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	<div class="section" id="liegroups">
	<h1>liegroups<a class="headerlink" href="#liegroups" title="Permalink to this headline">¶</a></h1>
	<p>The default implementation uses numpy as the backend linear algebra library.</p>
	<div class="section" id="so-2">
	<h2>SO(2)<a class="headerlink" href="#so-2" title="Permalink to this headline">¶</a></h2>
	<dl class="attribute">
	<dt id="liegroups.SO2">
	<code class="sig-prename descclassname">liegroups.</code><code class="sig-name descname">SO2</code><a class="headerlink" href="#liegroups.SO2" title="Permalink to this definition">¶</a></dt>
	<dd><p>alias of <a class="reference internal" href="#liegroups.numpy.so2.SO2Matrix" title="liegroups.numpy.so2.SO2Matrix"><code class="xref py py-class docutils literal notranslate"><span class="pre">liegroups.numpy.so2.SO2Matrix</span></code></a></p>
	</dd></dl>

	<dl class="class">
	<dt id="liegroups.numpy.so2.SO2Matrix">
	<em class="property">class </em><code class="sig-prename descclassname">liegroups.numpy.so2.</code><code class="sig-name descname">SO2Matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Rotation matrix in <span class="math notranslate nohighlight">\(SO(2)\)</span> using active (alibi) transformations.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}SO(2) &= \left\{ \mathbf{C} \in \mathbb{R}^{2 \times 2} ~\middle\|~ \mathbf{C}\mathbf{C}^T = \mathbf{1}, \det \mathbf{C} = 1 \right\} \\
	\mathfrak{so}(2) &= \left\{ \boldsymbol{\Phi} = \phi^\wedge \in \mathbb{R}^{2 \times 2} ~\middle\|~ \phi \in \mathbb{R} \right\}\end{split}\]</div>
	<dl class="field-list simple">
	<dt class="field-odd">Variables</dt>
	<dd class="field-odd"><ul class="simple">
	<li><p><strong>dim</strong> – Dimension of the rotation matrix.</p></li>
	<li><p><strong>dof</strong> – Underlying degrees of freedom (i.e., dimension of the tangent space).</p></li>
	<li><p><strong>mat</strong> – Storage for the rotation matrix <span class="math notranslate nohighlight">\(\mathbf{C}\)</span>.</p></li>
	</ul>
	</dd>
	</dl>
	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.adjoint">
	<code class="sig-name descname">adjoint</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.adjoint" title="Permalink to this definition">¶</a></dt>
	<dd><p>Adjoint matrix of the transformation.</p>
	<div class="math notranslate nohighlight">
	\[\text{Ad}(\mathbf{C}) = 1\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.as_matrix">
	<code class="sig-name descname">as_matrix</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.as_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the matrix representation of the rotation.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.so2.SO2Matrix.dim">
	<code class="sig-name descname">dim</code><em class="property"> = 2</em><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.dim" title="Permalink to this definition">¶</a></dt>
	<dd><p>Dimension of the transformation matrix.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.so2.SO2Matrix.dof">
	<code class="sig-name descname">dof</code><em class="property"> = 1</em><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.dof" title="Permalink to this definition">¶</a></dt>
	<dd><p>Underlying degrees of freedom (i.e., dimension of the tangent space).</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.dot">
	<code class="sig-name descname">dot</code><span class="sig-paren">(</span><em class="sig-param">other</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.dot" title="Permalink to this definition">¶</a></dt>
	<dd><p>Multiply another rotation or one or more vectors on the left.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.exp">
	<em class="property">classmethod </em><code class="sig-name descname">exp</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.exp" title="Permalink to this definition">¶</a></dt>
	<dd><p>Exponential map for <span class="math notranslate nohighlight">\(SO(2)\)</span>, which computes a transformation from a tangent vector:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{C}(\phi) =
	\exp(\phi^\wedge) =
	\cos \phi \mathbf{1} + \sin \phi 1^\wedge =
	\begin{bmatrix}
	\cos \phi & -\sin \phi \\
	\sin \phi & \cos \phi
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">log()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.from_angle">
	<em class="property">classmethod </em><code class="sig-name descname">from_angle</code><span class="sig-paren">(</span><em class="sig-param">angle_in_radians</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.from_angle" title="Permalink to this definition">¶</a></dt>
	<dd><p>Form a rotation matrix given an angle in radians.</p>
	<p>See <code class="xref py py-meth docutils literal notranslate"><span class="pre">exp()</span></code></p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.from_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">from_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em>, <em class="sig-param">normalize=False</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.from_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Create a rotation from a matrix (safe, but slower).</p>
	<p>Throws an error if mat is invalid and normalize=False.
