Measurement generation

Nyx allows custom implementations of a MeasurementDevice, which this MathSpec cannot cover. Therefore, this page focuses on the implementation for GroundStation: this is the structure used for the default orbit determination setups which rely on Earth based ground stations.

Range and range-rate calculation

The embeded ground stations are initialized with a latitude \(\phi\), a longitude \(\lambda\), a height \(h\), an elevation mask, a range noise level and a range-rate noise level.

The position of this ground station is converted into the frame of the spacecraft to generate a measurement between a spacecraft (\(r\), for receiver) and a ground station (\(t\), for transmitter). In the following, the subscript \(r\) corresponds to the radius of that orbital state and \(v\) the velocity vector of that state. Further, if a symbol has a frame associated to it, e.g. \(^{\text{SEZ}}\mathbf{\rho}\), then it is a vector (the bold font may not that visible).

The range in the IAU Earth fixed frame is then computed:

\[^{\text{IAU Earth}}\mathbf{\rho} = \mathbf{r}_r - \mathbf{t}_r\]

That range vector is then converted in the SEZ frame using the algorithm from Vallado, where \(R_i\) corresponds to a rotation by the \(i\)-th axis:

\[^{\text{SEZ}}\mathbf{\rho} = R_2\left(\frac \pi 2 - \phi\right)~R_3(\lambda)~\cdot~^{\text{IAU Earth}}\mathbf{\rho}\]

The elevation is then computed as follows:

\[ el= \sin^{-1}\left(\frac {^{\text{SEZ}}\mathbf{\rho_z}}{|^{\text{SEZ}}\mathbf{\rho}|}\right) \]

A Gaussian/normal PDF is sampled with the range and range-rate noises to noise-up the true range and range rate computations, respectively noted \(\mathcal{N}(\rho)\) and \(\mathcal{N}(\dot\rho)\). The range is computed trivially computed:

\[\rho = |^{\text{SEZ}}\mathbf{\rho}| + \mathcal{N}(\rho)\]

And the range-rate is computed as:

\[\dot\rho = ^{\text{SEZ}}\mathbf{\rho} \cdot \frac{(\mathbf{r}_v - \mathbf{t}_v)}{\rho} + \mathcal{N}(\dot\rho)\]

Measurement sensitivity matrix

In a Kalman filter, the sensitivity matrix, noted \(\tilde{H}\), relates the filter covariance, the filter gain, the measurements and the noise of the measurement. Like the state transition matrix, the sensitivity matrix is a partials matrix of size \(N\times M\), where \(N\) is the size of the measurement and \(M\) is the size of the state to be estimated. For example, if the measurement is the range \(\rho\) and the range-rate \(\dot{\rho}\), and the estimated state is the position \(\{x,y,z\}\) and velocity \(\{\dot x, \dot y, \dot z\}\), then the sensitivity matrix is written as follows.

\[\begin{equation} \label{sensitivity} \tilde H = \begin{bmatrix} \frac{\partial \rho}{\partial x} & \frac{\partial \rho}{\partial y} & \frac{\partial \rho}{\partial z} & \frac{\partial \rho}{\partial \dot x} & \frac{\partial \rho}{\partial \dot y} & \frac{\partial \rho}{\partial \dot z} \\ \frac{\partial \dot \rho}{\partial x} & \frac{\partial \dot \rho}{\partial y} & \frac{\partial \dot \rho}{\partial z} & \frac{\partial \dot \rho}{\partial \dot x} & \frac{\partial \dot \rho}{\partial \dot y} & \frac{\partial \dot \rho}{\partial \dot z} \\ \end{bmatrix} \end{equation}\]

Using the same methodology as previously, it is evident that the sensitivity matrix may be computed by simply defining a hyper-dual space whose size is equal to that of the state to be estimated. The equations which return the range \(\rho\) and the range-rate \(\dot\rho\) from an input state will then automatically also return the components of the sensitivity matrix.

Note

The ground stations currently do not support light-time corrections or tropospheric attenuations. This will be part of the 1.1.0 release.