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<!DOCTYPE html>
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<title>Transforms in 3D — Spatial Maths package documentation</title>
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Spatial Maths package
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<li class="toctree-l1"><a class="reference internal" href="intro.html">Introduction</a></li>
<li class="toctree-l1"><a class="reference internal" href="spatialmath.html">Class reference</a></li>
<li class="toctree-l1 current"><a class="reference internal" href="functions.html">Function reference</a><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="func_2d.html">Transforms in 2D</a></li>
<li class="toctree-l2 current"><a class="current reference internal" href="#">Transforms in 3D</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.angvec2r"><code class="docutils literal notranslate"><span class="pre">angvec2r()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.angvec2tr"><code class="docutils literal notranslate"><span class="pre">angvec2tr()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.angvelxform"><code class="docutils literal notranslate"><span class="pre">angvelxform()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.angvelxform_dot"><code class="docutils literal notranslate"><span class="pre">angvelxform_dot()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.delta2tr"><code class="docutils literal notranslate"><span class="pre">delta2tr()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.eul2jac"><code class="docutils literal notranslate"><span class="pre">eul2jac()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.eul2r"><code class="docutils literal notranslate"><span class="pre">eul2r()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.eul2tr"><code class="docutils literal notranslate"><span class="pre">eul2tr()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.exp2jac"><code class="docutils literal notranslate"><span class="pre">exp2jac()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.exp2r"><code class="docutils literal notranslate"><span class="pre">exp2r()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.exp2tr"><code class="docutils literal notranslate"><span class="pre">exp2tr()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.ishom"><code class="docutils literal notranslate"><span class="pre">ishom()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.isrot"><code class="docutils literal notranslate"><span class="pre">isrot()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.oa2r"><code class="docutils literal notranslate"><span class="pre">oa2r()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.oa2tr"><code class="docutils literal notranslate"><span class="pre">oa2tr()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.r2x"><code class="docutils literal notranslate"><span class="pre">r2x()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rodrigues"><code class="docutils literal notranslate"><span class="pre">rodrigues()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rot2jac"><code class="docutils literal notranslate"><span class="pre">rot2jac()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rotvelxform"><code class="docutils literal notranslate"><span class="pre">rotvelxform()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rotvelxform_inv_dot"><code class="docutils literal notranslate"><span class="pre">rotvelxform_inv_dot()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rotx"><code class="docutils literal notranslate"><span class="pre">rotx()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.roty"><code class="docutils literal notranslate"><span class="pre">roty()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rotz"><code class="docutils literal notranslate"><span class="pre">rotz()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rpy2jac"><code class="docutils literal notranslate"><span class="pre">rpy2jac()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rpy2r"><code class="docutils literal notranslate"><span class="pre">rpy2r()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.rpy2tr"><code class="docutils literal notranslate"><span class="pre">rpy2tr()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tr2adjoint"><code class="docutils literal notranslate"><span class="pre">tr2adjoint()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tr2angvec"><code class="docutils literal notranslate"><span class="pre">tr2angvec()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tr2delta"><code class="docutils literal notranslate"><span class="pre">tr2delta()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tr2eul"><code class="docutils literal notranslate"><span class="pre">tr2eul()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tr2jac"><code class="docutils literal notranslate"><span class="pre">tr2jac()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tr2rpy"><code class="docutils literal notranslate"><span class="pre">tr2rpy()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tr2x"><code class="docutils literal notranslate"><span class="pre">tr2x()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.tranimate"><code class="docutils literal notranslate"><span class="pre">tranimate()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.transl"><code class="docutils literal notranslate"><span class="pre">transl()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trexp"><code class="docutils literal notranslate"><span class="pre">trexp()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trinterp"><code class="docutils literal notranslate"><span class="pre">trinterp()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trinv"><code class="docutils literal notranslate"><span class="pre">trinv()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trlog"><code class="docutils literal notranslate"><span class="pre">trlog()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trnorm"><code class="docutils literal notranslate"><span class="pre">trnorm()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trotx"><code class="docutils literal notranslate"><span class="pre">trotx()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.troty"><code class="docutils literal notranslate"><span class="pre">troty()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trotz"><code class="docutils literal notranslate"><span class="pre">trotz()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trplot"><code class="docutils literal notranslate"><span class="pre">trplot()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.trprint"><code class="docutils literal notranslate"><span class="pre">trprint()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.x2r"><code class="docutils literal notranslate"><span class="pre">x2r()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transforms3d.x2tr"><code class="docutils literal notranslate"><span class="pre">x2tr()</span></code></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="func_nd.html">Transforms in ND</a></li>
<li class="toctree-l2"><a class="reference internal" href="func_quat.html">Quaternions</a></li>
<li class="toctree-l2"><a class="reference internal" href="func_symbolic.html">Symbolic computation</a></li>
<li class="toctree-l2"><a class="reference internal" href="func_vector.html">Vectors</a></li>
<li class="toctree-l2"><a class="reference internal" href="func_graphics.html">Graphics and animation</a></li>
<li class="toctree-l2"><a class="reference internal" href="func_args.html">Argument checking</a></li>
<li class="toctree-l2"><a class="reference internal" href="func_numeric.html">Numerical utility functions</a></li>
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<li class="toctree-l1"><a class="reference internal" href="indices.html">Indices</a></li>
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<section id="module-spatialmath.base.transforms3d">
<span id="transforms-in-3d"></span><h1>Transforms in 3D<a class="headerlink" href="#module-spatialmath.base.transforms3d" title="Permalink to this heading"></a></h1>
<p>These functions create and manipulate 3D rotation matrices and rigid-body
transformations as 3x3 SO(3) matrices and 4x4 SE(3) matrices respectively.
