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<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no">
<title>Estimation and Mapping using a Swarm of Resource-Constrained Robots</title>
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<meta name="author" content="Ragesh Kumar Ramachandran">
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<body>
<div class="reveal">
<div class="slides">
<!-- Title slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="z-index: 11;"><h3>Estimation and Mapping using a Swarm of Resource-Constrained Robots</h3></div>
<div>
<p style="font-weight: bold;"> Ragesh Kumar Ramachandran </p>
<p style="font-style: italic;">Advisor : Dr Spring Berman</p>
</div>
<div style="border: 5px solid #835C3B; border-style: groove;">
<p>Autonomous Collective Systems Laboratory</p>
</div>
<div style="position: absolute; width: 100px; left: 0px; top: 500px;">
<img src="figures/logos/nsf.png" alt="NSF logo" title="NSF logo" height="100" width="100">
</div>
<div style="position: absolute; width: 150px; right: 0px; top: 500px;">
<img src="figures/logos/darpa.png" alt="DARPA logo" title="DARPA logo" height="100" width="150">
</div>
<div style="position: absolute; width: 600px; left: 150px; top: 500px;">
<img src="figures/logos/asu.png" alt="ASU logo" title="ASU logo" height="100" width="600">
</div>
</section>
<!-- Motivation -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 28px;">
<h3>Motivation</h3>
<ul>
<li><p>Development of robot platforms that can be deployed in swarms to perform tasks autonomously over large spatial and temporal scales.</p></li>
<li><p class="fragment">Many potential applications for robotic swarms, including exploration, environmental monitoring, disaster response, search-and-rescue, mining, and intelligence-surveillance-reconnaissance, will require to operate in dynamic, uncertain and GPS denied environments</p></li>
<li><p class="fragment">Despite these limitations, it may be necessary for the swarm to characterize its surroundings, for instance to map obstacles, payloads for transport, or hazardous areas to avoid.</p></li>
<li><p class="fragment">Our current research progress and future work focus on novel estimation and mapping strategies using robotic swarms in these types of applications.</p></li>
</ul>
</section>
<!-- Overview -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 28px;">
<h3>Overview</h3>
<ul>
<li><p>An Advection-Diffusion-Reaction Based Approach to Mapping an Environmental Feature.</p></li>
<li><p>A Probabilistic Topological Approach to Quantifying Environmental Features.</p></li>
<li><p>A Probabilistic Approach to Constructing Topological Maps of an Environment.</p></li>
<li><p>Metric Map Construction using a Stochastic Robotic Swarm Leveraging Received Signal Strength.</p></li>
<li><p>Scalar Field Estimation by Large Networks with Partially Accessible Measurements.</p></li>
<li><p>Ongoing work</p></li>
</ul>
</section>
<!-- Title -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div class="heading_box">
<h3> An Advection-Diffusion-Reaction Based Approach to Mapping an Environmental Feature<sup>*</sup></h3>
</div>
<p align="left" style="font-size: 15px"><sup>*</sup>Ragesh K. Ramachandran, Karthik Elamvazhuthi, and Spring Berman. An optimal control approach to mapping GPS-denied environments using a stochastic robotic swarm. In Int’l. Symp. on Robotics Research (ISRR), 2015</p>
</section>
<!-- ISRR intro slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Problem Statement</h3>
<p>
Mapping a single feature of interest using a robotic swarm without communication or localization capabilities.
</p>
<div style="position: absolute; width: 460px; height: 350px; left: 0px;">
<img src="figures/ISRR_2015/intro/formation.png" alt="formation example" title="formation example" height="70%" width="100%">
</div>
<div style="position: absolute; width: 460px; height: 350px; right: 0px;">
<img src="figures/ISRR_2015/intro/deployment.png" alt="deployment example" title="deployment example" height="70%" width="100%">
</div>
</section>
<!-- ISRR intro Slide part 2-->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Robotic swarm models</h3>
<div style="font-size: 30px;">
<ul>
<li><p>Robotic swarms with stochastic behaviors.</p></li>
<li><p class="fragment">Macroscopic abstraction using partial differential equations</p></li>
<li><p class="fragment">Mapping formulated as an optimal control problem.</p></li>
</ul>
</div>
</section>
<!-- Related Work-->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Related Work</h3>
<div style="font-size: 30px;">
<ul>
<li><p>The most common approach in robotics is SLAM and probabilistic mapping.</p></li>
<li><p class="fragment">Mapping an environment using a robotic swarm is a relatively new area of research in the robotics.</p></li>
<li><p class="fragment">In <span style="color: #129cea">Dirafzoon et al.(2013)</span>a robotic swarm is used to identify the topological features of an environment from information about the times at which robots encounter other robots and environmental features.</p></li>
</ul>
</div>
</section>
<!-- Microscopic Model Slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Microscopic Model</h3>
<div style="font-size: 30px;">
<p>Robot State Changes (Chemical Reactions)</p>
<div style="position: absolute; width: 360px; height: 150px; left: 300px;">
<img src="figures/ISRR_2015/intro/passive.png" alt="passive example" title="passive example" height="70%" width="100%">
</div>
<div style="position: absolute; width: 960px; top: 300px; font-size: 30px;">
<p>Robot Motion Model</p>
<p>$$\textbf{X}_i(t + \Delta t) = \textbf{X}_i(t) + (\sqrt{2D \Delta t}) \mathbf{Z}(t) + \textbf{v}(t) \Delta t$$</p>
</div>
</div>
</section>
<!-- Microscopic Model Simulation Slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%">
<h3>Microscopic Model Simulation</h3>