	If normalize=True invalid matrices will be normalized to be valid.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.identity">
	<em class="property">classmethod </em><code class="sig-name descname">identity</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.identity" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the identity rotation.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.inv">
	<code class="sig-name descname">inv</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.inv" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the inverse rotation:</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{C}^{-1} = \mathbf{C}^T\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.inv_left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">inv_left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.inv_left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(2)\)</span> inverse left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{J}^{-1}(\phi) =
	\begin{cases}
	\mathbf{1} - \frac{1}{2} \phi^\wedge, & \text{if } \phi \text{ is small} \\
	\frac{\phi}{2} \cot \frac{\phi}{2} \mathbf{1} -
	\frac{\phi}{2} 1^\wedge, & \text{otherwise}
	\end{cases}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.is_valid_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">is_valid_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.is_valid_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Check if a matrix is a valid rotation matrix.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(2)\)</span> left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{J}(\phi) =
	\begin{cases}
	\mathbf{1} + \frac{1}{2} \phi^\wedge, & \text{if } \phi \text{ is small} \\
	\frac{\sin \phi}{\phi} \mathbf{1} -
	\frac{1 - \cos \phi}{\phi} 1^\wedge, & \text{otherwise}
	\end{cases}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.log">
	<code class="sig-name descname">log</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.log" title="Permalink to this definition">¶</a></dt>
	<dd><p>Logarithmic map for <span class="math notranslate nohighlight">\(SO(2)\)</span>, which computes a tangent vector from a transformation:</p>
	<div class="math notranslate nohighlight">
	\[\phi(\mathbf{C}) =
	\ln(\mathbf{C})^\vee =
	\text{atan2}(C_{1,0}, C_{0,0})\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">exp()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.normalize">
	<code class="sig-name descname">normalize</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.normalize" title="Permalink to this definition">¶</a></dt>
	<dd><p>Normalize the rotation matrix to ensure it is valid and
	negate the effect of rounding errors.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.perturb">
	<code class="sig-name descname">perturb</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.perturb" title="Permalink to this definition">¶</a></dt>
	<dd><p>Perturb the rotation in-place on the left by a vector in its local tangent space.</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{C} \gets \exp(\boldsymbol{\phi}^\wedge) \mathbf{C}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.to_angle">
	<code class="sig-name descname">to_angle</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.to_angle" title="Permalink to this definition">¶</a></dt>
	<dd><p>Recover the rotation angle in radians from the rotation matrix.</p>
	<p>See <code class="xref py py-meth docutils literal notranslate"><span class="pre">log()</span></code></p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.vee">
	<em class="property">classmethod </em><code class="sig-name descname">vee</code><span class="sig-paren">(</span><em class="sig-param">Phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.vee" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(2)\)</span> vee operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\phi = \boldsymbol{\Phi}^\vee\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">wedge()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so2.SO2Matrix.wedge">
	<em class="property">classmethod </em><code class="sig-name descname">wedge</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so2.SO2Matrix.wedge" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(2)\)</span> wedge operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\boldsymbol{\Phi} =
	\phi^\wedge =
	\begin{bmatrix}
	0 & -\phi \\
	\phi & 0
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">vee()</span></code>.</p>
	</dd></dl>

	</dd></dl>

	</div>
	<div class="section" id="se-2">
	<h2>SE(2)<a class="headerlink" href="#se-2" title="Permalink to this headline">¶</a></h2>
	<dl class="attribute">
	<dt id="liegroups.SE2">
	<code class="sig-prename descclassname">liegroups.</code><code class="sig-name descname">SE2</code><a class="headerlink" href="#liegroups.SE2" title="Permalink to this definition">¶</a></dt>
	<dd><p>alias of <a class="reference internal" href="#liegroups.numpy.se2.SE2Matrix" title="liegroups.numpy.se2.SE2Matrix"><code class="xref py py-class docutils literal notranslate"><span class="pre">liegroups.numpy.se2.SE2Matrix</span></code></a></p>
	</dd></dl>

	<dl class="class">
	<dt id="liegroups.numpy.se2.SE2Matrix">
	<em class="property">class </em><code class="sig-prename descclassname">liegroups.numpy.se2.</code><code class="sig-name descname">SE2Matrix</code><span class="sig-paren">(</span><em class="sig-param">rot</em>, <em class="sig-param">trans</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Homogeneous transformation matrix in <span class="math notranslate nohighlight">\(SE(2)\)</span> using active (alibi) transformations.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}SE(2) &= \left\{ \mathbf{T}=
	\begin{bmatrix}
	\mathbf{C} & \mathbf{r} \\
	\mathbf{0}^T & 1
	\end{bmatrix} \in \mathbb{R}^{3 \times 3} ~\middle\|~ \mathbf{C} \in SO(2), \mathbf{r} \in \mathbb{R}^2 \right\} \\
	\mathfrak{se}(2) &= \left\{ \boldsymbol{\Xi} =
	\boldsymbol{\xi}^\wedge \in \mathbb{R}^{3 \times 3} ~\middle\|~
	\boldsymbol{\xi}=
	\begin{bmatrix}
	\boldsymbol{\rho} \\ \phi
	\end{bmatrix} \in \mathbb{R}^3, \boldsymbol{\rho} \in \mathbb{R}^2, \phi \in \mathbb{R} \right\}\end{split}\]</div>
	<dl class="field-list simple">
	<dt class="field-odd">Variables</dt>
	<dd class="field-odd"><ul class="simple">
	<li><p><strong>dim</strong> – Dimension of the rotation matrix.</p></li>
	<li><p><strong>dof</strong> – Underlying degrees of freedom (i.e., dimension of the tangent space).</p></li>
	<li><p><strong>rot</strong> – Storage for the rotation matrix <span class="math notranslate nohighlight">\(\mathbf{C}\)</span>.</p></li>
	<li><p><strong>trans</strong> – Storage for the translation vector <span class="math notranslate nohighlight">\(\mathbf{r}\)</span>.</p></li>
	</ul>
