From a106331872a1e6024b54c83c4d17638278b83d72 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 3 Aug 2026 15:50:56 +0200 Subject: [PATCH] [AMR] Many learnings from exercises --- .../autonomous-mobile-robots-cheatsheet.pdf | Bin 686319 -> 761056 bytes .../autonomous-mobile-robots-cheatsheet.tex | 2 +- .../amr/parts/00_basics/02_trigonometry.tex | 2 + .../parts/00_basics/03_error-propagation.tex | 2 +- .../amr/parts/01_kinematics/01_forward.tex | 6 +-- .../amr/parts/01_kinematics/02_inverse.tex | 1 + .../01_kinematics/03_temporal-models.tex | 47 ++++++++++++++---- .../04_rigid-body-imu-kinematics.tex | 1 - .../parts/01_kinematics/06_wheeled-robot.tex | 1 - .../02_Sensors-Actuators/02_actuators.tex | 1 + .../00_linearization.tex | 4 ++ .../01_least-squares.tex | 1 + .../02_nonlinear-least-squares.tex | 15 +++--- .../03_bayes-filters.tex | 2 +- .../04_particle-filter.tex | 2 +- .../05_kalman-filter.tex | 2 +- .../06_extended-kalman-filter.tex | 19 ++++--- 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z)btK!ads4AlbU;X!QNtXbw_Us%*DH4|F}rQpE$?J!BF!Kw2z@H(Ain4=ApZN6G0?O z)0$<}TcFO=9!HDv%@zAuU6F@&6jN4rD-S0$#5l5_oQ;_G-AVmZu5yKvpZ%+ys;eD^A}QYPx(P zmr15#zupZ|FJgRsr&{qTG43!Wlf#Nsfg_dbTFMTQRpLy}(yV{5L0Tf9xPGd|s@vhM zP_h1{ag*6Y_hv0lAtyEP9=n87akDdj&PnY%wt^^G_pRe(H(8A)%Y|Mkq-@S5VKf2F1s=6C`Hz6;iXeE@pR4 zzB-UoH`V!oo(52p0$Mp+zH@i8GIRO|{ZnJ#xtqDUd%HcBR{ tol * np.linalg.norm(x): s = np.linalg.lstsq(DF(x), F(x))[0] - x = x-s; k += 1 # k opt for max iter + x = x-s; k += 1 # k optional for max iter return x, k \end{minted} -\rmvspace +} +\rmvspace[0.4] \bi{Levenberg-Marquardt} -Minimize $||F(x^{(k)}) + DF(x^{(k)})|| + \lambda||s||$. +Minimize $||F(x^{(k)}) + DF(x^{(k)})||^2 + \lambda||s||^2$, +with $s = \lambda_k DF(x^{(k)})$, $\lambda_k$ is called the \textit{learning rate} % TODO: Do we need these? \bi{Local Param.} diff --git a/electives/amr/parts/03_multi-sensor-estimation/03_bayes-filters.tex b/electives/amr/parts/03_multi-sensor-estimation/03_bayes-filters.tex index 0ea801d..014f3c8 100644 --- a/electives/amr/parts/03_multi-sensor-estimation/03_bayes-filters.tex +++ b/electives/amr/parts/03_multi-sensor-estimation/03_bayes-filters.tex @@ -1,6 +1,6 @@ \subsection{Bayes Filter (in DAG)} $\vec{x}_k^R$ state at time k, $\vec{z}_k^p$ dist. meas., $\vec{u}^p_k$ wheel odometry (= meas.). -Typ. care ab. curr. state: altern. pred. \& update. +Typically care about current state: alternate predict \& update. % TODO: Do we really need the below? % Init prev distr $\P[\vec{x}_0^R]$. % Pred: $\P[\vec{x}_k^R \divider \vec{u}_{1:k}^p, \vec{z}_{1 : k - 1}^d] = \int \P[\vec{x}_k^R \divider \vec{u}_{k}^p, \vec{x}_{k - 1}^d] diff --git a/electives/amr/parts/03_multi-sensor-estimation/04_particle-filter.tex b/electives/amr/parts/03_multi-sensor-estimation/04_particle-filter.tex index eafe105..ff08a3e 100644 --- a/electives/amr/parts/03_multi-sensor-estimation/04_particle-filter.tex +++ b/electives/amr/parts/03_multi-sensor-estimation/04_particle-filter.tex @@ -1,5 +1,5 @@ \subsection{Particle Filter} -Is a bayes filter approximating the state distribution with a set of random samples. +Is a bayes filter approximating the state distribution with a set of random samples. Dist. not necessarily unimodal. Update step: \begin{itemize} \item Apply Bayes rule $w'_{k, s} = \P[\vec{z}_i \divider \vec{x}_{k, s}] w_{k - 1, s}$ diff --git a/electives/amr/parts/03_multi-sensor-estimation/05_kalman-filter.tex b/electives/amr/parts/03_multi-sensor-estimation/05_kalman-filter.tex index abc1c98..5866d2f 100644 --- a/electives/amr/parts/03_multi-sensor-estimation/05_kalman-filter.tex +++ b/electives/amr/parts/03_multi-sensor-estimation/05_kalman-filter.tex @@ -3,7 +3,7 @@ Bayes Filter for Gauss. dist of R.V. \& \hl{linear meas. model.