<video width = "960px" height = "500px" src="videos/micro.mp4" type="video/mp4" controls>
</video>
</section>
<!-- Macroscopic Model Slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Macroscopic Model</h3>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
$$\frac{\partial u}{\partial t} = \nabla \cdot (D\nabla u - \mathbf{v}(t)u) - k K(\mathbf{x})u\ in\ L$$
$$L = \Omega \times [0,T],\ \Omega \subset \mathbb{R}^2$$
<p>No-flux Boundary Conditions : $\vec{n} \cdot (D\nabla u - \mathbf{v}(t)u) = 0 ~~~ on ~~\partial \Omega \times [0,T] $</p>
<p>Initial Condition : Gaussian density centered at a point $\mathbf{x}_0 \in \Omega$</p>
<p class="fragment">u(x,t) := Active robot density, D := Diffusion coefficient, $\mathbf{v}(t)$ := Velocity field</p>
<p class="fragment">k := Reaction rate constant, $K(\mathbf{x})$ := Feature Indicator Function (Map)</p>
</div>
</section>
<!-- Optimal Control Problem Slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Optimal Control Problem</h3>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<p style="color: rgb(0,255,0);">Objective Function</p>
$$\underset{(\mathbf{u},K(\mathbf{x})) \in U^N \times \Theta_{ad}}{\min}\mathbf{J}(\mathbf{u},K) = \sum_{i = 1}^{N} W_i J_i(u_i) + \frac{\lambda}{2}\|K(\mathbf{x})\|^2_{L^2(\Omega)}$$
$$J_i(u_i) = \frac{1}{2}\left \| \int_{\Omega} u_i(\mathbf{x},t) d\mathbf{x} - g_i(t)\right \|_{L^2([0,T])}^2$$
<p style="color: rgb(0,255,255);">Admissible control inputs</p>
$\Theta_{ad} = \{ K(\mathbf{x}) \in L^2(\Omega); ~K_{min} \leq K(\mathbf{x}) \leq K_{max} \}$
<p>With the macroscopic model as <span style="color: #1AE26A">Constraint equations</span></p>
</div>
<aside class="notes">
Solving PDE control problems are hard even in linear settings. This is non-linear and non-convex
</aside>
</section>
<!-- Lagrangian Slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Lagrangian Equation</h3>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 28px;">
<p style="font-style: italic;">The optimization problem is solved using a gradient descent approach</p>
$$\mathcal{L}(\mathbf{u},\mathbf{p},K) = \mathbf{J}(\mathbf{u},K) + \sum_{i = 1}^{N} \langle p_i, \Psi_i(u_i,K) \rangle$$
<p>$\Psi_i(u_i,K)$ is the $i^{th}$ constraint and $p_i$ is the adjoint variable. </p>
<h3 class="fragment">Gradient Equation</h3>
<p class="fragment">$$\mathbf{J}' = \sum_{i = 1}^{N} (\int_0^T(ku_ip_i)(\mathbf{x})dt) + \lambda K(\mathbf{x})$$</p>
<p class="fragment">$p_i$ is computed by solving the adjoint equation.</p>
</div>
</section>
<!-- Adjoint Equation Slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Adjoint Equation</h3>
<div style="position: absolute; width: 960px; height: 450px; top: 150px; font-size: 30px;">
$$-\frac{\partial p_i}{\partial t} = \nabla \cdot (D\nabla p_i +\mathbf{v}_i(t)p_i) - p_i k K(\vec{x}) - \nabla_{u_{i}} J_i(u_i) ~~\ in ~\ L$$
$$\vec{n}\cdot\nabla p_i = 0 ~~\ on\ ~\Gamma, $$
$$p_i(T) = 0, ~~~i = 1,...,N. $$
<p>The equations are solved backwards</p>
</div>
</section>
<!-- Simulation ISRR Slide -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Simulation Results</h3>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<div style="position: absolute; width: 300px; height: 450px; left: 0px; font-size: 30px;">
<img src="figures/ISRR_2015/rectangle_load/actual_map.png" alt="Actual Map" title="Actual Map" height="35%" width="100%">
<img src="figures/ISRR_2015/triangle/actual_map.png" alt="Actual Map" title="Actual Map" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Actual map</u></p></figcaption>
</div>
<div style="position: absolute; width: 300px; height: 450px; left: 330px; font-size: 30px;">
<img src="figures/ISRR_2015/rectangle_load/map_estimated.png" alt="Estimated map" title="Estimated map" height="35%" width="100%">
<img src="figures/ISRR_2015/triangle/map_estimated.png" alt="Estimated map" title="Estimated map" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Estimated map</u></p></figcaption>
</div>
<div style="position: absolute; width: 300px; height: 450px; right: 0px; font-size: 30px;">
<img src="figures/ISRR_2015/rectangle_load/error.png" alt="Absolute value of error" title="Absolute value of error" height="35%" width="100%">
<img src="figures/ISRR_2015/triangle/error.png" alt="Absolute value of error" title="Absolute value of error" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Absolute value of error</u></p></figcaption>
</div>
</div>
</section>
<!-- Simulation ISRR Slide 2 -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Simulation Results</h3>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<div style="position: absolute; width: 300px; height: 450px; left: 0px; font-size: 30px;">
<img src="figures/ISRR_2015/center_load8trials_part2/actual_map.png" alt="Actual Map" title="Actual Map" height="35%" width="100%">
<img src="figures/ISRR_2015/corner_combined6trails/actual_map.png" alt="Actual Map" title="Actual Map" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Actual map</u></p></figcaption>
</div>
<div style="position: absolute; width: 300px; height: 450px; left: 330px; font-size: 30px;">
<img src="figures/ISRR_2015/center_load8trials_part2/map_estimated.png" alt="Estimated map" title="Estimated map" height="35%" width="100%">
<img src="figures/ISRR_2015/corner_combined6trails/map_estimated.png" alt="Estimated map" title="Estimated map" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Estimated map</u></p></figcaption>
</div>
<div style="position: absolute; width: 300px; height: 450px; right: 0px; font-size: 30px;">
<img src="figures/ISRR_2015/center_load8trials_part2/error.png" alt="Absolute value of error" title="Absolute value of error" height="35%" width="100%">
<img src="figures/ISRR_2015/corner_combined6trails/error.png" alt="Absolute value of error" title="Absolute value of error" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Absolute value of error</u></p></figcaption>
</div>
</div>
<aside class="notes">
The case of small objects.
Also when they are located far from the origin of the swarm.
</aside>
</section>
<!-- Simulation ISRR Slide 3 -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Simulation Results</h3>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<div style="position: absolute; width: 300px; height: 450px; left: 0px; font-size: 30px;">
<img src="figures/ISRR_2015/inclined/actual_map.png" alt="Actual Map" title="Actual Map" height="35%" width="100%">
<img src="figures/ISRR_2015/l_shape/actual_map.png" alt="Actual Map" title="Actual Map" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Actual map</u></p></figcaption>
</div>
<div style="position: absolute; width: 300px; height: 450px; left: 330px; font-size: 30px;">
<img src="figures/ISRR_2015/inclined/map_estimated.png" alt="Estimated map" title="Estimated map" height="35%" width="100%">
<img src="figures/ISRR_2015/l_shape/map_estimated.png" alt="Estimated map" title="Estimated map" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Estimated map</u></p></figcaption>
</div>
<div style="position: absolute; width: 300px; height: 450px; right: 0px; font-size: 30px;">
<img src="figures/ISRR_2015/inclined/error.png" alt="Absolute value of error" title="Absolute value of error" height="35%" width="100%">
<img src="figures/ISRR_2015/l_shape/error.png" alt="Absolute value of error" title="Absolute value of error" height="35%" width="100%">
<figcaption><p style="font-size: 25px"><u>Absolute value of error</u></p></figcaption>
</div>
</div>
</section>
<!-- DARS 2016 *******************************************************************-->
<!-- Title -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div class="heading_box"><h3>A Probabilistic Topological Approach to Quantifying Environmental Features<sup>*</sup></h3></div>
<p align="left" style="font-size: 15px"><sup>*</sup>Ragesh K. Ramachandran, Sean Wilson, and Spring Berman. A probabilistic topological approach to feature identification using a stochastic robotic swarm. In To appear in the Int’l. Symposium on Distributed Autonomous Robotic Systems (DARS), 2016</p>
</section>
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Problem Statement</h3>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<div style="position: absolute; width: 500px; left: 20px">
<p>
We consider the problem of mapping a bounded, unknown, GPS-denied 2D environment using a swarm of N robots.
</p>
<p class="fragment" style="font-weight: bold; font-style: italic;">
The focus of the research is to find a scalable method to extract the map from the information obtained by resource constrained robots.
</p>
</div>
<div style="position: absolute; width: 500px; left: 500px">
<img src="figures/DARS_2016/experiment/4RobotSetup.png" alt="Experimental setup" title="Experimental setup" height="400" width="400">
</div>
</div>
</section>
<!-- Related work -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h2>Related Work</h2>
<ul>
<li><p>Our previous work, <span style="color: #129cea">Ramachandran et al.(2015)</span> uses a PDE based model of the swarm to solve the problem. But requires <span style="color: rgb(255,0,0);">large number of robots</span>.</p></li>
<li><p class="fragment"><span style="color: #129cea">Ramaithitima et al.(2016)</span> propose an algorithm that covers the free space of the environment with robots and then constructs an approximate generalized Voronoi graph. But requires <span style="color: rgb(255,0,0);">communication</span>.</p></l>
<li><p class="fragment">In <span style="color: #129cea">Dirafzoon et al.(2015)</span> persistent homology is applied on a point cloud of the domain’s free region based on encounter time with robots. But each robot requires <span style="color: rgb(255,0,0);">identification label</span>.</p></li>
</ul>
</div>
</section>
<!-- Current focus -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h2>Current focus</h2>
<p>
As a first step to solve this problem, we quantify the topological features (holes or obstacles) in the domain.
</p>
</div>
</section>
<!-- Types of Scenarios -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h2>Types of Scenarios</h2>
<p>
<span style="font-weight: bold; color: rgb(255,0,0);">Type I:</span> Robots receive accurate estimates of their global positions when they are close to the boundary of the domain.
</p>
<p>
<span style="font-weight: bold; color: rgb(255,0,0);">Type II:</span> Robots do not receive global position updates anywhere in the domain.
