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<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transformsNd.Ab2M"><code class="docutils literal notranslate"><span class="pre">Ab2M()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#spatialmath.base.transformsNd.det"><code class="docutils literal notranslate"><span class="pre">det()</span></code></a></li>
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<section id="module-spatialmath.base.transformsNd">
<span id="transforms-in-nd"></span><h1>Transforms in ND<a class="headerlink" href="#module-spatialmath.base.transformsNd" title="Permalink to this heading"></a></h1>
<p>This modules contains functions to operate on special matrices in 2D or 3D, for
example SE(n), SO(n), se(n) and so(n) where n is 2 or 3.</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.transformsNd.Ab2M">
<span class="sig-name descname"><span class="pre">Ab2M</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#Ab2M"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.Ab2M" title="Permalink to this definition"></a></dt>
<dd><p>Pack matrix and vector to matrix</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>A</strong> (<em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>) or </em><em>ndarray</em><em>(</em><em>2</em><em>,</em><em>2</em><em>)</em>) – square matrix</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em>(</em><em>3</em><em>) or </em><em>ndarray</em><em>(</em><em>2</em><em>)</em>) – translation vector</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(4,4) or 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">M</span> <span class="pre">=</span> <span class="pre">Ab2M(A,</span> <span class="pre">b)</span></code> is a matrix (N+1xN+1) formed from a matrix <code class="docutils literal notranslate"><span class="pre">R</span></code> (NxN) and a vector <code class="docutils literal notranslate"><span class="pre">t</span></code>
(Nx1). The bottom row is all zeros.</p>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">A</span></code> is 2x2 and <code class="docutils literal notranslate"><span class="pre">b</span></code> is 2x1, then <code class="docutils literal notranslate"><span class="pre">M</span></code> is 3x3</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">A</span></code> is 3x3 and <code class="docutils literal notranslate"><span class="pre">b</span></code> is 3x1, then <code class="docutils literal notranslate"><span class="pre">M</span></code> is 4x4</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="o">*</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]]</span><span class="o">.</span><span class="n">T</span>
<span class="gp">>>> </span><span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">Ab2M</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">array([[1., 2., 5.],</span>
<span class="go"> [3., 4., 6.],</span>
<span class="go"> [0., 0., 0.]])</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p>rt2tr, tr2rt, r2t</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.det">
<span class="sig-name descname"><span class="pre">det</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">m</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#det"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.det" title="Permalink to this definition"></a></dt>
<dd><p>Determinant of matrix</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>m</strong> (<span class="sphinx_autodoc_typehints-type"><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></span>) – any square matrix</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>determinant</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>float</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">det(v)</span></code> is the determinant of the matrix <code class="docutils literal notranslate"><span class="pre">m</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="o">*</span>
<span class="gp">>>> </span><span class="n">norm</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<span class="go">5.0</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 external" href="https://numpy.org/doc/stable/reference/generated/numpy.linalg.det.html#numpy.linalg.det" title="(in NumPy v2.2)"><code class="xref py py-func docutils literal notranslate"><span class="pre">det()</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.transformsNd.e2h">
<span class="sig-name descname"><span class="pre">e2h</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/transformsNd.html#e2h"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.e2h" title="Permalink to this definition"></a></dt>
<dd><p>Convert from Euclidean to homogeneous form</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>n</em><em>)</em><em>, </em><em>ndarray</em><em>(</em><em>n</em><em>,</em><em>m</em><em>)</em>) – Euclidean vector or matrix</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>homogeneous vector</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(n+1,m)</p>
</dd>
</dl>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">v</span></code> is an N-vector, return an (N+1)-column vector where a value of 1 has
been appended as the last element.</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">v</span></code> is a matrix (NxM), return a matrix (N+1xM), where each column has
been appended with a value of 1, ie. a row of ones has been appended to the 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="o">*</span>
<span class="gp">>>> </span><span class="n">e2h</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
<span class="go">array([[2.],</span>
<span class="go"> [4.],</span>
<span class="go"> [6.],</span>
<span class="go"> [1.]])</span>
<span class="gp">>>> </span><span class="n">e</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">e</span>
<span class="go">array([[1, 3, 5],</span>
<span class="go"> [2, 4, 6]])</span>
<span class="gp">>>> </span><span class="n">e2h</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<span class="go">array([[1., 3., 5.],</span>
<span class="go"> [2., 4., 6.],</span>
