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<article role="main">
<section id="acrobot">
<h1>Acrobot<a class="headerlink" href="#acrobot" title="Permalink to this heading">#</a></h1>
<figure class="align-default" id="id1">
<a class="reference internal image-reference" href="../../../_images/acrobot.gif"><img alt="../../../_images/acrobot.gif" src="../../../_images/acrobot.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">Discrete(3)</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([</span> <span class="pre">-1.</span> <span class="pre">-1.</span> <span class="pre">-1.</span> <span class="pre">-1.</span> <span class="pre">-12.566371</span> <span class="pre">-28.274334],</span> <span class="pre">[</span> <span class="pre">1.</span> <span class="pre">1.</span> <span class="pre">1.</span> <span class="pre">1.</span> <span class="pre">12.566371</span> <span class="pre">28.274334],</span> <span class="pre">(6,),</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;Acrobot-v1&quot;)</span></code></p></td>
</tr>
</tbody>
</table>
</div>
<section id="description">
<h2>Description<a class="headerlink" href="#description" title="Permalink to this heading">#</a></h2>
<p>The Acrobot environment is based on Sutton’s work in
<a class="reference external" href="https://papers.nips.cc/paper/1995/hash/8f1d43620bc6bb580df6e80b0dc05c48-Abstract.html">“Generalization in Reinforcement Learning: Successful Examples Using Sparse Coarse Coding”</a>
and <a class="reference external" href="http://www.incompleteideas.net/book/the-book-2nd.html">Sutton and Barto’s book</a>.
The system consists of two links connected linearly to form a chain, with one end of
the chain fixed. The joint between the two links is actuated. The goal is to apply
torques on the actuated joint to swing the free end of the linear chain above a
given height while starting from the initial state of hanging downwards.</p>
<p>As seen in the <strong>Gif</strong>: two blue links connected by two green joints. The joint in
between the two links is actuated. The goal is to swing the free end of the outer-link
to reach the target height (black horizontal line above system) by applying torque on
the actuator.</p>
</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 discrete, deterministic, and represents the torque applied on the actuated
joint between the two links.</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>Unit</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>0</p></td>
<td><p>apply -1 torque to the actuated joint</p></td>
<td><p>torque (N m)</p></td>
</tr>
<tr class="row-odd"><td><p>1</p></td>
<td><p>apply 0 torque to the actuated joint</p></td>
<td><p>torque (N m)</p></td>
</tr>
<tr class="row-even"><td><p>2</p></td>
<td><p>apply 1 torque to the actuated joint</p></td>
<td><p>torque (N m)</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">(6,)</span></code> that provides information about the
two rotational joint angles as well as their angular velocities:</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>Cosine of <code class="docutils literal notranslate"><span class="pre">theta1</span></code></p></td>
<td><p>-1</p></td>
<td><p>1</p></td>
</tr>
<tr class="row-odd"><td><p>1</p></td>
<td><p>Sine of <code class="docutils literal notranslate"><span class="pre">theta1</span></code></p></td>
<td><p>-1</p></td>
<td><p>1</p></td>
</tr>
<tr class="row-even"><td><p>2</p></td>
<td><p>Cosine of <code class="docutils literal notranslate"><span class="pre">theta2</span></code></p></td>
<td><p>-1</p></td>
<td><p>1</p></td>
</tr>
<tr class="row-odd"><td><p>3</p></td>
<td><p>Sine of <code class="docutils literal notranslate"><span class="pre">theta2</span></code></p></td>
<td><p>-1</p></td>
<td><p>1</p></td>
</tr>
<tr class="row-even"><td><p>4</p></td>
<td><p>Angular velocity of <code class="docutils literal notranslate"><span class="pre">theta1</span></code></p></td>
<td><p>~ -12.567 (-4 * pi)</p></td>
<td><p>~ 12.567 (4 * pi)</p></td>
</tr>
<tr class="row-odd"><td><p>5</p></td>
<td><p>Angular velocity of <code class="docutils literal notranslate"><span class="pre">theta2</span></code></p></td>
<td><p>~ -28.274 (-9 * pi)</p></td>
<td><p>~ 28.274 (9 * pi)</p></td>
</tr>
</tbody>
</table>
</div>
<p>where</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">theta1</span></code> is the angle of the first joint, where an angle of 0 indicates the first link is pointing directly
downwards.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">theta2</span></code> is <em><strong>relative to the angle of the first link.</strong></em>
An angle of 0 corresponds to having the same angle between the two links.</p></li>
</ul>
<p>The angular velocities of <code class="docutils literal notranslate"><span class="pre">theta1</span></code> and <code class="docutils literal notranslate"><span class="pre">theta2</span></code> are bounded at ±4π, and ±9π rad/s respectively.
