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As those nodes are expanded, their branch is removed from the graph and the search backtracks by expanding the next neighboring node of the start node until its deepest level and so on. +Depth-First Search stores only a single path from the start nodes for each node on the path, leading to a lower memory requirement. However, previously visited nodes might be visited and the algorithm might enter redundant paths. + +\begin{figure}[h] +\begin{center} +\includegraphics{fig15_8.PNG} +\caption{Depth First Search} +\end{center} +\end{figure} + +Breadth-First Search +Breadth-first begins with the start node and explores all of its neighboring nodes. Then, for each of these nodes, it explores all their unexplored neighbors and so on. +The search always returns the path with the fewest number of edges between the start and goal node. + +\begin{figure}[h] +\begin{center} +\includegraphics{fig15_9.PNG} +\caption{Breadth First Search} +\end{center} +\end{figure} + +Dijkstra’s algorithm (Best-First Search) +Dijkstra’s algorithm is similar to Breadth-First search, except that edge costs may assume any positive value and the search still guarantees solution optimality. Dijkstra’s algorithm uses the concept of the heap, a specialized tree-based data structure and selects the next q as: q = argminq2QC(q)) + +\subsection{Label Correcting Algorithm} +The label-correction algorithm is a general type of shortest path algorithm which includes very common graph search algorithms like breadth-first search, depth-first search, A*, Dijkstra’s algorithm as special cases. +The idea is to progressively discover shorter paths from the origin to every other node q, and to maintain the lowest cost path(from qI to q) found so far in a variable C(q). + +\begin{figure}[h] +\begin{center} +\includegraphics{fig15_10.PNG} +\caption{Label Correcting} +\end{center} +\end{figure} + +Algorithm Step +Step1. Remove a node q from the frontier queue and for each child q0 of q, execute step 2 +Step2. If C(q) + C(q, q0)  min(C(q0),UPPER), set C(q0) := C(q) + C(q, q0) and set q to be the parent of q0. In addition, if q0 6= qG, place q0 in frontier queue if it is not already there, while if q0 = qG, set UPPER to the new value C(q) + C(q, qG) +Step3. If the frontier queue is empty, terminate, else go to step 1. +Initialization : set the labels of all nodes to ¥, except for the label of the origin node, which is set to 0. + +\section{Correctness and Improvements} +Label correcting algorithms are guaranteed to find the shortest feasible path from an initial state to a goal state, given one exists. This attribute gives rise to the Correctness Theorem. +Theorem If a feasible path exists from q1 to qG, then the algorithm terminates in finite time with C(qG) equal to the optimal cost of traversal, C(qG). +However, computing this path can be computationally intensive especially in high-dimensional problems. This challenge arises from an explosion in the number of states expanded during the search procedure. Consider Fig 15.11, Depth-First Search and Breadth-First Search algorithms do not greedily select states from the priority queue for expansion. Hence, these algorithms search regions of the configuration space +relatively uniformly without consideration of the direction of the expanded state to the goal. Dijkstra’s algorithm improves on Depth-First Search and Breadth-First Search by greedily selecting states from the priority queue which prevents revisiting previously visited states. However, Dijkstra’s algorithm searches for paths uniformly around the initial state, leading to wasted effort. Concentrating the search phase in regions leading to the goal state can reduce the number of states expanded during the search, speeding up the motion planning process. + +\begin{figure}[h] +\begin{center} +\includegraphics{fig15_11.PNG} +\caption{Graph Search Algorithm} +\end{center} +\end{figure} + +\subsection{A*: Improving Dijkstra} +The A algorithm improves on Dijkstra by including a heuristic function to the optimal ”cost-to-arrival” when checking the children of the open node for addition to the priority queue. The heuristic function models the ”cost-to-go” to the goal. This function guides exploration towards the region of the configuration space close to the goal, leading to faster computations as less states are expanded. Thus, the test +condition changes from + +\[ C(q) + C(q, q0) \leq UPPER ]\ + +to + +\[ C(q) + C(q, q0) + h(q) \leq UPPER \] + +where h(q) is the optimal cost-to-go. The heuristic must be a positive underestimate of the true cost to the goal. The modification of the test condition reduces the number of nodes placed in the priority queue and still guarantees that an optimal path will be returned. Figure 15.12 illustrates A implemented on a grid. States in black represent + +obstacles while the light gray states represent nodes on the priority queue. States in dark gray represent opened nodes. The state with the minimum total estimated cost (including the heuristics cost) is selected from the priority queue for expansion. + +\begin{figure}[h] +\begin{center} +\includegraphics[scale=0.90]{fig15_12.PNG} +\caption{A* Algorithm} +\end{center} +\end{figure} -%% --- \printbibliography \subsubsection*{Contributors} -Winter 2019: [Your Names Here] \\ +Winter 2019: Warren Cheng, \\ Winter 2018: Ryan Richey, Hyoungju Seo, Ola Shorinwa, Josh Sullivan, Danning Sun \end{document} + From 56addf30d73223bf528f3e18d7471cf246bfac7c Mon Sep 17 00:00:00 2001 From: Warren Cheng Date: Fri, 8 Mar 2019 23:51:01 -0800 Subject: [PATCH 03/14] Made updates to section 15.1 --- lecture_15/main.tex | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/lecture_15/main.tex b/lecture_15/main.tex index 83f2925..cac1f95 100644 --- a/lecture_15/main.tex +++ b/lecture_15/main.tex @@ -78,10 +78,12 @@ \section{Introduction} \textbf{Robot motion planning}: compute a sequence of actions that drives a robot from an initial condition to a terminal condition while avoiding obstacles, satisfying kinematic and dynamic constraints, and possibly optimizing an objective function. +Some examples of common motion planning problems may include steering autonomous vehicles, controlling humanoid robots, protein docking, and surgery planning. + \subsection{ History } -The problem of motion planning was formally defined in the 1970s as mobile robots became more prevalent in industries. However, exact combinatorial solutions in discrete configuration spaces were not developed until the 1980s with the introduction of Breadth-First Search and Depth-First Search algorithms. Grid-based search algorithms such as A* and D* which was developed by Anthony Stentz followed in the mid-1990s [3]. These methods worked well for simple motion planning problems but were subject to the curse of dimensionality for planning in high-dimensional state spaces. To address this challenge, -sampling-based methods, notably Probability Road-Maps (PRM) and Rapidly-Exploring Random Trees (RRT), were developed in the late 1990s and deployed on real-time systems in the 2000s [4]. With the increasing complexity of robotic systems, solving motion planning problem s in real-time has become a significant challenge. Hence, current research has focused on speeding up sampling-based methods through lazy dynamic programming and massive parallelization of search algorithms. +The problem of motion planning was formally defined in the 1970s as mobile robots became more prevalent in industries. However, exact combinatorial solutions in discrete configuration spaces were not developed until the 1980s with the introduction of Breadth-First Search and Depth-First Search algorithms. Grid-based search algorithms such as A* and D* which was developed by Anthony Stentz followed in the mid-1990s \cite{bertsekas}. These methods worked well for simple motion planning problems but were subject to the curse of dimensionality for planning in high-dimensional state spaces. To address this challenge, +sampling-based methods, notably Probability Road-Maps (PRM) and Rapidly-Exploring Random Trees (RRT), were developed in the late 1990s and deployed on real-time systems in the 2000s \cite{lavalle}. With the increasing complexity of robotic systems, solving motion planning problems in real-time has become a significant challenge. Hence, current research has focused on speeding up sampling-based methods through lazy dynamic programming and massive parallelization of search algorithms. \section{Configuration-Space} In the context of robotic motion planning, we will first create a simple problem setup to define the variables of interest and introduce the proper nomenclature. First, consider a workspace defined in 2D, $ W \subseteq R^2$, which effectively describes the world that is occupied by the robot, the obstacles, and potentially other agents in the system. Within this workspace, we define the robot as a rigid polygon, and we introduce the set of obstacles $O \subset W$ as a subset of the workspace shown in Fig 15.1. 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zA75o{j71F`O#qal+{}zjTpz%^{~VjVU>^W9(#P^0Bqi~$Vd z29^&0bXNb0d;AlfsG($H`nS^s^$qYJ6q`T&UPMidY>Z7ljBKnN0Dl4*Fw2Rj>=ff1Xr0Y5LJ0o%vRU}ZGnGGSu=u$STb5C8(Q8nBxh zm>P4ja2YZ4>-_c9W*8ZQfRc+3=pe$&2&$onuu6f(YUcyu6d`2%`mhns1TE!Et0Z7j z%AjJ0gN2L*X+w;WLBd!3By||Sg3oBZ5R~2H2xowjPAKF+HCTX!7=I56XK5MqU9?