These matrices are represented as 2D NumPy arrays.</p>
<p>Vector arguments are what numpy refers to as <code class="docutils literal notranslate"><span class="pre">array_like</span></code> and can be a list,
tuple, numpy array, numpy row vector or numpy column vector.</p>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.angvec2r">
<span class="sig-name descname"><span class="pre">angvec2r</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">theta</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">unit</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rad'</span></span></em>, <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/base/transforms3d.html#angvec2r"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.angvec2r" title="Permalink to this definition"></a></dt>
<dd><p>Create an SO(3) rotation matrix from rotation angle and axis</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>theta</strong> (<em>float</em>) – rotation</p></li>
<li><p><strong>unit</strong> (<em>str</em>) – angular units: ‘rad’ [default], or ‘deg’</p></li>
<li><p><strong>v</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em>) – 3D rotation axis</p></li>
<li><p><strong>tol</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></span>) – Tolerance in units of eps for zero-rotation case, defaults to 20</p></li>
</ul>
</dd>
<dt class="field-even">Type<span class="colon">:</span></dt>
<dd class="field-even"><p>float</p>
</dd>
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>SO(3) rotation matrix</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p>ndarray(3,3)</p>
</dd>
<dt class="field-odd">Raises<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>ValueError</strong> – bad arguments</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">angvec2r(θ,</span> <span class="pre">V)</span></code> is an SO(3) orthonormal rotation matrix
equivalent to a rotation of <code class="docutils literal notranslate"><span class="pre">θ</span></code> about the vector <code class="docutils literal notranslate"><span class="pre">V</span></code>.</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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">angvec2r</span>
<span class="gp">>>> </span><span class="n">angvec2r</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</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="c1"># rotx(0.3)</span>
<span class="go">array([[ 1. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955],</span>
<span class="go"> [ 0. , 0.2955, 0.9553]])</span>
<span class="gp">>>> </span><span class="n">angvec2r</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</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="c1"># rotx(0)</span>
<span class="go">array([[1., 0., 0.],</span>
<span class="go"> [0., 1., 0.],</span>
<span class="go"> [0., 0., 1.]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">θ</span> <span class="pre">==</span> <span class="pre">0</span></code> then return identity matrix.</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">θ</span> <span class="pre">~=</span> <span class="pre">0</span></code> then <code class="docutils literal notranslate"><span class="pre">V</span></code> must have a finite length.</p></li>
</ul>
</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.base.transforms3d.angvec2tr" title="spatialmath.base.transforms3d.angvec2tr"><code class="xref py py-func docutils literal notranslate"><span class="pre">angvec2tr()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2angvec" title="spatialmath.base.transforms3d.tr2angvec"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2angvec()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>not supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.angvec2tr">
<span class="sig-name descname"><span class="pre">angvec2tr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">theta</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">unit</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rad'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#angvec2tr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.angvec2tr" title="Permalink to this definition"></a></dt>
<dd><p>Create an SE(3) pure rotation from rotation angle and axis</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>theta</strong> (<em>float</em>) – rotation</p></li>
<li><p><strong>unit</strong> (<em>str</em>) – angular units: ‘rad’ [default], or ‘deg’</p></li>
<li><p><strong>v</strong> (<em>: array_like</em><em>(</em><em>3</em><em>)</em>) – 3D rotation axis</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SE(3) transformation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(4,4)</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">angvec2tr(θ,</span> <span class="pre">V)</span></code> is an SE(3) homogeneous transformation matrix
equivalent to a rotation of <code class="docutils literal notranslate"><span class="pre">θ</span></code> about the vector <code class="docutils literal notranslate"><span class="pre">V</span></code>.</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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">angvec2tr</span>
<span class="gp">>>> </span><span class="n">angvec2tr</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</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="c1"># rtotx(0.3)</span>
<span class="go">array([[ 1. , 0. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955, 0. ],</span>
<span class="go"> [ 0. , 0.2955, 0.9553, 0. ],</span>
<span class="go"> [ 0. , 0. , 0. , 1. ]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">θ</span> <span class="pre">==</span> <span class="pre">0</span></code> then return identity matrix.</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">θ</span> <span class="pre">~=</span> <span class="pre">0</span></code> then <code class="docutils literal notranslate"><span class="pre">V</span></code> must have a finite length.</p></li>
<li><p>The translational part is zero.</p></li>
</ul>
</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.base.transforms3d.angvec2r" title="spatialmath.base.transforms3d.angvec2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">angvec2r()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2angvec" title="spatialmath.base.transforms3d.tr2angvec"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2angvec()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>not supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.angvelxform">
<span class="sig-name descname"><span class="pre">angvelxform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Γ</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">inverse</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">full</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">representation</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rpy/xyz'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#angvelxform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.angvelxform" title="Permalink to this definition"></a></dt>
<dd><p>DEPRECATED, use <a class="reference internal" href="#spatialmath.base.transforms3d.rotvelxform" title="spatialmath.base.transforms3d.rotvelxform"><code class="xref py py-func docutils literal notranslate"><span class="pre">rotvelxform()</span></code></a> instead</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.angvelxform_dot">
<span class="sig-name descname"><span class="pre">angvelxform_dot</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Γ</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Γd</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">full</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">representation</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rpy/xyz'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#angvelxform_dot"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.angvelxform_dot" title="Permalink to this definition"></a></dt>
