Issue 61

I’M CERTAIN ABOUT UNCERTAINTY: It may come as a surprise for me to tell you that despite all this work over 100’s of years, that astrodynamics is challenged by this pesky thing called uncertainty. Astrodynamics has been around for a long time. As far back as the 1600’s, folks like Kepler, Newton, Brahe, and others really got into being Astro geeks. Over time, countless equations, methods, algorithms and a lot of know-how have been developed and are applied today. All those precise equations and algorithms are awesome. Nothing wrong with them, it’s just that uncertainty is a constant companion for all who seek to determine orbits. This Jack’s Astro Corner article is going to explain why this uncertainty. Its important to know the challenges uncertainty brings and what our astro experts do about it.

SOME BUZZ WORDS OF ORBIT ESTIMATION: The estimation mathematics of orbit determination is a lot of matrix and vector math. In Astro math there are state vectors, the things we are trying to determine. For example, position and velocity of an object in space at a “now” time or at some future time. It’s 6 numbers, 3 are position in a coordinate frame like Earth Centered Inertial and 3 other numbers that are velocity in that frame. A state vector can also be the orbital elements. Sometimes we add to the state vector additional things to estimate like drag and solar radiation pressure and other odd things that effect the orbit path. A state vector is what we seek and the time of when it is current. Astro math has many equations that model orbital motion, the models we use to take a state vector and move it forward to some other time of interest. When an astro expert estimates and orbit using all sorts of nifty methods, they get the “here’s where we are” answer as well as and “here’s how fast we’re going.” They also get some additional info that is very helpful. They get numbers that describe the uncertainty of the “here’s where we are and how fast we are going” estimate. Along with the state vector we get an important matrix that captures the uncertainty, it’s called the Covariance Matrix. It is a set of numbers nicely arranged in a matrix that tells us a lot and is directly tied to this uncertainty I speak of. One more thing to foot stomp about this Astro math. Internal to the math are equations that relate what we are measuring via the sensor’s observations to the state vector we seek. We can’t directly measure a state vector, the pathway to finding it starts with the sensor’s observations. OK, enough of my astro ramblings, let’s look at two areas where uncertainty challenges us.

UNCERTAINTY IN THE SENSOR MEASUREMENTS/OBSERVATIONS: The sensors we use to track objects in space are amazing; radars, telescopes, electronic fences, radio receivers and some engineering technology that I don’t understand, but they work fabulously. Here’s a photo of the amazing and awesome Eglin space sensor in Florida.

They take measurements which are called observations. If you take a bunch of observations during an object passing in the field of view of the sensor you have a track. We use those time tagged observations as inputs to the orbit determination computers that host the equations, algorithms and much more. Guess what, all measurements from a sensor have some error inherent in them. Now error makes it seem like someone made a mistake, but no mistakes, it’s just the reality of we can’t measure things perfectly. We have measurement data that has some degree of uncertainty. There’s always a statistical plus or minus of what you are measuring. Sensors can also have a bias. The bias is an offset and it can vary with sensor temperature for example. For space sensors, a lot of work goes into determining these uncertainty parameters and developing a statistical summary that can be used as you apply those measurements. The observations have uncertainty and together with the sensor measurement we put that sensor uncertainty info in something called a noise matrix. We generally call them weights, or sigmas (standard deviation). They along with the actual measurement get used by the orbit determination equations and algorithms. So, right from the start we carry uncertainty into the math of orbit determination, that’s the way it is and we must use that information to ultimately determine the answer we seek, the state vector and the Covariance Matrix uncertainty data associated with that estimate. OK, hang on, we got more uncertainty coming!

