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\subsection{Probability}
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\subsection{Probability}
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\shortdefinition[Sum rule] $P(X) = \sum P(X, Y) = \sum P(X \cap Y)$
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\shortdefinition[Sum rule] $\P(X) = \sum \P(X, Y) = \sum \P(X \cap Y)$
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\shortdefinition[Prod] $P(X, Y) = P(X | Y) P(Y) = P(Y | X) P(X)$
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\shortdefinition[Prod] $\P(X \cap Y) = \P(X | Y) \P(Y) = \P(Y | X) P(X)$
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\shorttheorem[Bayes] $\displaystyle P(Y_i | X) = \frac{P(X | Y_i) P(Y_i)}{\sum_{j = 1}^n P(X | Y_j) P(Y_j)}$
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\shorttheorem[Bayes] $\displaystyle \P(Y_i | X) = \frac{\P(X | Y_i) \P(Y_i)}{\sum_{j = 1}^n \P(X | Y_j) \P(Y_j)}$
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\shortdefinition[Cont. Var] Sums become integrals\\
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\shortdefinition[Cont. Var] Sums become integrals\\
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e.g. $\sum_{X} P(X) = 1$ becomes $\int p(x) \dx = 1$
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e.g. $\sum_{X} \P(X) = 1$ becomes $\int \P(x) \dx = 1$
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\shortdefinition[Indep.] $x, y$ indep. iff $p(x, y) = p(x) p(y)$
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\shortdefinition[Indep.] $x, y$ indep. iff $\P(\cX \cap \cY) = \P(\cX) \P(\cY)$
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\shortdefinition[Cond. Indep.] iff $p(x, y | z) = p(x|z) p(y|z)$
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\shortdefinition[Cond. Indep.] iff $\P(\cX \cap \cY | \cZ) = \P(\cX | \cY) \P(\cY | \cZ)$
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\shortdefinition $E[\vec{x}] = \int_{-\8}^{\8} \vec{x} p(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$
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\shortdefinition $\E[\vec{x}] = \int_{-\8}^{\8} \vec{x} \P(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$
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\shortdefinition $\text{Cov}[x] = E[\vec{x} \vec{x}^\top] - E[\vec{x}]E[\vec{x}]^\top = \mat{\Sigma}$
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\shortdefinition $\text{Cov}[x] = \E[\vec{x} \vec{x}^\top] - \E[\vec{x}]\E[\vec{x}]^\top = \mat{\Sigma}$
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\shortdefinition[Gauss. Dist.] $\vec{x} \sim \cN(\vec{\mu}, \mat{\Sigma})$ ($\vec{\mu}$ mean, $\mat{\Sigma}$ cov.),\\
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\shortdefinition[Gauss. Dist.] $\vec{x} \sim \cN(\vec{\mu}, \mat{\Sigma})$ ($\vec{\mu}$ mean, $\mat{\Sigma}$ cov.),\\
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PDF: $p(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k |\mat{\Sigma}|}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$
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PDF: $f(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k \det(\mat{\Sigma})}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$
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$\Sigma^{-1}$ for $\Sigma$ diagonal, inverse of diag els (e.g. $\sigma^{-1}$)
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\subsection{Measurement models}
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\subsection{Measurement models}
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$\vec{z} = \vec{b}_C + s\mat{M} {_S}\vec{\omega} + \vec{b} + \vec{n} + \vec{o}$:
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$\vec{z} = \vec{b}_C + s\mat{M} {_S}\vec{\omega} + \vec{b} + \vec{n} + \vec{o}$:
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$\vec{b}_C$ const bias, $\vec{b}$ time bias, $\mat{M}$ missal., $\vec{n} \sim \cN(\vec{0}, \mat{R})$ noise, ${_S}\omega$ corr. meas., $\vec{o}$ other infl.
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$\vec{b}_C$ const bias, $\vec{b}$ time bias, $\mat{M}$ missal., $\vec{n} \sim \cN(\vec{0}, \mat{R})$ noise, ${_S}\omega$ corr. meas., $\vec{o}$ other infl.
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\hl{Finding}: Is in $W$-frame, so may need $\mat{T}_{BW}$ or $\mat{R}_{BW}$.
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Other model: $\vec{z} = \vec{h}(\vec{x}) + \vec{v}$, $\vec{h}(\vec{x})$ is pos of rob. dep. model
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\shortdefinition[Rot. Mat.]
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\shortdefinition[Rot. Mat.]
