[AMR] Finished update

2026-09-10 19:15:25 +02:00 · 2026-08-08 15:55:22 +02:00
parent c1bdbc64bf
commit cc5f111813
11 changed files with 35 additions and 12 deletions
@@ -26,3 +26,6 @@ Then apply Laplacian Operator $\nabla_\text{norm}^2 L = t\left( \frac{\partial^2
 \bi{Diff. of Gaussians} (DoG): $\Delta L = L(x, y, t) - L(x, y, kt)$

 \bi{SIFT Detector} \bi{(1)} Subsample + Blur \bi{(2)} DoG on each res. image \bi{(3)} Keypoints extrema in DoG pyramid
+
+\bi{BRISK} / binary descriptors: Compare pixel intensities at fixed sampling pattern around keypoint. Match by Hamming distance (very fast),
+\bi{SuperPoint}: CNN learns detector + descriptor
@@ -3,8 +3,11 @@
 \bi{PnP Problem} {\scriptsize Perspective $n$-Point}
 Find sol. for camera pose \textit{directly}

-\bi{RANSAC} {\scriptsize RANdom SAmpling Consensus} for find. outliers \& correct
+\bi{RANSAC} {\scriptsize RANdom SAmpling Consensus} Model estimation, for find. outliers \& correct.
 Also great for finding an initial guess for pose. (due to robustness)
+For $N$ iteration: sample min set, fit model, count inliers (reprojection error $<$ threshold $t$), keep best, optionally refit on inliers.
+More outliers $\Rightarrow$ more iterations.
+

 \bi{Stereo Triang.} Given two rays (known poses for points in 2D).
 Find good point in 3D. Fast sol: \bi{Midpoint Method}:
@@ -30,4 +30,14 @@ $l(o_j | \vec{x}_{R, 1 : k}, \vec{z}_{1 : k})$ =
 \shade{gray}{In 3D} 3D voxel $j$ as signed dist. $s$ and weight $w$, update:
 $\displaystyle s_k = \frac{w_{k - 1} s_{k - 1} + \tilde{s}_k}{w_{k - 1} + 1}$ with $w_k = \min(w_{\max}, w_{k - 1} + 1)$

-\bi{Impl.} Using HashTables or octree
+\bi{Implementation} Using Hash maps or octree (dense grid inefficient)
+
+
+\subsubsection{Iterative Closest Point}
+Build \textit{correspondences}: associate all live scan points $l_i$ to closest map points $m_i$. Error term:
+$\vec{e} = \vec{T}_{W L_l L_l} m_i - {_W}\vec{l}_i$.
+Minimize this via Gauss-Newton, then re-associate, iterate.
+
+\bi{Photometric}: $u_{KF} = \pi(\mat{T}_{WC_{KF}}^{-1} \mat{T}_{WC_l} \pi^{-1}(\vec{u}_{KF}, \mat{D}_{KF}[\vec{u}_{KF}]))$,
+error term $\vec{e} = \mat{I}_{KF}[\vec{u}_{KF}] - \mat{I}_L[\vec{u}_{L}]$,
+where all subscript $L$ are from live image, all subscript $KF$ key frame.
@@ -1,4 +1,5 @@
 \subsection{Proportional-Integral-Differential (PID)}
-$u(t) = k_p e(t) + k_i \int_{t_0}^{t} e(\tau) \dx \tau + k_d \frac{\dx e(t)}{\dx t}$, where params $k_p$ (curr), $k_i$ (long-term), $k_d$ (trend) reduce corresp. errors
+$u(t) = K_p e(t) + K_i \int_{t_0}^{t} e(\tau) \dx \tau + K_d \frac{\dx e(t)}{\dx t}$, where params $K_p$ (curr), $K_i$ (long-term), $K_d$ (trend) reduce corresp. errors,
+with $e = r - y$

 \bi{Drone Control} ${_B}\vec{F}$ and ${_B}\vec{M}$ as in sect. \ref{sec:rigid-body-dynamics}, use $\vec{u}' = {_B}\vec{M}$.
@@ -3,9 +3,14 @@

 \bi{Free C-Space}: $\cC_{\text{free}}$, \bi{C-Space Obstacles} $\cC_\text{obst}$ (occupied)

