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<section id="quaternion">
<h1>Quaternion<a class="headerlink" href="#quaternion" title="Permalink to this heading"></a></h1>
<dl class="py class">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Quaternion</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">s</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion" title="Permalink to this definition"></a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">BasePoseList</span></code></p>
<p>Quaternion class</p>
<p>A quaternion can be considered an ordered pair <span class="math notranslate nohighlight">\((s, \vec{v})\)</span>
where <span class="math notranslate nohighlight">\(s \in \mathbb{R}\)</span> is the <em>scalar</em> part and <span class="math notranslate nohighlight">\(\vec{v} = (v_x, v_y, v_z) \in \mathbb{R}^3\)</span>
is the <em>vector</em> part and is often written as</p>
<div class="math notranslate nohighlight">
\[\q = s \langle v_x, v_y, v_z \rangle\]</div>
<div class="graphviz"><img src="_images/inheritance-87601fd326d188805dd4e5ecf14b3e32af162805.png" alt="Inheritance diagram of spatialmath.quaternion.Quaternion" usemap="#inheritancefc2bec1728" class="inheritance graphviz" /></div>
<map id="inheritancefc2bec1728" name="inheritancefc2bec1728">
<area shape="rect" id="node4" href="#spatialmath.quaternion.Quaternion" target="_top" title="Quaternion class" alt="" coords="284,31,379,56"/>
</map><dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.Alloc">
<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Alloc</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#spatialmath.quaternion.Quaternion.Alloc" title="Permalink to this definition"></a></dt>
<dd><p>Construct an instance with N default values (BasePoseList superclass method)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>n</strong> (<em>int</em><em>, </em><em>optional</em>) – Number of values, defaults to 1</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p><span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">Self</span></code></span></p>
</dd>
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>pose instance with <code class="docutils literal notranslate"><span class="pre">n</span></code> default values</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">X.Alloc(N)</span></code> creates an instance of the pose class <code class="docutils literal notranslate"><span class="pre">X</span></code> with <code class="docutils literal notranslate"><span class="pre">N</span></code>
default values, ie. <code class="docutils literal notranslate"><span class="pre">len(X)</span></code> will be <code class="docutils literal notranslate"><span class="pre">N</span></code>.</p>
<p><code class="docutils literal notranslate"><span class="pre">X</span></code> can be considered a vector of pose objects, and those elements
can be referenced <code class="docutils literal notranslate"><span class="pre">X[i]</span></code> or assigned to <code class="docutils literal notranslate"><span class="pre">X[i]</span> <span class="pre">=</span> <span class="pre">...</span></code>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The default value depends on the pose class and is the result
of the empty constructor. For <code class="docutils literal notranslate"><span class="pre">SO2</span></code>,
<code class="docutils literal notranslate"><span class="pre">SE2</span></code>, <code class="docutils literal notranslate"><span class="pre">SO3</span></code>, <code class="docutils literal notranslate"><span class="pre">SE3</span></code> it is an identity matrix, for a
twist class <code class="docutils literal notranslate"><span class="pre">Twist2</span></code> or <code class="docutils literal notranslate"><span class="pre">Twist3</span></code> it is a zero vector,
for a <code class="docutils literal notranslate"><span class="pre">UnitQuaternion</span></code> or <code class="docutils literal notranslate"><span class="pre">Quaternion</span></code> it is a zero
vector.</p>
</div>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">Alloc</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">10</span>
</pre></div>
</div>
<p>where <code class="docutils literal notranslate"><span class="pre">X</span></code> is any of the SMTB classes.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.Empty">
<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Empty</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#spatialmath.quaternion.Quaternion.Empty" title="Permalink to this definition"></a></dt>
<dd><p>Construct an empty instance (BasePoseList superclass method)</p>
<dl class="field-list simple">
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">Self</span></code></span></p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>pose instance with zero values</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">Empty</span><span class="p">()</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">0</span>
</pre></div>
</div>
<p>where <code class="docutils literal notranslate"><span class="pre">X</span></code> is any of the SMTB classes.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.Pure">
<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Pure</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">v</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.Pure"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.Pure" title="Permalink to this definition"></a></dt>