} Initial state $\vec{x}_0 \sim \cN(\hat{\vec{x}}, \mat{P}_0)$, $\mat{P}_0$ previous covariance -\bi{Prediction} With linear state transition model ($\vec{u}_k$ odometry, $\vec{w}_k$ noise (covariance $\mat{Q}_k$)):\\ +\bi{Prediction} With lin. state trans. model ($\vec{u}_k$ wheel spd, $\vec{w}_k$ noise, covariance $\mat{Q}_k$). $\vec{x}_k = \mat{F}\vec{x}_{k - 1} + \mat{G}\vec{u}_k + \mat{L}\vec{w}_k$ with $\vec{w}_k \sim \cN(\vec{0}, \mat{Q}_k)$: \begin{itemize} \item \bi{Mean} $\hat{\vec{x}}_{k | k - 1} = \mat{F} \hat{\vec{x}}_{k - 1} + \mat{G} \vec{u}_k$ diff --git a/electives/amr/parts/03_multi-sensor-estimation/06_extended-kalman-filter.tex b/electives/amr/parts/03_multi-sensor-estimation/06_extended-kalman-filter.tex index afbc88e..50debe4 100644 --- a/electives/amr/parts/03_multi-sensor-estimation/06_extended-kalman-filter.tex +++ b/electives/amr/parts/03_multi-sensor-estimation/06_extended-kalman-filter.tex @@ -2,13 +2,16 @@ \hl{Non-l. state trans. model} $\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$ as above: \begin{itemize} \item \bi{Mean}: $\hat{\vec{x}}_{k | k - 1} = \vec{f}(\hat{\vec{x}}_{k - 1 | k - 1}, \vec{u}_k)$ - \item \bi{Covariance}: $\mat{P}_{k | k - 1} = \mat{F}_k \mat{P}_{k - 1 | k - 1} \mat{F}_k^\top + \mat{L}_k \mat{Q}_k \mat{L}_k^\top$\\ - With $\mat{F}_k$ linearisation $\frac{\partial \vec{f}}{\partial \vec{x}}$ and $\mat{L}_k$ lin. $\frac{\partial \vec{f}}{\partial \vec{w}}$ + \item \bi{Covariance}: $\mat{P}_{k | k - 1} = \mat{F}_k \mat{P}_{k - 1 | k - 1} \mat{F}_k^\top + \mat{L}_k \mat{Q}_k \mat{L}_k^\top$ \end{itemize} -\bi{Update} Non-Linear meas.: $\tilde{\vec{z}}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$: -\begin{itemize} - \item \bi{Meas. residual}: $\vec{y}_k = \tilde{\vec{z}}_k - \vec{h}(\hat{\vec{x}}_{k | k - 1})$ -\end{itemize} -Difference to KF: $\mat{H}$ becomes $\mat{H}_k$, and $\mat{H}^\top$ is $\mat{H}_k^\top$ +With $\mat{F}_k = \frac{\partial \vec{f}}{\partial \vec{x}}$ and $\mat{L}_k = \frac{\partial \vec{f}}{\partial \vec{w}}$ (computed as are Jacobians) -% TODO: Consider adding examples +\bi{Update} Non-Linear measurement: $\tilde{\vec{z}}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$: +\begin{itemize} + \item \bi{Measurement residual}: $\vec{y}_k = \tilde{\vec{z}}_k - \vec{h}(\hat{\vec{x}}_{k | k - 1})$ +\end{itemize} +Difference to KF: $\mat{H}$ becomes $\mat{H}_k$, and $\mat{H}^\top$ is $\mat{H}_k^\top$. +They are linearizations of $\vec{h}$, see \ref{sec:temporal-models} + +\shortexample D-D Rob, measurements: $\Delta \varphi_l, \varphi_r$, GPS pos, yaw angle. +% TODO: Finish this? diff --git a/electives/amr/parts/04_vision/01_bootstrapping.tex b/electives/amr/parts/04_vision/01_bootstrapping.tex index 0207a00..6b2d698 100644 --- a/electives/amr/parts/04_vision/01_bootstrapping.tex +++ b/electives/amr/parts/04_vision/01_bootstrapping.tex @@ -1,11 +1,10 @@ \subsection{Bootstrapping} -\bi{PnP Problem} {\scriptsize Persp. $n$-P.