A state of <code class="docutils literal notranslate"><span class="pre">[1,</span> <span class="pre">0,</span> <span class="pre">1,</span> <span class="pre">0,</span> <span class="pre">...,</span> <span class="pre">...]</span></code> indicates that both links are pointing downwards.</p>
</section>
<section id="rewards">
<h2>Rewards<a class="headerlink" href="#rewards" title="Permalink to this heading">#</a></h2>
<p>The goal is to have the free end reach a designated target height in as few steps as possible,
and as such all steps that do not reach the goal incur a reward of -1.
Achieving the target height results in termination with a reward of 0. The reward threshold is -100.</p>
</section>
<section id="starting-state">
<h2>Starting State<a class="headerlink" href="#starting-state" title="Permalink to this heading">#</a></h2>
<p>Each parameter in the underlying state (<code class="docutils literal notranslate"><span class="pre">theta1</span></code>, <code class="docutils literal notranslate"><span class="pre">theta2</span></code>, and the two angular velocities) is initialized
uniformly between -0.1 and 0.1. This means both links are pointing downwards with some initial stochasticity.</p>
</section>
<section id="episode-end">
<h2>Episode End<a class="headerlink" href="#episode-end" title="Permalink to this heading">#</a></h2>
<p>The episode ends if one of the following occurs:</p>
<ol class="arabic simple">
<li><p>Termination: The free end reaches the target height, which is constructed as:
<code class="docutils literal notranslate"><span class="pre">-cos(theta1)</span> <span class="pre">-</span> <span class="pre">cos(theta2</span> <span class="pre">+</span> <span class="pre">theta1)</span> <span class="pre">&gt;</span> <span class="pre">1.0</span></code></p></li>
<li><p>Truncation: Episode length is greater than 500 (200 for v0)</p></li>
</ol>
</section>
<section id="arguments">
<h2>Arguments<a class="headerlink" href="#arguments" title="Permalink to this heading">#</a></h2>
<p>No additional arguments are currently supported during construction.</p>
<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">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="s1">&#39;Acrobot-v1&#39;</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>
<p>By default, the dynamics of the acrobot follow those described in Sutton and Barto’s book
<a class="reference external" href="http://incompleteideas.net/book/11/node4.html">Reinforcement Learning: An Introduction</a>.
However, a <code class="docutils literal notranslate"><span class="pre">book_or_nips</span></code> parameter can be modified to change the pendulum dynamics to those described
in the original <a class="reference external" href="https://papers.nips.cc/paper/1995/hash/8f1d43620bc6bb580df6e80b0dc05c48-Abstract.html">NeurIPS paper</a>.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># To change the dynamics as described above</span>
<span class="n">env</span><span class="o">.</span><span class="n">unwrapped</span><span class="o">.</span><span class="n">book_or_nips</span> <span class="o">=</span> <span class="s1">&#39;nips&#39;</span>
</pre></div>
</div>
<p>See the following note for details:</p>
<blockquote>
<div><p>The dynamics equations were missing some terms in the NIPS paper which
are present in the book. R. Sutton confirmed in personal correspondence
that the experimental results shown in the paper and the book were
generated with the equations shown in the book.
However, there is the option to run the domain with the paper equations
by setting <code class="docutils literal notranslate"><span class="pre">book_or_nips</span> <span class="pre">=</span> <span class="pre">'nips'</span></code></p>
</div></blockquote>
</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: Maximum number of steps increased from 200 to 500. The observation space for v0 provided direct readings of
<code class="docutils literal notranslate"><span class="pre">theta1</span></code> and <code class="docutils literal notranslate"><span class="pre">theta2</span></code> in radians, having a range of <code class="docutils literal notranslate"><span class="pre">[-pi,</span> <span class="pre">pi]</span></code>. The v1 observation space as described here provides the
sine and cosine of each angle instead.</p></li>
<li><p>v0: Initial versions release (1.0.0) (removed from gymnasium for v1)</p></li>
</ul>
</section>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Permalink to this heading">#</a></h2>
<ul class="simple">
<li><p>Sutton, R. S. (1996). Generalization in Reinforcement Learning: Successful Examples Using Sparse Coarse Coding.
In D. Touretzky, M. C. Mozer, &amp; M. Hasselmo (Eds.), Advances in Neural Information Processing Systems (Vol. 8).
MIT Press. https://proceedings.neurips.cc/paper/1995/file/8f1d43620bc6bb580df6e80b0dc05c48-Paper.pdf</p></li>
<li><p>Sutton, R. S., Barto, A. G. (2018 ). Reinforcement Learning: An Introduction. The MIT Press.</p></li>
</ul>
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