<dd><p>Construct a pure quaternion from a vector</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>v</strong> (<em>3-element array_like</em>) – vector</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p><span class="sphinx_autodoc_typehints-type"><a class="reference internal" href="#spatialmath.quaternion.Quaternion" title="spatialmath.quaternion.Quaternion"><code class="xref py py-class docutils literal notranslate"><span class="pre">Quaternion</span></code></a></span></p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">Quaternion.Pure(v)</span></code> is a Quaternion with a zero scalar part and the
vector part set to <code class="docutils literal notranslate"><span class="pre">v</span></code>,
ie. <span class="math notranslate nohighlight">\(q = 0 \langle v_x, v_y, v_z \rangle\)</span></p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">Quaternion</span><span class="o">.</span><span class="n">Pure</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]))</span>
<span class="go"> 0.0000 < 1.0000, 2.0000, 3.0000 ></span>
</pre></div>
</div>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__add__">
<span class="sig-name descname"><span class="pre">__add__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">right</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__add__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__add__" title="Permalink to this definition"></a></dt>
<dd><p>Overloaded <code class="docutils literal notranslate"><span class="pre">+</span></code> operator</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>sum</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.quaternion.Quaternion" title="spatialmath.quaternion.Quaternion">Quaternion</a>, <a class="reference internal" href="3d_orient_unitquaternion.html#spatialmath.quaternion.UnitQuaternion" title="spatialmath.quaternion.UnitQuaternion">UnitQuaternion</a></p>
</dd>
<dt class="field-odd">Raises<span class="colon">:</span></dt>
<dd class="field-odd"><p>ValueError</p>
</dd>
</dl>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head" colspan="2"><p>Operands</p></th>
<th class="head" colspan="2"><p>Sum</p></th>
</tr>
<tr class="row-even"><th class="head"><p>left</p></th>
<th class="head"><p>right</p></th>
<th class="head"><p>type</p></th>
<th class="head"><p>result</p></th>
</tr>
</thead>
<tbody>
<tr class="row-odd"><td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-even"><td><p>Quaternion</p></td>
<td><p>UnitQuaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-odd"><td><p>Quaternion</p></td>
<td><p>scalar</p></td>
<td><p>Quaternion</p></td>
<td><p>add to each element</p></td>
</tr>
<tr class="row-even"><td><p>UnitQuaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-odd"><td><p>UnitQuaternion</p></td>
<td><p>UnitQuaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-even"><td><p>UnitQuaternion</p></td>
<td><p>scalar</p></td>
<td><p>Quaternion</p></td>
<td><p>add to each element</p></td>
</tr>
</tbody>
</table>
<p>Any other input combinations result in a ValueError.</p>
<p>Note that left and right can have a length greater than 1 in which case:</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>left</p></th>
<th class="head"><p>right</p></th>
<th class="head"><p>len</p></th>
<th class="head"><p>operation</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>1</p></td>
<td><p>1</p></td>
<td><p>1</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">sum</span> <span class="pre">=</span> <span class="pre">left</span> <span class="pre">+</span> <span class="pre">right</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>1</p></td>
<td><p>N</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">sum[i]</span> <span class="pre">=</span> <span class="pre">left</span> <span class="pre">+</span> <span class="pre">right[i]</span></code></p></td>
</tr>
<tr class="row-even"><td><p>N</p></td>
<td><p>1</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">sum[i]</span> <span class="pre">=</span> <span class="pre">left[i]</span> <span class="pre">+</span> <span class="pre">right</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>N</p></td>
<td><p>N</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">sum[i]</span> <span class="pre">=</span> <span class="pre">left[i]</span> <span class="pre">+</span> <span class="pre">right[i]</span></code></p></td>
</tr>
<tr class="row-even"><td><p>N</p></td>
<td><p>M</p></td>
<td><ul class="simple">
<li></li>
</ul>
</td>
<td><p><code class="docutils literal notranslate"><span class="pre">ValueError</span></code></p></td>
</tr>
</tbody>
</table>
<p>A scalar of length N is a list, tuple or numpy array.
A 3-vector of length N is a 3xN numpy array, where each column is a 3-vector.</p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="o">+</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">])</span>
<span class="go">Quaternion(array([ 6, 8, 10, 12]))</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">+</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="go">Quaternion([</span>
<span class="go"> array([2, 4, 6, 8]),</span>
<span class="go"> array([ 6, 8, 10, 12]) ])</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">+</span> <span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="go">Quaternion([</span>
<span class="go"> array([2, 4, 6, 8]),</span>
<span class="go"> array([10, 12, 14, 16]) ])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__eq__">
<span class="sig-name descname"><span class="pre">__eq__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">right</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__eq__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__eq__" title="Permalink to this definition"></a></dt>
