Single Point Positioning using Kalman Filtering

Algorithm - Extended Kalman Filter

• Prediction

  \begin{equation} \begin{aligned} \hat{\boldsymbol{x}}_{n \mid n-1} &=\boldsymbol{\Phi}_{n-1 \mid n-1} \cdot \hat{\boldsymbol{x}}_{n-1 \mid n-1} \\ \boldsymbol{P}_{n \mid n-1} &=\boldsymbol{\Phi}_{n-1 \mid n-1} \boldsymbol{P}_{n-1 \mid n-1} \boldsymbol{\Phi}_{n-1 \mid n-1}^{T}+\boldsymbol{Q} \end{aligned} \end{equation}

• Kalman Gain

  \begin{equation} \boldsymbol{K}_{n}=\boldsymbol{P}_{n \mid n-1} \boldsymbol{H}_{n}^{T}\left(\boldsymbol{H}_{n} \boldsymbol{P}_{n \mid n-1} \boldsymbol{H}_{n}^{T}+\boldsymbol{R}_{n}\right)^{-1} \end{equation}

• Update

  \begin{equation} \begin{aligned} \hat{\boldsymbol{x}}_{n \mid n}&=\hat{\boldsymbol{x}}_{n \mid n-1}+\boldsymbol{K}_{n}\left(\Delta \boldsymbol{y}_{n}-\boldsymbol{H}_{n} \left(\hat{\boldsymbol{x}}_{n \mid n-1} - \hat{\boldsymbol{x}}_{n-1 \mid n-1}\right)\right) \\ \boldsymbol{P}_{n \mid n}&=\left(\boldsymbol{I}-\boldsymbol{K}_{n} \boldsymbol{H}_{n}\right) \boldsymbol{P}_{n \mid n-1} \end{aligned} \end{equation}

Unknowns

Pseudorange and Doppler observations are processed together.

\begin{equation} \mathbf{x}^e = \begin{bmatrix} \text{Receiver Position [m]} \\ \text{Receiver Velocity [m]} \\ \hdashline \text{Clock error [m]} \\ \text{Clock drift [m]} \\\hdashline \text{Inter-system clock bias (System 2 to 1) [m]} \\ \text{Inter-system clock drift (System 2 to 1) [m]} \\ \vdots \\ \text{Inter-system clock bias (System o to 1) [m]} \\ \text{Inter-system clock drift (System o to 1) [m]} \\\hdashline \text{Inter-frequency bias (Frequency 2 to 1) [m]} \\ \vdots \\ \text{Inter-frequency bias (Frequency j to 1) [m]} \\ \end{bmatrix} \end{equation}

Process Model

System Model

\begin{equation} \overbrace{ \begin{bmatrix} { \mathbf{v}_r } \\ { \mathbf{a}_r } \\\hdashline { d\dot{t}_r } \\ { d\ddot{t}_r } \\\hdashline { d\dot{t}_{ISB}^{S_2 \rightarrow S_1} } \\ { d\ddot{t}_{ISB}^{S_2 \rightarrow S_1} } \\ { \vdots } \\ { d\dot{t}_{ISB}^{S_o \rightarrow S_1} } \\ { d\ddot{t}_{ISB}^{S_o \rightarrow S_1} } \\\hdashline { d\dot{t}_{IFB}^{F_2 \rightarrow F_1} } \\ { \vdots } \\ { d\dot{t}_{IFB}^{F_j \rightarrow F_1} } \\ \end{bmatrix} }^{\mathbf{\dot{x}}^e} = \overbrace{\left[\begin{array}{cc:cc:ccccc:ccc} \mathbf{0}_3 & \mathbf{I}_3 & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \dots & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \dots & \mathbf{0}_{3,1} \\ \mathbf{0}_3 & \mathbf{0}_3 & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \dots & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \dots & \mathbf{0}_{3,1} \\\hdashline \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 1 & 0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 \\\hdashline \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 1 & \dots & 0 & 0 & 0 & \dots & 0 \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & \dots & 0 & 1 & 0 & \dots & 0 \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 \\\hdashline \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 \\ \end{array}\right]}^{\mathbf{F}} \cdot \overbrace{ \begin{bmatrix} { \mathbf{r}_r } \\ { \mathbf{v}_r } \\\hdashline { dt_r} \\ { d\dot{t}_r} \\\hdashline { dt_{ISB}^{S_2 \rightarrow S_1} } \\ { d\dot{t}_{ISB}^{S_2 \rightarrow S_1} } \\ { \vdots } \\ { dt_{ISB}^{S_o \rightarrow S_1} } \\ { d\dot{t}_{ISB}^{S_o \rightarrow S_1} } \\\hdashline { dt_{IFB}^{F_2 \rightarrow F_1} } \\ { \vdots } \\ { dt_{IFB}^{F_j \rightarrow F_1} } \\ \end{bmatrix} }^{\mathbf{x}^e} + \mathbf{G} \cdot \boldsymbol{w} \end{equation}