</p>
</div>
</section>
<!-- Robot Motion Model -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h2>Robot Motion Model</h2>
<p>
The displacement of a robot over a time step $\Delta t$ is given by:
$$\mathbf{X}(t + \Delta t) = \mathbf{X}(t) + \mathbf{V}(t)\Delta t + \mathbf{W}(t)$$
</p>
<p align="left">
Where,
</p>
<p>
$$\mathbf{V}(t) = [v_x(t),\ v_y(t)]^T = [v\ \cos(\theta(t)), \ v\ \sin(\theta(t))]^T$$
$$\mathbf{X}(t) = [x(t),\ y(t)]^T$$
$\mathbf{W}(t) \in \mathbb{R}^2$ is a zero mean Gaussian random vector
</p>
</div>
</section>
<!-- Robot Data -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h2>Robot Data </h2>
<p>
Data that robot $j \in \{1,...,N\}$ obtains at time $t_k \in [0, T]$, $k \in \{1,...,K\}$, consists of the element $d^j_k = \{\mu^j_k, \Sigma^j_k\}$.
</p>
<p>
$\mu^j_k \in \mathbb{R}^{2}$ is the mean of the robot's estimate of its $(x,y)$ position.
</p>
<p>
$\Sigma^j_k \in \mathbb{R}^{2\times2}$ is the covariance matrix of its position estimate.
</p>
<p>
Both at time $t_k$
</p>
</div>
</section>
<!-- Processing the Data -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h2>Processing the Data</h2>
<ul>
<li><p>Discretize the domain into a high-resolution uniform grid. $m_i$ denotes a grid cell.</p></li>
<li><p class="fragment">Use the robots' data to assign each grid cell $m_i$ a probability $p^f_i$ of being free.</p> </li>
<li><p class="fragment">Integrate the Gaussian distribution with mean $\mu^j_k$ and covariance matrix $\Sigma^j_k$ over the cell region to obtain $p^f_i$.</p></li>
</ul>
</div>
<aside class="notes">
p<sup>f</sup><sub>i</sub> denote the probability that a j<sup>th</sup> robot visited the cell i based data d<sup>f</sup><sub>k</sub>.
</aside>
</section>
<!-- Processing the Data 2-->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h2>Processing the Data</h2>
<ul>
<li><p>Assign a score $s_i \in [0, \infty)$ to each grid cell $m_i$. $s_i = \frac{1}{\left | P_i \right |}\sum _{p \in P_i} \log\left ( \frac{1}{1 - p} \right )$. Where $P_i = \{p_{ijk} ~|~ p_{ijk} > \rho, \rho > 0\}$ and $p_{ijk}$ is computed using the $k^{th}$ data element of the $j^{th}$ robot.</p></li>
<li><p class="fragment">Finally, $p^f_i$ of each grid cell being free is computed as $p_{i}^{f} ~=~ 1 - (\exp(s_i))^{-1}$.</p></li>
</ul>
</div>
</section>
<!-- Probability map TYPE I -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3> Probability map for <span style="color: rgb(255,0,0);">TYPE I</span> environment in Simulation</h3>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/pdf_0_bounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/pdf_1_bounded.png" alt="actual map with 0 obstacle" title="Actual map" height="100%" width="100%"></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/pdf_2_bounded.png" alt="PDF with 1 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/pdf_3_bounded.png" alt="PDF with 1 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/pdf_4_bounded.png" alt="PDF with 1 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
</div>
<div style="position: relative; top: 25px"><h4>Actual Maps</h4></div>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/world.png" alt="actual map with 0 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/world.png" alt="actual map with 1 obstacle" title="Actual map" height="100%" width="100%"></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/world.png" alt="actual map with 1 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/world.png" alt="actual map with 1 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/world.png" alt="actual map with 1 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
</div>
<div style="position: relative; top: 25px;">
<p>30 point robots deployed for 200 seconds</p>
</div>
</div>
</section>
<!-- Probability map TYPE II -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3> Probability map for <span style="color: rgb(255,0,0);">TYPE II</span> environment in Simulation</h3>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/pdf_0_unbounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/pdf_1_unbounded.png" alt="PDF with 1 with 0 obstacle" title="Actual map" height="100%" width="100%"></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/pdf_2_unbounded.png" alt="PDF with 2 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/pdf_3_unbounded.png" alt="PDF with 3 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/pdf_4_unbounded.png" alt="PDF with 4 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
</div>
<div style="position: relative; top: 25px"><h4>Actual Maps</h4></div>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/world.png" alt="actual map with 0 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/world.png" alt="actual map with 1 obstacle" title="Actual map" height="100%" width="100%"></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/world.png" alt="actual map with 2 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/world.png" alt="actual map with 3 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/world.png" alt="actual map with 4 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
</div>
<div style="position: relative; top: 25px;">
<p>30 point robots deployed for 200 seconds</p>
</div>
</div>
</section>
<!-- Finding the Number of Features or Obstacles -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Finding the Number of Features or Obstacles</h3>
<ul>
<li><p>Point cloud is obtained by sampling a dense, uniformly random set of points from the domain based on a probability map of the domain and rejecting those points that are inside a grid cell whose $p^f_i$ is below a given threshold.</p></li>
<li><p class="fragment">Landmark points are selected from the point cloud using the sequential max-min algorithm.</p> </li>
<li><p class="fragment">Finally, a Rips complex based filtration is constructed from landmark points and barcode diagrams are extracted.</p></li>
<li><p class="fragment">The procedure was tested for Type I & Type II scenarios.</p></li>
</ul>
</div>
<aside class="notes">
The new planning technique based on measures could use this. Only Type II results shown in interest of time.
