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<p class="caption" role="heading"><span class="caption-text">Introduction</span></p>
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<article role="main">
<section class="tex2jax_ignore mathjax_ignore" id="pendulum">
<h1>Pendulum<a class="headerlink" href="#pendulum" title="Link to this heading">#</a></h1>
<figure class="align-default" id="id1">
<a class="reference internal image-reference" href="../../../_images/pendulum.gif"><img alt="../../../_images/pendulum.gif" src="../../../_images/pendulum.gif" style="width: 200px;" /></a>
</figure>
<p>This environment is part of the Classic Control environments which contains general information about the environment.</p>
<div class="table-wrapper colwidths-auto docutils container">
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p></p></th>
<th class="head"><p></p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Action Space</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">Box(-2.0,</span> <span class="pre">2.0,</span> <span class="pre">(1,),</span> <span class="pre">float32)</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>Observation Space</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">Box([-1.</span> <span class="pre">-1.</span> <span class="pre">-8.],</span> <span class="pre">[1.</span> <span class="pre">1.</span> <span class="pre">8.],</span> <span class="pre">(3,),</span> <span class="pre">float32)</span></code></p></td>
</tr>
<tr class="row-even"><td><p>import</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">gymnasium.make(&quot;Pendulum-v1&quot;)</span></code></p></td>
</tr>
</tbody>
</table>
</div>
<section id="description">
<h2>Description<a class="headerlink" href="#description" title="Link to this heading">#</a></h2>
<p>The inverted pendulum swingup problem is based on the classic problem in control theory.
The system consists of a pendulum attached at one end to a fixed point, and the other end being free.
The pendulum starts in a random position and the goal is to apply torque on the free end to swing it
into an upright position, with its center of gravity right above the fixed point.</p>
<p>The diagram below specifies the coordinate system used for the implementation of the pendulum’s
dynamic equations.</p>
<p><img alt="Pendulum Coordinate System" src="../../../_images/pendulum.png" /></p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">x-y</span></code>: cartesian coordinates of the pendulum’s end in meters.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">theta</span></code> : angle in radians.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">tau</span></code>: torque in <code class="docutils literal notranslate"><span class="pre">N</span> <span class="pre">m</span></code>. Defined as positive <em>counter-clockwise</em>.</p></li>
</ul>
</section>
<section id="action-space">
<h2>Action Space<a class="headerlink" href="#action-space" title="Link to this heading">#</a></h2>
<p>The action is a <code class="docutils literal notranslate"><span class="pre">ndarray</span></code> with shape <code class="docutils literal notranslate"><span class="pre">(1,)</span></code> representing the torque applied to free end of the pendulum.</p>
<div class="table-wrapper colwidths-auto docutils container">
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>Num</p></th>
<th class="head"><p>Action</p></th>
<th class="head"><p>Min</p></th>
<th class="head"><p>Max</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>0</p></td>
<td><p>Torque</p></td>
<td><p>-2.0</p></td>
<td><p>2.0</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="observation-space">
<h2>Observation Space<a class="headerlink" href="#observation-space" title="Link to this heading">#</a></h2>
<p>The observation is a <code class="docutils literal notranslate"><span class="pre">ndarray</span></code> with shape <code class="docutils literal notranslate"><span class="pre">(3,)</span></code> representing the x-y coordinates of the pendulum’s free
end and its angular velocity.</p>
<div class="table-wrapper colwidths-auto docutils container">
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>Num</p></th>
<th class="head"><p>Observation</p></th>
<th class="head"><p>Min</p></th>
<th class="head"><p>Max</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>0</p></td>
<td><p>x = cos(theta)</p></td>
<td><p>-1.0</p></td>
<td><p>1.0</p></td>
</tr>
<tr class="row-odd"><td><p>1</p></td>
<td><p>y = sin(theta)</p></td>
<td><p>-1.0</p></td>
<td><p>1.0</p></td>
</tr>
<tr class="row-even"><td><p>2</p></td>
<td><p>Angular Velocity</p></td>
<td><p>-8.0</p></td>
<td><p>8.0</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="rewards">
<h2>Rewards<a class="headerlink" href="#rewards" title="Link to this heading">#</a></h2>
<p>The reward function is defined as:</p>
<p><em>r = -(theta<sup>2</sup> + 0.1 * theta_dt<sup>2</sup> + 0.001 * torque<sup>2</sup>)</em></p>
<p>where <code class="docutils literal notranslate"><span class="pre">theta</span></code> is the pendulum’s angle normalized between <em>[-pi, pi]</em> (with 0 being in the upright position).