([17] Groves, ch. 9.4.2.2, eq. 9.148, p. 415)

State transition matrix

Higher order terms are zero, so the exact solution is

\begin{equation} \mathbf{\Phi} = \text{exp}(\mathbf{F} \tau_s) = \mathbf{I} + \mathbf{F}\tau_s \end{equation}

([17] Groves, ch. 9.4.2.2, eq. 9.150, p. 416)

Process noise covariance matrix

Clock modeling

• Phase drift \( \sigma_{c\phi} \) (random walk) and Frequency drift \( \sigma_{cf} \) (integrated random walk)

  \begin{equation} S_{c f}^a = \frac{\sigma^2\left(\delta \dot{\rho}_c^a\left(t+\tau_s\right)-\delta \dot{\rho}_c^a(t)\right)}{\tau_s} \quad S_{c \phi}^a = \frac{\sigma^2\left(\delta \rho_c^a\left(t+tau_s\right)-\delta \rho_c^a(t)-\delta \dot{\rho}_c^a(t) \tau_s\right)}{\tau_s} \end{equation}

• Typical values for a TCXO are \( S_{cf} \approx 0.04 \text{ m}^2 \text{s}^{-3} \) and \( S_{c\phi} \approx 0.01 \text{ m}^2 \text{s}^{-1} \)

Velocity change due to user motion (Variances)
\( \mathbf{S}_a^n = \begin{bmatrix} S_{aH} & 0 & 0 \\ 0 & S_{aH} & 0 \\ 0 & 0 & S_{aV} \\ \end{bmatrix} \)

\begin{equation} \begin{aligned} & S_{a H}=\frac{\sigma^2\left(v_{e b, N}^n\left(t+\tau_s\right)-v_{e b, N}^n(t)\right)}{\tau_s}=\frac{\sigma^2\left(v_{e b, E}^n\left(t+\tau_s\right)-v_{e b, E}^n(t)\right)}{\tau_s} \\ & S_{a V}=\frac{\sigma^2\left(v_{e b, D}^n\left(t+\tau_s\right)-v_{e b, D}^n(t)\right)}{\tau_s} \end{aligned} \end{equation}

• Suitable values for \( S_{aH} \) are around
  • \( 1 \text{ m}^2 \text{s}^{–3} \) for a pedestrian or ship,
  • \( 10 \text{ m}^2 \text{s}^{–3} \) for a car,
  • \( 100 \text{ m}^2 \text{s}^{–3} \) for a military aircraft.
• The vertical acceleration PSD \( S_{aV} \) is usually smaller

([17] Groves, ch. 9.4.2.2, p. 416-418)

Van Loan method

Noise input matrix

\begin{equation} \mathbf{G} = \text{diag} \left\{\begin{array}{cc:cc:ccccc:ccc} \,{ \mathbf{0}_3 } & { \mathbf{C}_n^e } & { 1 } & { 1 } & { 1 } & { 1 } & { \dots } & { 1 } & { 1 } & { 1 } & { \dots } & { 1 }\, \end{array}\right\}^T \end{equation}