} +\bi{PnP Problem} {\scriptsize Perspective $n$-Point} Find sol. for camera pose \textit{directly} \bi{RANSAC} {\scriptsize RANdom SAmpling Consensus} for find. outliers \& correct -% TODO: Do we need to know this algorithm / be able to implement it? -% -> likely not +Also great for finding an initial guess for pose. (due to robustness) \bi{Stereo Triang.} Given two rays (known poses for points in 2D). Find good point in 3D. Fast sol: \bi{Midpoint Method}: diff --git a/electives/amr/parts/05_planning-control/01_motion-planning-exploration/01_a-star-algorithm.tex b/electives/amr/parts/05_planning-control/01_motion-planning-exploration/01_a-star-algorithm.tex index d3a38bc..125fa35 100644 --- a/electives/amr/parts/05_planning-control/01_motion-planning-exploration/01_a-star-algorithm.tex +++ b/electives/amr/parts/05_planning-control/01_motion-planning-exploration/01_a-star-algorithm.tex @@ -3,6 +3,7 @@ Function \texttt{h(\dots)} is a lower bound of optimal cost. % TODO: If space constrained, move to below % Init vert. with \texttt{dist}, \texttt{totDistEst} to $\8$, \texttt{prev} \texttt{undef}, \texttt{start} \begin{algorithm} + \small \begin{algorithmic}[1] \Procedure{Astar}{\texttt{Graph, start, goal}} \For{each \texttt{v} in \texttt{Graph.Verticies}} diff --git a/electives/amr/parts/05_planning-control/01_motion-planning-exploration/02_rrt.tex b/electives/amr/parts/05_planning-control/01_motion-planning-exploration/02_rrt.tex index 94a8740..87f4943 100644 --- a/electives/amr/parts/05_planning-control/01_motion-planning-exploration/02_rrt.tex +++ b/electives/amr/parts/05_planning-control/01_motion-planning-exploration/02_rrt.tex @@ -1,5 +1,7 @@ +\newpage \subsubsection{Rapidly-Exploring Random Tree (RRT)} \begin{algorithm} + \small \begin{algorithmic}[1] \Procedure{RRT}{\texttt{start, goal}} \State \Call{insertVertex}{\texttt{start, Graph}} @@ -28,6 +30,7 @@ Returns a collision-free path as graph. Need nearest neighbour search. Extension to RRT* to make path better: \begin{algorithm} + \small \begin{algorithmic}[1] \State $X_{\text{near}} \gets$ \Call{neighbours}{Graph, $x_f$, $R$} \State $x_{\min} \gets$ \Call{neighbours}{Graph, $x_f$, $R$} diff --git a/electives/amr/parts/05_planning-control/02_learning-to-act/02_reinforcement-learning.tex b/electives/amr/parts/05_planning-control/02_learning-to-act/02_reinforcement-learning.tex index 959f6cc..eaaa800 100644 --- a/electives/amr/parts/05_planning-control/02_learning-to-act/02_reinforcement-learning.tex +++ b/electives/amr/parts/05_planning-control/02_learning-to-act/02_reinforcement-learning.tex @@ -21,7 +21,7 @@ Then update: $Q_{i + 1}(\vec{x}, \vec{u}) = (1 - \alpha)Q_i(\vec{x}, \vec{u}) + Called off-policy learning, needs exploration. Simplest is random actions ($\varepsilon$-greedy): $\varepsilon$ is prob. to act randomly, $1 - \varepsilon$ is prob. to act on pol. {\color{red} Space explored, still doing random stuff} -\bi{Approaches} +\bi{DL-Approaches} \textit{Model-based} (estimate trans. model, e.g. Dyna), Value-based (estimate val or $Q$-func and extract pol., e.g. Q-Learn), Actor-Critic (estim. val or $Q$ of curr. pol., improve pol., e.g. A3C, SAC),