<span class="go"> [1., 1., 1.]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The result is always a 2D array, a 1D input results in a column vector.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p>e2h</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.h2e">
<span class="sig-name descname"><span class="pre">h2e</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/transformsNd.html#h2e"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.h2e" title="Permalink to this definition"></a></dt>
<dd><p>Convert from homogeneous to Euclidean form</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>n</em><em>)</em><em>, </em><em>ndarray</em><em>(</em><em>n</em><em>,</em><em>m</em><em>)</em>) – homogeneous vector or matrix</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Euclidean vector</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(n-1), ndarray(n-1,m)</p>
</dd>
</dl>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">v</span></code> is an N-vector, return an (N-1)-column vector where the elements have
all been scaled by the last element of <code class="docutils literal notranslate"><span class="pre">v</span></code>.</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">v</span></code> is a matrix (NxM), return a matrix (N-1xM), where each column has
been scaled by its last element.</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="o">*</span>
<span class="gp">>>> </span><span class="n">h2e</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">array([[2.],</span>
<span class="go"> [4.],</span>
<span class="go"> [6.]])</span>
<span class="gp">>>> </span><span class="n">h2e</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="go">array([[1.],</span>
<span class="go"> [2.],</span>
<span class="go"> [3.]])</span>
<span class="gp">>>> </span><span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">h</span>
<span class="go">array([[1, 3, 5],</span>
<span class="go"> [2, 4, 6],</span>
<span class="go"> [1, 2, 1]])</span>
<span class="gp">>>> </span><span class="n">h2e</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
<span class="go">array([[1. , 1.5, 5. ],</span>
<span class="go"> [2. , 2. , 6. ]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The result is always a 2D array, a 1D input results in a column vector.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p>e2h</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.homtrans">
<span class="sig-name descname"><span class="pre">homtrans</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">p</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#homtrans"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.homtrans" title="Permalink to this definition"></a></dt>
<dd><p>Apply a homogeneous transformation to a Euclidean vector</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 array</em><em> (</em><em>n</em><em>,</em><em>n</em><em>)</em>) – homogeneous transformation</p></li>
<li><p><strong>p</strong> (<em>array_like</em><em>(</em><em>n-1</em><em>)</em><em>, </em><em>ndarray</em><em>(</em><em>n-1</em><em>,</em><em>m</em><em>)</em>) – Vector(s) to be transformed</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>transformed Euclidean vector(s)</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(n-1,m)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">homtrans(T,</span> <span class="pre">p)</span></code> applies the homogeneous transformation <code class="docutils literal notranslate"><span class="pre">T</span></code> to the Euclidean points
stored columnwise in the array <code class="docutils literal notranslate"><span class="pre">p</span></code>.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">homtrans(T,</span> <span class="pre">v)</span></code> as above but <code class="docutils literal notranslate"><span class="pre">v</span></code> is a 1D array considered to be a column vector, and the
retured value will be a column vector.</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="o">*</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">trotx</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">h2e</span><span class="p">(</span> <span class="n">T</span> <span class="o">@</span> <span class="n">e2h</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
<span class="go">array([[1. ],</span>
<span class="go"> [1.0241],</span>
<span class="go"> [3.457 ]])</span>
<span class="gp">>>> </span><span class="n">homtrans</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">array([[1. ],</span>
<span class="go"> [1.0241],</span>
<span class="go"> [3.457 ]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>If T is a homogeneous transformation defining the pose of {B} with respect to {A},
then the points are defined with respect to frame {B} and are transformed to be
with respect to frame {A}.</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.transformsNd.e2h" title="spatialmath.base.transformsNd.e2h"><code class="xref py py-func docutils literal notranslate"><span class="pre">e2h()</span></code></a> <a class="reference internal" href="#spatialmath.base.transformsNd.h2e" title="spatialmath.base.transformsNd.h2e"><code class="xref py py-func docutils literal notranslate"><span class="pre">h2e()</span></code></a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.isR">
<span class="sig-name descname"><span class="pre">isR</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">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/transformsNd.html#isR"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.isR" title="Permalink to this definition"></a></dt>
<dd><p>Test if matrix belongs to SO(n)</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>2</em><em>,</em><em>2</em><em>) or </em><em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>)</em>) – matrix to test</p></li>
<li><p><strong>tol</strong> (<em>float</em>) – tolerance in units of eps, defaults to 20</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>whether matrix is a proper orthonormal rotation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>bool</p>
</dd>
</dl>
<p>Checks orthogonality, ie. <span class="math notranslate nohighlight">\({\bf R} {\bf R}^T = {\bf I}\)</span> and <span class="math notranslate nohighlight">\(\det({\bf R}) > 0\)</span>.