</aside>
</section>
<!-- Rips complex based filtration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h2>Rips complex based filtration</h2>
<h3>Background (Algebraic Topology)</h3>
<ul>
<li><p> vectors $v_0, v_1, ..., v_k \in \mathbb{R}^n$ are <em>affinely independent</em> if the vectors $ v_1 - v_0, ..., v_k - v_0 $ are linearly independent.</p></li>
<li><p>A $k$-simplex $\sigma \subset \mathbb{R}^n$ can be defined as the convex hull of $k+1$ affinely independent vectors $\{v_0, v_1, ..., v_k\}$, called <em>vertices</em>. $\sigma = [v_0, v_1, ..., v_k]$</p></li>
<li><p>A face $\tau$ is the convex hull of a subset of $\{v_0, v_1, ..., v_k\}$</p></li>
<li><p>A simplicial complex $\kappa$ is defined as a finite collection of simplices $\sigma$.</p></li>
</ul>
</div>
<aside class="notes">
Ask if any know about point set topology, algebraic topology and group theory.
</aside>
</section>
<!-- Illustration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h2>Illustration</h2>
<img src="figures/background/simplical.png" alt="simplicial example" title="simplicial example" height="400" width="1000">
</div>
</section>
<!-- Background -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Background continued...</h3>
<ul>
<li><p> Rips complex is formed by choosing a parameter $\delta > 0$ and adding a $k$-simplex to the simplicial complex if every vertex in the simplex is within a distance $\delta$ from every other </p></li>
<li><p>If $f: \kappa \to \mathbb{R} $ is a function such that $\tau \leq \sigma$ implies that $f(\tau) \leq f(\sigma)$, then $f^{-1}((-\infty, \delta])$ is a simplicial complex denoted by $\kappa_\delta$ and $\delta_1 \leq \delta_2$ implies that $\kappa_{\delta_1} \subseteq \kappa_{\delta_2}$, yielding a filtration of simplicial complexes with $\delta$ as the filtration parameter.</p></li>
<li><p>A barcode diagram is a graphical representation used to identify the persistent topological features</p></li>
</ul>
</div>
<aside class="notes">f could be any function not necessarily distance function.</aside>
</section>
<!-- filtration with barcode -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Rips complex based filtration with barcode</h3>
<img src="figures/background/filtration_example.png" alt="homology example" title="homology example" height="500" width="800">
</div>
<aside class="notes"> Delta is the radius of the ball.</aside>
</section>
<!-- Point Cloud and Landmark Points (nested) -->
<section>
<!-- TYPE II -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3> Point Cloud and Landmark Points for <span style="color: rgb(255,0,0);">TYPE II</span> environment</h3>
<div style="position: relative; top: 0px"><h4>Point cloud</h4></div>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/point_cloud_0_unbounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/point_cloud_1_unbounded.png" alt="PDF with 1 with 0 obstacle" title="Actual map" height="100%" width="100%"></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/point_cloud_2_unbounded.png" alt="PDF with 2 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/point_cloud_3_unbounded.png" alt="PDF with 3 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/point_cloud_4_unbounded.png" alt="PDF with 4 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
</div>
<div style="position: relative; top: 25px"><h4>Landmark Points</h4></div>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/landmark_0_unbounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">No features</p></figcaption></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/landmark_1_unbounded.png" alt="PDF with 1 with 0 obstacle" title="Actual map" height="80%" width="100%">
<figcaption><p style="font-size: 25px">One feature</p></figcaption></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/Landmark_2_unbounded.png" alt="PDF with 2 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Two features</p></figcaption></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/landmark_3_unbounded.png" alt="PDF with 3 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Three features</p></figcaption></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/landmark_4_unbounded.png" alt="PDF with 4 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Four features</p></figcaption></div>
</div>
</div>
<aside class="notes"> Press down for type I</aside>
</section>
<!-- TYPE I -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3> Point Cloud and Landmark Points for <span style="color: rgb(255,0,0);">TYPE I</span> environment</h3>
<div style="position: relative; top: 0px"><h4>Point cloud</h4></div>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/point_cloud_0_bounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/point_cloud_1_bounded.png" alt="PDF with 1 with 0 obstacle" title="Actual map" height="100%" width="100%"></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/point_cloud_2_bounded.png" alt="PDF with 2 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/point_cloud_3_bounded.png" alt="PDF with 3 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/point_cloud_4_bounded.png" alt="PDF with 4 obstacle" title="$p^f_i$ contour plot" height="100%" width="100%"></div>
</div>
<div style="position: relative; top: 25px"><h4>Landmark Points</h4></div>
<div class="grid-container5">
<div class="item1"><img src="figures/DARS_2016/simulation/0/landmark_0_bounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">No features</p></figcaption></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/landmark_1_bounded.png" alt="PDF with 1 with 0 obstacle" title="Actual map" height="80%" width="100%">