	</dd>
	</dl>
	<dl class="attribute">
	<dt id="liegroups.numpy.se2.SE2Matrix.RotationType">
	<code class="sig-name descname">RotationType</code><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.RotationType" title="Permalink to this definition">¶</a></dt>
	<dd><p>alias of <a class="reference internal" href="#liegroups.numpy.so2.SO2Matrix" title="liegroups.numpy.so2.SO2Matrix"><code class="xref py py-class docutils literal notranslate"><span class="pre">liegroups.numpy.so2.SO2Matrix</span></code></a></p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.adjoint">
	<code class="sig-name descname">adjoint</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.adjoint" title="Permalink to this definition">¶</a></dt>
	<dd><p>Adjoint matrix of the transformation.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\text{Ad}(\mathbf{T}) =
	\begin{bmatrix}
	\mathbf{C} & 1^\wedge \mathbf{r} \\
	\mathbf{0}^T & 1
	\end{bmatrix}
	\in \mathbb{R}^{3 \times 3}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.as_matrix">
	<code class="sig-name descname">as_matrix</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.as_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the matrix representation of the rotation.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.se2.SE2Matrix.dim">
	<code class="sig-name descname">dim</code><em class="property"> = 3</em><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.dim" title="Permalink to this definition">¶</a></dt>
	<dd><p>Dimension of the transformation matrix.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.se2.SE2Matrix.dof">
	<code class="sig-name descname">dof</code><em class="property"> = 3</em><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.dof" title="Permalink to this definition">¶</a></dt>
	<dd><p>Underlying degrees of freedom (i.e., dimension of the tangent space).</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.dot">
	<code class="sig-name descname">dot</code><span class="sig-paren">(</span><em class="sig-param">other</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.dot" title="Permalink to this definition">¶</a></dt>
	<dd><p>Multiply another rotation or one or more vectors on the left.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.exp">
	<em class="property">classmethod </em><code class="sig-name descname">exp</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.exp" title="Permalink to this definition">¶</a></dt>
	<dd><p>Exponential map for <span class="math notranslate nohighlight">\(SE(2)\)</span>, which computes a transformation from a tangent vector:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{T}(\boldsymbol{\xi}) =
	\exp(\boldsymbol{\xi}^\wedge) =
	\begin{bmatrix}
	\exp(\phi ^\wedge) & \mathbf{J} \boldsymbol{\rho} \\
	\mathbf{0} ^ T & 1
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">log()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.from_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">from_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em>, <em class="sig-param">normalize=False</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.from_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Create a transformation from a matrix (safe, but slower).</p>
	<p>Throws an error if mat is invalid and normalize=False.
	If normalize=True invalid matrices will be normalized to be valid.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.identity">
	<em class="property">classmethod </em><code class="sig-name descname">identity</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.identity" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the identity transformation.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.inv">
	<code class="sig-name descname">inv</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.inv" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the inverse transformation:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{T}^{-1} =
	\begin{bmatrix}
	\mathbf{C}^T & -\mathbf{C}^T\mathbf{r} \\
	\mathbf{0}^T & 1
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.inv_left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">inv_left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.inv_left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(2)\)</span> inverse left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\mathcal{J}^{-1}(\boldsymbol{\xi})\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.is_valid_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">is_valid_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.is_valid_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Check if a matrix is a valid transformation matrix.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(2)\)</span> left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\mathcal{J}(\boldsymbol{\xi})\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.log">
	<code class="sig-name descname">log</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.log" title="Permalink to this definition">¶</a></dt>
	<dd><p>Logarithmic map for <span class="math notranslate nohighlight">\(SE(2)\)</span>, which computes a tangent vector from a transformation:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\boldsymbol{\xi}(\mathbf{T}) =
	\ln(\mathbf{T})^\vee =
	\begin{bmatrix}
	\mathbf{J} ^ {-1} \mathbf{r} \\
	\ln(\boldsymbol{C}) ^\vee
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">log()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.normalize">
	<code class="sig-name descname">normalize</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.normalize" title="Permalink to this definition">¶</a></dt>
	<dd><p>Normalize the transformation matrix to ensure it is valid and
	negate the effect of rounding errors.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.odot">
	<em class="property">classmethod </em><code class="sig-name descname">odot</code><span class="sig-paren">(</span><em class="sig-param">p</em>, <em class="sig-param">directional=False</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.odot" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(2)\)</span> odot operator as defined by Barfoot.</p>
	<p>This is the Jacobian of a vector</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{p} =
	\begin{bmatrix}
	sx \\ sy \\ sz \\ s
	\end{bmatrix} =
	\begin{bmatrix}
	\boldsymbol{\epsilon} \\ \eta
	\end{bmatrix}\end{split}\]</div>
	<p>with respect to a perturbation in the underlying parameters of <span class="math notranslate nohighlight">\(\mathbf{T}\)</span>.</p>
	<p>If <span class="math notranslate nohighlight">\(\mathbf{p}\)</span> is given in Euclidean coordinates and directional=False, the missing scale value <span class="math notranslate nohighlight">\(\eta\)</span> is assumed to be 1 and the Jacobian is 2x3. If directional=True, <span class="math notranslate nohighlight">\(\eta\)</span> is assumed to be 0:</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{p}^\odot =