For the first test we check that the norm of the residual is less than <code class="docutils literal notranslate"><span class="pre">tol</span> <span class="pre">*</span> <span class="pre">eps</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="o">*</span>
<span class="gp">>>> </span><span class="n">isR</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isR</span><span class="p">(</span><span class="n">rot2</span><span class="p">(</span><span class="mf">0.5</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isR</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</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>isrot2, isrot</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.iseye">
<span class="sig-name descname"><span class="pre">iseye</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">S</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/transformsNd.html#iseye"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.iseye" title="Permalink to this definition"></a></dt>
<dd><p>Test if matrix is identity</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>S</strong> (<em>ndarray</em><em>(</em><em>n</em><em>,</em><em>n</em><em>)</em>) – matrix to test</p></li>
<li><p><strong>tol</strong> (<em>float</em>) – tolerance in units of eps, defaults to 20</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>whether matrix is a proper skew-symmetric matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>bool</p>
</dd>
</dl>
<p>Check if matrix is an identity matrix.
We check that the sum of the absolute value of the residual is less than <code class="docutils literal notranslate"><span class="pre">tol</span> <span class="pre">*</span> <span class="pre">eps</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="o">*</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">iseye</span><span class="p">(</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="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]]))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">iseye</span><span class="p">(</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">2</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="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>isskew, isskewa</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.isskew">
<span class="sig-name descname"><span class="pre">isskew</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">S</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/transformsNd.html#isskew"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.isskew" title="Permalink to this definition"></a></dt>
<dd><p>Test if matrix belongs to so(n)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>S</strong> (<em>ndarray</em><em>(</em><em>2</em><em>,</em><em>2</em><em>) or </em><em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>)</em>) – matrix to test</p></li>
<li><p><strong>tol</strong> (<em>float</em>) – tolerance in units of eps, defaults to 20</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>whether matrix is a proper skew-symmetric matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>bool</p>
</dd>
</dl>
<p>Checks skew-symmetry, ie. <span class="math notranslate nohighlight">\({\bf S} + {\bf S}^T = {\bf 0}\)</span>.
We check that the norm of the residual is less than <code class="docutils literal notranslate"><span class="pre">tol</span> <span class="pre">*</span> <span class="pre">eps</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="o">*</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">isskew</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isskew</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isskew</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</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>isskewa</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.isskewa">
<span class="sig-name descname"><span class="pre">isskewa</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">S</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/transformsNd.html#isskewa"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.isskewa" title="Permalink to this definition"></a></dt>
<dd><p>Test if matrix belongs to se(n)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>S</strong> (<em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>) or </em><em>ndarray</em><em>(</em><em>4</em><em>,</em><em>4</em><em>)</em>) – matrix to test</p></li>
<li><p><strong>tol</strong> (<em>float</em>) – tolerance in units of eps, defaults to 20</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>whether matrix is a proper skew-symmetric matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>bool</p>
</dd>
</dl>
<p>Check if matrix is augmented skew-symmetric, ie. the top left (n-1xn-1) partition <code class="docutils literal notranslate"><span class="pre">S</span></code> is
skew-symmetric <span class="math notranslate nohighlight">\({\bf S} + {\bf S}^T = {\bf 0}\)</span>, and the bottom row is zero
We check that the norm of the residual is less than <code class="docutils literal notranslate"><span class="pre">tol</span> <span class="pre">*</span> <span class="pre">eps</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="o">*</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">isskewa</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isskewa</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span> <span class="c1"># this matrix is skew but not skewa</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isskewa</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</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="go">True</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p>isskew</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.r2t">