<dd><p>DEPRECATED, use <a class="reference internal" href="#spatialmath.base.transforms3d.rotvelxform" title="spatialmath.base.transforms3d.rotvelxform"><code class="xref py py-func docutils literal notranslate"><span class="pre">rotvelxform()</span></code></a> instead</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.delta2tr">
<span class="sig-name descname"><span class="pre">delta2tr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">d</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#delta2tr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.delta2tr" title="Permalink to this definition"></a></dt>
<dd><p>Convert differential motion to SE(3)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>Δ</strong> (<em>array_like</em><em>(</em><em>6</em><em>)</em>) – differential motion as a 6-vector</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SE(3) matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(4,4)</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">delta2tr(Δ)</span></code> is an SE(3) matrix representing differential
motion <span class="math notranslate nohighlight">\(\Delta = [\delta_x, \delta_y, \delta_z, \theta_x, \theta_y, \theta_z]\)</span>.</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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">delta2tr</span>
<span class="gp">>>> </span><span class="n">delta2tr</span><span class="p">([</span><span class="mf">0.001</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="mi">0</span><span class="p">,</span> <span class="mf">0.002</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="go">array([[ 1. , 0. , 0.002, 0.001],</span>
<span class="go"> [ 0. , 1. , 0. , 0. ],</span>
<span class="go"> [-0.002, 0. , 1. , 0. ],</span>
<span class="go"> [ 0. , 0. , 0. , 1. ]])</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Reference<span class="colon">:</span></dt>
<dd class="field-odd"><p>Robotics, Vision & Control for Python, Section 3.1, P. Corke, Springer 2023.</p>
</dd>
<dt class="field-even">Seealso<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.base.transforms3d.tr2delta" title="spatialmath.base.transforms3d.tr2delta"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2delta()</span></code></a></p>
</dd>
<dt class="field-odd">SymPy<span class="colon">:</span></dt>
<dd class="field-odd"><p>supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.eul2jac">
<span class="sig-name descname"><span class="pre">eul2jac</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">angles</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#eul2jac"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.eul2jac" title="Permalink to this definition"></a></dt>
<dd><p>Euler angle rate Jacobian</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>angles</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em>) – Euler angles (φ, θ, ψ)</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Jacobian matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3)</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">eul2jac(φ,</span> <span class="pre">θ,</span> <span class="pre">ψ)</span></code> is a Jacobian matrix (3x3) that maps ZYZ Euler angle
rates to angular velocity at the operating point specified by the Euler
angles φ, ϴ, ψ.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">eul2jac(𝚪)</span></code> as above but the Euler angles are taken from <code class="docutils literal notranslate"><span class="pre">𝚪</span></code> which
is a 3-vector with values (φ θ ψ).</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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">eul2jac</span>
<span class="gp">>>> </span><span class="n">eul2jac</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">])</span>
<span class="go">array([[ 0. , -0.0998, 0.1977],</span>
<span class="go"> [ 0. , 0.995 , 0.0198],</span>
<span class="go"> [ 1. , 0. , 0.9801]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>Used in the creation of an analytical Jacobian.</p></li>
<li><p>Angles in radians, rates in radians/sec.</p></li>
</ul>
</div>
<dl class="field-list simple">
<dt class="field-odd">Reference<span class="colon">:</span></dt>
<dd class="field-odd"><p>Robotics, Vision & Control for Python, Section 8.1.3, P. Corke, Springer 2023.</p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>supported</p>
</dd>
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="#spatialmath.base.transforms3d.angvelxform" title="spatialmath.base.transforms3d.angvelxform"><code class="xref py py-func docutils literal notranslate"><span class="pre">angvelxform()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.rpy2jac" title="spatialmath.base.transforms3d.rpy2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">rpy2jac()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.exp2jac" title="spatialmath.base.transforms3d.exp2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">exp2jac()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.eul2r">
<span class="sig-name descname"><span class="pre">eul2r</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">phi</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">theta</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">psi</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">unit</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rad'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#eul2r"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.eul2r" title="Permalink to this definition"></a></dt>
<dd><p>Create an SO(3) rotation matrix from Euler angles</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>phi</strong> (<em>float</em>) – Z-axis rotation</p></li>
<li><p><strong>theta</strong> (<em>float</em>) – Y-axis rotation</p></li>
<li><p><strong>psi</strong> (<em>float</em>) – Z-axis rotation</p></li>
<li><p><strong>unit</strong> (<em>str</em>) – angular units: ‘rad’ [default], or ‘deg’</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SO(3) rotation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3)</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">R</span> <span class="pre">=</span> <span class="pre">eul2r(φ,</span> <span class="pre">θ,</span> <span class="pre">ψ)</span></code> is an SO(3) orthonornal rotation
matrix equivalent to the specified Euler angles. These correspond
to rotations about the Z, Y, Z axes respectively.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">R</span> <span class="pre">=</span> <span class="pre">eul2r(EUL)</span></code> as above but the Euler angles are taken from
<code class="docutils literal notranslate"><span class="pre">EUL</span></code> which is a 3-vector with values (φ θ ψ).</p></li>
</ul>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">eul2r</span>
<span class="gp">>>> </span><span class="n">eul2r</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">)</span>
<span class="go">array([[ 0.9021, -0.3836, 0.1977],</span>
<span class="go"> [ 0.3875, 0.9216, 0.0198],</span>
<span class="go"> [-0.1898, 0.0587, 0.9801]])</span>
<span class="gp">>>> </span><span class="n">eul2r</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">])</span>
<span class="go">array([[ 0.9021, -0.3836, 0.1977],</span>
<span class="go"> [ 0.3875, 0.9216, 0.0198],</span>
<span class="go"> [-0.1898, 0.0587, 0.9801]])</span>
<span class="gp">>>> </span><span class="n">eul2r</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">],</span> <span class="n">unit</span><span class="o">=</span><span class="s1">'deg'</span><span class="p">)</span>
<span class="go">array([[ 0.7146, -0.6131, 0.3368],</span>
<span class="go"> [ 0.6337, 0.7713, 0.0594],</span>
<span class="go"> [-0.2962, 0.171 , 0.9397]])</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.base.transforms3d.rpy2r" title="spatialmath.base.transforms3d.rpy2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">rpy2r()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.eul2tr" title="spatialmath.base.transforms3d.eul2tr"><code class="xref py py-func docutils literal notranslate"><span class="pre">eul2tr()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2eul" title="spatialmath.base.transforms3d.tr2eul"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2eul()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.eul2tr">