	\begin{bmatrix}
	\eta \mathbf{1} & 1^\wedge \boldsymbol{\epsilon}
	\end{bmatrix}\]</div>
	<p>If <span class="math notranslate nohighlight">\(\mathbf{p}\)</span> is given in Homogeneous coordinates, the Jacobian is 3x3:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{p}^\odot =
	\begin{bmatrix}
	\eta \mathbf{1} & 1^\wedge \boldsymbol{\epsilon} \\
	\mathbf{0}^T & 0
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.perturb">
	<code class="sig-name descname">perturb</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.perturb" title="Permalink to this definition">¶</a></dt>
	<dd><p>Perturb the transformation in-place on the left by a vector in its local tangent space.</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{T} \gets \exp(\boldsymbol{\xi}^\wedge) \mathbf{T}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.vee">
	<em class="property">classmethod </em><code class="sig-name descname">vee</code><span class="sig-paren">(</span><em class="sig-param">Xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.vee" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(2)\)</span> vee operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\boldsymbol{\xi} = \boldsymbol{\Xi} ^\vee\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">wedge()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se2.SE2Matrix.wedge">
	<em class="property">classmethod </em><code class="sig-name descname">wedge</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se2.SE2Matrix.wedge" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(2)\)</span> wedge operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\boldsymbol{\Xi} =
	\boldsymbol{\xi} ^\wedge =
	\begin{bmatrix}
	\phi ^\wedge & \boldsymbol{\rho} \\
	\mathbf{0} ^ T & 0
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">vee()</span></code>.</p>
	</dd></dl>

	</dd></dl>

	</div>
	<div class="section" id="so-3">
	<h2>SO(3)<a class="headerlink" href="#so-3" title="Permalink to this headline">¶</a></h2>
	<dl class="attribute">
	<dt id="liegroups.SO3">
	<code class="sig-prename descclassname">liegroups.</code><code class="sig-name descname">SO3</code><a class="headerlink" href="#liegroups.SO3" title="Permalink to this definition">¶</a></dt>
	<dd><p>alias of <a class="reference internal" href="#liegroups.numpy.so3.SO3Matrix" title="liegroups.numpy.so3.SO3Matrix"><code class="xref py py-class docutils literal notranslate"><span class="pre">liegroups.numpy.so3.SO3Matrix</span></code></a></p>
	</dd></dl>

	<dl class="class">
	<dt id="liegroups.numpy.so3.SO3Matrix">
	<em class="property">class </em><code class="sig-prename descclassname">liegroups.numpy.so3.</code><code class="sig-name descname">SO3Matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Rotation matrix in <span class="math notranslate nohighlight">\(SO(3)\)</span> using active (alibi) transformations.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}SO(3) &= \left\{ \mathbf{C} \in \mathbb{R}^{3 \times 3} ~\middle\|~ \mathbf{C}\mathbf{C}^T = \mathbf{1}, \det \mathbf{C} = 1 \right\} \\
	\mathfrak{so}(3) &= \left\{ \boldsymbol{\Phi} = \boldsymbol{\phi}^\wedge \in \mathbb{R}^{3 \times 3} ~\middle\|~ \boldsymbol{\phi} = \phi \mathbf{a} \in \mathbb{R}^3, \phi = \Vert \boldsymbol{\phi} \Vert \right\}\end{split}\]</div>
	<dl class="field-list simple">
	<dt class="field-odd">Variables</dt>
	<dd class="field-odd"><ul class="simple">
	<li><p><strong>dim</strong> – Dimension of the rotation matrix.</p></li>
	<li><p><strong>dof</strong> – Underlying degrees of freedom (i.e., dimension of the tangent space).</p></li>
	<li><p><strong>mat</strong> – Storage for the rotation matrix <span class="math notranslate nohighlight">\(\mathbf{C}\)</span>.</p></li>
	</ul>
	</dd>
	</dl>
	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.adjoint">
	<code class="sig-name descname">adjoint</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.adjoint" title="Permalink to this definition">¶</a></dt>
	<dd><p>Adjoint matrix of the transformation.</p>
	<div class="math notranslate nohighlight">
	\[\text{Ad}(\mathbf{C}) = \mathbf{C}
	\in \mathbb{R}^{3 \times 3}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.as_matrix">
	<code class="sig-name descname">as_matrix</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.as_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the matrix representation of the rotation.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.so3.SO3Matrix.dim">
	<code class="sig-name descname">dim</code><em class="property"> = 3</em><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.dim" title="Permalink to this definition">¶</a></dt>
	<dd><p>Dimension of the transformation matrix.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.so3.SO3Matrix.dof">
	<code class="sig-name descname">dof</code><em class="property"> = 3</em><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.dof" title="Permalink to this definition">¶</a></dt>
	<dd><p>Underlying degrees of freedom (i.e., dimension of the tangent space).</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.dot">
	<code class="sig-name descname">dot</code><span class="sig-paren">(</span><em class="sig-param">other</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.dot" title="Permalink to this definition">¶</a></dt>
	<dd><p>Multiply another rotation or one or more vectors on the left.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.exp">
	<em class="property">classmethod </em><code class="sig-name descname">exp</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.exp" title="Permalink to this definition">¶</a></dt>
	<dd><p>Exponential map for <span class="math notranslate nohighlight">\(SO(3)\)</span>, which computes a transformation from a tangent vector:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{C}(\boldsymbol{\phi}) =
	\exp(\boldsymbol{\phi}^\wedge) =
	\begin{cases}
	\mathbf{1} + \boldsymbol{\phi}^\wedge, & \text{if } \phi \text{ is small} \\
	\cos \phi \mathbf{1} +
	(1 - \cos \phi) \mathbf{a}\mathbf{a}^T +
	\sin \phi \mathbf{a}^\wedge, & \text{otherwise}
	\end{cases}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">log()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.from_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">from_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em>, <em class="sig-param">normalize=False</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.from_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Create a rotation from a matrix (safe, but slower).</p>
	<p>Throws an error if mat is invalid and normalize=False.
