diff --git a/paper/sections/04-fusion.tex b/paper/sections/04-fusion.tex index 44bc5a90271f0ccdb50986989ce3687a4b18bcda..3ec986507d84ab3d8d2e0a6dee06df2ab1c0c4b8 100644 --- a/paper/sections/04-fusion.tex +++ b/paper/sections/04-fusion.tex @@ -85,7 +85,7 @@ v(k)-\hat{v}(k) \\ \end{bmatrix} , \end{equation} -where the attitude error vector $\epsilon (k)$ is defined as the the small (Euler) angle sequence that rotates the attitude vector $\hat{q}(k)$ into $q(k)$ \footnote{Formally $q(k)=\Gamma(\hat{q}(k), \epsilon (k))$, where $\Gamma(\hat{q}(k),\epsilon (k))\triangleq\{ q \in \mathbb{S}^3 | R_b^n (q(k)) = (I_3 - \left[\epsilon(k)\right]_{\times}) R_b^n (\hat{q}(k)) \}$ and $\left[a\right]_{\times}$ defines the antisymmetric matrix representation of $a$ for which $\left[a\right]_{\times} * b = a*b$.}. Next we can define the vector collecting all the quantities to be estimated and the vector that collects the disturbances as: +where the attitude error vector $\epsilon (k)$ is defined as the the small (Euler) angle sequence that rotates the attitude vector $\hat{q}(k)$ into $q(k)$ \footnote{Formally $q(k)=\Gamma(\hat{q}(k), \epsilon (k))$, where $\Gamma(\hat{q}(k),\epsilon (k))\triangleq\{ q \in \mathbb{S}^3 | R_b^n (q(k)) = (I_3 - \left[\epsilon(k)\right]_{\times}) R_b^n (\hat{q}(k)) \}$ and $\left[a\right]_{\times}$ defines the antisymmetric matrix representation of $a$ for which $\left[a\right]_{\times} \cdot b = a \cdot b$.}. Next we can define the vector collecting all the quantities to be estimated and the vector that collects the disturbances as: \begin{equation} \delta z(k) = @@ -108,7 +108,7 @@ where the attitude error vector $\epsilon (k)$ is defined as the the small (Eule The linearized state space model, that describes the evolution of $z(k)$ vector and is needed for defining how the covariance matrix of the estimation changes in time, is then: \begin{equation} - z(k) = \mathbf{F}(x(k),u(k))*z(k-1) + \mathbf{G}(x(k))*w(k) , + z(k) = \mathbf{F}(x(k),u(k)) \cdot z(k-1) + \mathbf{G}(x(k)) \cdot w(k) , \end{equation} where the state transition matrix $\mathbf{F}(x(k),u(k))$ and the noise gain matrix $\mathbf{G}(x(k))$ are defined as: