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<title>Exploration problem</title>
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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Exploration problem</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">For the exploration problem in recreational mathematics, see <a href="Jeep_problem" title="Jeep problem">jeep problem</a>.</div>
<p>In <a href="Robotics" title="Robotics">robotics</a>, the <b>exploration problem</b> deals with the use of a <a href="Robot" title="Robot">robot</a> to maximize the <a href="Knowledge" title="Knowledge">knowledge</a> over a particular area. The exploration problem arises in <a href="Robotic_mapping" title="Robotic mapping">robotic mapping</a> and <a href="Search_%26_rescue" class="mw-redirect" title="Search &amp; rescue">search &amp; rescue</a> situations, where an environment might be dangerous or inaccessible to humans.<sup id="cite_ref-Trun2005_1-0" class="reference"><a href="#cite_note-Trun2005-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>
<p>The exploration problem naturally arises in situations in which a robot is utilized to survey an area that is dangerous or inaccessible for humans. The field of robotic explorations draws from various fields of <a href="Information_gathering" class="mw-redirect" title="Information gathering">information gathering</a> and <a href="Decision_theory" title="Decision theory">decision theory</a>, and have been studied as far back as the 1950s.
</p><p>The earliest work in robotic exploration was done in the context of simple finite state automata known as bandits, where algorithms were designed to distinguish and map different states in a <a href="Finite-state_automaton" class="mw-redirect" title="Finite-state automaton">finite-state automaton</a>. Since then, the primary emphasis has been shifted to the robotics system development domain, where exploration-algorithms guided robot have been used to survey volcanos,<sup id="cite_ref-Bares1999_2-0" class="reference"><a href="#cite_note-Bares1999-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> search and rescue, and abandoned mines mapping.<sup id="cite_ref-Thrun2003_3-0" class="reference"><a href="#cite_note-Thrun2003-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Current state of the art system include advanced techniques on active localization, <a href="Simultaneous_localization_and_mapping" title="Simultaneous localization and mapping">simultaneous localization and mapping</a> (SLAM) based exploration, and multi-agent cooperative exploration.
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<div class="mw-heading mw-heading2"><h2 id="Information_gain">Information gain</h2></div>
<p>The key concept in the exploration problem is the notion of information gain, that is, the amount of knowledge acquired while pushing the frontiers. A probabilistic measure of information gain is defined by the entropy
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{p}(x)=-\int p(x)\log p(x)\,dx.}">
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<annotation encoding="application/x-tex">{\displaystyle H_{p}(x)=-\int p(x)\log p(x)\,dx.}</annotation>
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</math></span><img src="./b67ed0c45d62c17e1d417efa02835a4183b70710.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.946ex; height:5.676ex;" alt="{\displaystyle H_{p}(x)=-\int p(x)\log p(x)\,dx.}" loading="lazy"></span></dd></dl>
<p>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{p}(x)}">
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<annotation encoding="application/x-tex">{\displaystyle H_{p}(x)}</annotation>
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</math></span><img src="./014a574ccaa8f53d5696b5e799ba5577413bdfd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.129ex; height:3.009ex;" alt="{\displaystyle H_{p}(x)}" loading="lazy"></span> is maximized if <i>p</i> is a uniform distribution and minimized when <i>p</i> is a point mass distribution. By minimizing the <a href="Expected_value" title="Expected value">expected</a> entropy of belief, information gain is maximized as
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{b}(u)=H_{p}(x)-E_{z}\left[H_{b}(x'|z,u)\right].}">
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<annotation encoding="application/x-tex">{\displaystyle I_{b}(u)=H_{p}(x)-E_{z}\left[H_{b}(x'|z,u)\right].}</annotation>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Kidnapped_robot_problem" title="Kidnapped robot problem">Kidnapped robot problem</a></li>
<li><a href="Wake-up_robot_problem" title="Wake-up robot problem">Wake-up robot problem</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-Trun2005-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Trun2005_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFThrun,_S.Burgard,_W.Fox,_D.2005" class="citation book cs1"><a href="Sebastian_Thrun" title="Sebastian Thrun">Thrun, S.</a>; <a href="Wolfram_Burgard" title="Wolfram Burgard">Burgard, W.</a>; <a href="Dieter_Fox" title="Dieter Fox">Fox, D.</a> (2005). <i>Probabilistic Robotics</i>. Cambridge: <a href="MIT_Press" title="MIT Press">MIT Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-262-20162-9</bdi>.</cite></span>
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<li id="cite_note-Bares1999-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bares1999_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBares,_J.E.Wettergreen,_D.S.1999" class="citation journal cs1">Bares, J.E.; Wettergreen, D.S. (1999). "Dante II: Technical Description, Results, and Lessons Learned". <i>The International Journal of Robotics Research</i>. <b>18</b> (7): 621. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a>&nbsp;<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.8358">10.1.1.41.8358</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F02783649922066475">10.1177/02783649922066475</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&nbsp;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9772668">9772668</a>.</cite></span>
</li>
<li id="cite_note-Thrun2003-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Thrun2003_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFThrun,_S.Hahnel,_D.Ferguson,_D.Montemerlo,_M.2003" class="citation conference cs1">Thrun, S.; Hahnel, D.; Ferguson, D.; Montemerlo, M.; Triebel, R.; Burgard, W.; Baker, C.; Omohundro, Z.; Thayer, S.; Whittaker, W. (2003). "A system for volumetric robotic mapping of abandoned mines". <i>Robotics and Automation, 2003. Proceedings. ICRA'03. IEEE International Conference on</i>. Vol.&nbsp;3. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FROBOT.2003.1242260">10.1109/ROBOT.2003.1242260</a>.</cite></span>
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