	If normalize=True invalid matrices will be normalized to be valid.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.from_quaternion">
	<em class="property">classmethod </em><code class="sig-name descname">from_quaternion</code><span class="sig-paren">(</span><em class="sig-param">quat</em>, <em class="sig-param">ordering='wxyz'</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.from_quaternion" title="Permalink to this definition">¶</a></dt>
	<dd><p>Form a rotation matrix from a unit length quaternion.</p>
	<p>Valid orderings are ‘xyzw’ and ‘wxyz’.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{C} =
	\begin{bmatrix}
	1 - 2 (y^2 + z^2) & 2 (xy - wz) & 2 (wy + xz) \\
	2 (wz + xy) & 1 - 2 (x^2 + z^2) & 2 (yz - wx) \\
	2 (xz - wy) & 2 (wx + yz) & 1 - 2 (x^2 + y^2)
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.from_rpy">
	<em class="property">classmethod </em><code class="sig-name descname">from_rpy</code><span class="sig-paren">(</span><em class="sig-param">roll</em>, <em class="sig-param">pitch</em>, <em class="sig-param">yaw</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.from_rpy" title="Permalink to this definition">¶</a></dt>
	<dd><p>Form a rotation matrix from RPY Euler angles <span class="math notranslate nohighlight">\((\alpha, \beta, \gamma)\)</span>.</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{C} = \mathbf{C}_z(\gamma) \mathbf{C}_y(\beta) \mathbf{C}_x(\alpha)\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.identity">
	<em class="property">classmethod </em><code class="sig-name descname">identity</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.identity" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the identity rotation.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.inv">
	<code class="sig-name descname">inv</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.inv" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the inverse rotation:</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{C}^{-1} = \mathbf{C}^T\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.inv_left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">inv_left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.inv_left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(3)\)</span> inverse left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{J}^{-1}(\boldsymbol{\phi}) =
	\begin{cases}
	\mathbf{1} - \frac{1}{2} \boldsymbol{\phi}^\wedge, & \text{if } \phi \text{ is small} \\
	\frac{\phi}{2} \cot \frac{\phi}{2} \mathbf{1} +
	\left( 1 - \frac{\phi}{2} \cot \frac{\phi}{2} \right) \mathbf{a}\mathbf{a}^T -
	\frac{\phi}{2} \mathbf{a}^\wedge, & \text{otherwise}
	\end{cases}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.is_valid_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">is_valid_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.is_valid_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Check if a matrix is a valid rotation matrix.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(3)\)</span> left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{J}(\boldsymbol{\phi}) =
	\begin{cases}
	\mathbf{1} + \frac{1}{2} \boldsymbol{\phi}^\wedge, & \text{if } \phi \text{ is small} \\
	\frac{\sin \phi}{\phi} \mathbf{1} +
	\left(1 - \frac{\sin \phi}{\phi} \right) \mathbf{a}\mathbf{a}^T +
	\frac{1 - \cos \phi}{\phi} \mathbf{a}^\wedge, & \text{otherwise}
	\end{cases}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.log">
	<code class="sig-name descname">log</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.log" title="Permalink to this definition">¶</a></dt>
	<dd><p>Logarithmic map for <span class="math notranslate nohighlight">\(SO(3)\)</span>, which computes a tangent vector from a transformation:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\phi &= \frac{1}{2} \left( \mathrm{Tr}(\mathbf{C}) - 1 \right) \\
	\boldsymbol{\phi}(\mathbf{C}) &=
	\ln(\mathbf{C})^\vee =
	\begin{cases}
	\mathbf{1} - \boldsymbol{\phi}^\wedge, & \text{if } \phi \text{ is small} \\
	\left( \frac{1}{2} \frac{\phi}{\sin \phi} \left( \mathbf{C} - \mathbf{C}^T \right) \right)^\vee, & \text{otherwise}
	\end{cases}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">log()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.normalize">
	<code class="sig-name descname">normalize</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.normalize" title="Permalink to this definition">¶</a></dt>
	<dd><p>Normalize the rotation matrix to ensure it is valid and
	negate the effect of rounding errors.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.perturb">
	<code class="sig-name descname">perturb</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.perturb" title="Permalink to this definition">¶</a></dt>
	<dd><p>Perturb the rotation in-place on the left by a vector in its local tangent space.</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{C} \gets \exp(\boldsymbol{\phi}^\wedge) \mathbf{C}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.rotx">
	<em class="property">classmethod </em><code class="sig-name descname">rotx</code><span class="sig-paren">(</span><em class="sig-param">angle_in_radians</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.rotx" title="Permalink to this definition">¶</a></dt>
	<dd><p>Form a rotation matrix given an angle in rad about the x-axis.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{C}_x(\phi) =
	\begin{bmatrix}
	1 & 0 & 0 \\
	0 & \cos \phi & -\sin \phi \\
	0 & \sin \phi & \cos \phi
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.roty">
	<em class="property">classmethod </em><code class="sig-name descname">roty</code><span class="sig-paren">(</span><em class="sig-param">angle_in_radians</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.roty" title="Permalink to this definition">¶</a></dt>
	<dd><p>Form a rotation matrix given an angle in rad about the y-axis.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{C}_y(\phi) =
	\begin{bmatrix}
	\cos \phi & 0 & \sin \phi \\
	0 & 1 & 0 \\
	\sin \phi & 0 & \cos \phi