<figcaption><p style="font-size: 25px">One feature</p></figcaption></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/Landmark_2_bounded.png" alt="PDF with 2 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Two features</p></figcaption></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/landmark_3_bounded.png" alt="PDF with 3 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Three features</p></figcaption></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/landmark_4_bounded.png" alt="PDF with 4 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Four features</p></figcaption></div>
</div>
</div>
</section>
</section>
<!-- Barcode Diagrams(nested)-->
<section>
<!-- TYPE II -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Barcode Diagram <span style="color: rgb(255,0,0);">TYPE II</span> environments</h3>
<div class="grid-container5" style="grid: 450px / auto auto auto auto auto;">
<div class="item1"><img src="figures/DARS_2016/simulation/0/barcode_0_unbounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">No features</p></figcaption></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/barcode_1_unbounded.png" alt="PDF with 1 with 0 obstacle" title="Actual map" height="80%" width="100%">
<figcaption><p style="font-size: 25px">One feature</p></figcaption></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/barcode_2_unbounded.png" alt="PDF with 2 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Two features</p></figcaption></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/barcode_3_unbounded.png" alt="PDF with 3 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Three features</p></figcaption></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/barcode_4_unbounded.png" alt="PDF with 4 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Four features</p></figcaption></div>
</div>
</div>
<aside class="notes"> Press down for type I Barcode</aside>
</section>
<!-- TYPE I -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Barcode Diagram <span style="color: rgb(255,0,0);">TYPE I</span> environments</h3>
<div class="grid-container5" style="grid: 450px / auto auto auto auto auto;">
<div class="item1"><img src="figures/DARS_2016/simulation/0/barcode_0_bounded.png" alt="PDF with 0 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">No features</p></figcaption></div>
<div class="item2"><img src="figures/DARS_2016/simulation/1/barcode_1_bounded.png" alt="PDF with 1 with 0 obstacle" title="Actual map" height="80%" width="100%">
<figcaption><p style="font-size: 25px">One feature</p></figcaption></div>
<div class="item3"><img src="figures/DARS_2016/simulation/2/barcode_2_bounded.png" alt="PDF with 2 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Two features</p></figcaption></div>
<div class="item4"><img src="figures/DARS_2016/simulation/3/barcode_3_bounded.png" alt="PDF with 3 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Three features</p></figcaption></div>
<div class="item5"><img src="figures/DARS_2016/simulation/4/barcode_4_bounded.png" alt="PDF with 4 obstacle" title="$p^f_i$ contour plot" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Four features</p></figcaption></div>
</div>
</div>
</section>
</section>
<!-- varying time of deployment -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Effect of varying time of deployment on estimation</h3>
<img src="figures/DARS_2016/simulation/varying_time.png" alt="varying time" title="varying time" height="400px" width="600px">
<div style="position: relative; top: 0px;">
<p>30 point robots deployed for each simulation</p>
</div>
</div>
</section>
<!-- varying number of robots -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Effect of varying number of robots on estimation</h3>
<img src="figures/DARS_2016/simulation/varying_robots.png" alt="varying robots" title="varying robots" height="400px" width="600px">
<div style="position: relative; top: 0px;">
<p>Time of deployment for each simulation : 200 seconds</p>
</div>
</div>
</section>
<!-- Experiments TYPE II -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h2>Experiments on <span style="color: rgb(255,0,0);">TYPE II</span> environment</h2>
<h3>Experimental Setups</h3>
<div class="grid-container3">
<div class="item1"><img src="figures/DARS_2016/experiment/OneObject.png" alt="Experimental setup" title="Experimental setup" height="80%" width="100%">
<figcaption><p style="font-size: 25px">One feature</p></figcaption></div>
<div class="item2"><img src="figures/DARS_2016/experiment/TwoObject.png" alt="Experimental setup" title="Experimental setup" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Two features</p></figcaption></div>
<div class="item3"><img src="figures/DARS_2016/experiment/ThreeObject.png" alt="Experimental setup" title="Experimental setup" height="80%" width="100%">
<figcaption><p style="font-size: 25px">Three features</p></figcaption></div>
</div>
</div>
</section>
<!-- Pheeno robotic platform -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Introducing Our Robotic Platform <span style="font-style: italic; color: #3BDE17">Pheeno</span><sup>1</sup></h3>
<div style="position: absolute; left: 50px; height: 420px; width: 400px;">
<img src="figures/pheeno/pheeno1.png" alt="Pheeno Watching ducky" title="Pheeno Watching ducky" height="80%" width="100%">
<figcaption>Pheeno watching ducky</figcaption>
</div>
<div style="position: absolute; top:550px">
<p align="left" style="font-size: 15px"><sup>1</sup>Sean Wilson et al., IEEE R-AL, July 2016</p>
</div>
<div style="position: absolute; left: 470px; height: 420px; width: 400px;">
<img src="figures/pheeno/pheeno2.png" alt="Pheeno with gripper" title="Pheeno with gripper" height="80%" width="100%">
<figcaption>Pheeno holding ducky</figcaption>
</div>
</div>
</section>
<!-- Experiments Illustration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h2>An Experimental Illustration</h2>