<dd><p>Overloaded <code class="docutils literal notranslate"><span class="pre">==</span></code> operator</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>Equality of two operands</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p>bool or list of bool</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">q1</span> <span class="pre">==</span> <span class="pre">q2</span></code> is True if <code class="docutils literal notranslate"><span class="pre">q1`</span> <span class="pre">is</span> <span class="pre">elementwise</span> <span class="pre">equal</span> <span class="pre">to</span> <span class="pre">``q2</span></code>.</p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">=</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">q2</span> <span class="o">=</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">==</span> <span class="n">q1</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">==</span> <span class="n">q2</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">==</span> <span class="n">q1</span>
<span class="go">[True, False]</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">==</span> <span class="n">q2</span>
<span class="go">[False, True]</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">==</span> <span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="go">[True, True]</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="#spatialmath.quaternion.Quaternion.__ne__" title="spatialmath.quaternion.Quaternion.__ne__"><code class="xref py py-func docutils literal notranslate"><span class="pre">__ne__()</span></code></a> <a class="reference internal" href="func_quat.html#spatialmath.base.quaternions.qisequal" title="spatialmath.base.quaternions.qisequal"><code class="xref py py-func docutils literal notranslate"><span class="pre">qisequal()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__init__">
<span class="sig-name descname"><span class="pre">__init__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">s</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__init__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__init__" title="Permalink to this definition"></a></dt>
<dd><p>Construct a new quaternion</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>s</strong> (<em>float</em><em> or </em><em>ndarray</em><em>(</em><em>N</em><em>)</em>) – scalar part</p></li>
<li><p><strong>v</strong> (<em>ndarray</em><em>(</em><em>3</em><em>)</em><em>, </em><em>ndarray</em><em>(</em><em>Nx3</em><em>)</em>) – vector part</p></li>
</ul>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">Quaternion()</span></code> constructs a zero quaternion</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">Quaternion(s,</span> <span class="pre">v)</span></code> construct a new quaternion from the scalar <code class="docutils literal notranslate"><span class="pre">s</span></code>
and the vector <code class="docutils literal notranslate"><span class="pre">v</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">Quaternion(q)</span></code> construct a new quaternion from the 4-vector
<code class="docutils literal notranslate"><span class="pre">q</span> <span class="pre">=</span> <span class="pre">[s,</span> <span class="pre">v]</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">Quaternion([q1,</span> <span class="pre">q2</span> <span class="pre">..</span> <span class="pre">qN])</span></code> construct a new quaternion with <code class="docutils literal notranslate"><span class="pre">N</span></code>
values where each element is a 4-vector</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">Quaternion([Q1,</span> <span class="pre">Q2</span> <span class="pre">..</span> <span class="pre">QN])</span></code> construct a new quaternion with <code class="docutils literal notranslate"><span class="pre">N</span></code>
values where each element is a Quaternion instance</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">Quaternion(M)</span></code> construct a new quaternion with <code class="docutils literal notranslate"><span class="pre">N</span></code> values where
<code class="docutils literal notranslate"><span class="pre">Q</span></code> is a 4xN NumPy array.</p></li>
</ul>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">()</span>
<span class="go">Quaternion(array([0., 0., 0., 0.]))</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="go">Quaternion(array([1., 2., 3., 4.]))</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="go">Quaternion(array([1, 2, 3, 4]))</span>
<span class="gp">>>> </span><span class="n">q</span><span class="o">=</span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="go">2</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="go"> 1.0000 < 2.0000, 3.0000, 4.0000 ></span>
<span class="go"> 5.0000 < 6.0000, 7.0000, 8.0000 ></span>
</pre></div>
</div>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__mul__">
<span class="sig-name descname"><span class="pre">__mul__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">right</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__mul__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__mul__" title="Permalink to this definition"></a></dt>
<dd><p>Overloaded <code class="docutils literal notranslate"><span class="pre">*</span></code> operator</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>product</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.quaternion.Quaternion" title="spatialmath.quaternion.Quaternion">Quaternion</a></p>
</dd>
<dt class="field-odd">Raises<span class="colon">:</span></dt>
<dd class="field-odd"><p>ValueError</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">q1</span> <span class="pre">*</span> <span class="pre">q2</span></code> is the Hamilton product of two quaternions</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">q</span> <span class="pre">*</span> <span class="pre">s</span></code> is the scalar product, where <code class="docutils literal notranslate"><span class="pre">s</span></code> is a scalar</p></li>
