Merge pull request #375 from Gjacquenot/master

Fixed duplicated "of" keywords in two Jupyter notebooks

07-Kalman-Filter-Math.ipynb

@@ -170,7 +170,7 @@
     "\n",
     "We can do that only because $\\dot x = v$ is simplest differential equation possible. Almost all other in physical systems result in more complicated differential equation which do not yield to this approach. \n",
     "\n",
-    "*State-space* methods became popular around the time of the Apollo missions, largely due to the work of Dr. Kalman. The idea is simple. Model a system with a set of  $n^{th}$-order differential equations. Convert them into an equivalent set of first-order differential equations. Put them into the vector-matrix form used in the previous section: $\\dot{\\mathbf x} = \\mathbf{Ax} + \\mathbf{Bu}$. Once in this form we use of of several techniques to convert these linear differential equations into the recursive equation:\n",
+    "*State-space* methods became popular around the time of the Apollo missions, largely due to the work of Dr. Kalman. The idea is simple. Model a system with a set of  $n^{th}$-order differential equations. Convert them into an equivalent set of first-order differential equations. Put them into the vector-matrix form used in the previous section: $\\dot{\\mathbf x} = \\mathbf{Ax} + \\mathbf{Bu}$. Once in this form we use of several techniques to convert these linear differential equations into the recursive equation:\n",
     "\n",
     "$$ \\mathbf x_k = \\mathbf{Fx}_{k-1} + \\mathbf B_k\\mathbf u_k$$\n",
     "\n",

08-Designing-Kalman-Filters.ipynb

@@ -176,7 +176,7 @@
     "$$\\mathbf x = \n",
     "\\begin{bmatrix}x & \\dot x & y & \\dot y\\end{bmatrix}^\\mathsf T$$\n",
     "\n",
-    "There is nothing special about this organization. I could have used $\\begin{bmatrix}x & y & \\dot x &  \\dot y\\end{bmatrix}^\\mathsf T$ or something less logical. I just need to be consistent in the rest of the matrices. I like keeping positions and velocities next to each other because it keeps the covariances between positions and velocities in the same sub block of the covariance matrix. In my formulation `P[1,0]` contains the covariance of of $x$ and $\\dot x$. In the alternative formulation that covariance is at `P[2, 0]`. This gets worse as the number of dimension increases.\n",
+    "There is nothing special about this organization. I could have used $\\begin{bmatrix}x & y & \\dot x &  \\dot y\\end{bmatrix}^\\mathsf T$ or something less logical. I just need to be consistent in the rest of the matrices. I like keeping positions and velocities next to each other because it keeps the covariances between positions and velocities in the same sub block of the covariance matrix. In my formulation `P[1,0]` contains the covariance of $x$ and $\\dot x$. In the alternative formulation that covariance is at `P[2, 0]`. This gets worse as the number of dimension increases.\n",
     "\n",
     "Let's pause and address how you identify the hidden variables. This example is somewhat obvious because we've already worked through the 1D case, but other problems won't be obvious There is no easy answer to this question. The first thing to ask yourself is what is the interpretation of the first and second derivatives of the data from the sensors. We do that because obtaining the first and second derivatives is mathematically trivial if you are reading from the sensors using a fixed time step. The first derivative is just the difference between two successive readings. In our tracking case the first derivative has an obvious physical interpretation: the difference between two successive positions is velocity. \n",
     "\n",