<span class="sig-name descname"><span class="pre">r2t</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><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#r2t"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.r2t" title="Permalink to this definition"></a></dt>
<dd><p>Convert SO(n) to SE(n)</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>2</em><em>,</em><em>2</em><em>) or </em><em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>)</em>) – rotation matrix</p></li>
<li><p><strong>check</strong> (<em>bool</em>) – check if rotation matrix is valid (default False, no check)</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>homogeneous 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(4,4)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">T</span> <span class="pre">=</span> <span class="pre">r2t(R)</span></code> is an SE(2) or SE(3) homogeneous transform equivalent to an
SO(2) or SO(3) orthonormal rotation matrix <code class="docutils literal notranslate"><span class="pre">R</span></code> with a zero translational
component</p>
<ul class="simple">
<li><p>if <code class="docutils literal notranslate"><span class="pre">R</span></code> is 2x2 then <code class="docutils literal notranslate"><span class="pre">T</span></code> is 3x3: SO(2) → SE(2)</p></li>
<li><p>if <code class="docutils literal notranslate"><span class="pre">R</span></code> is 3x3 then <code class="docutils literal notranslate"><span class="pre">T</span></code> is 4x4: SO(3) → SE(3)</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="o">*</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">rot2</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R</span>
<span class="go">array([[ 0.9553, -0.2955],</span>
<span class="go"> [ 0.2955, 0.9553]])</span>
<span class="gp">>>> </span><span class="n">r2t</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
<span class="go">array([[ 0.9553, -0.2955, 0. ],</span>
<span class="go"> [ 0.2955, 0.9553, 0. ],</span>
<span class="go"> [ 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>t2r, rt2tr</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.rt2tr">
<span class="sig-name descname"><span class="pre">rt2tr</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">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><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#rt2tr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.rt2tr" title="Permalink to this definition"></a></dt>
<dd><p>Convert SO(n) and translation to SE(n)</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>2</em><em>) or </em><em>ndarray</em><em>(</em><em>3</em><em>)</em>) – SO(n) matrix</p></li>
<li><p><strong>t</strong> – translation vector</p></li>
<li><p><strong>check</strong> (<em>bool</em>) – check if SO(3) matrix is valid (default False, no check)</p></li>
</ul>
</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) or (3,3)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">T</span> <span class="pre">=</span> <span class="pre">rt2tr(R,</span> <span class="pre">t)</span></code> is a homogeneous transformation matrix (N+1xN+1) formed from an
orthonormal rotation matrix <code class="docutils literal notranslate"><span class="pre">R</span></code> (NxN) and a translation vector <code class="docutils literal notranslate"><span class="pre">t</span></code>
(Nx1).</p>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">R</span></code> is 2x2 and <code class="docutils literal notranslate"><span class="pre">t</span></code> is 2x1, then <code class="docutils literal notranslate"><span class="pre">T</span></code> is 3x3</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">R</span></code> is 3x3 and <code class="docutils literal notranslate"><span class="pre">t</span></code> is 3x1, then <code class="docutils literal notranslate"><span class="pre">T</span></code> is 4x4</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="o">*</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">rot2</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">rt2tr</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="go">array([[ 0.9553, -0.2955, 1. ],</span>
<span class="go"> [ 0.2955, 0.9553, 2. ],</span>
<span class="go"> [ 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>rt2m, tr2rt, r2t</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.skew">
<span class="sig-name descname"><span class="pre">skew</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/transformsNd.html#skew"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.skew" title="Permalink to this definition"></a></dt>
<dd><p>Create skew-symmetric metrix from vector</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>v</strong> (<em>array_like</em><em>(</em><em>1</em><em>) or </em><em>array_like</em><em>(</em><em>3</em><em>)</em>) – vector</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>skew-symmetric matrix in so(2) or so(3)</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(2,2) or ndarray(3,3)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">skew(V)</span></code> is a skew-symmetric matrix formed from the elements of <code class="docutils literal notranslate"><span class="pre">V</span></code>.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">len(V)</span></code> is 1 then <code class="docutils literal notranslate"><span class="pre">S</span></code> = <span class="math notranslate nohighlight">\(\left[ \begin{array}{cc} 0 & -v \\ v & 0 \end{array} \right]\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">len(V)</span></code> is 3 then <code class="docutils literal notranslate"><span class="pre">S</span></code> = <span class="math notranslate nohighlight">\(\left[ \begin{array}{ccc} 0 & -v_z & v_y \\ v_z & 0 & -v_x \\ -v_y & v_x & 0\end{array} \right]\)</span></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="o">*</span>