<span class="sig-name descname"><span class="pre">eul2tr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">phi</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">theta</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">psi</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">unit</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rad'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#eul2tr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.eul2tr" title="Permalink to this definition"></a></dt>
<dd><p>Create an SE(3) pure rotation matrix from Euler angles</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>phi</strong> (<em>float</em>) – Z-axis rotation</p></li>
<li><p><strong>theta</strong> (<em>float</em>) – Y-axis rotation</p></li>
<li><p><strong>psi</strong> (<em>float</em>) – Z-axis rotation</p></li>
<li><p><strong>unit</strong> (<em>str</em>) – angular units: ‘rad’ [default], or ‘deg’</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SE(3) transformation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(4,4)</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">R</span> <span class="pre">=</span> <span class="pre">eul2tr(PHI,</span> <span class="pre">θ,</span> <span class="pre">PSI)</span></code> is an SE(3) homogeneous transformation
matrix equivalent to the specified Euler angles. These correspond
to rotations about the Z, Y, Z axes respectively.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">R</span> <span class="pre">=</span> <span class="pre">eul2tr(EUL)</span></code> as above but the Euler angles are taken from
<code class="docutils literal notranslate"><span class="pre">EUL</span></code> which is a 3-vector with values
(PHI θ PSI).</p></li>
</ul>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">eul2tr</span>
<span class="gp">>>> </span><span class="n">eul2tr</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">)</span>
<span class="go">array([[ 0.9021, -0.3836, 0.1977, 0. ],</span>
<span class="go"> [ 0.3875, 0.9216, 0.0198, 0. ],</span>
<span class="go"> [-0.1898, 0.0587, 0.9801, 0. ],</span>
<span class="go"> [ 0. , 0. , 0. , 1. ]])</span>
<span class="gp">>>> </span><span class="n">eul2tr</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">])</span>
<span class="go">array([[ 0.9021, -0.3836, 0.1977, 0. ],</span>
<span class="go"> [ 0.3875, 0.9216, 0.0198, 0. ],</span>
<span class="go"> [-0.1898, 0.0587, 0.9801, 0. ],</span>
<span class="go"> [ 0. , 0. , 0. , 1. ]])</span>
<span class="gp">>>> </span><span class="n">eul2tr</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">],</span> <span class="n">unit</span><span class="o">=</span><span class="s1">'deg'</span><span class="p">)</span>
<span class="go">array([[ 0.7146, -0.6131, 0.3368, 0. ],</span>
<span class="go"> [ 0.6337, 0.7713, 0.0594, 0. ],</span>
<span class="go"> [-0.2962, 0.171 , 0.9397, 0. ],</span>
<span class="go"> [ 0. , 0. , 0. , 1. ]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>By default, the translational component is zero but it can be
set to a non-zero value.</p>
</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.base.transforms3d.rpy2tr" title="spatialmath.base.transforms3d.rpy2tr"><code class="xref py py-func docutils literal notranslate"><span class="pre">rpy2tr()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.eul2r" title="spatialmath.base.transforms3d.eul2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">eul2r()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2eul" title="spatialmath.base.transforms3d.tr2eul"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2eul()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.exp2jac">
<span class="sig-name descname"><span class="pre">exp2jac</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/base/transforms3d.html#exp2jac"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.exp2jac" title="Permalink to this definition"></a></dt>
<dd><p>Jacobian from exponential coordinate rates to angular velocity</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>array_like</em><em>(</em><em>3</em><em>)</em>) – Exponential coordinates</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Jacobian matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3)</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">exp2jac(v)</span></code> is a Jacobian matrix (3x3) that maps exponential coordinate
rates to angular velocity at the operating point <code class="docutils literal notranslate"><span class="pre">v</span></code>.</p></li>
</ul>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">exp2jac</span>
<span class="gp">>>> </span><span class="n">exp2jac</span><span class="p">([</span><span class="mf">0.3</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="go">array([[ 1. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9851, -0.1489],</span>
<span class="go"> [ 0. , 0.1489, 0.9851]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>Used in the creation of an analytical Jacobian.</p></li>
</ul>
</div>
<p>Reference:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">-</span> <span class="n">A</span> <span class="n">compact</span> <span class="n">formula</span> <span class="k">for</span> <span class="n">the</span> <span class="n">derivative</span> <span class="n">of</span> <span class="n">a</span> <span class="mi">3</span><span class="o">-</span><span class="n">D</span> <span class="n">rotation</span> <span class="ow">in</span>
<span class="n">exponential</span> <span class="n">coordinate</span>
<span class="n">Guillermo</span> <span class="n">Gallego</span><span class="p">,</span> <span class="n">Anthony</span> <span class="n">Yezzi</span>
<span class="n">https</span><span class="p">:</span><span class="o">//</span><span class="n">arxiv</span><span class="o">.</span><span class="n">org</span><span class="o">/</span><span class="n">pdf</span><span class="o">/</span><span class="mf">1312.0788</span><span class="n">v1</span><span class="o">.</span><span class="n">pdf</span>
<span class="o">-</span> <span class="n">Robot</span> <span class="n">Dynamics</span> <span class="n">Lecture</span> <span class="n">Notes</span>
<span class="n">Robotic</span> <span class="n">Systems</span> <span class="n">Lab</span><span class="p">,</span> <span class="n">ETH</span> <span class="n">Zurich</span><span class="p">,</span> <span class="mi">2018</span>
<span class="n">https</span><span class="p">:</span><span class="o">//</span><span class="n">ethz</span><span class="o">.</span><span class="n">ch</span><span class="o">/</span><span class="n">content</span><span class="o">/</span><span class="n">dam</span><span class="o">/</span><span class="n">ethz</span><span class="o">/</span><span class="n">special</span><span class="o">-</span><span class="n">interest</span><span class="o">/</span><span class="n">mavt</span><span class="o">/</span><span class="n">robotics</span><span class="o">-</span><span class="n">n</span><span class="o">-</span><span class="n">intelligent</span><span class="o">-</span><span class="n">systems</span><span class="o">/</span><span class="n">rsl</span><span class="o">-</span><span class="n">dam</span><span class="o">/</span><span class="n">documents</span><span class="o">/</span><span class="n">RobotDynamics2018</span><span class="o">/</span><span class="n">RD_HS2018script</span><span class="o">.</span><span class="n">pdf</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">SymPy<span class="colon">:</span></dt>
<dd class="field-odd"><p>supported</p>
</dd>