</ul>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head" colspan="2"><p>Multiplicands</p></th>
<th class="head" colspan="2"><p>Product</p></th>
</tr>
<tr class="row-even"><th class="head"><p>left</p></th>
<th class="head"><p>right</p></th>
<th class="head"><p>type</p></th>
<th class="head"><p>result</p></th>
</tr>
</thead>
<tbody>
<tr class="row-odd"><td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>Hamilton product</p></td>
</tr>
<tr class="row-even"><td><p>Quaternion</p></td>
<td><p>UnitQuaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>Hamilton product</p></td>
</tr>
<tr class="row-odd"><td><p>Quaternion</p></td>
<td><p>scalar</p></td>
<td><p>Quaternion</p></td>
<td><p>scalar product</p></td>
</tr>
</tbody>
</table>
<p>Any other input combinations result in a ValueError.</p>
<p>Note that left and right can have a length greater than 1 in which case:</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>left</p></th>
<th class="head"><p>right</p></th>
<th class="head"><p>len</p></th>
<th class="head"><p>operation</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>1</p></td>
<td><p>1</p></td>
<td><p>1</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">prod</span> <span class="pre">=</span> <span class="pre">left</span> <span class="pre">*</span> <span class="pre">right</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>1</p></td>
<td><p>N</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">prod[i]</span> <span class="pre">=</span> <span class="pre">left</span> <span class="pre">*</span> <span class="pre">right[i]</span></code></p></td>
</tr>
<tr class="row-even"><td><p>N</p></td>
<td><p>1</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">prod[i]</span> <span class="pre">=</span> <span class="pre">left[i]</span> <span class="pre">*</span> <span class="pre">right</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>N</p></td>
<td><p>N</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">prod[i]</span> <span class="pre">=</span> <span class="pre">left[i]</span> <span class="pre">*</span> <span class="pre">right[i]</span></code></p></td>
</tr>
<tr class="row-even"><td><p>N</p></td>
<td><p>M</p></td>
<td><ul class="simple">
<li></li>
</ul>
</td>
<td><p><code class="docutils literal notranslate"><span class="pre">ValueError</span></code></p></td>
</tr>
</tbody>
</table>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="o">*</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">])</span>
<span class="go">Quaternion(array([-60., 12., 30., 24.]))</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="o">*</span> <span class="mi">2</span>
<span class="go">Quaternion(array([2, 4, 6, 8]))</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">*</span> <span class="mi">2</span>
<span class="go">Quaternion([</span>
<span class="go"> array([2, 4, 6, 8]),</span>
<span class="go"> array([10, 12, 14, 16]) ])</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">*</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="go">Quaternion([</span>
<span class="go"> array([-28., 4., 6., 8.]),</span>
<span class="go"> array([-60., 20., 14., 32.]) ])</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="o">*</span> <span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="go">Quaternion([</span>
<span class="go"> array([-28., 4., 6., 8.]),</span>
<span class="go"> array([-60., 12., 30., 24.]) ])</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">*</span> <span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="go">Quaternion([</span>
<span class="go"> array([-28., 4., 6., 8.]),</span>
<span class="go"> array([-124., 60., 70., 80.]) ])</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p><code class="xref py py-func docutils literal notranslate"><span class="pre">__rmul__()</span></code> <code class="xref py py-func docutils literal notranslate"><span class="pre">__imul__()</span></code> <a class="reference internal" href="func_quat.html#spatialmath.base.quaternions.qqmul" title="spatialmath.base.quaternions.qqmul"><code class="xref py py-func docutils literal notranslate"><span class="pre">qqmul()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__ne__">
<span class="sig-name descname"><span class="pre">__ne__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">right</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__ne__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__ne__" title="Permalink to this definition"></a></dt>
<dd><p>Overloaded <code class="docutils literal notranslate"><span class="pre">!=</span></code> operator</p>
<dl class="field-list simple">
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>bool</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">q1</span> <span class="pre">!=</span> <span class="pre">q2</span></code> is True if <code class="docutils literal notranslate"><span class="pre">q`</span> <span class="pre">is</span> <span class="pre">elementwise</span> <span class="pre">not</span> <span class="pre">equal</span> <span class="pre">to</span> <span class="pre">``q2</span></code>.</p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">=</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">q2</span> <span class="o">=</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">!=</span> <span class="n">q1</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">!=</span> <span class="n">q2</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">!=</span> <span class="n">q1</span>
<span class="go">[False, True]</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">!=</span> <span class="n">q2</span>
