SLAM学习-李群和李代数相互转换公式
作者：互联网
在学习 十四讲时，李群和李代数相互转换公式为
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exp(\phi^{\wedge})=exp(\theta a^{\wedge})=cos\theta I+(1-cos\theta)aa^{T}+sin\theta a^{\wedge}
exp(ϕ∧)=exp(θa∧)=cosθI+(1−cosθ)aaT+sinθa∧
其证明过程可以参考四十讲
而在选在IMU 预积分中,李群和李代数相互转换公式为
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exp(\phi^{\wedge})= I+\frac{sin(||\phi ||)}{||\phi||}\phi^{\wedge}+\frac{1-cos(||\phi ||)}{||\phi||^{2}}(\phi^{\wedge})^{2}
exp(ϕ∧)=I+∣∣ϕ∣∣sin(∣∣ϕ∣∣)ϕ∧+∣∣ϕ∣∣21−cos(∣∣ϕ∣∣)(ϕ∧)2
这是两种等价的表示方式
下边我们简单证明下:
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\phi = \theta \vec{a} \\ a^{\wedge}a^{\wedge} = aa^{T}-I\\
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a∧a∧=aaT−I
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||\phi||=\theta\\ {\phi}^{\wedge} =\theta a^{\wedge}=||\phi||a^{\wedge}\\ a^{\wedge}=\frac{{\phi}^{\wedge} }{||\phi||}\\ ({\phi}^{\wedge})^{2} = \theta^{2}(aa^{T}-I)=||\phi||^{2}(aa^{T}-I)\\ aa^{T}=\frac{({\phi}^{\wedge})^{2}}{||\phi||^{2}}+I
∣∣ϕ∣∣=θϕ∧=θa∧=∣∣ϕ∣∣a∧a∧=∣∣ϕ∣∣ϕ∧(ϕ∧)2=θ2(aaT−I)=∣∣ϕ∣∣2(aaT−I)aaT=∣∣ϕ∣∣2(ϕ∧)2+I
带入
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exp(\phi^{\wedge})=exp(\theta a^{\wedge})=cos\theta I+(1-cos\theta)aa^{T}+sin\theta a^{\wedge}
exp(ϕ∧)=exp(θa∧)=cosθI+(1−cosθ)aaT+sinθa∧
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exp(\phi^{\wedge})=cos||\phi|| I+(1-cos||\phi||)*(\frac{({\phi}^{\wedge})^{2}}{||\phi||^{2}}+I)+sin||\phi|| *\frac{{\phi}^{\wedge} }{||\phi||}\\ =I+\frac{sin(||\phi ||)}{||\phi||}\phi^{\wedge}+\frac{1-cos(||\phi ||)}{||\phi||^{2}}(\phi^{\wedge})^{2}
exp(ϕ∧)=cos∣∣ϕ∣∣I+(1−cos∣∣ϕ∣∣)∗(∣∣ϕ∣∣2(ϕ∧)2+I)+sin∣∣ϕ∣∣∗∣∣ϕ∣∣ϕ∧=I+∣∣ϕ∣∣sin(∣∣ϕ∣∣)ϕ∧+∣∣ϕ∣∣21−cos(∣∣ϕ∣∣)(ϕ∧)2
标签：wedge,phi,李群,cos,SLAM,exp,theta,代数,sin 来源： https://blog.csdn.net/m0_54238665/article/details/122805107