<dt class="field-even">Seealso<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.base.transforms3d.rotvelxform" title="spatialmath.base.transforms3d.rotvelxform"><code class="xref py py-func docutils literal notranslate"><span class="pre">rotvelxform()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.eul2jac" title="spatialmath.base.transforms3d.eul2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">eul2jac()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.rpy2jac" title="spatialmath.base.transforms3d.rpy2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">rpy2jac()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.exp2r">
<span class="sig-name descname"><span class="pre">exp2r</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">w</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#exp2r"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.exp2r" title="Permalink to this definition"></a></dt>
<dd><p>Create an SO(3) rotation matrix from exponential coordinates</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>w</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em>) – exponential coordinate vector</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SO(3) rotation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad arguments</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">exp2r(w)</span></code> is an SO(3) orthonormal rotation matrix
equivalent to a rotation of <span class="math notranslate nohighlight">\(\| w \|\)</span> about the vector <span class="math notranslate nohighlight">\(\hat{w}\)</span>.</p>
<p>If <code class="docutils literal notranslate"><span class="pre">w</span></code> is zero then result is the identity matrix.</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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">exp2r</span><span class="p">,</span> <span class="n">rotx</span>
<span class="gp">>>> </span><span class="n">exp2r</span><span class="p">([</span><span class="mf">0.3</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="go">array([[ 1. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955],</span>
<span class="go"> [ 0. , 0.2955, 0.9553]])</span>
<span class="gp">>>> </span><span class="n">rotx</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span>
<span class="go">array([[ 1. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955],</span>
<span class="go"> [ 0. , 0.2955, 0.9553]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Exponential coordinates are also known as an Euler vector</p>
</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.base.transforms3d.angvec2r" title="spatialmath.base.transforms3d.angvec2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">angvec2r()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2angvec" title="spatialmath.base.transforms3d.tr2angvec"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2angvec()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>not supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.exp2tr">
<span class="sig-name descname"><span class="pre">exp2tr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">w</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#exp2tr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.exp2tr" title="Permalink to this definition"></a></dt>
<dd><p>Create an SE(3) pure rotation matrix from exponential coordinates</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>w</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em>) – exponential coordinate vector</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SO(3) rotation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad arguments</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">exp2r(w)</span></code> is an SO(3) orthonormal rotation matrix
equivalent to a rotation of <span class="math notranslate nohighlight">\(\| w \|\)</span> about the vector <span class="math notranslate nohighlight">\(\hat{w}\)</span>.</p>
<p>If <code class="docutils literal notranslate"><span class="pre">w</span></code> is zero then result is the identity matrix.</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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">exp2tr</span><span class="p">,</span> <span class="n">trotx</span>
<span class="gp">>>> </span><span class="n">exp2tr</span><span class="p">([</span><span class="mf">0.3</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="go">array([[ 1. , 0. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955, 0. ],</span>
<span class="go"> [ 0. , 0.2955, 0.9553, 0. ],</span>
<span class="go"> [ 0. , 0. , 0. , 1. ]])</span>
<span class="gp">>>> </span><span class="n">trotx</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span>
<span class="go">array([[ 1. , 0. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955, 0. ],</span>
<span class="go"> [ 0. , 0.2955, 0.9553, 0. ],</span>
<span class="go"> [ 0. , 0. , 0. , 1. ]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Exponential coordinates are also known as an Euler vector</p>
</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.base.transforms3d.angvec2r" title="spatialmath.base.transforms3d.angvec2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">angvec2r()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2angvec" title="spatialmath.base.transforms3d.tr2angvec"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2angvec()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>not supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.ishom">
<span class="sig-name descname"><span class="pre">ishom</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">T</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">False</span></span></em>, <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/base/transforms3d.html#ishom"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.ishom" title="Permalink to this definition"></a></dt>
<dd><p>Test if matrix belongs to SE(3)</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>T</strong> (<em>numpy</em><em>(</em><em>4</em><em>,</em><em>4</em><em>)</em>) – SE(3) matrix to test</p></li>
<li><p><strong>check</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">bool</span></code></span>) – check validity of rotation submatrix</p></li>
<li><p><strong>tol</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></span>) – Tolerance in units of eps for rotation submatrix check, defaults to 20</p></li>
</ul>
</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>
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>whether matrix is an SE(3) homogeneous transformation matrix</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">ishom(T)</span></code> is True if the argument <code class="docutils literal notranslate"><span class="pre">T</span></code> is of dimension 4x4</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">ishom(T,</span> <span class="pre">check=True)</span></code> as above, but also checks orthogonality of the
rotation sub-matrix and validitity of the bottom row.</p></li>
</ul>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">ishom</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</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="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</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="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</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="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">ishom</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</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="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</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="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span> <span class="c1"># invalid SE(3)</span>
<span class="gp">>>> </span><span class="n">ishom</span><span class="p">(</span><span class="n">T</span><span class="p">)</span> <span class="c1"># a quick check says it is an SE(3)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">ishom</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="c1"># but if we check more carefully...</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">ishom</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