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.rotz">
	<em class="property">classmethod </em><code class="sig-name descname">rotz</code><span class="sig-paren">(</span><em class="sig-param">angle_in_radians</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.rotz" title="Permalink to this definition">¶</a></dt>
	<dd><p>Form a rotation matrix given an angle in rad about the z-axis.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{C}_z(\phi) =
	\begin{bmatrix}
	\cos \phi & -\sin \phi & 0 \\
	\sin \phi & \cos \phi & 0 \\
	0 & 0 & 1
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.to_quaternion">
	<code class="sig-name descname">to_quaternion</code><span class="sig-paren">(</span><em class="sig-param">ordering='wxyz'</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.to_quaternion" title="Permalink to this definition">¶</a></dt>
	<dd><p>Convert a rotation matrix to a unit length quaternion.</p>
	<p>Valid orderings are ‘xyzw’ and ‘wxyz’.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.to_rpy">
	<code class="sig-name descname">to_rpy</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.to_rpy" title="Permalink to this definition">¶</a></dt>
	<dd><p>Convert a rotation matrix to RPY Euler angles <span class="math notranslate nohighlight">\((\alpha, \beta, \gamma)\)</span>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.vee">
	<em class="property">classmethod </em><code class="sig-name descname">vee</code><span class="sig-paren">(</span><em class="sig-param">Phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.vee" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(3)\)</span> vee operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\phi = \boldsymbol{\Phi}^\vee\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">wedge()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.so3.SO3Matrix.wedge">
	<em class="property">classmethod </em><code class="sig-name descname">wedge</code><span class="sig-paren">(</span><em class="sig-param">phi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.so3.SO3Matrix.wedge" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SO(3)\)</span> wedge operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\boldsymbol{\Phi} =
	\boldsymbol{\phi}^\wedge =
	\begin{bmatrix}
	0 & -\phi_3 & \phi_2 \\
	\phi_3 & 0 & -\phi_1 \\
	-\phi_2 & \phi_1 & 0
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">vee()</span></code>.</p>
	</dd></dl>

	</dd></dl>

	</div>
	<div class="section" id="se-3">
	<h2>SE(3)<a class="headerlink" href="#se-3" title="Permalink to this headline">¶</a></h2>
	<dl class="attribute">
	<dt id="liegroups.SE3">
	<code class="sig-prename descclassname">liegroups.</code><code class="sig-name descname">SE3</code><a class="headerlink" href="#liegroups.SE3" title="Permalink to this definition">¶</a></dt>
	<dd><p>alias of <a class="reference internal" href="#liegroups.numpy.se3.SE3Matrix" title="liegroups.numpy.se3.SE3Matrix"><code class="xref py py-class docutils literal notranslate"><span class="pre">liegroups.numpy.se3.SE3Matrix</span></code></a></p>
	</dd></dl>

	<dl class="class">
	<dt id="liegroups.numpy.se3.SE3Matrix">
	<em class="property">class </em><code class="sig-prename descclassname">liegroups.numpy.se3.</code><code class="sig-name descname">SE3Matrix</code><span class="sig-paren">(</span><em class="sig-param">rot</em>, <em class="sig-param">trans</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Homogeneous transformation matrix in <span class="math notranslate nohighlight">\(SE(3)\)</span> using active (alibi) transformations.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}SE(3) &= \left\{ \mathbf{T}=
	\begin{bmatrix}
	\mathbf{C} & \mathbf{r} \\
	\mathbf{0}^T & 1
	\end{bmatrix} \in \mathbb{R}^{4 \times 4} ~\middle\|~ \mathbf{C} \in SO(3), \mathbf{r} \in \mathbb{R}^3 \right\} \\
	\mathfrak{se}(3) &= \left\{ \boldsymbol{\Xi} =
	\boldsymbol{\xi}^\wedge \in \mathbb{R}^{4 \times 4} ~\middle\|~
	\boldsymbol{\xi}=
	\begin{bmatrix}
	\boldsymbol{\rho} \\ \boldsymbol{\phi}
	\end{bmatrix} \in \mathbb{R}^6, \boldsymbol{\rho} \in \mathbb{R}^3, \boldsymbol{\phi} \in \mathbb{R}^3 \right\}\end{split}\]</div>
	<dl class="field-list simple">
	<dt class="field-odd">Variables</dt>
	<dd class="field-odd"><ul class="simple">
	<li><p><strong>dim</strong> – Dimension of the rotation matrix.</p></li>
	<li><p><strong>dof</strong> – Underlying degrees of freedom (i.e., dimension of the tangent space).</p></li>
	<li><p><strong>rot</strong> – Storage for the rotation matrix <span class="math notranslate nohighlight">\(\mathbf{C}\)</span>.</p></li>
	<li><p><strong>trans</strong> – Storage for the translation vector <span class="math notranslate nohighlight">\(\mathbf{r}\)</span>.</p></li>
	</ul>
	</dd>
	</dl>
	<dl class="attribute">
	<dt id="liegroups.numpy.se3.SE3Matrix.RotationType">
	<code class="sig-name descname">RotationType</code><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.RotationType" title="Permalink to this definition">¶</a></dt>
	<dd><p>alias of <a class="reference internal" href="#liegroups.numpy.so3.SO3Matrix" title="liegroups.numpy.so3.SO3Matrix"><code class="xref py py-class docutils literal notranslate"><span class="pre">liegroups.numpy.so3.SO3Matrix</span></code></a></p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.adjoint">
	<code class="sig-name descname">adjoint</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.adjoint" title="Permalink to this definition">¶</a></dt>
	<dd><p>Adjoint matrix of the transformation.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\text{Ad}(\mathbf{T}) =
	\begin{bmatrix}
	\mathbf{C} & \mathbf{r}^\wedge\mathbf{C} \\
	\mathbf{0} & \mathbf{C}
	\end{bmatrix}
	\in \mathbb{R}^{6 \times 6}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.as_matrix">
	<code class="sig-name descname">as_matrix</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.as_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the matrix representation of the rotation.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.curlyvee">