Noise scale matrix

\begin{equation} \mathbf{W} = \text{diag} \left\{\begin{array}{cc:cc:ccccc:ccc} \,{ \mathbf{0}_3 } & {\mathbf{S}_a^n} & {\sigma_{c\phi}^2} & {\sigma_{cf}^2} & {{\sigma_{ISB,\phi}^{S_2 \rightarrow S_1}}^2} & {{\sigma_{ISB,f}^{S_2 \rightarrow S_1}}^2} & { \dots } & {{\sigma_{ISB,\phi}^{S_o \rightarrow S_1}}^2} & {{\sigma_{ISB,f}^{S_o \rightarrow S_1}}^2} & {{\sigma_{IFB}^{F_2 \rightarrow F_1}}^2} & { \dots } & {{\sigma_{IFB}^{F_j \rightarrow F_1}}^2}\, \end{array}\right\}^T \end{equation}

Van Loan algorithm

1. Form a \( 2n \times 2n \) matrix called \( \mathbf{A} \) ( \( n \) is the dimension of \( \mathbf{x} \) and \( \mathbf{W} \) is the power spectral density of the noise \( W(t) \))

   \begin{equation} \mathbf{A} = \begin{bmatrix} -\mathbf{F} & \mathbf{G} \mathbf{W} \mathbf{G}^T \\ \mathbf{0} & \mathbf{F}^T \end{bmatrix} \Delta t \end{equation}

2. Calculate the exponential of \( \mathbf{A} \)

   \begin{equation} \mathbf{B} = \text{expm}(\mathbf{A}) = \left[ \begin{array}{c:c} \dots & \mathbf{\Phi}^{-1} \mathbf{Q} \\[2mm] \hdashline \mathbf{0} & \mathbf{\Phi}^T \end{array} \right] = \left[ \begin{array}{c:c} \mathbf{B}_{11} & \mathbf{B}_{12} \\[2mm] \hdashline \mathbf{B}_{21} & \mathbf{B}_{22} \end{array} \right] \end{equation}

3. Calculate the state transition matrix \( \mathbf{\Phi} \) as

   \begin{equation} \mathbf{\Phi} = \mathbf{B}_{22}^T \end{equation}

4. Calculate the process noise covariance matrix \( \mathbf{Q} \) as

   \begin{equation} \mathbf{Q} = \mathbf{\Phi} \mathbf{B}_{12} \end{equation}

Uses GUI input values for \( S_{aH} \), \( S_{cf} \) and \( S_{c\phi} \).

Groves

\begin{equation} \mathbf{Q} = \left[\begin{array}{cc:cc} \frac{1}{3} S_a^\gamma \tau_s^3 & \frac{1}{2} S_a^\gamma \tau_s^2 & 0_{3,1} & 0_{3,1} \\ \frac{1}{2} S_a^\gamma \tau_s^2 & S_a^\gamma \tau_s & 0_{3,1} & 0_{3,1} \\ \hdashline 0_{1,3} & 0_{1,3} & S_{c \phi}^a \tau_s+\frac{1}{3} S_{c f}^a \tau_s^3 & \frac{1}{2} S_{c f}^a \tau_s^2 \\ 0_{1,3} & 0_{1,3} & \frac{1}{2} S_{c f}^a \tau_s^2 & S_{c f}^a \tau_s \end{array}\right] \end{equation}

Uses calculated values for \( S_{aH} \), \( S_{cf} \) and \( S_{c\phi} \).

([17] Groves, ch. 9.4.2.2, eq. 9.152, p. 417-418)

The inter-system errors and drifts are assumed constant. Note: Groves does not estimate an inter-system drift ([17] Groves, Appendix G.8, p. G-23 - G-24), but we do for all models.