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{\scriptsize
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{\scriptsize
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$\mat{R}_{z}(\psi)$ (Yaw),
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$\mat{R}_{z}(\psi)$ (Yaw),
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$\mat{R}_y(\theta)$ (Pitch),
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$\mat{R}_y(\theta)$ (Pitch),
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$\mat{R}_x(\varphi)$ (Roll)\\
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$\mat{R}_x(\varphi)$ (Roll)\\
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$\begin{bmatrix}
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$\begin{bmatrix}
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c(\psi) & -s(\psi) & 0 \\
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c(\psi) & -s(\psi) & 0 \\
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s(\psi) & c(\psi) & 0 \\
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s(\psi) & c(\psi) & 0 \\
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0 & 0 & 1
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0 & 0 & 1
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\end{bmatrix};
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\end{bmatrix};
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\begin{bmatrix}
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\begin{bmatrix}
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c(\theta) & 0 & s(\theta) \\
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c(\theta) & 0 & s(\theta) \\
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0 & 1 & 0 \\
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0 & 1 & 0 \\
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-s(\theta) & 0 & c(\theta) \\
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-s(\theta) & 0 & c(\theta) \\
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\end{bmatrix};
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\end{bmatrix};
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\begin{bmatrix}
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\begin{bmatrix}
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1 & 0 & 0 \\
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1 & 0 & 0 \\
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0 & c(\varphi) & -s(\varphi) \\
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0 & c(\varphi) & -s(\varphi) \\
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0 & s(\varphi) & c(\varphi)
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0 & s(\varphi) & c(\varphi)
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\end{bmatrix}$
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\end{bmatrix}$
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}
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}
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\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
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\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
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\shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
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\shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
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$\begin{smallmatrix}
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$\begin{smallmatrix}
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\psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
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\psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
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\theta = \arcsin(-R_{31})\\
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\theta = \arcsin(-R_{31})\\
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\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
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\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
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\end{smallmatrix}$
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\end{smallmatrix}$
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{\scriptsize
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{\scriptsize
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$[\vec{n}]^\times = \begin{bmatrix}
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$[\vec{n}]^\times = \begin{bmatrix}
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0 & -a_3 & a_2 \\
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0 & -a_3 & a_2 \\
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a_3 & 0 & -a_1 \\
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a_3 & 0 & -a_1 \\
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-a_2 & a_1 & 0
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-a_2 & a_1 & 0
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\end{bmatrix}$
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\end{bmatrix}$
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}
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}
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\shortdefinition[Rot. Vec]
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\shortdefinition[Angle-Axis]
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$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
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$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
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$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$
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$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
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To quat: $\vec{q} = [\vec{n}, \alpha]$
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\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
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\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
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$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$).
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$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$).
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$\vec{q} = \begin{bmatrix}
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$\vec{q} = \begin{bmatrix}
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\vec{v}(\vec{q}), a(\vec{q})
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\vec{v}(\vec{q}), a(\vec{q})
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\end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')
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\end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')
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\bi{Mult} {\scriptsize
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\bi{Mult} {\scriptsize
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$\vec{q} \otimes \vec{p} = \begin{bmatrix}
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$\vec{q} \otimes \vec{p} = \begin{bmatrix}
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a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
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a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
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a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
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a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
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\end{bmatrix}$
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\end{bmatrix}$
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}
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}
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\bi{To Rot Mat} {\scriptsize
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\bi{To Rot Mat} {\scriptsize
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$R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
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$R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
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}
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}
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\shortdefinition[Transf. M]
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\shortdefinition[Transf. M]
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{\scriptsize
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{\scriptsize
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$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
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$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
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$\mat{T}_{AB} = \begin{bmatrix}
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$\mat{T}_{AB} = \begin{bmatrix}
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\mat{R}_{AB} & {_A}\vec{t}_B \\
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\mat{R}_{AB} & {_A}\vec{t}_B \\
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\mat{0}_{1\times 3} & 1
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\mat{0}_{1\times 3} & 1
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\end{bmatrix};
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\end{bmatrix};
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\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
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\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
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\begin{bmatrix}
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\begin{bmatrix}
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\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
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\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
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\mat{0}_{1 \times 3} & 1
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\mat{0}_{1 \times 3} & 1
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\end{bmatrix}$
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\end{bmatrix}$
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}
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}
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\subsection{Forward Kinematics (FK)}
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\subsection{Forward Kinematics (FK)}
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$\mat{T}_{WB_n}(\theta) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
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$\mat{T}_{WB_n}(\vec{\theta}) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
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For 2R system:
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For 2R system:
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${_W}\vec{t}_{WE} =$
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${_W}\vec{t}_{WE} =$
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{\scriptsize
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{\scriptsize
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