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notebooks/covariance_propagation.html

@@ -433,8 +433,8 @@ <h1>Covariance Propagation</h1>
   <section class="tex2jax_ignore mathjax_ignore" id="covariance-propagation">
 <h1>Covariance Propagation<a class="headerlink" href="#covariance-propagation" title="Permalink to this heading">#</a></h1>
 <p>In this notebook, we discuss how to use <code class="docutils literal notranslate"><span class="pre">dsgp4</span></code> to apply the first order approximation for propagating a covariance matrix:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-c8044e60-b713-471f-98bd-81ef7ef0038b">
-<span class="eqno">(1)<a class="headerlink" href="#equation-c8044e60-b713-471f-98bd-81ef7ef0038b" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-eca2413e-2c8f-4e2d-8339-c31bb62147be">
+<span class="eqno">(1)<a class="headerlink" href="#equation-eca2413e-2c8f-4e2d-8339-c31bb62147be" title="Permalink to this equation">#</a></span>\[\begin{equation}
 P_{\pmb{x}_f}=\dfrac{\partial \pmb{x}}{\partial \textrm{TLE}_0}  P_{\textrm{TLE}_{0}}\dfrac{\partial \pmb{x}}{\partial \textrm{TLE}_0}^T\text{,}
 \end{equation}\]</div>
 <p>where <span class="math notranslate nohighlight">\(\textrm{TLE}_0\)</span> is the initial TLE, at time <span class="math notranslate nohighlight">\(t_0\)</span>, and <span class="math notranslate nohighlight">\(\pmb{x}\)</span> is the state vector in Cartesian TEME, at a certain propagation time <span class="math notranslate nohighlight">\(t_f\)</span>.</p>

notebooks/covariance_transformation.html

@@ -463,13 +463,13 @@ <h2>Covariance Transformation<a class="headerlink" href="#covariance-transformat
 <li><p>first convert the RTN covariance into TEME</p></li>
 <li><p>then leverage <code class="docutils literal notranslate"><span class="pre">dsgp4</span></code> to transform the covariance matrix from position and velocity coordinates to TLE elements, leveraging the similarity transformation:</p></li>
 </ul>
-<div class="amsmath math notranslate nohighlight" id="equation-25e3e1c9-885d-4672-9ae0-2224141200b2">
-<span class="eqno">(1)<a class="headerlink" href="#equation-25e3e1c9-885d-4672-9ae0-2224141200b2" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-d6a26e58-d87c-4bff-9c02-e7a4f53a58d6">
+<span class="eqno">(1)<a class="headerlink" href="#equation-d6a26e58-d87c-4bff-9c02-e7a4f53a58d6" title="Permalink to this equation">#</a></span>\[\begin{equation}
 P_{y} = m P_x m^T\text{,}
 \end{equation}\]</div>
 <p>where:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-d759b85d-089c-4d30-b35f-4550165fa735">
-<span class="eqno">(2)<a class="headerlink" href="#equation-d759b85d-089c-4d30-b35f-4550165fa735" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-289da3fc-6ae6-47ac-b337-b6157b4ff662">
+<span class="eqno">(2)<a class="headerlink" href="#equation-289da3fc-6ae6-47ac-b337-b6157b4ff662" title="Permalink to this equation">#</a></span>\[\begin{equation}
 m_{ij}=\dfrac{\partial y_i}{\partial x_j}
 \end{equation}\]</div>
 <ul class="simple">