<span class="gp">>>> </span><span class="n">skew</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">array([[ 0., -2.],</span>
<span class="go"> [ 2., 0.]])</span>
<span class="gp">>>> </span><span class="n">skew</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="go">array([[ 0, -3, 2],</span>
<span class="go"> [ 3, 0, -1],</span>
<span class="go"> [-2, 1, 0]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>This is the inverse of the function <code class="docutils literal notranslate"><span class="pre">vex()</span></code>.</p></li>
<li><p>These are the generator matrices for the Lie algebras so(2) and so(3).</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.transformsNd.vex" title="spatialmath.base.transformsNd.vex"><code class="xref py py-func docutils literal notranslate"><span class="pre">vex()</span></code></a> <a class="reference internal" href="#spatialmath.base.transformsNd.skewa" title="spatialmath.base.transformsNd.skewa"><code class="xref py py-func docutils literal notranslate"><span class="pre">skewa()</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.transformsNd.skewa">
<span class="sig-name descname"><span class="pre">skewa</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/transformsNd.html#skewa"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.skewa" title="Permalink to this definition"></a></dt>
<dd><p>Create augmented skew-symmetric metrix from vector</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>v</strong> (<em>array_like</em><em>(</em><em>3</em><em>)</em><em>, </em><em>array_like</em><em>(</em><em>6</em><em>)</em>) – vector</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>augmented skew-symmetric matrix in se(2) or se(3)</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3,3) or ndarray(4,4)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">skewa(V)</span></code> is an augmented skew-symmetric matrix formed from the elements of <code class="docutils literal notranslate"><span class="pre">V</span></code>.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">len(V)</span></code> is 3 then S = <span class="math notranslate nohighlight">\(\left[ \begin{array}{ccc} 0 & -v_3 & v_1 \\ v_3 & 0 & v_2 \\ 0 & 0 & 0 \end{array} \right]\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">len(V)</span></code> is 6 then S = <span class="math notranslate nohighlight">\(\left[ \begin{array}{cccc} 0 & -v_6 & v_5 & v_1 \\ v_6 & 0 & -v_4 & v_2 \\ -v_5 & v_4 & 0 & v_3 \\ 0 & 0 & 0 & 0 \end{array} \right]\)</span></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="o">*</span>
<span class="gp">>>> </span><span class="n">skewa</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="go">array([[ 0., -3., 1.],</span>
<span class="go"> [ 3., 0., 2.],</span>
<span class="go"> [ 0., 0., 0.]])</span>
<span class="gp">>>> </span><span class="n">skewa</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
<span class="go">array([[ 0., -6., 5., 1.],</span>
<span class="go"> [ 6., 0., -4., 2.],</span>
<span class="go"> [-5., 4., 0., 3.],</span>
<span class="go"> [ 0., 0., 0., 0.]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>This is the inverse of the function <code class="docutils literal notranslate"><span class="pre">vexa()</span></code>.</p></li>
<li><p>These are the generator matrices for the Lie algebras se(2) and se(3).</p></li>
<li><p>Map twist vectors in 2D and 3D space to se(2) and se(3).</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.transformsNd.vexa" title="spatialmath.base.transformsNd.vexa"><code class="xref py py-func docutils literal notranslate"><span class="pre">vexa()</span></code></a> <a class="reference internal" href="#spatialmath.base.transformsNd.skew" title="spatialmath.base.transformsNd.skew"><code class="xref py py-func docutils literal notranslate"><span class="pre">skew()</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.transformsNd.t2r">
<span class="sig-name descname"><span class="pre">t2r</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><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#t2r"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.t2r" title="Permalink to this definition"></a></dt>
<dd><p>Convert SE(n) to SO(n)</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>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>) or </em><em>ndarray</em><em>(</em><em>4</em><em>,</em><em>4</em><em>)</em>) – homogeneous transformation matrix</p></li>
<li><p><strong>check</strong> (<em>bool</em>) – check if rotation matrix is valid (default False, no check)</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>rotation matrix</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(2,2) or ndarray(3,3)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">R</span> <span class="pre">=</span> <span class="pre">T2R(T)</span></code> is the orthonormal rotation matrix component of homogeneous
transformation matrix <code class="docutils literal notranslate"><span class="pre">T</span></code></p>
<ul class="simple">