=) zza$TMYR>^IjRUa(>=qo0b(D-pQB@Y<)Qc6k`pE17x|kkSsp(zYaR4P}a_uDZB=SXi zAx0_$wd-(DkZx6M%0HZZ{@F+e@W1-F{I6#HKTU#la5S)YbhS4zh5car!IF`Q$&17O EAChHHeE 1 : suboptimal or greedy A* variant \subsection{Various graph search algorithms} @@ -262,13 +258,13 @@ \subsection{Label Correcting Algorithm} Algorithm Step Step1. Remove a node q from the frontier queue and for each child q0 of q, execute step 2 -Step2. If C(q) + C(q, q0)  min(C(q0),UPPER), set C(q0) := C(q) + C(q, q0) and set q to be the parent of q0. In addition, if q0 6= qG, place q0 in frontier queue if it is not already there, while if q0 = qG, set UPPER to the new value C(q) + C(q, qG) +Step2. If C(q) + C(q, q0) min(C(q0),UPPER), set C(q0) := C(q) + C(q, q0) and set q to be the parent of q0. In addition, if q0 6= qG, place q0 in frontier queue if it is not already there, while if q0 = qG, set UPPER to the new value C(q) + C(q, qG) Step3. If the frontier queue is empty, terminate, else go to step 1. -Initialization : set the labels of all nodes to ¥, except for the label of the origin node, which is set to 0. +Initialization : set the labels of all nodes to, except for the label of the origin node, which is set to 0. \section{Correctness and Improvements} Label correcting algorithms are guaranteed to find the shortest feasible path from an initial state to a goal state, given one exists. This attribute gives rise to the Correctness Theorem. -Theorem If a feasible path exists from q1 to qG, then the algorithm terminates in finite time with C(qG) equal to the optimal cost of traversal, C(qG). +Theorem If a feasible path exists from q1 to qG, then the algorithm terminates in finite time with C(qG) equal to the optimal cost of traversal, C(qG). However, computing this path can be computationally intensive especially in high-dimensional problems. This challenge arises from an explosion in the number of states expanded during the search procedure. Consider Fig 15.11, Depth-First Search and Breadth-First Search algorithms do not greedily select states from the priority queue for expansion. Hence, these algorithms search regions of the configuration space relatively uniformly without consideration of the direction of the expanded state to the goal. Dijkstra’s algorithm improves on Depth-First Search and Breadth-First Search by greedily selecting states from the priority queue which prevents revisiting previously visited states. However, Dijkstra’s algorithm searches for paths uniformly around the initial state, leading to wasted effort. Concentrating the search phase in regions leading to the goal state can reduce the number of states expanded during the search, speeding up the motion planning process. @@ -280,16 +276,16 @@ \section{Correctness and Improvements} \end{figure} \subsection{A*: Improving Dijkstra} -The A algorithm improves on Dijkstra by including a heuristic function to the optimal ”cost-to-arrival” when checking the children of the open node for addition to the priority queue. The heuristic function models the ”cost-to-go” to the goal. This function guides exploration towards the region of the configuration space close to the goal, leading to faster computations as less states are expanded. Thus, the test +The A* algorithm improves on Dijkstra by including a heuristic function to the optimal ”cost-to-arrival” when checking the children of the open node for addition to the priority queue. The heuristic function models the ”cost-to-go” to the goal. This function guides exploration towards the region of the configuration space close to the goal, leading to faster computations as less states are expanded. Thus, the test condition changes from -\[ C(q) + C(q, q0) \leq UPPER ]\ +\[ C(q) + C(q, q0) \leq UPPER \] to \[ C(q) + C(q, q0) + h(q) \leq UPPER \] -where h(q) is the optimal cost-to-go. The heuristic must be a positive underestimate of the true cost to the goal. The modification of the test condition reduces the number of nodes placed in the priority queue and still guarantees that an optimal path will be returned. Figure 15.12 illustrates A implemented on a grid. States in black represent +where h(q) is the optimal cost-to-go. The heuristic must be a positive underestimate of the true cost to the goal. The modification of the test condition reduces the number of nodes placed in the priority queue and still guarantees that an optimal path will be returned. Figure 15.12 illustrates A* implemented on a grid. States in black represent obstacles while the light gray states represent nodes on the priority queue. States in dark gray represent opened nodes. The state with the minimum total estimated cost (including the heuristics cost) is selected from the priority queue for expansion. @@ -305,7 +301,7 @@ \subsection{A*: Improving Dijkstra} \printbibliography \subsubsection*{Contributors} -Winter 2019: Warren Cheng, \\ +Winter 2019: Warren Cheng, Junwu Zhang\\ Winter 2018: Ryan Richey, Hyoungju Seo, Ola Shorinwa, Josh Sullivan, Danning Sun \end{document} From f549c0c6e32df876f88fa8759ea2bb82dc70fa48 Mon Sep 17 00:00:00 2001 From: Junwu Zhang Date: Sat, 9 Mar 2019 16:31:22 -0800 Subject: [PATCH 05/14] updated section 15.3 --- lecture_15/main.pdf | Bin 555732 -> 559262 bytes lecture_15/main.tex | 23 +++++++++++++++-------- 2 files changed, 15 insertions(+), 8 deletions(-) diff --git a/lecture_15/main.pdf b/lecture_15/main.pdf index 236ef841ab1f3c1e0ce4815bfbc3e4e7461fd55a..645dda7d99ebfaa4d44eb813a2b743b77933f9ed 100644 GIT binary patch delta 35231 zcmZs?Q+OuOvaXvyw%xIlj%}x7+qSJv{@Au{+jcs(t&VN&o@?%PZuYsWfy?pK_`a&Q 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zWjyMO!^F{MQ(RvRa(a*W*jb$$z{2pPVliW|sfO0wA01WC%1wg}i<6{<5u5YwbXH>D}qX z)Kb|HB@Gk$gB}>9A=zO?i_8T5g(JF;NC#=>3btAW}AbN<1 zyoD}9ib_Zb8Xx9*U=k1&r^GBCvTHYWHco0nRXawGPU@aNA|zR2Qp2sm|fH6s7jJ);1us`+25@ry9># zwdTPm-^bm>v~Zv$eQwWsn!D5-Nn3D-{)r z&00P=-P`%jjE+4teAcJGG+!3y!2&A21_bHkFt3?d{<%EyK&vd9-2Zdhuujf~j?V6m Vre?7J6-Z2M?672HVhZB0{{w-PV<7+l diff --git a/lecture_15/main.tex b/lecture_15/main.tex index 2183c7d..1e5ec6f 100644 --- a/lecture_15/main.tex +++ b/lecture_15/main.tex @@ -121,7 +121,7 @@ \subsection{ Planning in C-Space } Compute the continuous path: t : [0, 1] ) fq 2 CjR(q) \ O = Æg with t(0) = qI and t(1) = qG In general, we refer to the set fq 2 CjR(q) \ O = Æg as Cf ree, which represents all possible combinations of the robot’s degrees of freedom which are collision-free. A key fact here is that, while a robot occupies physical dimensions in the real space, the robotic motion is described by a simple point in the configuration space that captures the translation degrees of freedom (of the center of mass for example) and the rotational degrees of freedom of the robot. In this way, mapping to the configuration space poses the motion planning problem as a simpler problem of finding a path for a point. An illustration of this problem is highlighted in Figure 15.3. The physical dimensionality of the robot is intrinsically captured in the definition of the configuration space (in particular, in Cf ree). This property represents a key advantage of moving to the configuration space for robotic motion planning, as it allows the same general planning problem formulation (i.e., path planning for a point) to be equally applicable to myriad dynamical systems. -\begin{figure}[h] +\begin{figure}[H] \begin{center} \includegraphics{fig15_3.PNG} \caption{Robotic motion planning as a point path planning problem in C-space.} @@ -131,22 +131,29 @@ \subsection{ Planning in C-Space } We now have a formal mathematical definition for the robot motion planning problem. In order to simplify the procedure, we recast the problem from the real/physical space to the configuration space. The result is that the motion planning problem now becomes a path planning problem for a single point. We have not yet, however, set up an architecture for obtaining the point path solution. In the following sections, several solution methods will be highlighted which leverage the aforementioned advantages of working in the configuration space for motion planning. \section{Combinatorial Planning} -The goal of combinatorial planning is to find paths through the configuration space without needing approximations as shown in Figure 15.4. For low dimensional convex problems, this type of planning either provides \textit{exact} solutions or none if no feasible path exists. This can allow for better performance in some cases , compared to approximation-based methods. +The goal of combinatorial planning is to find paths through the configuration space without needing approximations. For example, Figure \ref{fig:15_4} represents a point robot in a plane. The key idea is to compute a roadmap, which is a graph in which each vertex is a configuration in $C_\text{free}$ and each edge is a path through $C_\text{free}$ that connects a pair of vertices. + +For low dimensional convex problems, this type of planning either provides \textit{exact} solutions or none if no feasible path exists. This can allow for better performance in some cases, compared to approximation-based methods. \begin{figure}[H] \begin{center} -\includegraphics{fig15_4.PNG} +\includegraphics[width=0.6\textwidth]{fig15_4.PNG} \caption{Combinatorial Planning} +\label{fig:15_4} \end{center} \end{figure} -Combinatorial Planning refers to the use of all combinations of possible states in defining the robot configuration $q$. For example, consider the configurations used above in setting up the configuration-space: $q = (xt, yt, q)$. In this case, the $x$ and $y$ position and the heading q define a unique configuration. +Combinatorial Planning refers to the use of all combinations of possible states in defining the robot configuration $q$. For example, consider the configurations used above in setting up the configuration-space: $q = (xt, yt, q)$. In this case, the $x$ and $y$ position and the heading $\theta$ define a unique configuration. + +Combinatorial motion planning approaches construct a roadmap, a graph $G$ which should preserve: + +\begin{enumerate} + \item \textbf{Accessiblity}: It is always possible to connect some $q$ to the roadmap. For instance, $q_I$ can be connected to $s_1$ and $q_IG$ can be connected to $s_2$ on the roadmap. + \item \textbf{Connectivity}: If there is a path between two configurations $q_I$ and $q_G$, then there exists a path on the roadmap between $s_1$ and $s_2$. +\end{enumerate} -Combinatorial motion planning approaches construct a roadmap, a graph G with conditions: -1. Accessiblity: From any configuration it must always be possible to connect to the graph. For instance, q1 can be connected to s1 on the roadmap. +The roadmap essentially provides a discrete representation of the continuous problem without losing any of the original connectivity information needed to solve it. -2. Connectivity: If there is a path between two configurations q1 and q2, then there exists a path on the roadmap between s1 and s2. -The roadmap essentially provides a discrete representation of the continuous problem. Any original connectivity information is maintained across the mapping. The typical approach involves cell decomposition. There are requirements that the cells be easy to traverse, the decomposition be easy to compute, and the cell adjacencies be straightforward to determine. \section{Cell Decomposition} From d636d7993c4d2c62b0f260b0be94744d0b26a4a8 Mon Sep 17 00:00:00 2001 From: Junwu Zhang Date: Sat, 9 Mar 2019 16:33:37 -0800 Subject: [PATCH 06/14] updated section 15.3 --- lecture_15/main.pdf | Bin 559262 -> 559255 bytes lecture_15/main.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/lecture_15/main.pdf b/lecture_15/main.pdf index 645dda7d99ebfaa4d44eb813a2b743b77933f9ed..9df3fbcc85186f2b0294f1828923eb0797d96da3 100644 GIT binary patch delta 2609 zcmV-13eNSOks_CoBCzu6mwUqjDu1n6Pmkle4ZqK)$k72Nde%Q%UZ98fcHb6Tv_K0C z&_mmU<7g6f5<43!yF2gGmlS1N&Uj}Q*kTTnDT<^h@<%H3+r!;2AC*~EtSUsYdU#w_ z`Kr{M38hyLyVbw4zuLxMFKccxyWWV>WL&>nZ=}-M`|h-LZSA{@tBn|mlw=||D=iH(IVVFu zx;?l;2TxQS>}CpmQScYOfPdBW{Ge5nyp2+ftAga+?2MrXl>$aUARJ5CL zxEzkhS*2sd;AZes&>b&74KP(Qm5+D-c_};~ILd*6tv8yh>|=MqsDG20{Z^mOM|%%n zI@r*=osDZ`s1is{t>}pg-rCWu-gawIW?$AYk6XgwVeGB{V7cIP&&0!Ox#BsKhI8aBSs|+7w12|b$-#v>Ul0lb2YVyyw~?V_y5xaDGXWZz zs-i-_;9bc$SF42Set+qbm?(=u)ARpZQ}I1biw;t(D*37$ANE|;N@$HRI*ZyzKvNO<<>M@Tle`INjI_02)INAaDS-*F5z5UF9R3uK6TSE z&LfBgRp)a6j=DM6epJ=_%eC<{V~A^as?YZ``uq4DPh4`3KgKKgoa z&4Fb0hwgfe&OQg2?GZ(w%n>NOqti=KF$v)d6^$9F6;!VnX_B*R7ffk2X!^BjY=Bwk zmqI{mCdC+NP=7;|fq+F^)jY#+#j!qqT}!~KwXCw|NK3#QzCu0F!JS}w*4ieD=)2QW zeDe~)M!yy^h6y>IVvtYpr_&okSXMm3G8zpRTDOf|0uv53R0@8p_6ANkf+f`2inu;4 z7+0_gyp}LEbxZ2EcJ#z1-dqKZfYcFyW89*L?2uyJ5`XSVR`Su(ezak-R0S(_8AD+) z>!8J0B}_kn6H7h;+{1N1)tynw^LrfdG0D_h1PGLyHtue#4G0> znY#7{%+l~7aLFYgt_k3xcR?nnI+( z;avLz&40Gme+@uj+qe%t^Q;@&j(LCa(xU?4IWOOec?x;+YRGH8N|=^|N#sm(^YhVE zKu4_8Xu>|L%+YJ@lK<>C*Zc4JklH^;MP9(=iFq%y5EMh$PM%)~)QV~GBMEALBtgxa z3Ce#iC!j z(SNB+A#wfl*)>5gd?XzS5TLxe_Djg_D6%KE#vbE_orU+*(5hj7sT7JH-hwD{n9Jz! 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For example, consider the configurations used above in setting up the configuration-space: $q = (xt, yt, q)$. In this case, the $x$ and $y$ position and the heading $\theta$ define a unique configuration. +Combinatorial Planning refers to the use of all combinations of possible states in defining the robot configuration $q$. For example, consider the configurations used above in setting up the configuration-space: $q = (x_t, y_t, \theta)$. In this case, the $x$ and $y$ position and the heading $\theta$ define a unique configuration. Combinatorial motion planning approaches construct a roadmap, a graph $G$ which should preserve: \begin{enumerate} - \item \textbf{Accessiblity}: It is always possible to connect some $q$ to the roadmap. For instance, $q_I$ can be connected to $s_1$ and $q_IG$ can be connected to $s_2$ on the roadmap. + \item \textbf{Accessiblity}: It is always possible to connect some $q$ to the roadmap. For instance, $q_I$ can be connected to $s_1$ and $q_G$ can be connected to $s_2$ on the roadmap. \item \textbf{Connectivity}: If there is a path between two configurations $q_I$ and $q_G$, then there exists a path on the roadmap between $s_1$ and $s_2$. \end{enumerate} From 523d321c017a537402db3ca539fa3371d3f79acd Mon Sep 17 00:00:00 2001 From: Theo Diamandis Date: Sun, 10 Mar 2019 13:08:43 -0700 Subject: [PATCH 07/14] section 15.6 figs --- lecture_15/main.pdf | Bin 559255 -> 442130 bytes lecture_15/main.tex | 177 ++++++++++++++++++++++++++++++++++++++++---- 2 files changed, 164 insertions(+), 13 deletions(-) diff --git a/lecture_15/main.pdf b/lecture_15/main.pdf index 9df3fbcc85186f2b0294f1828923eb0797d96da3..5ee52bcc7c9fba1ccdd6520a3be72611ad7f9f58 100644 GIT binary patch delta 92410 zcmZU)b8seI*Y=$=wrwX9+xEn^ZChuoiEZ1qZQGh~l8KG?y6*dV-umi$|603tcUM>K z>RRi!j$?IN0%_tGE;5Cpm;^mD13N6m{PM^;ELQ?DIw(7HQbHg#utZBHX}cYvcdl+; zDJN51tfz(Z@237=b=}M8vPq~dsMg_g^J+<*Wb^H>M*y@UeM}0M8g~nH05LQuk&yct 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\bibliography{./lecture_15_biblio.bib} +\usepackage{tikz,pgfplots} +\usepackage{xcolor, colortbl} +\usepackage{array} + \usepackage{graphics} \usepackage{graphicx} \usepackage{subcaption} @@ -269,36 +273,183 @@ \subsection{Label Correcting Algorithm} Step3. If the frontier queue is empty, terminate, else go to step 1. Initialization : set the labels of all nodes to, except for the label of the origin node, which is set to 0. + \section{Correctness and Improvements} Label correcting algorithms are guaranteed to find the shortest feasible path from an initial state to a goal state, given one exists. This attribute gives rise to the Correctness Theorem. -Theorem If a feasible path exists from q1 to qG, then the algorithm terminates in finite time with C(qG) equal to the optimal cost of traversal, C(qG). -However, computing this path can be computationally intensive especially in high-dimensional problems. This challenge arises from an explosion in the number of states expanded during the search procedure. Consider Fig 15.11, Depth-First Search and Breadth-First Search algorithms do not greedily select states from the priority queue for expansion. Hence, these algorithms search regions of the configuration space -relatively uniformly without consideration of the direction of the expanded state to the goal. Dijkstra’s algorithm improves on Depth-First Search and Breadth-First Search by greedily selecting states from the priority queue which prevents revisiting previously visited states. However, Dijkstra’s algorithm searches for paths uniformly around the initial state, leading to wasted effort. Concentrating the search phase in regions leading to the goal state can reduce the number of states expanded during the search, speeding up the motion planning process. +\paragraph{Theorem} If a feasible path exists from $q_I$ to $q_G$, then the algorithm terminates in finite time with $C(q_G)$ equal to the optimal cost of traversal, $C^*(q_G)$.\\ + +However, computing this path can be computationally intensive especially in high-dimensional problems. This challenge arises from an explosion in the number of states expanded during the search procedure. \\ +Consider Fig 15.11, Depth-First Search and Breadth-First Search algorithms do not greedily select states from the priority queue for expansion. Hence, these algorithms search regions of the configuration space +relatively uniformly without consideration of the direction of the expanded state to the goal. Dijkstra's algorithm improves on Depth-First Search and Breadth-First Search by greedily selecting states from the priority queue which prevents revisiting previously visited states. However, Dijkstra's algorithm searches for paths uniformly around the initial state, leading to wasted effort. Concentrating the search phase in regions leading to the goal state can reduce the number of states expanded during the search, speeding up the motion planning process. \begin{figure}[h] \begin{center} -\includegraphics{fig15_11.PNG} +%\includegraphics{fig15_11.PNG} +\begin{tikzpicture}[scale=0.9, transform shape] +\draw (0,0) circle (0.75); +% \draw (0,0) circle (1.0); +\draw (0,0) circle (1.5); +% \draw (0,0) circle (2.0); +\draw (0,0) circle (2.25); +\draw (0,0) circle (3.0); + +\filldraw[yellow] (0,0) circle (2pt); +\draw (0,0) node[anchor=west] {$s_{\text{start}}$}; +\filldraw[green] (3,0) circle (2pt); +\draw (3,0) node[anchor=west] {$s_{\text{end}}$}; + +\draw (-4,0) node[anchor=west,color=red] {Wasted Effort?