	<em class="property">classmethod </em><code class="sig-name descname">curlyvee</code><span class="sig-paren">(</span><em class="sig-param">Psi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.curlyvee" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(3)\)</span> curlyvee operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\boldsymbol{\xi} =
	\boldsymbol{\Psi}^\curlyvee\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">curlywedge()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.curlywedge">
	<em class="property">classmethod </em><code class="sig-name descname">curlywedge</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.curlywedge" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(3)\)</span> curlywedge operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\boldsymbol{\Psi} =
	\boldsymbol{\xi}^\curlywedge =
	\begin{bmatrix}
	\boldsymbol{\phi}^\wedge & \boldsymbol{\rho}^\wedge \\
	\mathbf{0} & \boldsymbol{\phi}^\wedge
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">curlyvee()</span></code>.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.se3.SE3Matrix.dim">
	<code class="sig-name descname">dim</code><em class="property"> = 4</em><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.dim" title="Permalink to this definition">¶</a></dt>
	<dd><p>Dimension of the transformation matrix.</p>
	</dd></dl>

	<dl class="attribute">
	<dt id="liegroups.numpy.se3.SE3Matrix.dof">
	<code class="sig-name descname">dof</code><em class="property"> = 6</em><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.dof" title="Permalink to this definition">¶</a></dt>
	<dd><p>Underlying degrees of freedom (i.e., dimension of the tangent space).</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.dot">
	<code class="sig-name descname">dot</code><span class="sig-paren">(</span><em class="sig-param">other</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.dot" title="Permalink to this definition">¶</a></dt>
	<dd><p>Multiply another rotation or one or more vectors on the left.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.exp">
	<em class="property">classmethod </em><code class="sig-name descname">exp</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.exp" title="Permalink to this definition">¶</a></dt>
	<dd><p>Exponential map for <span class="math notranslate nohighlight">\(SE(3)\)</span>, which computes a transformation from a tangent vector:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{T}(\boldsymbol{\xi}) =
	\exp(\boldsymbol{\xi}^\wedge) =
	\begin{bmatrix}
	\exp(\boldsymbol{\phi}^\wedge) & \mathbf{J} \boldsymbol{\rho} \\
	\mathbf{0} ^ T & 1
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">log()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.from_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">from_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em>, <em class="sig-param">normalize=False</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.from_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Create a transformation from a matrix (safe, but slower).</p>
	<p>Throws an error if mat is invalid and normalize=False.
	If normalize=True invalid matrices will be normalized to be valid.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.identity">
	<em class="property">classmethod </em><code class="sig-name descname">identity</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.identity" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the identity transformation.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.inv">
	<code class="sig-name descname">inv</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.inv" title="Permalink to this definition">¶</a></dt>
	<dd><p>Return the inverse transformation:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{T}^{-1} =
	\begin{bmatrix}
	\mathbf{C}^T & -\mathbf{C}^T\mathbf{r} \\
	\mathbf{0}^T & 1
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.inv_left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">inv_left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.inv_left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(3)\)</span> inverse left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathcal{J}^{-1}(\boldsymbol{\xi}) =
	\begin{bmatrix}
	\mathbf{J}^{-1} & -\mathbf{J}^{-1} \mathbf{Q} \mathbf{J}^{-1} \\
	\mathbf{0} & \mathbf{J}^{-1}
	\end{bmatrix}\end{split}\]</div>
	<p>with <span class="math notranslate nohighlight">\(\mathbf{J}^{-1}\)</span> as in <code class="xref py py-meth docutils literal notranslate"><span class="pre">liegroups.SO3.inv_left_jacobian()</span></code> and <span class="math notranslate nohighlight">\(\mathbf{Q}\)</span> as in <code class="xref py py-meth docutils literal notranslate"><span class="pre">left_jacobian_Q_matrix()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.is_valid_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">is_valid_matrix</code><span class="sig-paren">(</span><em class="sig-param">mat</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.is_valid_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>Check if a matrix is a valid transformation matrix.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.left_jacobian">
	<em class="property">classmethod </em><code class="sig-name descname">left_jacobian</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.left_jacobian" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(3)\)</span> left Jacobian.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathcal{J}(\boldsymbol{\xi}) =
	\begin{bmatrix}
	\mathbf{J} & \mathbf{Q} \\
	\mathbf{0} & \mathbf{J}
	\end{bmatrix}\end{split}\]</div>
	<p>with <span class="math notranslate nohighlight">\(\mathbf{J}\)</span> as in <code class="xref py py-meth docutils literal notranslate"><span class="pre">liegroups.SO3.left_jacobian()</span></code> and <span class="math notranslate nohighlight">\(\mathbf{Q}\)</span> as in <code class="xref py py-meth docutils literal notranslate"><span class="pre">left_jacobian_Q_matrix()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.left_jacobian_Q_matrix">