<span class="go">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="func_nd.html#spatialmath.base.transformsNd.isR" title="spatialmath.base.transformsNd.isR"><code class="xref py py-func docutils literal notranslate"><span class="pre">isR()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.isrot" title="spatialmath.base.transforms3d.isrot"><code class="xref py py-func docutils literal notranslate"><span class="pre">isrot()</span></code></a> <a class="reference internal" href="func_2d.html#spatialmath.base.transforms2d.ishom2" title="spatialmath.base.transforms2d.ishom2"><code class="xref py py-func docutils literal notranslate"><span class="pre">ishom2()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.isrot">
<span class="sig-name descname"><span class="pre">isrot</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">R</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">False</span></span></em>, <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/base/transforms3d.html#isrot"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.isrot" title="Permalink to this definition"></a></dt>
<dd><p>Test if matrix belongs to SO(3)</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>R</strong> (<em>numpy</em><em>(</em><em>3</em><em>,</em><em>3</em><em>)</em>) – SO(3) matrix to test</p></li>
<li><p><strong>check</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">bool</span></code></span>) – check validity of rotation submatrix</p></li>
<li><p><strong>tol</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></span>) – Tolerance in units of eps for rotation matrix test, defaults to 20</p></li>
</ul>
</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>
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>whether matrix is an SO(3) rotation matrix</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">isrot(R)</span></code> is True if the argument <code class="docutils literal notranslate"><span class="pre">R</span></code> is of dimension 3x3</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">isrot(R,</span> <span class="pre">check=True)</span></code> as above, but also checks orthogonality of the
rotation matrix.</p></li>
</ul>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">isrot</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</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="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</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="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</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="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">isrot</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</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="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">isrot</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">R</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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="mi">1</span><span class="p">]])</span> <span class="c1"># invalid SO(3)</span>
<span class="gp">>>> </span><span class="n">isrot</span><span class="p">(</span><span class="n">R</span><span class="p">)</span> <span class="c1"># a quick check says it is an SO(3)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isrot</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="c1"># but if we check more carefully...</span>
<span class="go">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="func_nd.html#spatialmath.base.transformsNd.isR" title="spatialmath.base.transformsNd.isR"><code class="xref py py-func docutils literal notranslate"><span class="pre">isR()</span></code></a> <a class="reference internal" href="func_2d.html#spatialmath.base.transforms2d.isrot2" title="spatialmath.base.transforms2d.isrot2"><code class="xref py py-func docutils literal notranslate"><span class="pre">isrot2()</span></code></a>, <a class="reference internal" href="#spatialmath.base.transforms3d.ishom" title="spatialmath.base.transforms3d.ishom"><code class="xref py py-func docutils literal notranslate"><span class="pre">ishom()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.oa2r">
<span class="sig-name descname"><span class="pre">oa2r</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">o</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#oa2r"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.oa2r" title="Permalink to this definition"></a></dt>
<dd><p>Create SO(3) rotation matrix from two vectors</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>o</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em>) – 3D vector parallel to Y- axis</p></li>
<li><p><strong>a</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-data docutils literal notranslate"><span class="pre">Union</span></code>[<code class="xref py py-class docutils literal notranslate"><span class="pre">List</span></code>, <code class="xref py py-data docutils literal notranslate"><span class="pre">Tuple</span></code>[<code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code>, <code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code>, <code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code>], <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.2)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>[<code class="xref py py-data docutils literal notranslate"><span class="pre">Any</span></code>, <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.dtype.html#numpy.dtype" title="(in NumPy v2.2)"><code class="xref py py-class docutils literal notranslate"><span class="pre">dtype</span></code></a>[<code class="xref py py-class docutils literal notranslate"><span class="pre">TypeVar</span></code>(<code class="docutils literal notranslate"><span class="pre">ScalarType</span></code>, bound= <a class="reference external" href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.generic" title="(in NumPy v2.2)"><code class="xref py py-class docutils literal notranslate"><span class="pre">generic</span></code></a>, covariant=True)]]]</span>) – 3D vector parallel to the Z-axis</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SO(3) rotation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3)</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">T</span> <span class="pre">=</span> <span class="pre">oa2tr(O,</span> <span class="pre">A)</span></code> is an SO(3) orthonormal rotation matrix for a frame
defined in terms of vectors parallel to its Y- and Z-axes with respect to a
reference frame. In robotics these axes are respectively called the
orientation and approach vectors defined such that R = [N O A] and N = O x
A.</p>
<p>Steps:</p>
<blockquote>
<div><ol class="arabic simple">
<li><p>N’ = O x A</p></li>
<li><p>O’ = A x N</p></li>
<li><p>normalize N’, O’, A</p></li>
<li><p>stack horizontally into rotation matrix</p></li>
</ol>
</div></blockquote>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">oa2r</span>
<span class="gp">>>> </span><span class="n">oa2r</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="c1"># Y := Y, Z := -Z</span>
<span class="go">array([[-1., 0., 0.],</span>
<span class="go"> [ 0., 1., 0.],</span>
<span class="go"> [ 0., 0., -1.]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>The A vector is the only guaranteed to have the same direction in the
resulting rotation matrix</p></li>
<li><p>O and A do not have to be unit-length, they are normalized</p></li>
<li><p>O and A do not have to be orthogonal, so long as they are not parallel</p></li>
<li><p>The vectors O and A are parallel to the Y- and Z-axes of the
equivalent coordinate frame.</p></li>
</ul>
</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.base.transforms3d.oa2tr" title="spatialmath.base.transforms3d.oa2tr"><code class="xref py py-func docutils literal notranslate"><span class="pre">oa2tr()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>not supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.oa2tr">
