HPOP Covariance
Select Compute Covariance to include covariance in an HPOP satellite's propagation.
About covariance
HPOP can propagate the state error covariance matrix as it is propagating ephemeris. This matrix represents the uncertainty of the satellite's position and velocity. The satellite's position covariance submatrix can then be reported and graphed. You can also display the submatrix in the 3D Graphics window over time.
State error covariance growth is usually dominated by the initial position and velocity covariance. This covariance is typically generated by orbit determination software. Any errors between the estimated state and the actual state tend to grow over time, especially in the along-velocity direction. This is a direct effect of the dominant two-body force, since the period of the orbital motion is on its initial state. Eccentric orbits also show an oscillation in uncertainty that superposes the growth trend, whose amplitude is larger for larger eccentricity. Uncertainty increases near perigee, where the speed of the satellite is largest in its orbital cycle. Uncertainty decreases near apogee, where the speed is the smallest.
A second contributor to state error covariance growth is force model mis-modeling. For example, the force model environment that you simulate in HPOP does not model the actual forces experienced by the satellite. When using Kalman or related filters, force model mis-modeling contributes to covariance growth through process noise models. These models aim to characterize the uncertainty of the modeling itself. Because HPOP does not have all the data needed to do this sophisticated analysis, it uses a simpler scheme. This scheme - called consider analysis - accounts for some force model mis-modeling.
Consider analysis adds contributions of force model mis-modeling to the state error covariance propagation. It does this by treating constant parameters in the force model as being uncertain themselves, instead. The uncertainty in the parameter value leads to uncertainty in the force evaluation, which then contributes to the state error covariance.
Set up the covariance matrix
To apply covariance, you must first define the initial state error covariance matrix. Because the matrix is symmetric, you only need to define the lower triangle. The matrix naturally decomposes as:
where Pzz is the 6x6 symmetric position/velocity covariance, Pwz is an nx6 matrix representing the cross-correlation between consider parameters and the position and velocity covariance, and Pww represents the consider parameter covariance matrix (which is also symmetric). Here n is the number of consider parameters being used.
Frame
The Pzz covariance matrix can be input in any of the following frames:
- J2000
- TrueOfDate
- LVLH
- Frenet coordinate frame
Representation
Gravity
Define the number of terms to use in the gravity field when propagating the state error covariance matrix. These values should be consistent with the values that you enter in the Force Model Properties dialog box. In particular, if the number of terms has been set to anything other than 0x0, then the largest degree that you define for propagating covariance should be two or higher. A 4x4 field for covariance propagation is typically sufficient, even when the ephemeris uses a higher order field.
Consider analysis
Select Include Consider Analysis to include consider parameters when propagating covariance. To include a parameter, select Use in its table row. The Value column corresponds to the self-covariance of that parameter (i.e. the value that appears as the diagonal element of Pww); the other columns are for entering cross-correlations between that parameter and position/velocity covariance. While the Value column must be non-negative, the other columns can be positive or negative. But, the overall state error covariance matrix must be positive definite; thus, the cross correlations cannot be too large compared to the diagonal elements of Pzz. Usually, this cross correlation is ignored (i.e. values are set to zero for the position/velocity cross-correlation).
All Varieties of HPOP make the following consider parameters available:
- Drag -- the consider parameter is D, where D = Cd * DragArea/Mass
- SRP -- the consider parameter is K, where K = Cr * SRPArea/Mass
Consider cross correlation
In most cases, you will not want to model cross-correlation among the consider parameters themselves, i.e. Pww is modeled as a diagonal matrix. But, you can choose to model cross-correlation effects as well using the third grid. These effects are computed in the frame that you selected for the covariance matrix.
Select Include Consider Cross Correlation and then click the Add and Remove buttons to insert or delete rows to represent each non-zero cross-correlation. For each row, choose the two consider parameters (double-click in the Row or Column cells to choose the correct parameter) that have the correlation, and enter the cross-correlation.
About the Frenet coordinate frame
The directions that constitute the Frenet coordinate frame are provided in the following table:
| Symbol | Name | Description |
|---|---|---|
|
R
|
Radial | Along line from center of Earth through satellite position |
|
I
|
In-track | Normal to radial direction, towards inertial velocity |
|
C
|
Cross-track | Along the orbit angular momentum vector (R x I) |
|
T
|
Tangential | Along the inertial velocity direction |
|
N
|
Normal | Normal to tangential direction, towards radial direction |
The Frenet Coordinate frame is an axes definition of (N)(T)(C). When you select the Frenet frame, the local frame of reference for Yaw Pitch Roll (YPR) is (T)(-C)(-N).
Rotations are performed in the sequence with the nominal thrust direction, specified with a (0,0,0) rotation, being along the X axis of the local frame. The nominal thrust direction is in the (T) direction for the Frenet frame.