Improved Hybrid Navigation for Space Transportation

Abstract

This work presents a tightly coupled hybrid navigation system for space transportation applications. The tightly integrated set-up, selected for its robustness and design flexibility, is here updated with GPS Pseudoranges and Time-Differenced GPS Carrier Phases (TDCP) to promote fast-dynamics estimation. The receiver clock errors affecting both observables are analysed and modelled. Tropospheric delay-rate is found to cause major disturbance to the TDCP during atmospheric ascent. A robust correction scheme for this effect is devised. Performance is evaluated using real GPS measurements.

Appendix

Parkinson et al. [15] suggests the following coarse tropospheric delay model, which does not require in situ atmospheric measurements nor the use of look-up tables:

$$\begin{aligned} \displaystyle h_{\tau _\mathrm {T},i,k}\left( {\mathbf x}_k\right) = {\textstyle \frac{1}{c}}\,m(\mathrm {E} _{i,k})\varDelta (\mathrm {h} _{\mathsf {a},k})\;, \end{aligned}$$

(63)

$$\begin{aligned}&\text {with}&\displaystyle m(\mathrm {E} _{i,k})= \frac{1.0121}{\sin \mathrm {E} _{i,k}+0.0121}\;, \quad \varDelta (\mathrm {h} _{\mathsf {a},k})=2.4405\;e^{-0.133\times 10^{-3}\;\mathrm {h} _{\mathsf {a},k}}\;. \end{aligned}$$

(64)

The Zenit delay $\varDelta$ is a function of the receiver antenna altitude $\mathrm {h} _{\mathsf {a},k}$, and the mapping function m depends on the satellite apparent elevation $\mathrm {E} _{i,k}$. c is the speed of light.

The sensitivity vector of this model to the error-state vector can be given as

$$\begin{aligned} \displaystyle {\mathbf h}_{\tau _\mathrm {T},i,k}=\frac{\partial h_{\tau _\mathrm {T},i,k}\left( {\mathbf x}_k\right) }{\partial \varvec{\delta } {\mathbf x}_k} = -{\textstyle \frac{1}{c}}\,m(\mathrm {E} _{i,k})\varDelta (\mathrm {h} _{\mathsf {a},k})\begin{bmatrix}0.133\times 10^{-3}\\ m(\mathrm {E} _{i,k})\end{bmatrix}^{\mathsf {T}}\begin{bmatrix}\frac{\partial \mathrm {h} _{\mathsf {a},k}}{\partial \varvec{\delta } \!{\mathbf x}_k}\\ \frac{\partial \sin {\mathrm {E} _{i,k}}}{\partial \varvec{\delta } \!{\mathbf x}_k}\end{bmatrix}\;, \end{aligned}$$

(65)

$$\begin{aligned}&\text {with}&\displaystyle \mathrm {h} _{\mathsf {a},k}=\left\| {\mathbf r}_{\mathsf {a},k}^E\right\| -\mathrm {R} _\oplus \;, \qquad \sin \mathrm {E} _{i,k} = \left( {\mathbf {e}}_{\mathsf {a},k}^E\right) ^{\mathsf {T}}{\mathbf {e}}_{\rho \!,i,k}^E\;, \end{aligned}$$

(66)

where $\mathrm {R} _\oplus$ is the Earth radius, ${\mathbf r}_{\mathsf {a},k}^E$ is the receiver antenna position, ${\mathbf {e}}_{\mathsf {a},k}^E$ is the unit direction from the ECEF origin to the receiver antenna, and ${\mathbf {e}}_{\rho \!,i,k}^E$, as in (45), is the unit range vector. The non-null partial derivatives in (65) are then

$$\begin{aligned} \frac{\partial \sin {\mathrm {E} _{i,k}}}{\partial \varvec{\delta } {\mathbf r}_k^E}&= \frac{\hat{\mathbf {e}}_{\mathsf {a},k}^E}{\Vert \hat{\mathbf r}_{\mathsf {a},k}^E\Vert }^{\mathsf {T}}\bigl [\hat{\mathbf {e}}_{\rho \!,i,k}^E \times \bigr ]^2-\frac{\hat{\mathbf {e}}_{\rho \!,i,k}^E}{\hat{\rho }_{i,k}}^{\mathsf {T}}\bigl [\hat{\mathbf {e}}_{\mathsf {a},k}^E \times \bigr ]^2\;, \qquad \frac{\partial \mathrm {h} _{\mathsf {a},k}}{\partial \varvec{\delta } {\mathbf r}_k^E} = \left( \hat{\mathbf {e}}_{\mathsf {a},k}^E\right) ^{\mathsf {T}}\;,\end{aligned}$$

(67)

$$\begin{aligned} \frac{\partial \sin {\mathrm {E} _{i,k}}}{\partial \varvec{\delta } \varvec{\theta } _k^E}&= \frac{\partial \sin {\mathrm {E} _{i,k}}}{\partial \varvec{\delta } {\mathbf r}_k^E}\hat{\mathbf C}_{B_k}^{E_k}\bigl [{\mathbf {l}}_\mathsf {a}^B \times \bigr ]\;, \qquad \frac{\partial \mathrm {h} _{\mathsf {a},k}}{\partial \varvec{\delta } \varvec{\theta } _k^E} = \frac{\partial \mathrm {h} _{\mathsf {a},k}}{\partial \varvec{\delta } {\mathbf r}_k^E}\hat{\mathbf C}_{B_k}^{E_k}\bigl [{\mathbf {l}}_\mathsf {a}^B \times \bigr ]\;. \end{aligned}$$

(68)