}; + +\end{tikzpicture} \caption{Graph Search Algorithm} \end{center} \end{figure} \subsection{A*: Improving Dijkstra} -The A* algorithm improves on Dijkstra by including a heuristic function to the optimal ”cost-to-arrival” when checking the children of the open node for addition to the priority queue. The heuristic function models the ”cost-to-go” to the goal. This function guides exploration towards the region of the configuration space close to the goal, leading to faster computations as less states are expanded. Thus, the test +Dijkstra's algorithm orders nodes added to the priority queue by optimal ``cost-to-arrival." The A* algorithm improves on Dijkstra by adding a heuristic function that models cost to go to the goal to the optimal ``cost-to-arrival," i.e. A* orders by ``cost-to-arrival" + ``cost-to-go". This heuristic function guides exploration towards the region of the configuration space close to the goal, leading to faster computations as less states are expanded. Thus, the test condition changes from +\[ C(q) + C(q, q') \leq \text{UPPER} \] +to +\[ C(q) + C(q, q') + h(q') \leq \text{UPPER} \] +where $h(q)$ is the optimal cost-to-go. The heuristic must be a positive underestimate of the true cost to the goal. This modification of the test condition reduces the number of nodes placed in the priority queue and still guarantees that an optimal path will be returned. -\[ C(q) + C(q, q0) \leq UPPER \] +Figure 15.12 illustrates A* implemented on a grid. Black states represent obstacles. Yellow states represent nodes on the priority queue. Blue represent opened nodes. Green states indicate the final path. The state with the minimum total estimated cost (including the heuristics cost) is selected from the priority queue for expansion. -to -\[ C(q) + C(q, q0) + h(q) \leq UPPER \] +\newcolumntype{C}[1]{% + >{\vbox to 5ex\bgroup\vfill\centering}% + m{#1}% + <{\egroup}} -where h(q) is the optimal cost-to-go. The heuristic must be a positive underestimate of the true cost to the goal. The modification of the test condition reduces the number of nodes placed in the priority queue and still guarantees that an optimal path will be returned. Figure 15.12 illustrates A* implemented on a grid. States in black represent +\newcommand{\stategh}[2]{ +\vspace{0.1cm}g={#1}\\ \vspace{0.3cm}h={#2} +} -obstacles while the light gray states represent nodes on the priority queue. States in dark gray represent opened nodes. The state with the minimum total estimated cost (including the heuristics cost) is selected from the priority queue for expansion. +\newcommand{\stateghblank}[2]{ +} -\begin{figure}[h] +\newcommand{\stateghopen}[2]{ +\cellcolor{blue}\vspace{0.1cm}g={#1}\\ \vspace{0.3cm}h={#2} +} + +\newcommand{\stateghpq}[2]{ +\cellcolor{yellow}\vspace{0.1cm}g={#1}\\ \vspace{0.3cm}h={#2} +} + +\newcommand{\stateghpath}[2]{ +\cellcolor{green}\vspace{0.1cm}g={#1}\\ \vspace{0.3cm}h={#2} +} + +\begin{figure}[h!] \begin{center} -\includegraphics[scale=0.90]{fig15_12.PNG} +% \includegraphics[scale=0.90]{fig15_12.PNG} +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + goal & \cellcolor{black} & \stateghpq{1.4}{2.0} & \stateghpq{1.0}{3.0} \tabularnewline + \hline + \stateghblank{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{blue}{start} \tabularnewline + \hline + \stateghblank{3.4}{2.0} & \stateghblank{2.4}{2.4} & \stateghpq{1.4}{2.8} & \stateghpq{1.0}{3.8} \tabularnewline + \hline + \stateghblank{3.8}{3.0} & \stateghblank{2.8}{3.4} & \stateghblank{2.4}{3.8} & \stateghblank{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +} +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + goal & \cellcolor{black} & \stateghopen{1.4}{2.0} & \stategh{1.0}{3.0} \tabularnewline + \hline + \stateghblank{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{blue}{start} \tabularnewline + \hline + \stateghblank{3.4}{2.0} & \stateghblank{2.4}{2.4} & \stategh{1.4}{2.8} & \stategh{1.0}{3.8} \tabularnewline + \hline + \stateghblank{3.8}{3.0} & \stateghblank{2.8}{3.4} & \stateghblank{2.4}{3.8} & \stateghblank{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +} +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + goal & \cellcolor{black} & \stateghopen{1.4}{2.0} & \stateghopen{1.0}{3.0} \tabularnewline + \hline + \stateghblank{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{blue}{start} \tabularnewline + \hline + \stateghblank{3.4}{2.0} & \stateghblank{2.4}{2.4} & \stategh{1.4}{2.8} & \stategh{1.0}{3.8} \tabularnewline + \hline + \stateghblank{3.8}{3.0} & \stateghblank{2.8}{3.4} & \stateghblank{2.4}{3.8} & \stateghblank{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +} +\vspace{0.3cm} +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + goal & \cellcolor{black} & \stateghopen{1.4}{2.0} & \stateghopen{1.0}{3.0} \tabularnewline + \hline + \stateghblank{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{blue}{start} \tabularnewline + \hline + \stateghblank{3.4}{2.0} & \stateghpq{2.4}{2.4} & \stateghopen{1.4}{2.8} & \stategh{1.0}{3.8} \tabularnewline + \hline + \stateghblank{3.8}{3.0} & \stateghpq{2.8}{3.4} & \stateghpq{2.4}{3.8} & \stateghpq{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +}\\ +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + goal & \cellcolor{black} & \stateghopen{1.4}{2.0} & \stateghopen{1.0}{3.0} \tabularnewline + \hline + \stateghpq{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{blue}{start} \tabularnewline + \hline + \stateghpq{3.4}{2.0} & \stateghopen{2.4}{2.4} & \stateghopen{1.4}{2.8} & \stategh{1.0}{3.8} \tabularnewline + \hline + \stateghpq{3.8}{3.0} & \stategh{2.8}{3.4} & \stategh{2.4}{3.8} & \stategh{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +} +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + \cellcolor{yellow}goal & \cellcolor{black} & \stateghopen{1.4}{2.0} & \stateghopen{1.0}{3.0} \tabularnewline + \hline + \stateghopen{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{blue}{start} \tabularnewline + \hline + \stategh{3.4}{2.0} & \stateghopen{2.4}{2.4} & \stateghopen{1.4}{2.8} & \stategh{1.0}{3.8} \tabularnewline + \hline + \stategh{3.8}{3.0} & \stategh{2.8}{3.4} & \stategh{2.4}{3.8} & \stategh{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +} +\vspace{0.3cm} +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + \cellcolor{blue}goal & \cellcolor{black} & \stateghopen{1.4}{2.0} & \stateghopen{1.0}{3.0} \tabularnewline + \hline + \stateghopen{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{blue}{start} \tabularnewline + \hline + \stategh{3.4}{2.0} & \stateghopen{2.4}{2.4} & \stateghopen{1.4}{2.8} & \stategh{1.0}{3.8} \tabularnewline + \hline + \stategh{3.8}{3.0} & \stategh{2.8}{3.4} & \stategh{2.4}{3.8} & \stategh{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +} +\resizebox{.24\columnwidth}{!}{ +\begin{tabular}{|C{1cm}|C{1cm}|C{1cm}|C{1cm}|} + \hline + \cellcolor{green}goal & \cellcolor{black} & \stateghopen{1.4}{2.0} & \stateghopen{1.0}{3.0} \tabularnewline + \hline + \stateghpath{3.8}{1.0} & \cellcolor{black}& \cellcolor{black} & \cellcolor{green}{start} \tabularnewline + \hline + \stategh{3.4}{2.0} & \stateghpath{2.4}{2.4} & \stateghpath{1.4}{2.8} & \stategh{1.0}{3.8} \tabularnewline + \hline + \stategh{3.8}{3.0} & \stategh{2.8}{3.4} & \stategh{2.4}{3.8} & \stategh{2.8}{4.2}\tabularnewline + \hline +\end{tabular} +} + \caption{A* Algorithm} \end{center} \end{figure} @@ -308,7 +459,7 @@ \subsection{A*: Improving Dijkstra} \printbibliography \subsubsection*{Contributors} -Winter 2019: Warren Cheng, Junwu Zhang\\ +Winter 2019: Warren Cheng, Junwu Zhang, Theo Diamandis\\ Winter 2018: Ryan Richey, Hyoungju Seo, Ola Shorinwa, Josh Sullivan, Danning Sun \end{document} From f81bf8618711117dfe4e184fc4ad5ba5d402f6e0 Mon Sep 17 00:00:00 2001 From: Theo Diamandis Date: Sun, 10 Mar 2019 15:15:48 -0700 Subject: [PATCH 08/14] added section, fixed wording --- lecture_15/main.pdf | Bin 442130 -> 443782 bytes lecture_15/main.tex | 20 ++++++++++++++------ 2 files changed, 14 insertions(+), 6 deletions(-) diff --git a/lecture_15/main.pdf b/lecture_15/main.pdf index 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This challenge arises from an explosion in the number of states expanded during the search procedure. \\ -Consider Fig 15.11, Depth-First Search and Breadth-First Search algorithms do not greedily select states from the priority queue for expansion. Hence, these algorithms search regions of the configuration space +Consider Fig \ref{fig:graph_search}, Depth-First Search and Breadth-First Search algorithms do not greedily select states from the priority queue for expansion. Hence, these algorithms search regions of the configuration space relatively uniformly without consideration of the direction of the expanded state to the goal. Dijkstra's algorithm improves on Depth-First Search and Breadth-First Search by greedily selecting states from the priority queue which prevents revisiting previously visited states. However, Dijkstra's algorithm searches for paths uniformly around the initial state, leading to wasted effort. Concentrating the search phase in regions leading to the goal state can reduce the number of states expanded during the search, speeding up the motion planning process. \begin{figure}[h] @@ -302,18 +302,19 @@ \section{Correctness and Improvements} \end{tikzpicture} \caption{Graph Search Algorithm} +\label{fig:graph_search} \end{center} \end{figure} \subsection{A*: Improving Dijkstra} -Dijkstra's algorithm orders nodes added to the priority queue by optimal ``cost-to-arrival." The A* algorithm improves on Dijkstra by adding a heuristic function that models cost to go to the goal to the optimal ``cost-to-arrival," i.e. A* orders by ``cost-to-arrival" + ``cost-to-go". This heuristic function guides exploration towards the region of the configuration space close to the goal, leading to faster computations as less states are expanded. Thus, the