<div class="grid-container3">
<div class="item1"><img src="figures/DARS_2016/experiment/4RobotSetup.png" alt="Experimental setup" title="Experimental setup" height="92%" width="100%"><figcaption>Experimental setup</figcaption></div>
<div class="item2"><img src="figures/DARS_2016/experiment/pdf.png" alt="Experimental Result pdf" title="Experimental result pdf" height="80" width="100%">
<figcaption>Probability map</figcaption>
<img src="figures/DARS_2016/experiment/pointcloud.png" alt="Experimental Result pointcloud" title="Experimental result pointcloud" height="80" width="100%">
<figcaption>Point cloud</figcaption>
<img src="figures/DARS_2016/experiment/landmark.png" alt="Experimental Result landmark" title="Experimental result landmark" height="80" width="100%">
<figcaption>Landmark points</figcaption>
</div>
<div class="item3"><img src="figures/DARS_2016/experiment/barcode.png" alt="Experimental Barcode" title="Experimental Result Barcode" height="92%" width="100%">Barcode</div>
</div>
<div style="position: relative; top: 30px; left: 0px; height: 40px; width: 520px; font-size: 20px;">
<p style="text-align: left; color: #10E834">*4 robots deployed for 180 seconds</p>
</div>
</div>
</section>
<!-- Experiments Validation of Estimation -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Experimental Validation of Estimation</h3>
<div style="position: absolute; left: 50px; height: 420px; width: 400px;">
<img src="figures/DARS_2016/experiment/expt_varying_robots.png" alt="Experimental setup" title="Experimental setup" height="80%" width="100%">
<figcaption>Varying number of robots with deployment time of 180s</figcaption>
</div>
<div style="position: absolute; left: 470px; height: 420px; width: 400px;">
<img src="figures/DARS_2016/experiment/expt_varying_time.png" alt="Experimental setup" title="Experimental setup" height="80%" width="100%">
<figcaption>Varying time of deployment with 4 robots</figcaption>
</div>
</div>
</section>
<!-- RAL 2017 -->
<!-- Title -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div class="heading_box">
<h3>A Probabilistic Approach to Constructing Topological Maps of an Environment<sup>*</sup></h3>
</div>
<p align="left" style="font-size: 15px"><sup>*</sup>Ragesh K. Ramachandran, Sean. Wilson, and Spring. Berman. A probabilistic approach to automated construction of topological maps using a stochastic robotic swarm. IEEE Robotics and Automation Letters, 2(2):616–623, April 2017.</p>
</section>
<!-- Problem Statement -->
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<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h3> Problem Statement</h3>
<p>
We consider the problem of generating a topological map of a bounded, unknown, GPS-denied 2D environment using data collected by a swarm of N robots.
</p>
</div>
</section>
<!-- Topological Map -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Topological Map</h3>
<p>Topological maps are sparse roadmap representations of environments that can be used to identify collision-free trajectories for robots to navigate through a domain</p>
<p class="fragment">The topological map constructed here is in the form of a <span style="font-style: italic;">Generalized Voronoi Diagram (GVD)</span>. Due to the computational complexity of computing exact GVDs, algorithms have been developed to generate <span style="font-style: italic;">approximate GVDs (AGVDs)</span>
</p>
<p class="fragment">A Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane.
</p>
</div>
</section>
<!-- Voronoi Diagram Illustration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Voronoi Diagram Illustration</h3>
<div style="position: absolute; width: 800px; height: 450px; left: 80px;">
<img src="figures/background/Euclidean_Voronoi_diagram.png" alt="Euclidean_Voronoi_diagram example" title="Euclidean_Voronoi_diagram" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Voronoi Diagram based on Euclidean distance<sup>1</sup></p></figcaption>
<p align="left" style="font-size: 15px"><sup>1</sup>Taken from Wikipedia</p>
</div>
</div>
</section>
<!-- Topological Map Illustration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Topological Map Illustration</h3>
<div style="position: absolute; width: 800px; height: 450px; left: 80px;">
<img src="figures/background/top-voronoi-diagram.jpg" alt="top-voronoi-diagram example" title="top-voronoi-diagram" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Topological map of a domain with three obstacles<sup>1</sup></p></figcaption>
<p align="left" style="font-size: 15px"><sup>1</sup>Taken from Dr.Markus Haid blog</p>
</div>
</div>
</section>
<!-- extension of DARS -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h2>Extension of the Previous work</h2>
<h3>What more can be done?</h3>
<ul>
<li><p class="fragment fade-up">How to find an automated procedure to threshold the density function in order to isolate the obstacle region?</p></li>
<li><p class="fragment fade-up">Can we build a topological map of the domain?</p></li>
</ul>
</div>
</section>
<!-- COMPUTE THRESHOLD AND IDENTIFY NUMBER OF OBSTACLES -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="font-size: 30px;">
<h3>Compute Threshold And Identify Number Of Obstacles</h3>
<ul>
<li><p>A filtration is generated by the thresholding the probability map and constructing a simplicial complex at each level and choosing $\delta$ = 1 − threshold ($\alpha$) as the <span style="font-style: italic;">filtration parameter</span></p></li>
<li><p class="fragment fade-up">The barcode generated out of this filtration can be used to compute optimal threshold and identify number of obstacle.</p></li>
</ul>
</div>
</section>
<!-- Density Function Thresholding Based Filtration Illustration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Density Function Thresholding Based on Filtration</h3>