<span class="go">[True, False]</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="#spatialmath.quaternion.Quaternion.__ne__" title="spatialmath.quaternion.Quaternion.__ne__"><code class="xref py py-func docutils literal notranslate"><span class="pre">__ne__()</span></code></a> <a class="reference internal" href="func_quat.html#spatialmath.base.quaternions.qisequal" title="spatialmath.base.quaternions.qisequal"><code class="xref py py-func docutils literal notranslate"><span class="pre">qisequal()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__pow__">
<span class="sig-name descname"><span class="pre">__pow__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__pow__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__pow__" title="Permalink to this definition"></a></dt>
<dd><p>Overloaded <code class="docutils literal notranslate"><span class="pre">**</span></code> operator</p>
<dl class="field-list simple">
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>Quaternion instance</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">q</span> <span class="pre">**</span> <span class="pre">N</span></code> computes the product of <code class="docutils literal notranslate"><span class="pre">q</span></code> with itself <code class="docutils literal notranslate"><span class="pre">N-1</span></code> times, where <code class="docutils literal notranslate"><span class="pre">N</span></code> must be
an integer. If <a href="#id1"><span class="problematic" id="id2">``</span></a>N``<0 the result is conjugated.</p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">-28.0000 < 4.0000, 6.0000, 8.0000 ></span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="o">**</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go"> 1.0000 < -2.0000, -3.0000, -4.0000 ></span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">-28.0000 < 4.0000, 6.0000, 8.0000 ></span>
<span class="go">-124.0000 < 60.0000, 70.0000, 80.0000 ></span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="func_quat.html#spatialmath.base.quaternions.qpow" title="spatialmath.base.quaternions.qpow"><code class="xref py py-func docutils literal notranslate"><span class="pre">qpow()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__sub__">
<span class="sig-name descname"><span class="pre">__sub__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">right</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__sub__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__sub__" title="Permalink to this definition"></a></dt>
<dd><p>Overloaded <code class="docutils literal notranslate"><span class="pre">-</span></code> operator</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>difference</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.quaternion.Quaternion" title="spatialmath.quaternion.Quaternion">Quaternion</a>, <a class="reference internal" href="3d_orient_unitquaternion.html#spatialmath.quaternion.UnitQuaternion" title="spatialmath.quaternion.UnitQuaternion">UnitQuaternion</a></p>
</dd>
<dt class="field-odd">Raises<span class="colon">:</span></dt>
<dd class="field-odd"><p>ValueError</p>
</dd>
</dl>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head" colspan="2"><p>Operands</p></th>
<th class="head" colspan="2"><p>Difference</p></th>
</tr>
<tr class="row-even"><th class="head"><p>left</p></th>
<th class="head"><p>right</p></th>
<th class="head"><p>type</p></th>
<th class="head"><p>result</p></th>
</tr>
</thead>
<tbody>
<tr class="row-odd"><td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-even"><td><p>Quaternion</p></td>
<td><p>UnitQuaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-odd"><td><p>Quaternion</p></td>
<td><p>scalar</p></td>
<td><p>Quaternion</p></td>
<td><p>subtract from each element</p></td>
</tr>
<tr class="row-even"><td><p>UnitQuaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-odd"><td><p>UnitQuaternion</p></td>
<td><p>UnitQuaternion</p></td>
<td><p>Quaternion</p></td>
<td><p>elementwise sum</p></td>
</tr>
<tr class="row-even"><td><p>UnitQuaternion</p></td>
<td><p>scalar</p></td>
<td><p>Quaternion</p></td>
<td><p>subtract from each element</p></td>
</tr>
</tbody>
</table>
<p>Any other input combinations result in a ValueError.</p>
<p>Note that left and right can have a length greater than 1 in which case:</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>left</p></th>
<th class="head"><p>right</p></th>
<th class="head"><p>len</p></th>
<th class="head"><p>operation</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>1</p></td>
<td><p>1</p></td>
<td><p>1</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">diff</span> <span class="pre">=</span> <span class="pre">left</span> <span class="pre">-</span> <span class="pre">right</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>1</p></td>
<td><p>N</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">diff[i]</span> <span class="pre">=</span> <span class="pre">left</span> <span class="pre">-</span> <span class="pre">right[i]</span></code></p></td>
</tr>
<tr class="row-even"><td><p>N</p></td>
<td><p>1</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">diff[i]</span> <span class="pre">=</span> <span class="pre">left[i]</span> <span class="pre">-</span> <span class="pre">right</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>N</p></td>
<td><p>N</p></td>
<td><p>N</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">diff[i]</span> <span class="pre">=</span> <span class="pre">left[i]</span> <span class="pre">-</span> <span class="pre">right[i]</span></code></p></td>
</tr>
<tr class="row-even"><td><p>N</p></td>
<td><p>M</p></td>
<td><ul class="simple">
<li></li>
</ul>
</td>
<td><p><code class="docutils literal notranslate"><span class="pre">ValueError</span></code></p></td>
</tr>
</tbody>
</table>
<p>A scalar of length N is a list, tuple or numpy array.