-\bi{Collision checker}: $c(\vec{x}) : \cC \rightarrow \{ 0, 1 \}$ 
+\bi{Collision checker}: $c(\vec{x}) : \cC \rightarrow \{ 0, 1 \}$

 \bi{Visibility graph}: Connect corners, goal outside obstacles

 \bi{Voronoi Diagram}: Edges at max. dist. from obst. (benefit: safer paths).
 Also tends to be faster than Dijkstra.
+
+\bi{Discrete} {\scriptsize via graph/grid}: {\color{ForestGreen} complete solution}, {\color{red} Curse of dimensions}
+
+\bi{Continuous} {\scriptsize probabilistic}: {\color{ForestGreen} High-D okay, differential constraints ok},
+{\color{red} Probabilistic, depending on setup may not find correct sol. or run forever}
@@ -1,3 +1,4 @@
+\newpage
 \subsubsection{Rapidly-Exploring Random Tree (RRT)}
 \begin{algorithm}
    \small
@@ -45,3 +46,4 @@ Extension to RRT* to make path better:
 \end{algorithm}

 \bi{Informed RRT*} extension: Once conn. betw. start and goal found, restrict sampling to (hyper)ellipsoid.
+$b = 0.5 \sqr{d^2 - ||x_s - x_g||^2}$.
@@ -2,7 +2,7 @@
 \bi{Dynamic Window Approach} Assume: Robot moves inst. on circ. arcs $(v, \omega)$.
 Compute arcs with coll. {\color{ForestGreen} Accounts for Kino-Dyn}, {\color{red} Cost func prone to loc. min, assumes static obj}

-\bi{Vel. Obst.} Assume: Robot in str. line $(v_x, v_y)$. Comp pos with coll.
+\bi{Velocity Obstacles} Assume: Robot in str. line $(v_x, v_y)$. Comp pos with coll.
    {\color{ForestGreen} Vel. of obj}, {\color{red} prone to local optima, no kino-dyn.}

 \bi{Potential Field Methods} Define {\color{gray} \textit{repulsive}} and {\color{purple} attractive} potential $c = {\color{purple} c_{\text{att}}} + {\color{gray} c_{\text{rep}}}$.
@@ -10,5 +10,4 @@ With e.g. ${\color{purple} c_\text{att}} = \frac{1}{2} k_\text{att} ||\vec{x} -
        \frac{1}{2} k_\text{rep} \left( \frac{1}{\rho(\vec{x})} + \frac{1}{\rho_{\lim}} \right) & \rho \leq \rho_{\lim} \\
        0                                                                                       & \text{else}
    \end{cases}$
-
 {\color{ForestGreen} Simple control laws}, {\color{red} may trap in loc. min., no diff. const, no guar. to avoid coll}
@@ -16,16 +16,16 @@ Repeat until conv. to $V^*$ ($\tco{|\cU||\cX|^2}$ per iter). Optimal policy:
    \vec{\pi}^*(\vec{x}) = \text{argmax}_{\vec{u}} Q(\vec{x}, \vec{u})
 \]

-Using policy iter:
+Using \bi{policy iter}:
 \begin{algorithm}
    \begin{algorithmic}[1]
        \State Choose $\vec{\pi}_0(\vec{x})$
        \While{\textit{policy} has not converged}
-        \Repeat $V_{i + 1}^{\vec{\pi}_j}(\vec{x}) = Q(\vec{x}, \vec{\pi}(\vec{x}))$ $\forall \vec{x}$ and \textit{fixed} pol. $\vec{\pi}_j$
+            \Repeat $V_{i + 1}^{\vec{\pi}_j}(\vec{x}) = Q(\vec{x}, \vec{\pi}(\vec{x}))$ $\forall \vec{x}$ and \textit{fixed} pol. $\vec{\pi}_j$
            \Until{values converge}
        \EndWhile
        \State One step: $\vec{\pi}_{j + 1}(\vec{x}) = \text{argmax}_{\vec{u}} Q(\vec{x}. \vec{u})$ with $V_i = V_{i + 1}^{\pi_j}$
    \end{algorithmic}
 \end{algorithm}

-Model-based learning uses empirical models of $\cT$ and $\cR$
+\bi{Model-based} learning uses empirical models of $\cT$ and $\cR$