MAINT: fix formula.

doc/book/kalman_filter.Rmd

@@ -13,7 +13,7 @@ predicts the next state of the object but it also provides how wrong we could be
 in our prediction. How wrong we could be is quantified in terms of covariance
 matrix?
 
-So when we predict the next state of the object, we actually predict a Gaussian
-distribution.
+So when we predict the next state of the object, we actually predict a
+**Gaussian distribution** at each time step.
 
 Let's start reminding ourselves the equations.

doc/book/kalman_filter/theory.Rmd

@@ -62,9 +62,8 @@ where
 
 \begin{equation}
   \mathbf{H}_k = \begin{bmatrix}
-    \mathbf{I}_4 & \mathbf{0}_{4 \times 8} \\
-    \mathbf{0}_{8 \times 4} & \mathbf{0}_{8 \times 8} \\
-\end{bmatrix}
+    \mathbf{I}_4 & \mathbf{0}_{4 \times 8}
+  \end{bmatrix}
 \end{equation}
 
 is called the observation model in the Kalman filter
@@ -102,6 +101,18 @@ obtain is from the temporal integration as in classical Newtonian mechanics:
     \frac{1}{2} \ddot{\mathbf{b}}_k (t_{k+1} - t_k)^2
 \end{equation}
 
+And so the matrix $\mathbf{F}_k$ is
+
+\begin{equation}
+\mathbf{F}_k = \begin{bmatrix}
+  \mathbf{I}_4 & \Delta_k \mathbf{I}_4 & \frac{\Delta_k^2}{2} \mathbf{I}_4 \\
+  \mathbf{0}_4 & \mathbf{I}_4 & \Delta_k \mathbf{I}_4 \\
+  \mathbf{0}_4 & \mathbf{0}_4 & \mathbf{I}_4
+  \end{bmatrix},
+\end{equation}
+where $\Delta_k = t_{k+1} - t_k$.
+
+
 Unless we are driving a car autonomously trying to keep a minimum distance with
 every pedestrian in a busy street, we can ignore $\mathbf{B}_k$ by setting it to
 $\mathbf{0}$.