test +Dijkstra's algorithm orders nodes added to the frontier queue (priority queue) by optimal ``cost-to-arrival" $C(q)$. The A* algorithm improves on Dijkstra by adding a heuristic function $h(q)$ that models cost to go to the goal, i.e. A* orders by ``cost-to-arrival" + (approximate) ``cost-to-go." This heuristic function, for example, distance to goal, guides exploration towards the region of the configuration space close to the goal, leading to faster computations as less states are expanded. Thus, the test condition changes from \[ C(q) + C(q, q') \leq \text{UPPER} \] to \[ C(q) + C(q, q') + h(q') \leq \text{UPPER} \] -where $h(q)$ is the optimal cost-to-go. The heuristic must be a positive underestimate of the true cost to the goal. This modification of the test condition reduces the number of nodes placed in the priority queue and still guarantees that an optimal path will be returned. +where $h(q)$ is a heuristic for the optimal ``cost-to-go" from the next state $q'$ to the goal state. The heuristic must be a positive \textit{underestimate} of the true cost to the goal. This modification of the test condition reduces the number of nodes placed in the frontier queue and still guarantees that an optimal path will be returned. In practice, it leads to much faster runtimes. -Figure 15.12 illustrates A* implemented on a grid. Black states represent obstacles. Yellow states represent nodes on the priority queue. Blue represent opened nodes. Green states indicate the final path. The state with the minimum total estimated cost (including the heuristics cost) is selected from the priority queue for expansion. +Figure \ref{fig:astar_example} illustrates A* implemented on a grid. Black states represent obstacles. Yellow states represent nodes on the frontier queue. Blue represent opened nodes. Green states indicate the final path. The state with the minimum total estimated cost (including the heuristics cost) is selected from the frontier queue for expansion. \newcolumntype{C}[1]{% @@ -449,11 +450,18 @@ \subsection{A*: Improving Dijkstra} \hline \end{tabular} } - -\caption{A* Algorithm} +\caption{A* Algorithm Example} +\label{fig:astar_example} \end{center} \end{figure} +\subsection{Benefits and Drawbacks of Graph-based Approach} +The graph-based approaches are simple, easy to use, and fast for some problems. However, this approach presents two drawbacks. +\begin{enumerate} +\item These algorithms are resolution dependent, and this resolution is decided a priori. We might find that this resolution is not fine enough to uncover a path or the best path for the original motion planning problem. +\item These approaches ignore many robot degrees of freedom. Usually, we need to construct a grid with number of dimensions equal to the number of degrees of freedom of our robot. Thus, the number of gird cells increase exponentially, which also increases the number of neighbors of each vertex exponentially in the number of degrees of freedom. Thus, the graph-based approaches become less effective for high dimensional problems. + +\end{enumerate} \printbibliography From 27f1992bcdc265ed7e502b067b9e0294e9dc27eb Mon Sep 17 00:00:00 2001 From: Theo Diamandis Date: Sun, 10 Mar 2019 16:26:25 -0700 Subject: [PATCH 09/14] added astar alg --- lecture_15/main.pdf | Bin 443782 -> 466959 bytes lecture_15/main.tex | 91 +++++++++++++++++++++++++++++--------------- 2 files changed, 61 insertions(+), 30 deletions(-) diff --git a/lecture_15/main.pdf b/lecture_15/main.pdf index 7a635b6d0a045f23603dc223a4ddabf4f2cbb269..92f39ec66826313f273ddae91ab8a553e5992dbb 100644 GIT binary patch delta 79342 zcmZs?RZyNy6D^AK;_hz2-GjSBAh^4``-8i?yF-xRZUKV3I|O%kIN!f_ovL$jE_!NJ z*W9e0nXX>lHDS>N@qL8#qd^dC?8&q6G(f1vWZXU%V$Tt+Z&P6Ga1!I{rCFdN-w-zA z3^us8rOyvrDzqgaHg`3%m6=vHjy?H;*qlL`QbL7N=v!=@GX6)?i+`(^=a`pjgDf1@ z1#;p6Hq zxY;|8tIztG7mv>7+-lvRZFt^NZ*#|$9q@;qtFrafW}Cz#8{if_sF$J8|0ge@TO|$O zuUcEwvCi*%`ujNC=p2du 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Changed wording. 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[PATCH 11/14] configuration space formatting fixed --- lecture_15/main.tex | 31 ++++++++++++++++--------------- 1 file changed, 16 insertions(+), 15 deletions(-) diff --git a/lecture_15/main.tex b/lecture_15/main.tex index 1009e51..3e4a407 100644 --- a/lecture_15/main.tex +++ b/lecture_15/main.tex @@ -93,11 +93,11 @@ \subsection{ History } sampling-based methods, notably Probability Road-Maps (PRM) and Rapidly-Exploring Random Trees (RRT), were developed in the late 1990s and deployed on real-time systems in the 2000s \cite{lavalle}. With the increasing complexity of robotic systems, solving motion planning problems in real-time has become a significant challenge. Hence, current research has focused on speeding up sampling-based methods through lazy dynamic programming and massive parallelization of search algorithms. \section{Configuration-Space} -In the context of robotic motion planning, we will first create a simple problem setup to define the variables of interest and introduce the proper nomenclature. First, consider a workspace defined in 2D, $ W \subseteq R^2$, which effectively describes the world that is occupied by the robot, the obstacles, and potentially other agents in the system. Within this workspace, we define the robot as a rigid polygon, and we introduce the set of obstacles $O \subset W$ as a subset of the workspace shown in Fig 15.1. +In the context of robotic motion planning, we will first create a simple problem setup to define the variables of interest and introduce the proper nomenclature. First, consider a \textit{workspace} defined in 2D, $ \mathcal{W} \subseteq \mathbb{R}^2$, which effectively describes the world that is occupied by the robot, the obstacles, and potentially other agents in the system. Within this workspace, we define the robot as a rigid polygon, and we introduce the set of obstacles $\mathcal{O} \subset \mathcal{W}$ as a subset of the workspace shown in Fig 15.1. The rigid robot can be translated or rotated according to its own kinematic constraints. With these definitions established, we can qualitatively define the robotic motion planning problem as: -\begin{figure}[h] +\begin{figure}[ht!] \begin{center} \includegraphics{fig15_1.PNG} \caption{2D Workspace} @@ -106,31 +106,32 @@ \section{Configuration-Space} \textbf{Problem Statement}: Compute the sequence of movements to go from a given initial placement of the robot to a desired goal placement such that the robot never touches the obstacle regions. -While the robot motion planning problem is described in a physically intuitive ”real-world” setting (e.g., the 2D planar workspace described above), it is vital to map the problem into another space, denoted as the configuration (C-) space. The configuration space is simply the space containing the degrees of freedom of the robot. Returning to our simple example highlighted above, the configuration space contains -the translational degrees of freedom and the rotational degrees of freedom of the robot. In this way, the 2D path planning problem in physical space is cast as a 3D path planning problem in the configuration space, with two dimensions describe the translational modes of the trajectory and one degree of freedom capturing the rotational mode of the trajectory in the planar workspace. +While the robot motion planning problem is described in a physically intuitive ”real-world” setting (e.g., the 2D planar workspace described above), it is vital to map the problem into another space, denoted as the \textit{configuration} ($\mathcal{C-}$) space. The configuration space is simply the space containing the degrees of freedom of the robot. Returning to our simple example highlighted above, the configuration space contains +the translational degrees of freedom and the rotational degrees of freedom of the robot. In this way, the 2D path planning problem in physical space is cast as a 3D path planning problem in the configuration space, with two dimensions describing the translational modes of the trajectory and one degree of freedom capturing the rotational mode of the trajectory in the planar workspace. -Formally, the degrees of freedom for our rigid polygon robot, R R2, is the set of generalized coordinates, $q = (xt, yt, q)$, where (xt, yt) R2 are the translational coordinates, and q is the rotational coordinate. Accordingly, the configuration space consists of every combination of q which yields a unique robot placement (in this case a subset of R3). An important distinction must be made for the generalized coordinate describing the rotational degree of freedom; namely, q and q2pk, k2 Z describe equivalent attitudes. The configuration space in this case is actually in R2 S1, where S1 is the manifold which contains the angular displacement variable and includes the aforementioned ”equivalence” between angles that differ by integer multiples of 2p. If we relax the planar restriction in our working example, the angular orientation coordinates of the robot would instead be in the manifold S3. The importance of this distinction is highlighted with a simple example: consider a robot that has a current heading of q = 3p/2 rads, and we want to reconfigure the robot to heading qg = 0 subject to the constraint of avoiding a C-space obstacle at q = p. The geometry of this problem is highlighted in Figure 15.2. Now, if the equivalence between the angles 0 and 2p is not established