A 3-vector of length N is a 3xN numpy array, where each column is a 3-vector.</p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="o">-</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">])</span>
<span class="go">Quaternion(array([-4, -4, -4, -4]))</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">-</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="go">Quaternion([</span>
<span class="go"> array([0, 0, 0, 0]),</span>
<span class="go"> array([4, 4, 4, 4]) ])</span>
<span class="gp">>>> </span><span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="o">-</span> <span class="n">Quaternion</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="go">Quaternion([</span>
<span class="go"> array([0, 0, 0, 0]),</span>
<span class="go"> array([0, 0, 0, 0]) ])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.__truediv__">
<span class="sig-name descname"><span class="pre">__truediv__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.__truediv__"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.__truediv__" title="Permalink to this definition"></a></dt>
<dd></dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.append">
<span class="sig-name descname"><span class="pre">append</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">item</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#spatialmath.quaternion.Quaternion.append" title="Permalink to this definition"></a></dt>
<dd><p>Append a value to an instance (BasePoseList superclass method)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>x</strong> (<a class="reference internal" href="#spatialmath.quaternion.Quaternion" title="spatialmath.quaternion.Quaternion"><em>Quaternion</em></a><em> or </em><em>UnitQuaternion instance</em>) – the value to append</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – incorrect type of appended object</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><span class="sphinx_autodoc_typehints-type"><code class="xref py py-obj docutils literal notranslate"><span class="pre">None</span></code></span></p>
</dd>
</dl>
<p>Appends the argument to the object’s internal list of values.</p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">Alloc</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">10</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">())</span> <span class="c1"># append to the list</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">11</span>
</pre></div>
</div>
<p>where <code class="docutils literal notranslate"><span class="pre">X</span></code> is any of the SMTB classes.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.arghandler">
<span class="sig-name descname"><span class="pre">arghandler</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">arg</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">convertfrom</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">()</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#spatialmath.quaternion.Quaternion.arghandler" title="Permalink to this definition"></a></dt>
<dd><p>Standard constructor support (BasePoseList superclass method)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>arg</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-data docutils literal notranslate"><span class="pre">Any</span></code></span>) – initial value</p></li>
<li><p><strong>convertfrom</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-data docutils literal notranslate"><span class="pre">Tuple</span></code></span>) – list of classes to accept and convert from</p></li>
<li><p><strong>check</strong> (<em>bool</em>) – check value is valid, defaults to True</p></li>
</ul>
</dd>
<dt class="field-even">Type<span class="colon">:</span></dt>
<dd class="field-even"><p>tuple of typles</p>
</dd>
<dt class="field-odd">Raises<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>ValueError</strong> – bad type passed</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p><span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">bool</span></code></span></p>
</dd>
</dl>
<p>The value <code class="docutils literal notranslate"><span class="pre">arg</span></code> can be any of:</p>
<ol class="arabic simple">
<li><p>None, an identity value is created</p></li>
<li><p>a numpy.ndarray of the appropriate shape and value which is valid for the subclass</p></li>
<li><p>a list whose elements all meet the criteria above</p></li>
<li><p>an instance of the subclass</p></li>
<li><p>a list whose elements are all singelton instances of the subclass</p></li>
</ol>
<p>For cases 2 and 3, a NumPy array or a list of NumPy array is passed.
Each NumPyarray is tested for validity (if <code class="docutils literal notranslate"><span class="pre">check</span></code> is False a cursory
check of shape is made, if <code class="docutils literal notranslate"><span class="pre">check</span></code> is True the numerical value is
inspected) and converted to the required internal format by the
<code class="docutils literal notranslate"><span class="pre">_import</span></code> method. The default <code class="docutils literal notranslate"><span class="pre">_import</span></code> method calls the <code class="docutils literal notranslate"><span class="pre">isvalid</span></code>
method for checking. This mechanism allows equivalent forms to be
passed, ie. 6x1 or 4x4 for an se(3).</p>
<p>If <code class="docutils literal notranslate"><span class="pre">self</span></code> is an instance of class <code class="docutils literal notranslate"><span class="pre">A</span></code>, and an instance of class
<code class="docutils literal notranslate"><span class="pre">B</span></code> is passed and <code class="docutils literal notranslate"><span class="pre">B</span></code> is an element of the <code class="docutils literal notranslate"><span class="pre">convertfrom</span></code> argument,
then <code class="docutils literal notranslate"><span class="pre">B.A()</span></code> will be invoked to perform the type conversion.</p>
<p>Examples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">SE3</span><span class="p">()</span>
<span class="n">SE3</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
<span class="n">SE3</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">)])</span>
<span class="n">SE3</span><span class="p">(</span><span class="n">SE3</span><span class="p">())</span>
<span class="n">SE3</span><span class="p">([</span><span class="n">SE3</span><span class="p">(),</span> <span class="n">SE3</span><span class="p">()])</span>