<li><p>if <code class="docutils literal notranslate"><span class="pre">T</span></code> is 3x3 then <code class="docutils literal notranslate"><span class="pre">R</span></code> is 2x2: SE(2) → SO(2)</p></li>
<li><p>if <code class="docutils literal notranslate"><span class="pre">T</span></code> is 4x4 then <code class="docutils literal notranslate"><span class="pre">R</span></code> is 3x3: SE(3) → SO(3)</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="o">*</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">trot2</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">T</span>
<span class="go">array([[ 0.9553, -0.2955, 1. ],</span>
<span class="go"> [ 0.2955, 0.9553, 2. ],</span>
<span class="go"> [ 0. , 0. , 1. ]])</span>
<span class="gp">>>> </span><span class="n">t2r</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="go">array([[ 0.9553, -0.2955],</span>
<span class="go"> [ 0.2955, 0.9553]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Any translational component of T is lost.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p>r2t, tr2rt</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.tr2rt">
<span class="sig-name descname"><span class="pre">tr2rt</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><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#tr2rt"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.tr2rt" title="Permalink to this definition"></a></dt>
<dd><p>Convert SE(n) to SO(n) and translation</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>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>) or </em><em>ndarray</em><em>(</em><em>4</em><em>,</em><em>4</em><em>)</em>) – SE(n) matrix</p></li>
<li><p><strong>check</strong> (<em>bool</em>) – check if SO(3) submatrix is valid (default False, no check)</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>SO(n) matrix and translation vector</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>tuple: (ndarray(2,2), ndarray(2)) or (ndarray(3,3), ndarray(3))</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p>(R,t) = tr2rt(T) splits a homogeneous transformation matrix (NxN) into an orthonormal
rotation matrix R (MxM) and a translation vector T (Mx1), where N=M+1.</p>
<ul class="simple">
<li><p>if <code class="docutils literal notranslate"><span class="pre">T</span></code> is 3x3 - in SE(2) - then <code class="docutils literal notranslate"><span class="pre">R</span></code> is 2x2 and <code class="docutils literal notranslate"><span class="pre">t</span></code> is 2x1.</p></li>
<li><p>if <code class="docutils literal notranslate"><span class="pre">T</span></code> is 4x4 - in SE(3) - then <code class="docutils literal notranslate"><span class="pre">R</span></code> is 3x3 and <code class="docutils literal notranslate"><span class="pre">t</span></code> is 3x1.</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="o">*</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">trot2</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">T</span>
<span class="go">array([[ 0.9553, -0.2955, 1. ],</span>
<span class="go"> [ 0.2955, 0.9553, 2. ],</span>
<span class="go"> [ 0. , 0. , 1. ]])</span>
<span class="gp">>>> </span><span class="n">R</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">tr2rt</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R</span>
<span class="go">array([[ 0.9553, -0.2955],</span>
<span class="go"> [ 0.2955, 0.9553]])</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">array([1., 2.])</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Seealso<span class="colon">:</span></dt>
<dd class="field-odd"><p>rt2tr, tr2r</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="spatialmath.base.transformsNd.vex">
<span class="sig-name descname"><span class="pre">vex</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">s</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><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#vex"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.vex" title="Permalink to this definition"></a></dt>
<dd><p>Convert skew-symmetric matrix to vector</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>s</strong> (<em>ndarray</em><em>(</em><em>2</em><em>,</em><em>2</em><em>) or </em><em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>)</em>) – skew-symmetric matrix</p></li>
<li><p><strong>check</strong> (<em>bool</em>) – check if matrix is skew symmetric (default False, no check)</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>vector of unique values</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(1) or ndarray(3)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">vex(S)</span></code> is the vector which has the corresponding skew-symmetric matrix <code class="docutils literal notranslate"><span class="pre">S</span></code>.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">S</span></code> is 2x2 - so(2) case - where <code class="docutils literal notranslate"><span class="pre">S</span></code> <span class="math notranslate nohighlight">\(= \left[ \begin{array}{cc} 0 & -v \\ v & 0 \end{array} \right]\)</span> then return <span class="math notranslate nohighlight">\([v]\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">S</span></code> is 3x3 - so(3) case - where <code class="docutils literal notranslate"><span class="pre">S</span></code> <span class="math notranslate nohighlight">\(= \left[ \begin{array}{ccc} 0 & -v_z & v_y \\ v_z & 0 & -v_x \\ -v_y & v_x & 0\end{array} \right]\)</span> then return <span class="math notranslate nohighlight">\([v_x, v_y, v_z]\)</span>.</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="o">*</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">skew</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">[[ 0. -2.]</span>