in the definition of the configuration space, the robot would not be able to traverse a collision-free path to the desired heading in the configuration space (see red trajectory). Instead, since the configuration space is defined with respect to S1, the robot is able to achieve the desired heading (see green trajectory). +Formally, the degrees of freedom for our rigid polygon robot, $\mathcal{R} \subset \mathbb{R}^2$, is the set of generalized coordinates, $q = (x_t, y_t, \theta)$, where $(x_t, y_t) \subset \mathbb{R}^2$ are the translational coordinates, and $\theta$ is the rotational coordinate. Accordingly, the configuration space consists of every combination of q which yields a unique robot placement (in this case a subset of $\mathbb{R}^3$). An important distinction must be made for the generalized coordinate describing the rotational degree of freedom; namely, $\theta$ and $\theta \pm 2\pi k, k \in \mathbb{Z}$ describe equivalent attitudes. The configuration space in this case is actually in $\mathbb{R}^2 \times \mathcal{S}^1$, where $\mathcal{S}^1$ is the manifold which contains the angular displacement variable and includes the aforementioned ”equivalence” between angles that differ by integer multiples of $2\pi$. If we relax the planar restriction in our working example, the angular orientation coordinates of the robot would instead be in the manifold $\mathcal{S}^3$. The importance of this distinction is highlighted with a simple example: consider a robot that has a current heading of $\theta = 3\pi/2$ rads, and we want to reconfigure the robot to heading $\theta_g = 0$ subject to the constraint of avoiding a $\mathcal{C}$-space obstacle at $\theta = \pi$. The geometry of this problem is highlighted in Figure 15.2. Now, if the equivalence between the angles 0 and $2\pi$ is not established in the definition of the configuration space, the robot would not be able to traverse a collision-free path to the desired heading in the configuration space (see red trajectory). Instead, since the configuration space is defined with respect to $\mathcal{S}^1$, the robot is able to achieve the desired heading (see green trajectory). -\begin{figure}[h] +\begin{figure}[ht!] \begin{center} -\includegraphics{fig15_2.PNG} -\caption{Example trajectory planning where the rotational degree of freedom is not described on S1 (red) and where it is described on S1 (green)} +\includegraphics[width=0.75\textwidth]{fig15_2.PNG} +\caption{Example trajectory planning where the rotational degree of freedom is not described on $\mathcal{S}^1$ (red) and where it is described on $\mathcal{S}^1$ (green)} \end{center} \end{figure} Note that for our simple planar workspace example, the dimensions of the physical space and the configuration space are quite similar. Instead, it is not uncommon for complicated dynamical systems (e.g., a robotic arm mounted on a spacecraft) to have configuration spaces of much higher dimension than the physical space counterpart. In these cases, mapping from physical space to configuration space is extremely vital to the motion planning problem as it allows all constraints of the complex kinematic system to be accounted for with higher-dimensional information (i.e., more coordinates). -\subsection{ Planning in C-Space } -Now that we have introduced the necessary nomenclature and addressed the important distinction between motion planning in the real (physical) space and the configuration space, we can return to our prior problem statement and re-write it in rigorous mathematical detail. Let the set of points in the workspace that are occupied by the robot in configuration q be denoted by R(q) W. Furthermore, we can make the intuitive definition of collision-free motion as that set of all robot configurations which do not intersect the obstacle regions: $R(q)\cap O = \emptyset$. Then the formal problem statement can be re-written as: +\subsection{ Planning in $\mathcal{C}$-Space } +Now that we have introduced the necessary nomenclature and addressed the important distinction between motion planning in the real (physical) space and the configuration space, we can return to our prior problem statement and re-write it in rigorous mathematical detail. Let the set of points in the workspace that are occupied by the robot in configuration $q$ be denoted by $\mathcal{R}(q) \subset \mathcal{W}$. Furthermore, we can make the intuitive definition of collision-free motion as that set of all robot configurations which do not intersect the obstacle regions: $\mathcal{R}(q)\cap \mathcal{O} = \emptyset$. Then the formal problem statement can be re-written as: -Problem Statement, Revisited: -Compute the continuous path: t : [0, 1] ) fq 2 CjR(q) \ O = Æg with t(0) = qI and t(1) = qG In general, we refer to the set fq 2 CjR(q) \ O = Æg as Cf ree, which represents all possible combinations of the robot’s degrees of freedom which are collision-free. A key fact here is that, while a robot occupies physical dimensions in the real space, the robotic motion is described by a simple point in the configuration space that captures the translation degrees of freedom (of the center of mass for example) and the rotational degrees of freedom of the robot. In this way, mapping to the configuration space poses the motion planning problem as a simpler problem of finding a path for a point. An illustration of this problem is highlighted in Figure 15.3. The physical dimensionality of the robot is intrinsically captured in the definition of the configuration space (in particular, in Cf ree). This property represents a key advantage -of moving to the configuration space for robotic motion planning, as it allows the same general planning problem formulation (i.e., path planning for a point) to be equally applicable to myriad dynamical systems. +\textbf{Problem Statement, Revisited:} \\ +Compute the continuous path: $\tau : [0, 1] \Rightarrow \{q \in \mathcal{C}|\mathcal{R}(q) \cap \mathcal{O} = \emptyset\}$ with $\tau(0) = q_1$ and $\tau(1) = q_G$ -\begin{figure}[H] + In general, we refer to the set $\{q \in \mathcal{C} | \mathcal{R}(q) \cap \mathcal{O} = \emptyset\}$ as $\mathcal{C}_{free}$, which represents all possible combinations of the robot’s degrees of freedom which are collision-free. A key fact here is that, while a robot occupies physical dimensions in the real space, the robotic motion is described by a simple point in the configuration space that captures the translation degrees of freedom (of the center of mass for example) and the rotational degrees of freedom of the robot. In this way, mapping to the configuration space poses the motion planning problem as a simpler problem of finding a path for a point. An illustration of this problem is highlighted in Figure 15.3. The physical dimensionality of the robot is intrinsically captured in the definition of the configuration space (in particular, in $\mathcal{C}_{free}$). This property represents a key advantage of moving to the configuration space for robotic motion planning, as it allows the same general planning problem formulation (i.e., path planning for a point) to be equally applicable to myriad dynamical systems. + +\begin{figure}[ht!] \begin{center} -\includegraphics{fig15_3.PNG} +\includegraphics[width=0.4\textwidth]{fig15_3.PNG} \caption{Robotic motion planning as a point path planning problem in C-space.} \end{center} \end{figure} From fcbdd6b3ae09eed552bee0fb094ebc3bf0bdcb66 Mon Sep 17 00:00:00 2001 From: Laura Date: Mon, 11 Mar 2019 23:21:02 -0700 Subject: [PATCH 12/14] configuration space --- lecture_15/forbidden_space.jpeg | Bin 0 -> 66318 bytes .../forbidden_space_robot_obstacle.jpeg | Bin 0 -> 67526 bytes lecture_15/main.tex | 25 +++++++++++++----- 3 files changed, 19 insertions(+), 6 deletions(-) create mode 100644 lecture_15/forbidden_space.jpeg create mode 100644 lecture_15/forbidden_space_robot_obstacle.jpeg diff --git 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HcmV?d00001 diff --git a/lecture_15/main.tex b/lecture_15/main.tex index 3e4a407..3f1fd90 100644 --- a/lecture_15/main.tex +++ b/lecture_15/main.tex @@ -106,20 +106,18 @@ \section{Configuration-Space} \textbf{Problem Statement}: Compute the sequence of movements to go from a given initial placement of the robot to a desired goal placement such that the robot never touches the obstacle regions. -While the robot motion planning problem is described in a physically intuitive ”real-world” setting (e.g., the 2D planar workspace described above), it is vital to map the problem into another space, denoted as the \textit{configuration} ($\mathcal{C-}$) space. The configuration space is simply the space containing the degrees of freedom of the robot. Returning to our simple example highlighted above, the configuration space contains -the translational degrees of freedom and the rotational degrees of freedom of the robot. In this way, the 2D path planning problem in physical space is cast as a 3D path planning problem in the configuration space, with two dimensions describing the translational modes of the trajectory and one degree of freedom capturing the rotational mode of the trajectory in the planar workspace. +While the robot motion planning problem is described in a physically intuitive ”real-world” setting (e.g., the 2D planar workspace described above), it is vital to map the problem into another space, denoted as the \textit{configuration} ($\mathcal{C-}$) space. The