<span class="sig-name descname"><span class="pre">oa2tr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">o</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#oa2tr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.oa2tr" title="Permalink to this definition"></a></dt>
<dd><p>Create SE(3) pure rotation from two vectors</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>o</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em>) – 3D vector parallel to Y- axis</p></li>
<li><p><strong>a</strong> (<span class="sphinx_autodoc_typehints-type"><code class="xref py py-data docutils literal notranslate"><span class="pre">Union</span></code>[<code class="xref py py-class docutils literal notranslate"><span class="pre">List</span></code>, <code class="xref py py-data docutils literal notranslate"><span class="pre">Tuple</span></code>[<code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code>, <code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code>, <code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code>], <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.2)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>[<code class="xref py py-data docutils literal notranslate"><span class="pre">Any</span></code>, <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.dtype.html#numpy.dtype" title="(in NumPy v2.2)"><code class="xref py py-class docutils literal notranslate"><span class="pre">dtype</span></code></a>[<code class="xref py py-class docutils literal notranslate"><span class="pre">TypeVar</span></code>(<code class="docutils literal notranslate"><span class="pre">ScalarType</span></code>, bound= <a class="reference external" href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.generic" title="(in NumPy v2.2)"><code class="xref py py-class docutils literal notranslate"><span class="pre">generic</span></code></a>, covariant=True)]]]</span>) – 3D vector parallel to the Z-axis</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SE(3) transformation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(4,4)</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">T</span> <span class="pre">=</span> <span class="pre">oa2tr(O,</span> <span class="pre">A)</span></code> is an SE(3) homogeneous transformation matrix for a
frame defined in terms of vectors parallel to its Y- and Z-axes with respect
to a reference frame. In robotics these axes are respectively called the
orientation and approach vectors defined such that R = [N O A] and N = O x
A.</p>
<p>Steps:</p>
<blockquote>
<div><ol class="arabic simple">
<li><p>N’ = O x A</p></li>
<li><p>O’ = A x N</p></li>
<li><p>normalize N’, O’, A</p></li>
<li><p>stack horizontally into rotation matrix</p></li>
</ol>
</div></blockquote>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="n">oa2tr</span>
<span class="gp">>>> </span><span class="n">oa2tr</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="c1"># Y := Y, Z := -Z</span>
<span class="go">array([[-1., 0., 0., 0.],</span>
<span class="go"> [ 0., 1., 0., 0.],</span>
<span class="go"> [ 0., 0., -1., 0.],</span>
<span class="go"> [ 0., 0., 0., 1.]])</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.base.transforms3d.oa2r" title="spatialmath.base.transforms3d.oa2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">oa2r()</span></code></a></p>
</dd>
<dt class="field-even">SymPy<span class="colon">:</span></dt>
<dd class="field-even"><p>not supported</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.r2x">
<span class="sig-name descname"><span class="pre">r2x</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">R</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">representation</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rpy/xyz'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#r2x"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.r2x" title="Permalink to this definition"></a></dt>
<dd><p>Convert SO(3) matrix to angular representation</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>R</strong> (<em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>)</em>) – SO(3) rotation matrix</p></li>
<li><p><strong>representation</strong> (<em>str</em><em>, </em><em>optional</em>) – rotational representation, defaults to “rpy/xyz”</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>angular representation</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3)</p>
</dd>
</dl>
<p>Convert an SO(3) rotation matrix to a minimal rotational representation
<span class="math notranslate nohighlight">\(\vec{\Gamma} \in \mathbb{R}^3\)</span>.</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">representation</span></code></p></th>
<th class="head"><p>Rotational representation</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">"rpy/xyz"</span></code> <code class="docutils literal notranslate"><span class="pre">"arm"</span></code></p></td>
<td><p>RPY angular rates in XYZ order (default)</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">"rpy/zyx"</span></code> <code class="docutils literal notranslate"><span class="pre">"vehicle"</span></code></p></td>
<td><p>RPY angular rates in XYZ order</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">"rpy/yxz"</span></code> <code class="docutils literal notranslate"><span class="pre">"camera"</span></code></p></td>
<td><p>RPY angular rates in YXZ order</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">"eul"</span></code></p></td>
<td><p>Euler angular rates in ZYZ order</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">"exp"</span></code></p></td>
<td><p>exponential coordinate rates</p></td>
</tr>
</tbody>
</table>
<dl class="field-list simple">
<dt class="field-odd">SymPy<span class="colon">:</span></dt>
<dd class="field-odd"><p>supported</p>
</dd>
<dt class="field-even">Seealso<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.base.transforms3d.x2r" title="spatialmath.base.transforms3d.x2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">x2r()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2rpy" title="spatialmath.base.transforms3d.tr2rpy"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2rpy()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.tr2eul" title="spatialmath.base.transforms3d.tr2eul"><code class="xref py py-func docutils literal notranslate"><span class="pre">tr2eul()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.trlog" title="spatialmath.base.transforms3d.trlog"><code class="xref py py-func docutils literal notranslate"><span class="pre">trlog()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.rodrigues">
<span class="sig-name descname"><span class="pre">rodrigues</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">w</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">theta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#rodrigues"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.rodrigues" title="Permalink to this definition"></a></dt>
<dd><p>Rodrigues’ formula for 3D rotation</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>w</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em>) – rotation vector</p></li>
<li><p><strong>theta</strong> (<em>float</em><em> or </em><em>None</em>) – rotation angle</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SO(3) matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3)</p>
</dd>
</dl>
<p>Compute Rodrigues’ formula for a rotation matrix given a rotation axis
and angle.</p>
<div class="math notranslate nohighlight">
\[\mat{R} = \mat{I}_{3 \times 3} + \sin \theta \skx{\hat{\vec{v}}} + (1 - \cos \theta) \skx{\hat{\vec{v}}}^2\]</div>
<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.base</span><span class="w"> </span><span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">rodrigues</span><span class="p">([</span><span class="mi">1</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="mf">0.3</span><span class="p">)</span>