<div style="font-size: 30px;">
<div style="position: absolute; width: 460px; height: 420px; left: 0px;">
<img src="figures/RAL_2017/2/world.png" alt="World" title="World" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Actual Domain</p></figcaption>
</div>
<div style="position: absolute; width: 460px; height: 420px; right: 0px;">
<img src="figures/RAL_2017/2/pdf_map.png" alt="PDF map" title="PDF map" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Computed density function </p></figcaption>
</div>
</div>
</section>
<!-- Density Function Thresholding Based Filtration Illustration 2-->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Density Function Thresholding Based on Filtration</h3>
<h4>Simplicial complex for each filtration value</h4>
<div style="font-size: 30px;">
<div style="position: absolute; width: 460px; height: 420px; left: 0px;">
<img src="figures/RAL_2017/2/filtration/fil_02.png" alt="filtration" title="filtration" height="70%" width="100%">
<figcaption><p style="font-size: 25px">$\delta$ = 0.2</p></figcaption>
</div>
<div style="position: absolute; width: 460px; height: 420px; right: 0px;">
<img src="figures/RAL_2017/2/filtration/fil_03.png" alt="filtration" title="filtration" height="70%" width="100%">
<figcaption><p style="font-size: 25px">$\delta$ = 0.3</p></figcaption>
</div>
</div>
</section>
<!-- Density Function Thresholding Based Filtration Illustration 3-->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Density Function Thresholding Based on Filtration</h3>
<h4>Simplicial complex for each filtration value</h4>
<div style="font-size: 30px;">
<div style="position: absolute; width: 460px; height: 420px; left: 0px;">
<img src="figures/RAL_2017/2/filtration/fil_04.png" alt="filtration" title="filtration" height="70%" width="100%">
<figcaption><p style="font-size: 25px">$\delta$ = 0.4</p></figcaption>
</div>
<div style="position: absolute; width: 460px; height: 420px; right: 0px;">
<img src="figures/RAL_2017/2/filtration/fil_05.png" alt="filtration" title="filtration" height="70%" width="100%">
<figcaption><p style="font-size: 25px">$\delta$ = 0.5</p></figcaption>
</div>
</div>
</section>
<!-- Density Function Thresholding Based Filtration Illustration 4-->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>Density Function Thresholding Based on Filtration</h3>
<h4>Simplicial complex for each filtration value</h4>
<div style="font-size: 30px;">
<div style="position: absolute; width: 460px; height: 420px; left: 0px;">
<img src="figures/RAL_2017/2/filtration/fil_06.png" alt="filtration" title="filtration" height="70%" width="100%">
<figcaption><p style="font-size: 25px">$\delta$ = 0.6</p></figcaption>
</div>
<div style="position: absolute; width: 460px; height: 420px; right: 0px;">
<img src="figures/RAL_2017/2/filtration/fil_08.png" alt="filtration" title="filtration" height="70%" width="100%">
<figcaption><p style="font-size: 25px">$\delta$ = 0.8</p></figcaption>
</div>
</div>
</section>
<!-- The Barcode From The Filtration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<h3>The Barcode From The Filtration</h3>
<div style=" font-size: 30px;">
<img src="figures/RAL_2017/2/barcode.png" alt="barcode" title="barcode" height="300px" width="400px">
<figcaption>Barcode for the filtration. $\delta_{opt} = 0.6\ or\ \alpha_{opt} = 0.4$. Number of obstacles = 2, with one connected component in the domain </figcaption>
</div>
</section>
<!-- Obstacle Segmentation -->
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<div style="font-size: 30px;">
<h3>Obstacle Segmentation</h3>
<ul>
<li><p class="fragment "> The grid cells with $p^f_i < \alpha_{opt}$ belong to an obstacle.</p></li>
<li><p class="fragment "> Classify the grid cells in each obstacle and identify its boundary.</p></li>
<li><p class="fragment "> A graph is constructed with obstacle grid cell centers as vertices and the cells are classified by performing a BFS on this graph</p></li>
<li><p class="fragment "> The boundaries of each obstacle are identified as the elements in graph that belong to the same obstacle and have fewer than four neighbors.</p></li>
</ul>
</div>
</section>
<!-- Obstacle Segmentation Illustration -->
<section data-background-image="figures/background/black_maroon.png" data-background-size="100% 100%" data-background-interactive>
<div style="position: absolute; width: 960px; height: 450px; left: 0px; font-size: 30px;">
<h3>Obstacle Segmentation Illustration</h3>
<div style="position: absolute; width: 460px; height: 450px; left: 0px;">
<img src="figures/RAL_2017/3/world.png" alt="world" title="world" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Actual Domain</p></figcaption>
</div>
<div style="position: absolute; width: 460px; height: 450px; right: 0px;">
<img src="figures/RAL_2017/3/pdf_map.png" alt="filtration" title="filtration" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Density function computed</p></figcaption>
</div>
</div>
</section>
<!-- Obstacle Segmentation Illustration 2-->
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<div style="font-size: 30px;">
<h3>Obstacle Segmentation Illustration</h3>
<div style="position: absolute; width: 460px; height: 450px; left: 0px;">
<img src="figures/RAL_2017/3/barcode.png" alt="Barcode" title="Barcode" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Barcode</p></figcaption>
</div>
<div style="position: absolute; width: 460px; height: 450px; right: 0px;">
<img src="figures/RAL_2017/3/segmented.png" alt="segmented" title="segmented" height="70%" width="100%">
<figcaption><p style="font-size: 25px">Segmented Obstacles and their Boundaries. The boundary is shown in black.</p></figcaption>
</div>
</div>
</section>