configuration space is simply the space containing the degrees of freedom of the robot. The degrees of freedom of the robot is defined as the smallest number, $d \geq 1$, of coordinates required to fully represent the configuration of the robot. Returning to our simple example highlighted above, the configuration space contains the translational degrees of freedom and the rotational degrees of freedom of the robot, totaling $d=3$ degrees of freedom. In this way, the 2D path planning problem in physical space is cast as a 3D path planning problem in the configuration space, with two dimensions describing the translational modes of the trajectory and one degree of freedom capturing the rotational mode of the trajectory in the planar workspace. Formally, the degrees of freedom for our rigid polygon robot, $\mathcal{R} \subset \mathbb{R}^2$, is the set of generalized coordinates, $q = (x_t, y_t, \theta)$, where $(x_t, y_t) \subset \mathbb{R}^2$ are the translational coordinates, and $\theta$ is the rotational coordinate. Accordingly, the configuration space consists of every combination of q which yields a unique robot placement (in this case a subset of $\mathbb{R}^3$). An important distinction must be made for the generalized coordinate describing the rotational degree of freedom; namely, $\theta$ and $\theta \pm 2\pi k, k \in \mathbb{Z}$ describe equivalent attitudes. The configuration space in this case is actually in $\mathbb{R}^2 \times \mathcal{S}^1$, where $\mathcal{S}^1$ is the manifold which contains the angular displacement variable and includes the aforementioned ”equivalence” between angles that differ by integer multiples of $2\pi$. If we relax the planar restriction in our working example, the angular orientation coordinates of the robot would instead be in the manifold $\mathcal{S}^3$. The importance of this distinction is highlighted with a simple example: consider a robot that has a current heading of $\theta = 3\pi/2$ rads, and we want to reconfigure the robot to heading $\theta_g = 0$ subject to the constraint of avoiding a $\mathcal{C}$-space obstacle at $\theta = \pi$. The geometry of this problem is highlighted in Figure 15.2. Now, if the equivalence between the angles 0 and $2\pi$ is not established in the definition of the configuration space, the robot would not be able to traverse a collision-free path to the desired heading in the configuration space (see red trajectory). Instead, since the configuration space is defined with respect to $\mathcal{S}^1$, the robot is able to achieve the desired heading (see green trajectory). \begin{figure}[ht!] \begin{center} -\includegraphics[width=0.75\textwidth]{fig15_2.PNG} +\includegraphics[width=0.8\textwidth]{fig15_2.PNG} \caption{Example trajectory planning where the rotational degree of freedom is not described on $\mathcal{S}^1$ (red) and where it is described on $\mathcal{S}^1$ (green)} \end{center} \end{figure} -Note that for our simple planar workspace example, the dimensions of the physical space and the configuration space are quite similar. Instead, it is not uncommon for complicated dynamical systems (e.g., a robotic arm mounted on a spacecraft) to have configuration spaces of much higher dimension than the physical space counterpart. In these cases, mapping from physical space to configuration space is extremely -vital to the motion planning problem as it allows all constraints of the complex kinematic system to be accounted for with higher-dimensional information (i.e., more coordinates). +Note that for our simple planar workspace example, the dimensions of the physical space and the configuration space are quite similar. Instead, it is not uncommon for complicated dynamical systems (e.g., a robotic arm mounted on a spacecraft) to have configuration spaces of much higher dimension than the physical space counterpart. In these cases, mapping from physical space to configuration space is extremely vital to the motion planning problem as it allows all constraints of the complex kinematic system to be accounted for with higher-dimensional information (i.e., more coordinates). \subsection{ Planning in $\mathcal{C}$-Space } Now that we have introduced the necessary nomenclature and addressed the important distinction between motion planning in the real (physical) space and the configuration space, we can return to our prior problem statement and re-write it in rigorous mathematical detail. Let the set of points in the workspace that are occupied by the robot in configuration $q$ be denoted by $\mathcal{R}(q) \subset \mathcal{W}$. Furthermore, we can make the intuitive definition of collision-free motion as that set of all robot configurations which do not intersect the obstacle regions: $\mathcal{R}(q)\cap \mathcal{O} = \emptyset$. Then the formal problem statement can be re-written as: @@ -127,7 +125,22 @@ \subsection{ Planning in $\mathcal{C}$-Space } \textbf{Problem Statement, Revisited:} \\ Compute the continuous path: $\tau : [0, 1] \Rightarrow \{q \in \mathcal{C}|\mathcal{R}(q) \cap \mathcal{O} = \emptyset\}$ with $\tau(0) = q_1$ and $\tau(1) = q_G$ - In general, we refer to the set $\{q \in \mathcal{C} | \mathcal{R}(q) \cap \mathcal{O} = \emptyset\}$ as $\mathcal{C}_{free}$, which represents all possible combinations of the robot’s degrees of freedom which are collision-free. A key fact here is that, while a robot occupies physical dimensions in the real space, the robotic motion is described by a simple point in the configuration space that captures the translation degrees of freedom (of the center of mass for example) and the rotational degrees of freedom of the robot. In this way, mapping to the configuration space poses the motion planning problem as a simpler problem of finding a path for a point. An illustration of this problem is highlighted in Figure 15.3. The physical dimensionality of the robot is intrinsically captured in the definition of the configuration space (in particular, in $\mathcal{C}_{free}$). This property represents a key advantage of moving to the configuration space for robotic motion planning, as it allows the same general planning problem formulation (i.e., path planning for a point) to be equally applicable to myriad dynamical systems. + In general, we refer to the set $\{q \in \mathcal{C} | \mathcal{R}(q) \cap \mathcal{O} = \emptyset\}$ as $\mathcal{C}_{free}$, which represents all possible combinations of the robot’s degrees of freedom which are collision-free. For a simple 2 degree of freedom circle robot and triangle obstacle, as seen in Figure 15.3a, the forbidden space can be constructed by drawing a line through the center every robot configuration that just touches the obstacle. It can be seen that a forbidden robot state will be any state in which the center of the robot is inside the forbidden space. + + \begin{figure} + \centering + \begin{subfigure}[b]{0.35\textwidth} + \includegraphics[width=\textwidth]{forbidden_space_robot_obstacle} + \caption{Simple 2 DOF robot and obstacle in $\mathbb{R}^2$} + \end{subfigure} + \begin{subfigure}[b]{0.3\textwidth} + \includegraphics[width=\textwidth]{forbidden_space} + \caption{Forbidden space of simple robot and obstacle.} + \end{subfigure} +\end{figure} + + + A key fact here is that, while a robot occupies physical dimensions in the real space, the robotic motion is described by a simple point in the configuration space that captures the translation degrees of freedom (of the center of mass for example) and the rotational degrees of freedom of the robot. In this way, mapping to the configuration space poses the motion planning problem as a simpler problem of finding a path for a point. An illustration of this problem is highlighted in Figure 15.4. The physical dimensionality of the robot is intrinsically captured in the definition of the configuration space (in particular, in $\mathcal{C}_{free}$). This property represents a key advantage of moving to the configuration space for robotic motion planning, as it allows the same general planning problem formulation (i.e., path planning for a point) to be equally applicable to myriad dynamical systems. \begin{figure}[ht!] \begin{center} From 56ac231bb897ea4424e813b7eff3ac0ae1ce312b Mon Sep 17 00:00:00 2001 From: Kaushik Ram Sadagopan Date: Tue, 12 Mar 2019 02:07:31 -0700 Subject: [PATCH 13/14] Changes to Section 15.5 --- lecture_15/fig15_10.PNG | Bin 54156 -> 179164 bytes lecture_15/fig15_8.PNG | Bin 52519 -> 54327 bytes lecture_15/fig15_9.PNG | Bin 49608 -> 54274 bytes lecture_15/main.pdf | Bin 471222 -> 685173 bytes lecture_15/main.tex | 94 +++++++++++++++++++++++----------------- 5 files changed, 54 insertions(+), 40 deletions(-) diff --git a/lecture_15/fig15_10.PNG b/lecture_15/fig15_10.PNG index 6ec495c6f94f92c089cad3768685fe07d23eff69..f85059f0265447e1115dbfd3a26190491393e238 100644 GIT binary patch literal 179164 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z_`7b!j1vyV(3H(NQ+9s$xAn%_V@j{gcjF=a8C#-puS{1{Lx*U#Ng_?_i<+hrFbwf} zAa>byUJwQVvU#GR69hn}B=c1i0xe8Rif2=lrIH}EAXh2Gz8(qNuJ8}%T~LNq;wvdb zAp13|jM#uWj~b-GJHPm&;iIYy+4Qfj63XPpU+#>(v#G^J(3$ttVk92=zo}Y)Adx*< wJ~a2hgL}x31p(dv?>!