	<em class="property">classmethod </em><code class="sig-name descname">left_jacobian_Q_matrix</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.left_jacobian_Q_matrix" title="Permalink to this definition">¶</a></dt>
	<dd><p>The <span class="math notranslate nohighlight">\(\mathbf{Q}\)</span> matrix used to compute <span class="math notranslate nohighlight">\(\mathcal{J}\)</span> in <code class="xref py py-meth docutils literal notranslate"><span class="pre">left_jacobian()</span></code> and <span class="math notranslate nohighlight">\(\mathcal{J}^{-1}\)</span> in <code class="xref py py-meth docutils literal notranslate"><span class="pre">inv_left_jacobian()</span></code>.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{Q}(\boldsymbol{\xi}) =
	\frac{1}{2}\boldsymbol{\rho}^\wedge &+
	\left( \frac{\phi - \sin \phi}{\phi^3} \right)
	\left(
	\boldsymbol{\phi}^\wedge \boldsymbol{\rho}^\wedge +
	\boldsymbol{\rho}^\wedge \boldsymbol{\phi}^\wedge +
	\boldsymbol{\phi}^\wedge \boldsymbol{\rho}^\wedge \boldsymbol{\phi}^\wedge
	\right) \\ &+
	\left( \frac{\phi^2 + 2 \cos \phi - 2}{2 \phi^4} \right)
	\left(
	\boldsymbol{\phi}^\wedge \boldsymbol{\phi}^\wedge \boldsymbol{\rho}^\wedge +
	\boldsymbol{\rho}^\wedge \boldsymbol{\phi}^\wedge \boldsymbol{\phi}^\wedge -
	3 \boldsymbol{\phi}^\wedge \boldsymbol{\rho}^\wedge \boldsymbol{\phi}^\wedge
	\right) \\ &+
	\left( \frac{2 \phi - 3 \sin \phi + \phi \cos \phi}{2 \phi^5} \right)
	\left(
	\boldsymbol{\phi}^\wedge \boldsymbol{\rho}^\wedge \boldsymbol{\phi}^\wedge \boldsymbol{\phi}^\wedge +
	\boldsymbol{\phi}^\wedge \boldsymbol{\phi}^\wedge \boldsymbol{\rho}^\wedge \boldsymbol{\phi}^\wedge
	\right)\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.log">
	<code class="sig-name descname">log</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.log" title="Permalink to this definition">¶</a></dt>
	<dd><p>Logarithmic map for <span class="math notranslate nohighlight">\(SE(3)\)</span>, which computes a tangent vector from a transformation:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\boldsymbol{\xi}(\mathbf{T}) =
	\ln(\mathbf{T})^\vee =
	\begin{bmatrix}
	\mathbf{J} ^ {-1} \mathbf{r} \\
	\ln(\boldsymbol{C}) ^\vee
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">exp()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.normalize">
	<code class="sig-name descname">normalize</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.normalize" title="Permalink to this definition">¶</a></dt>
	<dd><p>Normalize the transformation matrix to ensure it is valid and
	negate the effect of rounding errors.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.odot">
	<em class="property">classmethod </em><code class="sig-name descname">odot</code><span class="sig-paren">(</span><em class="sig-param">p</em>, <em class="sig-param">directional=False</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.odot" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(3)\)</span> odot operator as defined by Barfoot.</p>
	<p>This is the Jacobian of a vector</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{p} =
	\begin{bmatrix}
	sx \\ sy \\ sz \\ s
	\end{bmatrix} =
	\begin{bmatrix}
	\boldsymbol{\epsilon} \\ \eta
	\end{bmatrix}\end{split}\]</div>
	<p>with respect to a perturbation in the underlying parameters of <span class="math notranslate nohighlight">\(\mathbf{T}\)</span>.</p>
	<p>If <span class="math notranslate nohighlight">\(\mathbf{p}\)</span> is given in Euclidean coordinates and directional=False, the missing scale value <span class="math notranslate nohighlight">\(\eta\)</span> is assumed to be 1 and the Jacobian is 3x6. If directional=True, <span class="math notranslate nohighlight">\(\eta\)</span> is assumed to be 0:</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{p}^\odot =
	\begin{bmatrix}
	\eta \mathbf{1} & -\boldsymbol{\epsilon}^\wedge
	\end{bmatrix}\]</div>
	<p>If <span class="math notranslate nohighlight">\(\mathbf{p}\)</span> is given in Homogeneous coordinates, the Jacobian is 4x6:</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\mathbf{p}^\odot =
	\begin{bmatrix}
	\eta \mathbf{1} & -\boldsymbol{\epsilon}^\wedge \\
	\mathbf{0}^T & \mathbf{0}^T
	\end{bmatrix}\end{split}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.perturb">
	<code class="sig-name descname">perturb</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.perturb" title="Permalink to this definition">¶</a></dt>
	<dd><p>Perturb the transformation in-place on the left by a vector in its local tangent space.</p>
	<div class="math notranslate nohighlight">
	\[\mathbf{T} \gets \exp(\boldsymbol{\xi}^\wedge) \mathbf{T}\]</div>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.vee">
	<em class="property">classmethod </em><code class="sig-name descname">vee</code><span class="sig-paren">(</span><em class="sig-param">Xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.vee" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(3)\)</span> vee operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\boldsymbol{\xi} = \boldsymbol{\Xi} ^\vee\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">wedge()</span></code>.</p>
	</dd></dl>

	<dl class="method">
	<dt id="liegroups.numpy.se3.SE3Matrix.wedge">
	<em class="property">classmethod </em><code class="sig-name descname">wedge</code><span class="sig-paren">(</span><em class="sig-param">xi</em><span class="sig-paren">)</span><a class="headerlink" href="#liegroups.numpy.se3.SE3Matrix.wedge" title="Permalink to this definition">¶</a></dt>
	<dd><p><span class="math notranslate nohighlight">\(SE(3)\)</span> wedge operator as defined by Barfoot.</p>
	<div class="math notranslate nohighlight">
	\[\begin{split}\boldsymbol{\Xi} =
	\boldsymbol{\xi} ^\wedge =
	\begin{bmatrix}
	\boldsymbol{\phi} ^\wedge & \boldsymbol{\rho} \\
	\mathbf{0} ^ T & 0
	\end{bmatrix}\end{split}\]</div>
	<p>This is the inverse operation to <code class="xref py py-meth docutils literal notranslate"><span class="pre">vee()</span></code>.</p>
	</dd></dl>

	</dd></dl>

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