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b/electives/amr/parts/00_basics/00_probability.tex @@ -1,20 +1,22 @@ \subsection{Probability} -\shortdefinition[Sum rule] $P(X) = \sum P(X, Y) = \sum P(X \cap Y)$ +\shortdefinition[Sum rule] $\P(X) = \sum \P(X, Y) = \sum \P(X \cap Y)$ -\shortdefinition[Prod] $P(X, Y) = P(X | Y) P(Y) = P(Y | X) P(X)$ +\shortdefinition[Prod] $\P(X \cap Y) = \P(X | Y) \P(Y) = \P(Y | X) P(X)$ -\shorttheorem[Bayes] $\displaystyle P(Y_i | X) = \frac{P(X | Y_i) P(Y_i)}{\sum_{j = 1}^n P(X | Y_j) P(Y_j)}$ +\shorttheorem[Bayes] $\displaystyle \P(Y_i | X) = \frac{\P(X | Y_i) \P(Y_i)}{\sum_{j = 1}^n \P(X | Y_j) \P(Y_j)}$ \shortdefinition[Cont. Var] Sums become integrals\\ -e.g. $\sum_{X} P(X) = 1$ becomes $\int p(x) \dx = 1$ +e.g. $\sum_{X} \P(X) = 1$ becomes $\int \P(x) \dx = 1$ -\shortdefinition[Indep.] $x, y$ indep. iff $p(x, y) = p(x) p(y)$ +\shortdefinition[Indep.] $x, y$ indep. iff $\P(\cX \cap \cY) = \P(\cX) \P(\cY)$ -\shortdefinition[Cond. Indep.] iff $p(x, y | z) = p(x|z) p(y|z)$ +\shortdefinition[Cond. Indep.] iff $\P(\cX \cap \cY | \cZ) = \P(\cX | \cY) \P(\cY | \cZ)$ -\shortdefinition $E[\vec{x}] = \int_{-\8}^{\8} \vec{x} p(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$ +\shortdefinition $\E[\vec{x}] = \int_{-\8}^{\8} \vec{x} \P(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$ -\shortdefinition $\text{Cov}[x] = E[\vec{x} \vec{x}^\top] - E[\vec{x}]E[\vec{x}]^\top = \mat{\Sigma}$ +\shortdefinition $\text{Cov}[x] = \E[\vec{x} \vec{x}^\top] - \E[\vec{x}]\E[\vec{x}]^\top = \mat{\Sigma}$ \shortdefinition[Gauss. Dist.] $\vec{x} \sim \cN(\vec{\mu}, \mat{\Sigma})$ ($\vec{\mu}$ mean, $\mat{\Sigma}$ cov.),\\ -PDF: $p(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k |\mat{\Sigma}|}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$ +PDF: $f(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k \det(\mat{\Sigma})}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$ + +$\Sigma^{-1}$ for $\Sigma$ diagonal, inverse of diag els (e.g. $\sigma^{-1}$) diff --git a/electives/amr/parts/00_basics/01_measurement-models.tex b/electives/amr/parts/00_basics/01_measurement-models.tex index 3118abb..f2f798d 100644 --- a/electives/amr/parts/00_basics/01_measurement-models.tex +++ b/electives/amr/parts/00_basics/01_measurement-models.tex @@ -1,3 +1,6 @@ \subsection{Measurement models} $\vec{z} = \vec{b}_C + s\mat{M} {_S}\vec{\omega} + \vec{b} + \vec{n} + \vec{o}$: $\vec{b}_C$ 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. + +\hl{Finding}: Is in $W$-frame, so may need $\mat{T}_{BW}$ or $\mat{R}_{BW}$. +Other model: $\vec{z} = \vec{h}(\vec{x}) + \vec{v}$, $\vec{h}(\vec{x})$ is pos of rob. dep. model diff --git a/electives/amr/parts/01_kinematics/00_intro.tex b/electives/amr/parts/01_kinematics/00_intro.tex index 09bcd1a..ef027dc 100644 --- a/electives/amr/parts/01_kinematics/00_intro.tex +++ b/electives/amr/parts/01_kinematics/00_intro.tex @@ -8,24 +8,24 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb \shortdefinition[Rot. Mat.] {\scriptsize - $\mat{R}_{z}(\psi)$ (Yaw), - $\mat{R}_y(\theta)$ (Pitch), - $\mat{R}_x(\varphi)$ (Roll)\\ - $\begin{bmatrix} - c(\psi) & -s(\psi) & 0 \\ - s(\psi) & c(\psi) & 0 \\ - 0 & 0 & 1 - \end{bmatrix}; - \begin{bmatrix} - c(\theta) & 0 & s(\theta) \\ - 0 & 1 & 0 \\ - -s(\theta) & 0 & c(\theta) \\ - \end{bmatrix}; - \begin{bmatrix} - 1 & 0 & 0 \\ - 0 & c(\varphi) & -s(\varphi) \\ - 0 & s(\varphi) & c(\varphi) - \end{bmatrix}$ + $\mat{R}_{z}(\psi)$ (Yaw), + $\mat{R}_y(\theta)$ (Pitch), + $\mat{R}_x(\varphi)$ (Roll)\\ + $\begin{bmatrix} + c(\psi) & -s(\psi) & 0 \\ + s(\psi) & c(\psi) & 0 \\ + 0 & 0 & 1 + \end{bmatrix}; + \begin{bmatrix} + c(\theta) & 0 & s(\theta) \\ + 0 & 1 & 0 \\ + -s(\theta) & 0 & c(\theta) \\ + \end{bmatrix}; + \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & c(\varphi) & -s(\varphi) \\ + 0 & s(\varphi) & c(\varphi) + \end{bmatrix}$ } \shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$ @@ -36,52 +36,53 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb \shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$. $\begin{smallmatrix} - \psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\ - \theta = \arcsin(-R_{31})\\ - \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\ - \end{smallmatrix}$ + \psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\ + \theta = \arcsin(-R_{31})\\ + \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\ + \end{smallmatrix}$ {\scriptsize -$[\vec{n}]^\times = \begin{bmatrix} - 0 & -a_3 & a_2 \\ - a_3 & 0 & -a_1 \\ - -a_2 & a_1 & 0 - \end{bmatrix}$ -} + $[\vec{n}]^\times = \begin{bmatrix} + 0 & -a_3 & a_2 \\ + a_3 & 0 & -a_1 \\ + -a_2 & a_1 & 0 + \end{bmatrix}$ + } -\shortdefinition[Rot. Vec] +\shortdefinition[Angle-Axis] $\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\ -$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$ +$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\ +To quat: $\vec{q} = [\vec{n}, \alpha]$ \shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\ $i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$). $\vec{q} = \begin{bmatrix} - \vec{v}(\vec{q}), a(\vec{q}) - \end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'') + \vec{v}(\vec{q}), a(\vec{q}) + \end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'') \bi{Mult} {\scriptsize - $\vec{q} \otimes \vec{p} = \begin{bmatrix} - a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\ - a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p}) - \end{bmatrix}$ + $\vec{q} \otimes \vec{p} = \begin{bmatrix} + a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\ + a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p}) + \end{bmatrix}$ } \bi{To Rot Mat} {\scriptsize - $R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$ + $R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$ } \shortdefinition[Transf. M] {\scriptsize - $\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\ - $\mat{T}_{AB} = \begin{bmatrix} - \mat{R}_{AB} & {_A}\vec{t}_B \\ - \mat{0}_{1\times 3} & 1 - \end{bmatrix}; - \mat{T}_{BA} = \mat{T}_{AB}^{-1} = - \begin{bmatrix} - \mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\ - \mat{0}_{1 \times 3} & 1 - \end{bmatrix}$ + $\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\ + $\mat{T}_{AB} = \begin{bmatrix} + \mat{R}_{AB} & {_A}\vec{t}_B \\ + \mat{0}_{1\times 3} & 1 + \end{bmatrix}; + \mat{T}_{BA} = \mat{T}_{AB}^{-1} = + \begin{bmatrix} + \mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\ + \mat{0}_{1 \times 3} & 1 + \end{bmatrix}$ } diff --git a/electives/amr/parts/01_kinematics/01_forward.tex b/electives/amr/parts/01_kinematics/01_forward.tex index bbf1e70..252748a 100644 --- a/electives/amr/parts/01_kinematics/01_forward.tex +++ b/electives/amr/parts/01_kinematics/01_forward.tex @@ -1,5 +1,5 @@ \subsection{Forward Kinematics (FK)} -$\mat{T}_{WB_n}(\theta) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\ +$\mat{T}_{WB_n}(\vec{\theta}) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\ For 2R system: ${_W}\vec{t}_{WE} =$ {\scriptsize