<span class="n">Twist3</span><span class="p">(</span><span class="n">SE3</span><span class="p">())</span>
</pre></div>
</div>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.binop">
<span class="sig-name descname"><span class="pre">binop</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">right</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">op</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">op2</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">list1</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#spatialmath.quaternion.Quaternion.binop" title="Permalink to this definition"></a></dt>
<dd><p>Perform binary operation</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>left</strong> (<em>BasePoseList subclass</em>) – left operand</p></li>
<li><p><strong>right</strong> (<em>BasePoseList subclass</em><em>, </em><em>scalar</em><em> or </em><em>array</em>) – right operand</p></li>
<li><p><strong>op</strong> (<em>callable</em>) – binary operation</p></li>
<li><p><strong>op2</strong> (<em>callable</em>) – binary operation</p></li>
<li><p><strong>list1</strong> (<em>bool</em>) – return single array as a list, default True</p></li>
</ul>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – arguments are not compatible</p>
</dd>
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>list of values</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p>list</p>
</dd>
</dl>
<p>The is a helper method for implementing binary operation with overloaded
operators such as <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">*</span> <span class="pre">Y</span></code> where <code class="docutils literal notranslate"><span class="pre">X</span></code> and <code class="docutils literal notranslate"><span class="pre">Y</span></code> are both subclasses
of <code class="docutils literal notranslate"><span class="pre">BasePoseList</span></code>. Each operand has a list of one or more
values and this methods computes a list of result values according to:</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head" colspan="2"><p>Inputs</p></th>
<th class="head" colspan="2"><p>Output</p></th>
</tr>
<tr class="row-even"><th class="head"><p>len(left)</p></th>
<th class="head"><p>len(right)</p></th>
<th class="head"><p>len</p></th>
<th class="head"><p>operation</p></th>
</tr>
</thead>
<tbody>
<tr class="row-odd"><td><p>1</p></td>
<td><p>1</p></td>
<td><p>1</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">ret</span> <span class="pre">=</span> <span class="pre">op(left,</span> <span class="pre">right)</span></code></p></td>
</tr>
<tr class="row-even"><td><p>1</p></td>
<td><p>M</p></td>
<td><p>M</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">ret[i]</span> <span class="pre">=</span> <span class="pre">op(left,</span> <span class="pre">right[i])</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>M</p></td>
<td><p>1</p></td>
<td><p>M</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">ret[i]</span> <span class="pre">=</span> <span class="pre">op(left[i],</span> <span class="pre">right)</span></code></p></td>
</tr>
<tr class="row-even"><td><p>M</p></td>
<td><p>M</p></td>
<td><p>M</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">ret[i]</span> <span class="pre">=</span> <span class="pre">op(left[i],</span> <span class="pre">right[i])</span></code></p></td>
</tr>
</tbody>
</table>
<p>The arguments to <code class="docutils literal notranslate"><span class="pre">op</span></code> are the internal numeric values, ie. as returned
by the <code class="docutils literal notranslate"><span class="pre">._A</span></code> property.</p>
<p>The result is always a list, except for the first case above and
<code class="docutils literal notranslate"><span class="pre">list1</span></code> is <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p>
<p>If the right operand is not a <code class="docutils literal notranslate"><span class="pre">BasePoseList</span></code> subclass, but is a numeric
scalar or array then then <code class="docutils literal notranslate"><span class="pre">op2</span></code> is invoked</p>
<p>For example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">X</span><span class="o">.</span><span class="n">_binop</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
</pre></div>
</div>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>Input</p></th>
<th class="head" colspan="2"><p>Output</p></th>
</tr>
<tr class="row-even"><th class="head"><p>len(left)</p></th>
<th class="head"><p>len</p></th>
<th class="head"><p>operation</p></th>
</tr>
</thead>
<tbody>
<tr class="row-odd"><td><p>1</p></td>
<td><p>1</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">ret</span> <span class="pre">=</span> <span class="pre">op2(left,</span> <span class="pre">right)</span></code></p></td>
</tr>
<tr class="row-even"><td><p>M</p></td>
<td><p>M</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">ret[i]</span> <span class="pre">=</span> <span class="pre">op2(left[i],</span> <span class="pre">right)</span></code></p></td>
</tr>
</tbody>
</table>
<p>There is no check on the shape of <code class="docutils literal notranslate"><span class="pre">right</span></code> if it is an array.
The result is always a list, except for the first case above and
<code class="docutils literal notranslate"><span class="pre">list1</span></code> is <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.clear">
<span class="sig-name descname"><span class="pre">clear</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><span class="pre">None</span> <span class="pre">--</span> <span class="pre">remove</span> <span class="pre">all</span> <span class="pre">items</span> <span class="pre">from</span> <span class="pre">S</span></span></span><a class="headerlink" href="#spatialmath.quaternion.Quaternion.clear" title="Permalink to this definition"></a></dt>
<dd></dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.conj">
<span class="sig-name descname"><span class="pre">conj</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.conj"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.conj" title="Permalink to this definition"></a></dt>
<dd><p>Conjugate of quaternion</p>
<dl class="field-list simple">
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>Quaternion instance</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">q.conj()</span></code> is the quaternion <code class="docutils literal notranslate"><span class="pre">q</span></code> with the vector part negated, ie.