notebooks/gradient_based_optimization.html

@@ -446,26 +446,26 @@ <h1>Gradient Based Optimization<a class="headerlink" href="#gradient-based-optim
 <h2>Problem description:<a class="headerlink" href="#problem-description" title="Permalink to this heading">#</a></h2>
 <p>We have a TLE at a given time, which we call TLE<span class="math notranslate nohighlight">\(_{0}\)</span>, and we look for a TLE at a future observation time (<span class="math notranslate nohighlight">\(t_{obs}\)</span>): TLE<span class="math notranslate nohighlight">\(_{t}\)</span>.</p>
 <p>We can propagate the state from <span class="math notranslate nohighlight">\(t_0 \rightarrow t_{obs}\)</span>, and obtain the state at <span class="math notranslate nohighlight">\(t_{obs}\)</span>. In general, we define the state (i.e., position and velocity), as:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-6fa82ddc-de84-45ea-aa8d-35256d2282b7">
-<span class="eqno">(1)<a class="headerlink" href="#equation-6fa82ddc-de84-45ea-aa8d-35256d2282b7" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-22618c33-f304-41ef-8977-5bfa34286ba2">
+<span class="eqno">(1)<a class="headerlink" href="#equation-22618c33-f304-41ef-8977-5bfa34286ba2" title="Permalink to this equation">#</a></span>\[\begin{equation}
 \vec{x}(t)=[x(t), y(t), z(t), \dot{x}(t), \dot{y}(t), \dot{z}(t)]^T
 \end{equation}\]</div>
 <p>We then have: TLE<span class="math notranslate nohighlight">\(_0\)</span>, <span class="math notranslate nohighlight">\(\vec{x}(t_0)\)</span>, and <span class="math notranslate nohighlight">\(\vec{x}(t_{obs})\)</span>, but we want to find TLE<span class="math notranslate nohighlight">\(_{obs}\)</span>. That is, the TLE at the observation time, that when propagated with SGP4 at its time, it corresponds to that <span class="math notranslate nohighlight">\(\vec{x}(t_{obs})\)</span>. In general, this means that we are able to invert from the state to the TLE, at any given time.</p>
 <p>In order to do this, we formulate the problem as looking for the minimum of a function of a free variables vector (i.e., <span class="math notranslate nohighlight">\(\vec{y}\)</span>) <span class="math notranslate nohighlight">\(F(\vec{y})\)</span>, where this function defines the difference between the given state propagated from TLE<span class="math notranslate nohighlight">\(_0\)</span> at <span class="math notranslate nohighlight">\(t_{obs}\)</span>, and the state generated from the free variables that make a TLE which is then propagated at its current time: TLE<span class="math notranslate nohighlight">\((\vec{y})(t_{0}\rightarrow t_{obs})\)</span>. So we can reformulate the problem as:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-2b33d79d-7fb3-4565-a1d4-6e2b943d05a8">
-<span class="eqno">(2)<a class="headerlink" href="#equation-2b33d79d-7fb3-4565-a1d4-6e2b943d05a8" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-975fab1a-b031-40e7-bf48-72af9cdd3e32">
+<span class="eqno">(2)<a class="headerlink" href="#equation-975fab1a-b031-40e7-bf48-72af9cdd3e32" title="Permalink to this equation">#</a></span>\[\begin{align}
 \textrm{given}: &amp; \ \textrm{TLE}_0, \vec{x}_0\\
 \textrm{find}: &amp; \ \vec{y}\\
 \textrm{that minimize}: &amp; F(\vec{y})=|SGP4(\textrm{TLE}(\vec{y}),t_{obs})-\vec{x}(t_{obs})| =|\vec{\tilde{x}}(t_{obs})-\vec{x}(t_{obs})|
 \end{align}\]</div>
 <p>We can do this via Newton method, by updating an initial guess <span class="math notranslate nohighlight">\(y_{0}\)</span> until convergence. Where the update is done as follows:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-4ed7f12a-c312-44d1-afab-5cad14a222e1">
-<span class="eqno">(3)<a class="headerlink" href="#equation-4ed7f12a-c312-44d1-afab-5cad14a222e1" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-407a56de-9a6c-4d92-a9e6-1141bc281998">
+<span class="eqno">(3)<a class="headerlink" href="#equation-407a56de-9a6c-4d92-a9e6-1141bc281998" title="Permalink to this equation">#</a></span>\[\begin{equation}
 y_{k+1}=y_{k}-DF^{-1}(y_k)F(y_k)
 \end{equation}\]</div>
 <p>with <span class="math notranslate nohighlight">\(DF\)</span> the Jacobian of <span class="math notranslate nohighlight">\(F\)</span> with respect to <span class="math notranslate nohighlight">\(y_k\)</span>. We can easily see that this Jacobian is made of the following elements:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-3640c82f-0798-4039-b6c5-05a1c33bf208">
-<span class="eqno">(4)<a class="headerlink" href="#equation-3640c82f-0798-4039-b6c5-05a1c33bf208" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-6d5db151-b228-471d-b535-f18d0426ba1c">
+<span class="eqno">(4)<a class="headerlink" href="#equation-6d5db151-b228-471d-b535-f18d0426ba1c" title="Permalink to this equation">#</a></span>\[\begin{equation}
 DF_{ij}=\dfrac{\partial \tilde{x}_{i}}{\partial y_{j}}|_{y_k}
 \end{equation}\]</div>
 <p>where <span class="math notranslate nohighlight">\(\tilde{x}_{i} \in [\tilde{x}_1,\tilde{x}_2,\tilde{x}_3,\tilde{x}_4,\tilde{x}_5,\tilde{x}_6]=[\tilde{x},\tilde{y},\tilde{z},\tilde{\dot{x}},\tilde{\dot{y}},\tilde{\dot{z}}]\)</span>; and <span class="math notranslate nohighlight">\(y_i \in [no_{kozai}, ecco, inclo, mo, argpo, nodeo, n_{dot},n_{ddot},B^*]\)</span>.</p>