Measurement Model

Measurement innovation

See Measurement innovation

Measurement sensitivity matrix

See Design matrix / Measurement sensitivity matrix

In detail for Kalman Filtering

\begin{equation} \overbrace {\begin{bmatrix} { {p}_{r}^{1,S1,F1}} \\[0.5em] { {p}_{r}^{1,S1,F2}} \\[0.5em] { {p}_{r}^{2,S2}} \\[0.5em] \dots \\[0.4em] \hdashline { {\dot{p}}_{r}^{1,S1}} \\[0.5em] { {\dot{p}}_{r}^{2,S2}} \\[0.5em] \dots \\[0.4em] \end{bmatrix}}^{\mathbf{z}} =\overbrace{\left[\begin{array}{ccc:ccc:cc:cc:c} \frac{- {x^s} + {x_r}}{{\rho_r^s}} & \frac{- {y^s} + {y_r}}{{\rho_r^s}} & \frac{- {z^s} + {z_r}}{{\rho_r^s}} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\[0.3em] \frac{- {x^s} + {x_r}}{{\rho_r^s}} & \frac{- {y^s} + {y_r}}{{\rho_r^s}} & \frac{- {z^s} + {z_r}}{{\rho_r^s}} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\[0.3em] \frac{- {x^s} + {x_r}}{{\rho_r^s}} & \frac{- {y^s} + {y_r}}{{\rho_r^s}} & \frac{- {z^s} + {z_r}}{{\rho_r^s}} & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\[0.3em] \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\[0.3em] \hdashline - \frac{{v^{s,x}} - {v_{r,x}}}{{\rho_r^s}} & - \frac{{v^{s,y}} - {v_{r,y}}}{{\rho_r^s}} & - \frac{{v^{s,z}} - {v_{r,z}}}{{\rho_r^s}} & - \frac{{x^s} - {x_r}}{{\rho_r^s}} & - \frac{{y^s} - {y_r}}{{\rho_r^s}} & - \frac{{z^s} - {z_r}}{{\rho_r^s}} & 0 & 1 & 0 & 0 & 0\\[0.3em] - \frac{{v^{s,x}} - {v_{r,x}}}{{\rho_r^s}} & - \frac{{v^{s,y}} - {v_{r,y}}}{{\rho_r^s}} & - \frac{{v^{s,z}} - {v_{r,z}}}{{\rho_r^s}} & - \frac{{x^s} - {x_r}}{{\rho_r^s}} & - \frac{{y^s} - {y_r}}{{\rho_r^s}} & - \frac{{z^s} - {z_r}}{{\rho_r^s}} & 0 & 1 & 0 & 1 & 0\\[0.3em] \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \end{array}\right]}^{\mathbf{H}} \cdot \overbrace{\begin{bmatrix} {x_r} \\ {y_r} \\ {z_r} \\ \hdashline {v_{r,x}} \\ {v_{r,y}} \\ {v_{r,z}} \\ \hdashline { dt_r} \\ { d\dot{t}_r} \\ \hdashline { dt_{ISB}^{S_2 \rightarrow S_1} } \\ { d\dot{t}_{ISB}^{S_2 \rightarrow S_1} } \\\hdashline { dt_{IFB}^{F_2 \rightarrow F_1} } \\ \end{bmatrix}}^{\mathbf{x}} \end{equation}

Which can be written with \( {\mathbf{u}_{r,v}^s} = \frac{{\mathbf{v}^s} - {\mathbf{v}_r}}{ \rho_r^s} \) as

\begin{equation} \mathbf{H} = \left[\begin{array}{ccc:ccc:cc:cc:c} - {\text{u}_{r,x}^s} & - {\text{u}_{r,y}^s} & - {\text{u}_{r,z}^s} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\[0.3em] - {\text{u}_{r,x}^s} & - {\text{u}_{r,y}^s} & - {\text{u}_{r,z}^s} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\[0.3em] - {\text{u}_{r,x}^s} & - {\text{u}_{r,y}^s} & - {\text{u}_{r,z}^s} & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\[0.3em] \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\ \hdashline - {\text{u}_{r,v_x}^s} & - {\text{u}_{r,v_y}^s} & - {\text{u}_{r,v_z}^s} & - {\text{u}_{r,x}^s} & - {\text{u}_{r,y}^s} & - {\text{u}_{r,z}^s} & 0 & 1 & 0 & 0 & 0\\[0.3em] - {\text{u}_{r,v_x}^s} & - {\text{u}_{r,v_y}^s} & - {\text{u}_{r,v_z}^s} & - {\text{u}_{r,x}^s} & - {\text{u}_{r,y}^s} & - {\text{u}_{r,z}^s} & 0 & 1 & 0 & 1 & 0\\[0.3em] \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \end{array}\right] \end{equation}