<span class="go"> [ 2. 0.]]</span>
<span class="gp">>>> </span><span class="n">vex</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">array([2.])</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">skew</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">[[ 0 -3 2]</span>
<span class="go"> [ 3 0 -1]</span>
<span class="go"> [-2 1 0]]</span>
<span class="gp">>>> </span><span class="n">vex</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">array([1., 2., 3.])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>This is the inverse of the function <code class="docutils literal notranslate"><span class="pre">skew()</span></code>.</p></li>
<li><p>Only rudimentary checking (zero diagonal) is done to ensure that the matrix
is actually skew-symmetric.</p></li>
<li><p>The function takes the mean of the two elements that correspond to each unique
element of the matrix.</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.transformsNd.skew" title="spatialmath.base.transformsNd.skew"><code class="xref py py-func docutils literal notranslate"><span class="pre">skew()</span></code></a> <a class="reference internal" href="#spatialmath.base.transformsNd.vexa" title="spatialmath.base.transformsNd.vexa"><code class="xref py py-func docutils literal notranslate"><span class="pre">vexa()</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.transformsNd.vexa">
<span class="sig-name descname"><span class="pre">vexa</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Omega</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><span class="sig-paren">)</span><a class="reference internal" href="_modules/spatialmath/base/transformsNd.html#vexa"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#spatialmath.base.transformsNd.vexa" title="Permalink to this definition"></a></dt>
<dd><p>Convert skew-symmetric matrix to vector</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>s</strong> (<em>ndarray</em><em>(</em><em>3</em><em>,</em><em>3</em><em>) or </em><em>ndarray</em><em>(</em><em>4</em><em>,</em><em>4</em><em>)</em>) – augmented skew-symmetric matrix</p></li>
<li><p><strong>check</strong> (<em>bool</em>) – check if matrix is skew symmetric part is valid (default False, no check)</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>vector of unique values</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray(3) or ndarray(6)</p>
</dd>
<dt class="field-even">Raises<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>ValueError</strong> – bad argument</p>
</dd>
</dl>
<p><code class="docutils literal notranslate"><span class="pre">vexa(S)</span></code> is the vector which has the corresponding augmented skew-symmetric matrix <code class="docutils literal notranslate"><span class="pre">S</span></code>.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">S</span></code> is 3x3 - se(2) case - where <code class="docutils literal notranslate"><span class="pre">S</span></code> <span class="math notranslate nohighlight">\(= \left[ \begin{array}{ccc} 0 & -v_3 & v_1 \\ v_3 & 0 & v_2 \\ 0 & 0 & 0 \end{array} \right]\)</span> then return <span class="math notranslate nohighlight">\([v_1, v_2, v_3]\)</span>.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">S</span></code> is 4x4 - se(3) case - where <code class="docutils literal notranslate"><span class="pre">S</span></code> <span class="math notranslate nohighlight">\(= \left[ \begin{array}{cccc} 0 & -v_6 & v_5 & v_1 \\ v_6 & 0 & -v_4 & v_2 \\ -v_5 & v_4 & 0 & v_3 \\ 0 & 0 & 0 & 0 \end{array} \right]\)</span> then return <span class="math notranslate nohighlight">\([v_1, v_2, v_3, v_4, v_5, v_6]\)</span>.</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="o">*</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">skewa</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">[[ 0. -3. 1.]</span>
<span class="go"> [ 3. 0. 2.]</span>
<span class="go"> [ 0. 0. 0.]]</span>
<span class="gp">>>> </span><span class="n">vexa</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">array([1., 2., 3.])</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">skewa</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">[[ 0. -6. 5. 1.]</span>
<span class="go"> [ 6. 0. -4. 2.]</span>
<span class="go"> [-5. 4. 0. 3.]</span>
<span class="go"> [ 0. 0. 0. 0.]]</span>
<span class="gp">>>> </span><span class="n">vexa</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="go">array([1., 2., 3., 4., 5., 6.])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<ul class="simple">
<li><p>This is the inverse of the function <code class="docutils literal notranslate"><span class="pre">skewa</span></code>.</p></li>
<li><p>Only rudimentary checking (zero diagonal) is done to ensure that the matrix
is actually skew-symmetric.</p></li>
<li><p>The function takes the mean of the two elements that correspond to each unique
element of the matrix.</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.transformsNd.skewa" title="spatialmath.base.transformsNd.skewa"><code class="xref py py-func docutils literal notranslate"><span class="pre">skewa()</span></code></a> <a class="reference internal" href="#spatialmath.base.transformsNd.vex" title="spatialmath.base.transformsNd.vex"><code class="xref py py-func docutils literal notranslate"><span class="pre">vex()</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>
</section>
</div>
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