Homework in Chapter 3 of slam course of dark blue vision -- derivative application of T6 common functions

0. introduction

The number of homework this time is the most difficult , It took a long time , Record it after you finish .

1. subject & answer

• among (6.2) Yes $e^x$ Taylor's expansion （ In fact, I'm more used to the expansion of infinite series ）
• (6.3) Yes BCH The formula （《SLAM14 speak 》P82） and T5 Conclusion
  

2. Summary

(6.3) In the use of BCH In formula ,$R_1{R}_2$ To convert to Lie algebraic form $ln(R_1{R}_2{)}^{\vee }$

• (6.5) Chinese formula $$R{a}^{\wedge }{R}^T=(Ra{)}^{\wedge }$$ want Make $R=R^T$（ It took me a long time to think of this bend , And then you can use $R_2$ be able to raise a certain amount of money etc. BCH The form required by the formula ） have to
  $$R^T{a}^{\wedge }R=(R^{T}a{)}^{\wedge }$$
• Pay special attention to Jacobian matrix is $J_l$ still $J_r$ It has nothing to do with whether the given disturbance is a left disturbance or a right disturbance , yes $J_l$ still $J_r$ depends BCH Is the small quantity in the formula on the left or right , A small quantity on the left is $J_l$, A small quantity on the right is $J_r$.(6.5) Medium is for $R_2$ Left disturbance , But the resulting Jacobian is $J_r$, Of course , It can also be in (6.5) The first 2 Line with $R_1^T{R}_1$ Gather together
  $$\begin{gathered} \underset{{\phi }_2\to 0}{\lim }\frac{ln(R_{1}exp({\phi }_2^{\wedge })R_1^T{R}_1{R}_2{)}^{\vee }-ln(R_1{R}_2{)}^{\vee }}{{\phi }_2} \\ =\underset{{\phi }_2\to 0}{\lim }\frac{ln(exp((R_1{\phi }_2{)}^{\wedge })R_1{R}_2{)}^{\vee }-ln(R_1{R}_2{)}^{\vee }}{{\phi }_2} \\ \approx (BCHJaneturn,mostendjunctionfruit)J_l(ln(R_1{R}_2{)}^{\vee }{)}^{-1}{R}_1 \end{gathered}$$
  It uses $J_l$（ I haven't verified this result yet , Wait a moment to verify it and see if it's right .）
• According to the teaching assistant, in VIO Commonly used $J_r$, Later on , Handwriting VIO It has been arranged .