_1+\epsilon b_1\\ a_2+\epsilon b_2\\ a_3+\epsilon b_3\\ a_4+\epsilon b_4\end{array}\right]\circ \left[\begin{array}{c}{c}_1+\epsilon d_1\\ c_2+\epsilon d_2\\ c_3+\epsilon d_3\\ c_4+\epsilon d_4\end{array}\right]=\left[\begin{array}{c}{a}_1{c}_1-a_2{c}_2-a_3{c}_3-a_4{c}_4+\epsilon (b_1{c}_1+a_1{d}_1-b_2{c}_2-a_2{d}_2-b_3{c}_3-a_3{d}_3-b_4{c}_4-a_4{d}_4)\\ \left[\begin{array}{c}{c}_1{a}_2+\epsilon (c_1{b}_2+d_1{a}_2)\\ c_1{a}_3+\epsilon (c_1{b}_3+d_1{a}_3)\\ c_1{a}_4+\epsilon (c_1{b}_4+d_1{a}_4)\end{array}\right]+\left[\begin{array}{c}{a}_1{c}_2+\epsilon (a_1{d}_2+b_1{c}_2)\\ a_1{c}_3+\epsilon (a_1{d}_3+b_1{c}_3)\\ a_1{c}_4+\epsilon (a_1{d}_4+b_1{c}_4)\end{array}\right]+\left[\begin{array}{ccc}0& -(a_4+\epsilon b_4)& a_3+\epsilon b_3\\ a_4+\epsilon b_4& 0& -(a_2+\epsilon b_2)\\ -(a_3+\epsilon b_3)& a_2+\epsilon b_2& 0\end{array}\right]\left[\begin{array}{c}{c}_2+\epsilon d_2\\ c_3+\epsilon d_3\\ c_4+\epsilon d_4\end{array}\right]\end{array}\right]$$

再进行整理，可以得到
$$\left[\begin{array}{c}{a}_1+\epsilon b_1\\ a_2+\epsilon b_2\\ a_3+\epsilon b_3\\ a_4+\epsilon b_4\end{array}\right]\circ \left[\begin{array}{c}{c}_1+\epsilon d_1\\ c_2+\epsilon d_2\\ c_3+\epsilon d_3\\ c_4+\epsilon d_4\end{array}\right]=\left[\begin{array}{c}{a}_1{c}_1-a_2{c}_2-a_3{c}_3-a_4{c}_4+\epsilon (b_1{c}_1+a_1{d}_1-b_2{c}_2-a_2{d}_2-b_3{c}_3-a_3{d}_3-b_4{c}_4-a_4{d}_4)\\ \left[\begin{array}{c}{c}_1{a}_2+a_1{c}_2+\epsilon (c_1{b}_2+d_1{a}_2+a_1{d}_2+b_1{c}_2)\\ c_1{a}_3+a_1{c}_3+\epsilon (c_1{b}_3+d_1{a}_3+a_1{d}_3+b_1{c}_3)\\ c_1{a}_4+a_1{c}_4+\epsilon (c_1{b}_4+d_1{a}_4+a_1{d}_4+b_1{c}_4)\end{array}\right]+\left[\begin{array}{c}-(a_4+\epsilon b_4)(c_3+\epsilon d_3)+(a_3+\epsilon b_3)(c_4+\epsilon d_4)\\ (a_4+\epsilon b_4)(c_2+\epsilon d_2)-(a_2+\epsilon b_2)(c_4+\epsilon d_4)\\ -(a_3+\epsilon b_3)(c_2+\epsilon d_2)+(a_2+\epsilon b_2)(c_3+\epsilon d_3)\end{array}\right]\end{array}\right]$$

---

总结

对偶四元数的乘法与四元数的乘法形式一致。