([17] Groves, ch. 9.4.2.2, eq. 9.163, p. 420)

Measurement noise covariance matrix

See NAV::GnssMeasurementErrorModel

Initialization

• State \( x \): from Single Point Positioning solution using weighted Least-squares estimation
• Covariance matrix of state \( P \):
  • if more observations than unknowns: SPP solution using LSE
  • otherwise: GUI input

Inter system time difference reference system change

Case 1: New Sat System with higher priority found (GPS > GAL > GLO > ...)

Old state:

• Observed GAL \( S_2 \), GLO \( S_3 \) and BDS \( S_4 \)
• Reference system: GAL \( S_2 \) New state:
• Additionally observed GPS \( S_1 \)
• Reference system: GAL \( S_2 \) -> GPS \( S_1 \)

First we need to wait one epoch, in order to estimate the new inter system differences \( { dt_{ISB}^{S_1 \rightarrow S_2}} \) and \( { d\dot{t}_{ISB}^{S_1 \rightarrow S_2}} \) Then we can transform the state with a matrix \( \mathbf{D} \) with \( \mathbf{x}_{\text{new}} = \mathbf{D} \cdot \mathbf{x}_{\text{old}} \)

\begin{equation} \begin{bmatrix} { \mathbf{r}_{r} } \\ { \mathbf{v}_{r} } \\ { dt_r} \\ { d\dot{t}_r} \\ \hdashline { dt_{ISB}^{S_3 \rightarrow S_1}} = { dt^{S_3}} - { dt^{S_1}} \\ { d\dot{t}_{ISB}^{S_3 \rightarrow S_1}} = { d\dot{t}^{S_3}} - { d\dot{t}^{S_1}} \\ { dt_{ISB}^{S_4 \rightarrow S_1}} = { dt^{S_4}} - { dt^{S_1}} \\ { d\dot{t}_{ISB}^{S_4 \rightarrow S_1}} = { d\dot{t}^{S_4}} - { d\dot{t}^{S_1}} \\ { dt_{ISB}^{S_2 \rightarrow S_1}} = { dt^{S_2}} - { dt^{S_1}} \\ { d\dot{t}_{ISB}^{S_2 \rightarrow S_1}} = { d\dot{t}^{S_2}} - { d\dot{t}^{S_1}} \\ \hdashline { dt_{IFB}^{F_2 \rightarrow F_1}} \end{bmatrix}_{\text{new}} = \overbrace{ \left[\begin{array}{cccc:cccccc:c} \mathbf{I}_3 & \mathbf{0}_3 & 0 & 0 & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & 0 \\ \mathbf{0}_3 & \mathbf{I}_3 & 0 & 0 & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & 0 \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\[0.1em] \hdashline \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\[0.3em] \hdashline \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\[0.3em] \end{array}\right] }^{\mathbf{D}} \cdot \begin{bmatrix} { \mathbf{r}_{r} } \\ { \mathbf{v}_{r} } \\ { dt_r} \\ { d\dot{t}_r} \\ \hdashline { dt_{ISB}^{S_3 \rightarrow S_2}} = { dt^{S_3}} - { dt^{S_2}} \\ { d\dot{t}_{ISB}^{S_3 \rightarrow S_2}} = { d\dot{t}^{S_3}} - { d\dot{t}^{S_2}} \\ { dt_{ISB}^{S_4 \rightarrow S_2}} = { dt^{S_4}} - { dt^{S_2}} \\ { d\dot{t}_{ISB}^{S_4 \rightarrow S_2}} = { d\dot{t}^{S_4}} - { d\dot{t}^{S_2}} \\ { dt_{ISB}^{S_1 \rightarrow S_2}} = { dt^{S_1}} - { dt^{S_2}} \\ { d\dot{t}_{ISB}^{S_1 \rightarrow S_2}} = { d\dot{t}^{S_1}} - { d\dot{t}^{S_2}} \\ \hdashline { dt_{IFB}^{F_2 \rightarrow F_1}} \end{bmatrix}_{\text{old}} \end{equation}

