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<article role="main">
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<section id="pendulum">
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||
<h1>Pendulum<a class="headerlink" href="#pendulum" title="Permalink 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>
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</figure>
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<p>This environment is part of the Classic Control environments which contains general information about the environment.</p>
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<div class="table-wrapper colwidths-auto docutils container">
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<table class="docutils align-default">
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<thead>
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<th class="head"><p></p></th>
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</thead>
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<tbody>
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<tr class="row-even"><td><p>Action Space</p></td>
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||
<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>
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</tr>
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||
<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>
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||
</tr>
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<tr class="row-even"><td><p>import</p></td>
|
||
<td><p><code class="docutils literal notranslate"><span class="pre">gymnasium.make("Pendulum-v1")</span></code></p></td>
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</tr>
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</tbody>
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</table>
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<section id="description">
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<h2>Description<a class="headerlink" href="#description" title="Permalink 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="Permalink 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="Permalink 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="Permalink 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">$</span>    <span class="pre">heta$</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="Permalink 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="Permalink 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="Permalink to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p><code class="docutils literal notranslate"><span class="pre">g</span></code>: acceleration of gravity measured in <em>(m s<sup>-2</sup>)</em> used to calculate the pendulum dynamics.
|
||
The default value is g = 10.0 .</p></li>
|
||
</ul>
|
||
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">gymnasium</span> <span class="k">as</span> <span class="nn">gym</span>
|
||
<span class="n">gym</span><span class="o">.</span><span class="n">make</span><span class="p">(</span><span class="s1">'Pendulum-v1'</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="mf">9.81</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>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>
|
||
</section>
|
||
<section id="version-history">
|
||
<h2>Version History<a class="headerlink" href="#version-history" title="Permalink 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 (1.0.0)</p></li>
|
||
</ul>
|
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<li><a class="reference internal" href="#">Pendulum</a><ul>
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<li><a class="reference internal" href="#action-space">Action Space</a></li>
|
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<li><a class="reference internal" href="#observation-space">Observation Space</a></li>
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