<span class="go">array([[ 1. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955],</span>
<span class="go"> [ 0. , 0.2955, 0.9553]])</span>
<span class="gp">>>> </span><span class="n">rodrigues</span><span class="p">([</span><span class="mf">0.3</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="go">array([[ 1. , 0. , 0. ],</span>
<span class="go"> [ 0. , 0.9553, -0.2955],</span>
<span class="go"> [ 0. , 0.2955, 0.9553]])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.rot2jac">
<span class="sig-name descname"><span class="pre">rot2jac</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">R</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">representation</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rpy/xyz'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#rot2jac"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.rot2jac" title="Permalink to this definition"></a></dt>
<dd><p>DEPRECATED, use <a class="reference internal" href="#spatialmath.base.transforms3d.rotvelxform" title="spatialmath.base.transforms3d.rotvelxform"><code class="xref py py-func docutils literal notranslate"><span class="pre">rotvelxform()</span></code></a> instead</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transforms3d.rotvelxform">
<span class="sig-name descname"><span class="pre">rotvelxform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Γ</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">inverse</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">full</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">representation</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rpy/xyz'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transforms3d.html#rotvelxform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transforms3d.rotvelxform" title="Permalink to this definition"></a></dt>
<dd><p>Rotational velocity transformation</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>𝚪</strong> (<em>array_like</em><em>(</em><em>3</em><em>) or </em><em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>)</em>) – angular representation or rotation matrix</p></li>
<li><p><strong>representation</strong> (<em>str</em><em>, </em><em>optional</em>) – defaults to ‘rpy/xyz’</p></li>
<li><p><strong>inverse</strong> (<em>bool</em>) – compute mapping from analytical rates to angular velocity</p></li>
<li><p><strong>full</strong> (<em>bool</em>) – return 6x6 transform for spatial velocity</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>rotation rate transformation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3) or ndarray(6,6)</p>
</dd>
</dl>
<p>Computes the transformation from analytical rates
<span class="math notranslate nohighlight">\(\dvec{x}\)</span> where the rotational part is expressed as the rate of change in
some angular representation to spatial velocity <span class="math notranslate nohighlight">\(\omega\)</span>, where
rotation rate is expressed as angular velocity.</p>
<div class="math notranslate nohighlight">
\[\vec{\omega} = \mat{A}(\Gamma) \dvec{x}\]</div>
<p>where <span class="math notranslate nohighlight">\(\mat{A}\)</span> is a 3x3 matrix and <span class="math notranslate nohighlight">\(\Gamma \in
\mathbb{R}^3\)</span> is a minimal angular representation.</p>
<p><span class="math notranslate nohighlight">\(\mat{A}(\Gamma)\)</span> is a function of the rotational representation
which can be specified by the parameter <code class="docutils literal notranslate"><span class="pre">𝚪</span></code> as a 1D array, or by
an SO(3) rotation matrix which will be converted to the <code class="docutils literal notranslate"><span class="pre">representation</span></code>.</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">representation</span></code></p></th>
<th class="head"><p>Rotational representation</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">"rpy/xyz"</span></code> <code class="docutils literal notranslate"><span class="pre">"arm"</span></code></p></td>
<td><p>RPY angular rates in XYZ order (default)</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">"rpy/zyx"</span></code> <code class="docutils literal notranslate"><span class="pre">"vehicle"</span></code></p></td>
<td><p>RPY angular rates in XYZ order</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">"rpy/yxz"</span></code> <code class="docutils literal notranslate"><span class="pre">"camera"</span></code></p></td>
<td><p>RPY angular rates in YXZ order</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">"eul"</span></code></p></td>
<td><p>Euler angular rates in ZYZ order</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">"exp"</span></code></p></td>
<td><p>exponential coordinate rates</p></td>
</tr>
</tbody>
</table>
<p>If <code class="docutils literal notranslate"><span class="pre">inverse==True</span></code> return <span class="math notranslate nohighlight">\(\mat{A}^{-1}\)</span> computed using
a closed-form solution rather than matrix inverse.</p>
<p>If <code class="docutils literal notranslate"><span class="pre">full=True</span></code> a block diagonal 6x6 matrix is returned which transforms analytic
velocity to spatial velocity.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Similar to <a class="reference internal" href="#spatialmath.base.transforms3d.eul2jac" title="spatialmath.base.transforms3d.eul2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">eul2jac()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.rpy2jac" title="spatialmath.base.transforms3d.rpy2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">rpy2jac()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.exp2jac" title="spatialmath.base.transforms3d.exp2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">exp2jac()</span></code></a>
with <code class="docutils literal notranslate"><span class="pre">full=False</span></code>.</p>
</div>
<p>The analytical Jacobian is</p>
<div class="math notranslate nohighlight">
\[\mat{J}_a(q) = \mat{A}^{-1}(\Gamma)\, \mat{J}(q)\]</div>
<p>where <span class="math notranslate nohighlight">\(\mat{A}\)</span> is computed with <code class="docutils literal notranslate"><span class="pre">inverse==True</span></code> and <code class="docutils literal notranslate"><span class="pre">full=True</span></code>.</p>
<p>Reference:</p>
<blockquote>
<div><ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">symbolic/angvelxform.ipynb</span></code> in this Toolbox</p></li>
<li><p>Robot Dynamics Lecture Notes, Robotic Systems Lab, ETH Zurich, 2018
<a class="reference external" href="https://ethz.ch/content/dam/ethz/special-interest/mavt/robotics-n-intelligent-systems/rsl-dam/documents/RobotDynamics2018/RD_HS2018script.pdf">https://ethz.ch/content/dam/ethz/special-interest/mavt/robotics-n-intelligent-systems/rsl-dam/documents/RobotDynamics2018/RD_HS2018script.pdf</a></p></li>
</ul>
</div></blockquote>
<dl class="field-list simple">
<dt class="field-odd">SymPy<span class="colon">:</span></dt>
<dd class="field-odd"><p>supported</p>
</dd>
<dt class="field-even">Seealso<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference internal" href="#spatialmath.base.transforms3d.rotvelxform_inv_dot" title="spatialmath.base.transforms3d.rotvelxform_inv_dot"><code class="xref py py-func docutils literal notranslate"><span class="pre">rotvelxform_inv_dot()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.eul2jac" title="spatialmath.base.transforms3d.eul2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">eul2jac()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.rpy2r" title="spatialmath.base.transforms3d.rpy2r"><code class="xref py py-func docutils literal notranslate"><span class="pre">rpy2r()</span></code></a> <a class="reference internal" href="#spatialmath.base.transforms3d.exp2jac" title="spatialmath.base.transforms3d.exp2jac"><code class="xref py py-func docutils literal notranslate"><span class="pre">exp2jac()</span></code></a></p>
</dd>
</dl>
</dd></dl>