Case 2: No observation for reference system within an epoch

Old state:

• Observed GPS \( S_1 \), GAL \( S_2 \), GLO \( S_3 \) and BDS \( S_4 \)
• Reference system: GPS \( S_1 \) New state:
• No observations for GPS \( S_1 \)
• Reference system: GPS \( S_1 \) -> GAL \( S_2 \)

We can transform the state with a matrix \( \mathbf{D} \) with \( \mathbf{x}_{\text{new}} = \mathbf{D} \cdot \mathbf{x}_{\text{old}} \)

\begin{equation} \begin{bmatrix} { \mathbf{r}_{r} } \\ { \mathbf{v}_{r} } \\ { dt_r} \\ { d\dot{t}_r} \\ \hdashline \enclose{updiagonalstrike}{{ dt_{ISB}^{S_2 \rightarrow S_2}} = 0} \\ \enclose{updiagonalstrike}{{ d\dot{t}_{ISB}^{S_2 \rightarrow S_2}} = 0} \\ { dt_{ISB}^{S_3 \rightarrow S_2}} = { dt^{S_3}} - { dt^{S_2}} \\ { d\dot{t}_{ISB}^{S_3 \rightarrow S_2}} = { d\dot{t}^{S_3}} - { d\dot{t}^{S_2}} \\ { dt_{ISB}^{S_4 \rightarrow S_2}} = { dt^{S_4}} - { dt^{S_2}} \\ { d\dot{t}_{ISB}^{S_4 \rightarrow S_2}} = { d\dot{t}^{S_4}} - { d\dot{t}^{S_2}} \\\hdashline { dt_{IFB}^{F_2 \rightarrow F_1}} \end{bmatrix}_{\text{new}} = \overbrace{ \left[\begin{array}{cccc:cccccc:c} \mathbf{I}_3 & \mathbf{0}_3 & 0 & 0 & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & 0 \\ \mathbf{0}_3 & \mathbf{I}_3 & 0 & 0 & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & \mathbf{0}_{3,1} & 0 \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1em] \hdashline \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 \\[0.3em] \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 \\[0.3em] \hdashline \mathbf{0}_{1,3} & \mathbf{0}_{1,3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\[0.3em] \end{array}\right] }^{\mathbf{D}} \cdot \begin{bmatrix} { \mathbf{r}_{r} } \\ { \mathbf{v}_{r} } \\ { dt_r} \\ { d\dot{t}_r} \\ \hdashline { dt_{ISB}^{S_2 \rightarrow S_1}} = { dt^{S_2}} - { dt^{S_1}} \\ { d\dot{t}_{ISB}^{S_2 \rightarrow S_1}} = { d\dot{t}^{S_2}} - { d\dot{t}^{S_1}} \\ { dt_{ISB}^{S_3 \rightarrow S_1}} = { dt^{S_3}} - { dt^{S_1}} \\ { d\dot{t}_{ISB}^{S_3 \rightarrow S_1}} = { d\dot{t}^{S_3}} - { d\dot{t}^{S_1}} \\ { dt_{ISB}^{S_4 \rightarrow S_1}} = { dt^{S_4}} - { dt^{S_1}} \\ { d\dot{t}_{ISB}^{S_4 \rightarrow S_1}} = { d\dot{t}^{S_4}} - { d\dot{t}^{S_1}} \\\hdashline { dt_{IFB}^{F_2 \rightarrow F_1}} \end{bmatrix}_{\text{old}} \end{equation}

Error propagation

After a transformation, the covariance matrix has also to be adapted by error propagation

\begin{equation} \mathbf{P}_{\text{new}} = \mathbf{D} \cdot \mathbf{P}_{\text{old}} \cdot \mathbf{D}^T \end{equation}