Based on the above equation, the minimum reward that can be obtained is
<em>-(pi<sup>2</sup> + 0.1 * 8<sup>2</sup> + 0.001 * 2<sup>2</sup>) = -16.2736044</em>,
while the maximum reward is zero (pendulum is upright with zero velocity and no torque applied).</p>
</section>
<section id="starting-state">
<h2>Starting State<a class="headerlink" href="#starting-state" title="Link to this heading">#</a></h2>
<p>The starting state is a random angle in <em>[-pi, pi]</em> and a random angular velocity in <em>[-1,1]</em>.</p>
</section>
<section id="episode-truncation">
<h2>Episode Truncation<a class="headerlink" href="#episode-truncation" title="Link to this heading">#</a></h2>
<p>The episode truncates at 200 time steps.</p>
</section>
<section id="arguments">
<h2>Arguments<a class="headerlink" href="#arguments" title="Link to this heading">#</a></h2>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">g</span></code>: .</p></li>
</ul>
<p>Pendulum has two parameters for <code class="docutils literal notranslate"><span class="pre">gymnasium.make</span></code> with <code class="docutils literal notranslate"><span class="pre">render_mode</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span></code> representing
the acceleration of gravity measured in <em>(m s<sup>-2</sup>)</em> used to calculate the pendulum dynamics.
The default value is <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">=</span> <span class="pre">10.0</span></code>.
On reset, the <code class="docutils literal notranslate"><span class="pre">options</span></code> parameter allows the user to change the bounds used to determine the new random state.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">gymnasium</span> <span class="k">as</span> <span class="nn">gym</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">env</span> <span class="o">=</span> <span class="n">gym</span><span class="o">.</span><span class="n">make</span><span class="p">(</span><span class="s2">&quot;Pendulum-v1&quot;</span><span class="p">,</span> <span class="n">render_mode</span><span class="o">=</span><span class="s2">&quot;rgb_array&quot;</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="mf">9.81</span><span class="p">)</span> <span class="c1"># default g=10.0</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">env</span>
<span class="go">&lt;TimeLimit&lt;OrderEnforcing&lt;PassiveEnvChecker&lt;PendulumEnv&lt;Pendulum-v1&gt;&gt;&gt;&gt;&gt;</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">env</span><span class="o">.</span><span class="n">reset</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="mi">123</span><span class="p">,</span> <span class="n">options</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;low&quot;</span><span class="p">:</span> <span class="o">-</span><span class="mf">0.7</span><span class="p">,</span> <span class="s2">&quot;high&quot;</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">})</span> <span class="c1"># default low=-0.6, high=-0.5</span>
<span class="go">(array([ 0.4123625 , 0.91101986, -0.89235795], dtype=float32), {})</span>
</pre></div>
</div>
</section>
<section id="version-history">
<h2>Version History<a class="headerlink" href="#version-history" title="Link to this heading">#</a></h2>
<ul class="simple">
<li><p>v1: Simplify the math equations, no difference in behavior.</p></li>
<li><p>v0: Initial versions release</p></li>
</ul>
</section>
</section>
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d="M8 0C3.58 0 0 3.58 0 8c0 3.54 2.29 6.53 5.47 7.59.4.07.55-.17.55-.38 0-.19-.01-.82-.01-1.49-2.01.37-2.53-.49-2.69-.94-.09-.23-.48-.94-.82-1.13-.28-.15-.68-.52-.01-.53.63-.01 1.08.58 1.23.82.72 1.21 1.87.87 2.33.66.07-.52.28-.87.51-1.07-1.78-.2-3.64-.89-3.64-3.95 0-.87.31-1.59.82-2.15-.08-.2-.36-1.02.08-2.12 0 0 .67-.21 2.2.82.64-.18 1.32-.27 2-.27.68 0 1.36.09 2 .27 1.53-1.04 2.2-.82 2.2-.82.44 1.1.16 1.92.08 2.12.51.56.82 1.27.82 2.15 0 3.07-1.87 3.75-3.65 3.95.29.25.54.73.54 1.48 0 1.07-.01 1.93-.01 2.2 0 .21.15.46.55.38A8.013 8.013 0 0 0 16 8c0-4.42-3.58-8-8-8z">
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<aside class="toc-drawer">
<div class="toc-sticky toc-scroll">
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<span class="toc-title">
On this page
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<div class="toc-tree">
<ul>
<li><a class="reference internal" href="#">Pendulum</a><ul>
<li><a class="reference internal" href="#description">Description</a></li>
<li><a class="reference internal" href="#action-space">Action Space</a></li>
<li><a class="reference internal" href="#observation-space">Observation Space</a></li>
<li><a class="reference internal" href="#rewards">Rewards</a></li>
<li><a class="reference internal" href="#starting-state">Starting State</a></li>
<li><a class="reference internal" href="#episode-truncation">Episode Truncation</a></li>
<li><a class="reference internal" href="#arguments">Arguments</a></li>
<li><a class="reference internal" href="#version-history">Version History</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div>
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<script src="../../../_static/documentation_options.js?v=ed34540e"></script>
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<script src="../../../_static/scripts/furo.js?v=7660844c"></script>
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link: "https://farama.org/about"
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link: "https://farama.org/project_standards",
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name: "Donate",
link: "https://farama.org/donations"
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};
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// innerHtml doenst evaluate scripts, we need to add them dynamically
Array.from(container.querySelectorAll("script")).forEach(oldScript => {
const newScript = document.createElement("script");
Array.from(oldScript.attributes).forEach(attr => newScript.setAttribute(attr.name, attr.value));
newScript.appendChild(document.createTextNode(oldScript.innerHTML));
oldScript.parentNode.replaceChild(newScript, oldScript);
});
});
} else {
console.warn("Unable to load versioning menu", response);
}
});
</script>
</body>
</html>
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