>0`L9&~CuajkXLrYcFAfVk0~-S?3@NFooEXgi0FG!CzW@LL diff --git a/lecture_15/main.tex b/lecture_15/main.tex index 3f1fd90..ad324ee 100644 --- a/lecture_15/main.tex +++ b/lecture_15/main.tex @@ -228,71 +228,85 @@ \subsection{3D Vertical Decomposition} \end{figure} \section{Graph-Search Algorithms} -Suppose that our environment map has been converted into a connectivity graph using one of the graph generation methods presented earlier. Whatever map representation is chosen, the goal of path planning is to find the best path in the map’s connectivity graph between the start and the goal, where best refers to the selected optimization criteria. In this section, we present several search algorithms that have become -quite popular in mobile robotics. - - +Suppose our environment map has been converted into a connectivity graph using one of the graph generation methods presented earlier. Whatever map representation we choose, the goal of path planning is to find the best path in the map’s connectivity graph between the start and the goal, where best refers to the selected optimization criteria. In this section, we present several popular search algorithms in mobile robotics. \subsection{Concepts in graph search} -We introduce the concepts which are all function of the node n (and an adjacent node n0). -g(n) : Path cost(accumulated cost from the start node to any given node n) -c(n, n0) : Edge traversal cost(cost from a node n to an adjacent node n0) -h(n) : Heuristic cost(expected cost from a node n to the goal node) -f (n) : Expected total cost(from start to goal via state n) -f (n) = g(n) + eh(n), where e is a parameter that assumes algorithm-dependent values. +We introduce the concepts which are all functions of the node $n$ (and an adjacent node $n'$): +\begin{itemize} + \item $g(n)$: Path cost (accumulated cost from the start node to any given node $n$). + + \item $C(n, n')$: Edge traversal cost (cost from a node $n$ to an adjacent node $n'$). + + \item $h(n)$: Heuristic cost (expected cost from a node $n$ to the goal node). + + \item$f(n)$: Expected total cost (from start to goal via state $n$). + +And $f(n) = g(n) + \epsilon h(n)$, where $\epsilon$ is a parameter that assumes algorithm-dependent values. +\end{itemize} \subsection{Discriminator graph search} -Heuristic function unemployed -– Uniform traversal cost simpler form, obtain faster execution speeds -ex) depth-first, breadth-first -– Nonuniform traversal cost higher algorithmic complexity -ex) Dijkstra’s algorithm -- Heuristic function employed incorporates additional information about the problem set and thus often allows for faster convergence of the search query. -ex) e = 1 : optimal A* algorithm, e > 1 : suboptimal or greedy A* variant +\begin{itemize} + +\item No heuristic function employed + \begin{itemize} + \item[-] Uniform traversal cost with a simpler form and obtains faster execution speeds. \\ + e.g.) depth-first, breadth-first \\ + \item[-] Nonuniform traversal cost with a higher algorithmic complexity.\\ + e.g.) Dijkstra’s algorithm + \end{itemize} + + +\item Heuristic function employed + \begin{itemize} + \item[-] Incorporates additional information about the problem set and thus often allows for faster convergence of the search query. \\ + e.g.) $e = 1$: optimal A* algorithm, $e > 1$: sub-optimal or greedy A* variant +\end{itemize} +\end{itemize} \subsection{Various graph search algorithms} -Depth-First Search -Depth-first search expands each node up to the deepest level of the graph, until the node has no more successors. As those nodes are expanded, their branch is removed from the graph and the search backtracks by expanding the next neighboring node of the start node until its deepest level and so on. -Depth-First Search stores only a single path from the start nodes for each node on the path, leading to a lower memory requirement. However, previously visited nodes might be visited and the algorithm might enter redundant paths. +\subsubsection{Depth-First Search (DFS)} +Depth-first search expands each node up to the deepest level of the graph, until the node has no more successors. As those nodes are expanded, their branch is removed from the graph and the search backtracks by expanding the next neighboring node of the start node until its deepest level and so on. DFS stores only a single path from the start nodes for each node on the path, leading to a lower memory requirement. However, previously visited nodes might be visited and the algorithm might enter redundant paths. \begin{figure}[h!] \begin{center} -\includegraphics{fig15_8.PNG} +\includegraphics[width=0.5\textwidth]{fig15_8.PNG} \caption{Depth First Search} \end{center} \end{figure} -Breadth-First Search -Breadth-first begins with the start node and explores all of its neighboring nodes. Then, for each of these nodes, it explores all their unexplored neighbors and so on. -The search always returns the path with the fewest number of edges between the start and goal node. +\subsubsection{Breadth-First Search (BFS)} +Breadth-first search begins with the start node and explores all of its neighboring nodes. Then for each of these nodes, it explores all their unexplored neighbors and so on. This search always returns the path with the fewest number of edges between the start and goal node. \begin{figure}[h!] \begin{center} -\includegraphics{fig15_9.PNG} +\includegraphics[width=0.5\textwidth]{fig15_9.PNG} \caption{Breadth First Search} \end{center} \end{figure} -Dijkstra’s algorithm (Best-First Search) -Dijkstra’s algorithm is similar to Breadth-First search, except that edge costs may assume any positive value and the search still guarantees solution optimality. Dijkstra’s algorithm uses the concept of the heap, a specialized tree-based data structure and selects the next q as: q = argminq2QC(q)) +\subsubsection{Dijkstra’s Algorithm (Shortest Path First)} +Dijkstra’s algorithm is similar to Breadth-first search, except that edge costs may assume any positive value and the search still guarantees solution optimality. Dijkstra’s algorithm uses the concept of the heap, a specialized tree-based data structure and selects the next $q$ as: $q = \textrm{arg} \min_{q \in Q} C(q)$ \subsection{Label Correcting Algorithm} -The label-correction algorithm is a general type of shortest path algorithm which includes very common graph search algorithms like breadth-first search, depth-first search, A*, Dijkstra’s algorithm as special cases. -The idea is to progressively discover shorter paths from the origin to every other node q, and to maintain the lowest cost path(from qI to q) found so far in a variable C(q). - +The label-correction algorithm is a general type of shortest path algorithm which includes common graph search algorithms like breadth-first search, depth-first search, A*, Dijkstra’s algorithm as special cases. +The idea is to progressively discover shorter paths from the origin to every other node $q$, and to maintain the lowest cost path (from $q_I$ to $q$) found so far in a variable $C(q)$. +\vspace{3em} \begin{figure}[h] \begin{center} -\includegraphics{fig15_10.PNG} -\caption{Label Correcting} +\includegraphics[width=0.8\textwidth]{fig15_10.PNG} +\caption{Label Correcting Algorithm} \end{center} \end{figure} - -Algorithm Step -Step1. Remove a node q from the frontier queue and for each child q0 of q, execute step 2 -Step2. If C(q) + C(q, q0) min(C(q0),UPPER), set C(q0) := C(q) + C(q, q0) and set q to be the parent of q0. In addition, if q0 6= qG, place q0 in frontier queue if it is not already there, while if q0 = qG, set UPPER to the new value C(q) + C(q, qG) -Step3. If the frontier queue is empty, terminate, else go to step 1. -Initialization : set the labels of all nodes to, except for the label of the origin node, which is set to 0. - +\newpage +\textbf{Algorithm}: +\vspace{-0.7em} +\begin{itemize} + \item \textbf{Step 1}: Remove a node q from the frontier queue and for each child q0 of q, execute step 2. + \item \textbf{Step 2}: If $ C(q) + C(q, q') \leq \min(C(q'),\textrm{UPPER})$, set $ C(q') := C(q) + C(q, q')$ and set $q$ to be the parent of $q'$. In addition, if $q' \neq q_G$, place $q'$ in frontier queue if it is not already there, while if $q' = q_G$, set UPPER to the new value $C(q) + C(q, q_G)$ + \item \textbf{Step 3}: If the frontier queue is empty, terminate, else go to step 1. + \item \textbf{Initialization}: Set the labels of all nodes to $\infty$, except the label of the origin node which is set to 0. +\end{itemize} +\vspace{-1em} \section{Correctness and Improvements} Label correcting algorithms are guaranteed to find the shortest feasible path from an initial state to a goal state, given one exists. This attribute gives rise to the Correctness Theorem. \vspace{-0.5cm} @@ -515,7 +529,7 @@ \subsection{Benefits and Drawbacks of Graph-based Approaches} \printbibliography \subsubsection*{Contributors} -Winter 2019: Warren Cheng, Junwu Zhang, Theo Diamandis, Jeremy Crowley\\ +Winter 2019: Warren Cheng, Junwu Zhang, Theo Diamandis, Jeremy Crowley, Kaushik Ram Sadagopan\\ Winter 2018: Ryan Richey, Hyoungju Seo, Ola Shorinwa, Josh Sullivan, Danning Sun \end{document} From 15bdabc37ffae45b68061bfc528fec917696e1c7 Mon Sep 17 00:00:00 2001 From: Warren Cheng Date: Tue, 12 Mar 2019 13:24:44 -0700 Subject: [PATCH 14/14] Added new intro and further reading section --- lecture_15/main.pdf | Bin 685173 -> 688058 bytes lecture_15/main.tex | 41 +++++++++++++++++++++++++++++++++-------- 2 files changed, 33 insertions(+), 8 deletions(-) diff --git a/lecture_15/main.pdf b/lecture_15/main.pdf index 662aa24fae7beeb57e21c620b20df44e01959040..266f6d6b467b6aea0f91c777650255bfdc66e623 100644 GIT binary patch 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