# Problem of process noise of satellite orbit determination

1. May 1, 2005

### whigg_wang

During my work of space vehicle orbit determination, I have been confronted with a difficulty that hasn't been solved until now. When calculating the transition matrix that translate the process noises to the state parameters of position and velocity,how can I work out the partial derivatives of the process noise with respect to the position and the velocity parameters?May I take the atmospheric drag model as the process noise explicit expression,if I take air drag acceleration as a process noise? WAITING for your helpful answer.

2. May 2, 2005

### whigg_wang

Doesn't any one knows the solution?

3. May 2, 2005

### SpaceTiger

Staff Emeritus
Are you trying to do Kalman filtering? It seems to me that atmospheric drag would provide a steady deceleration and that term would have to be included in your calculation of the state variables. I think process noise has to be some kind of random process. This could indeed be cause by fluctuations in the atmosphere, but perhaps you could give us some more detail on the "atmospheric drag model" you're referring to.

4. May 2, 2005

### Billy T

I don't know if it is helpful or not but before GPS, the Applied Physic Lab of JHU, where I worked developed for US Navy a series of navigational satellites that were immune to atmospheric drag, solar radiation pressure, Etc.

There was a "proof mass" - a small sphere (1.3 cm diameter as recall, but that is not important) of a platinum/gold mixture that was carefully adjusted to be neither para nor dia magnetic.

The main satellite had very fine impulse "thrusters" (teflon ablation in electric pulse discharges) that could "fly" the main satellite to keep the "proof mass" exactly (less than mm) at the gravitational center of the satellite. It was expensive to build, because everything had to be accurately weighed and located precisely to calculate the exact location of the C of G of the main satellite body - a location of the proof mass at any other location would produce an unknown gravitational force on the proof mass in addition to those of the Earth, moon, sun etc.

Some parts of this work (at least their results) were highly classified. - after months of observations (years? I was only slightly envolved.) one could describe the Earth's gravitational field with many terms of the Tessler harmonics. Even how many is terms is classified and very few people knew their values after the first dozen or so, back in the IBM era with serious stress between USSR and US.

The first of these satellite was called "Triad." Because the production cost could be reduced greatly, Triad was replaced by "Discos" which only very accurately controlled the in track component of the gravitational interaction between the proof mass and the main satellite body. There were a lot of these satellites launched as the navy want to be sure one would be available (above the horizon) when a IBM submarine briefly stuck a detectable antenna up. Unlike GPS, by curve fitting the Doppler shift of a very stable oscillator in the satellite, in about five minutes, you can tell exactly where you are on Earth with only one satellite - Navy liked this as many might die in exo-atmospheric nuclear blasts by EMP, as GPS still would. Sub could not accurately hit its targets if it does not know exactly were it is at launch and with sub-sized multiple bombs on each IBM of the period, this was important.

5. May 3, 2005

### whigg_wang

to space tiger:
The atmosphere drag model i have adopted is :
a=-[Cp(A/M)V(VectorV)]/2
where C is the coefficient of atmosphere drag
p is the atmosphere density
V is the velocity (scale value)
VectorV is a velocity (vector value), the satellite velocity with respect to atmosphere

but i don't know confirmly whether treating the atmosphere drag as process noise is correct.

yes,i want to do filtering, SRIF(square root information filter) instead of the classic kalman filter

6. May 3, 2005

7. May 3, 2005

### SpaceTiger

Staff Emeritus
I should emphasize that this is the first I've even heard of such things as Kalman filtering, square root information filtering, or process noise, but it seems you're dealing with some variation on the equation:

$$\vec{x'}=A\vec{x}+\vec{n}$$

where x is the vector of position and velocity at some time (t), x' is the position and velocity at some later time, n is the process noise, and A is the matrix that evolves the state variables (position and velocity). From your post, your x vector seems to be two-dimensional (position and velocity). However, in order to take an acceleration into account (like, say, atmospheric drag), you would have to make it three-dimensional and include the appropriate terms in the matrix, A. Noise processes, on the other hand, have to be drawn from some kind of random variable with an underlying probability distribution. The equation you gave is not a probability distribution, but rather an equation that gives exact values of the deceleration at a given time.

8. May 3, 2005

### Nereid

Staff Emeritus
Welcome to Physics Forums, whigg_wang!
Maybe a nitpicky question, but in what sense of 'correct' are you asking? I mean, are you asking about the tractability of the analyses? the robustness of any conclusions?

I'm moving this to Celestial Mechanics - may get more attention there.

(BTW, that you got a reply within ~12 hours of your first post is pretty good, don't you think? I mean, when you compare PF with other physics fora on the internet).

9. May 5, 2005

### whigg_wang

To SpaceTiger:
SIRF is a kind of filtering method. Position and velocity are all vector variants,which consist of three direction components. Accelerations are not included in the state of X estimated, and the process noise(here,we take atmosphere drag acceleration as the process noise),which is treated as first order of Gauss-Markov process is part of accelerations. The determinant part of the process noise is also included in the state of X.

10. May 5, 2005

### whigg_wang

To Nereid:
After reading relevant papers,I found many researchers have successfully accomplished the work in which atmosphere drag acceleration is treated as process noise. But the authors have not mentioned their solution in detail.
I posted this question in the sub-area of Celestial Mechanics a few days ago. But I didn't know why it was locked.
Yes, it's pretty good to get timely information.So i will look through the interesting topics almost every day.

11. May 5, 2005

### SpaceTiger

Staff Emeritus
This means that you're assuming something is fluctuating with zero mean. In the context of an atmospheric drag model, it could be wind currents, density fluctuations, etc. but a steady deceleration due to the motion of the satellite would not be "noise" in any sense that I've ever heard. If you're looking for numbers giving the magnitude of these fluctuations, you've almost certainly come to the wrong place.

12. May 11, 2005

### whigg_wang

13. May 11, 2005

### enigma

Staff Emeritus
I locked it because it was a doublepost.

It is difficult to have a coherent conversation when there are two discussions ongoing, so we require that threads be posted in only one subforum.

14. Jul 8, 2005

### SGT

If you don't have enough information about your process, you may model the unknown states as process noise. In your case you know the expression for atmospheric drag, so you should use it in your model and not consider it as noise.
Because drag is a nonlinear function of velocity, you should use a nonlinear filter such as the EKF for your estimation.