<span class="math notranslate nohighlight">\(q = s \langle -v_x, -v_y, -v_z \rangle\)</span></p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">Quaternion</span><span class="o">.</span><span class="n">Pure</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span><span class="o">.</span><span class="n">conj</span><span class="p">())</span>
<span class="go"> 0.0000 < -1.0000, -2.0000, -3.0000 ></span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="func_quat.html#spatialmath.base.quaternions.qconj" title="spatialmath.base.quaternions.qconj"><code class="xref py py-func docutils literal notranslate"><span class="pre">qconj()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.exp">
<span class="sig-name descname"><span class="pre">exp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">20</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.exp"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.exp" title="Permalink to this definition"></a></dt>
<dd><p>Exponential of quaternion</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>tol</strong> (<em>float</em><em>, </em><em>optional</em>) – Tolerance when checking for pure quaternion, in multiples of eps, defaults to 20</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p>Quaternion instance</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">q.exp()</span></code> is the exponential of the quaternion <code class="docutils literal notranslate"><span class="pre">q</span></code>, ie.</p>
<div class="math notranslate nohighlight">
\[e^s \cos \| v \|, \langle e^s \frac{\vec{v}}{\| \vec{v} \|} \sin \| \vec{v} \| \rangle\]</div>
<p>For a pure quaternion with vector value <span class="math notranslate nohighlight">\(\vec{v}\)</span> the the result
is a unit quaternion equivalent to a rotation defined by
<span class="math notranslate nohighlight">\(2\vec{v}\)</span> intepretted as an Euler vector, that is, parallel to
the axis of rotation and whose norm is the magnitude of rotation.</p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">spatialmath</span><span class="w"> </span><span class="kn">import</span> <span class="n">Quaternion</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">math</span><span class="w"> </span><span class="kn">import</span> <span class="n">pi</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">Quaternion</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">exp</span><span class="p">())</span>
<span class="go"> 1.6939 < -0.7896, -1.1843, -1.5791 ></span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">Quaternion</span><span class="o">.</span><span class="n">Pure</span><span class="p">([</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">exp</span><span class="p">())</span> <span class="c1"># result is a UnitQuaternion</span>
<span class="go"> 0.7071 << 0.7071, 0.0000, 0.0000 >></span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">exp</span><span class="p">()</span><span class="o">.</span><span class="n">angvec</span><span class="p">())</span>
<span class="go">(1.5707963267948963, array([1., 0., 0.]))</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Reference<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://en.wikipedia.org/wiki/Quaternion#Exponential,_logarithm,_and_power_functions">Wikipedia</a></p>
</dd>
<dt class="field-even">Seealso<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.quaternion.Quaternion.log" title="spatialmath.quaternion.Quaternion.log"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Quaternion.log()</span></code></a> <a class="reference internal" href="3d_orient_unitquaternion.html#spatialmath.quaternion.UnitQuaternion.log" title="spatialmath.quaternion.UnitQuaternion.log"><code class="xref py py-meth docutils literal notranslate"><span class="pre">UnitQuaternion.log()</span></code></a> <a class="reference internal" href="3d_orient_unitquaternion.html#spatialmath.quaternion.UnitQuaternion.AngVec" title="spatialmath.quaternion.UnitQuaternion.AngVec"><code class="xref py py-meth docutils literal notranslate"><span class="pre">UnitQuaternion.AngVec()</span></code></a> <a class="reference internal" href="3d_orient_unitquaternion.html#spatialmath.quaternion.UnitQuaternion.EulerVec" title="spatialmath.quaternion.UnitQuaternion.EulerVec"><code class="xref py py-meth docutils literal notranslate"><span class="pre">UnitQuaternion.EulerVec()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.extend">
<span class="sig-name descname"><span class="pre">extend</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">iterable</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#spatialmath.quaternion.Quaternion.extend" title="Permalink to this definition"></a></dt>
<dd><p>Extend sequence of values in an instance (BasePoseList superclass method)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>x</strong> (<em>instance</em><em> of </em><em>same type</em>) – the value to extend</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – incorrect type of appended object</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><span class="sphinx_autodoc_typehints-type"><code class="xref py py-obj docutils literal notranslate"><span class="pre">None</span></code></span></p>
</dd>
</dl>
<p>Appends the argument’s values to the object’s internal list of values.</p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">Alloc</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">10</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">Alloc</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span> <span class="c1"># extend the list</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">15</span>
</pre></div>
</div>
<p>where <code class="docutils literal notranslate"><span class="pre">X</span></code> is any of the SMTB classes.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.inner">
<span class="sig-name descname"><span class="pre">inner</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/quaternion.html#Quaternion.inner"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.quaternion.Quaternion.inner" title="Permalink to this definition"></a></dt>
<dd><p>Inner product of quaternions</p>
<dl class="field-list simple">
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>float</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">q1.inner(q2)</span></code> is the dot product of the equivalent vectors,
ie. <code class="docutils literal notranslate"><span class="pre">numpy.dot(q1.vec,</span> <span class="pre">q2.vec)</span></code>.
The value of <code class="docutils literal notranslate"><span class="pre">q.inner(q)</span></code> is the same as <code class="docutils literal notranslate"><span class="pre">q.norm</span> <span class="pre">**</span> <span class="pre">2</span></code>.</p>
<p>Example:</p>
<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="func_quat.html#spatialmath.base.quaternions.qinner" title="spatialmath.base.quaternions.qinner"><code class="xref py py-func docutils literal notranslate"><span class="pre">qinner()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="spatialmath.quaternion.Quaternion.insert">