UNCERTAINTY AND SIMPLICITY IN THE MATH MODELLING OF ORBITS: Let’s now look at the equations of orbit determination from an uncertainty perspective. So, did Kepler and all those old dudes leave something out? Why is there uncertainty in the math? That’s where our Astro experts have to make some calls or decisions to try to help minimize the uncertainty of their math equations that use the sensor measurements and “noise” data. They have to make math models or set of equations that can take the observations and mathematically mash on them and get the state vector answer. That’s not easy. We might assume an orbit path is a circle for the model, but the object in question is not exactly in a circle, so our model may be off a little. So, our modelling of the orbit path and all the forces acting on the object (i.e., big gravity, atmospheric drag, Earth’s odd gravity, the tugging of the Sun, Moon and other gravitational objects, the solar wind, there’s a lot more) must be made into an equation that might not fully represent what Mother Nature is dishing out. My astro friend and mentor TS Kelso adds and important point to amplify the uncertainty of the math models, he says “Force models aren’t so much uncertain but, as with any model, are simplifications of reality. The choice of a specific force model strikes a balance between keeping things like computation time reasonable and getting usable results for whatever the primary task is. And, of course, there are those cases where it appears we simply don’t understand the science as well as we would like to. So, there is simplifications and uncertainty in the models we use to propagate the state vector and also propagate the error or uncertainty we start with. The uncertainty grows over time because the equations don’t perfectly predict what will really happen to the object in space. Are we not trying hard enough? No, the efforts to model orbits are heroic and constantly being worked. We have uncertainty in the inputs (observations) and now we have uncertainty in the math models that must be simplified to work on the computers nicely. Ugh, is this bad news?

Fear not-we deal with it smartly and wisely. Like the sensor uncertainty data, we also have uncertainty data the way we model the orbit equations. That data (which is stashed away in something called a Process Noise Matrix, don’t you love these nifty terms?) can be used to numerically estimate the uncertainty at the time of each observation and know the uncertainty as we propagate to a future time of interest and how it grows. The state vector answer may be off a little. But in the techniques, we use to estimate the orbit give us something really awesome called the Covariance Matrix. This Covariance info comes as companion of the state vector estimate we get a numerical feel for what the bounds of confidence are on that estimate. What can we do with that uncertainty numerical info? Let’s look at that next.

LET’S MEET THE COVARIANCE MATRIX (UNCERTAINTY STORE HOUSE OF INFORMATION): We all want the answer; where am I? Where am I going to be and how fast will I be going? That’s important but so is the how confident am I in that answer? How much can I trust the answer? That’s where along with the answer you also need the covariance matrix information which is laden with good info to better understand the uncertainty of your state vector. Let’s look at position in the ECI coordinate frame. The state vector answer tells us “You are here and going this fast”, a moving dot in space in the ECI frame. BUT, the covariance data contained in the covariance matrix will help you understand the “it’s not a dot.” The Covariance Matrix give you info that ultimately with a little math (square root and multiplication is all you got to do) can give you the plus or minus numbers for some statistical confidence associated with the “here’s where you are” estimate.

THE FOOTBALL BALL OF UNCERTAINTY: Allow me to introduce what we can do with this “plus or minus” info to get a visual idea of the uncertainty volume surrounding the state vector “dot.”. Our Astro smarties can convert this representation of uncertainty and tell you the position of the space object using what’s called error ellipsoid. It’s like a football or rugby ball shaped volume around that dot. Say what? Football? Rugby ball? At the time we estimate the “where am I and how fast am I going?” state we also estimate the error ellipsoid. Then as time marches on, we will see the error ellipsoid actually grows in size until we get some observations to re-accomplish the orbit estimate and shrink this uncertainty volume down. The way we can get that back to small is take more observations and redo the orbit determination. So, think of a “Rugby Ball of Uncertainty,” that’s what Covariance Matrix helps us understand. Here’s an illustration from Aerospace Corporation’s Crosslink magazine issue Fall 2015. It shows the concept of error ellipsoid and in this case, they are overlapping in a conjunction situation. Hummmm, that might be a problem. That issue of Crosslink is dedicated to Understanding Space Debris, it’s a must read if you are in this space business! Here’s the link: https://aerospace.org/paper/crosslink-fall-2015

WRAP IT UP JACK: Astrodynamics rocks! Sensors aren’t perfect, but we do have some good insight into just how uncertain their measurements are. The math models we use for orbit determination are awesome and comprised of huge equations, but the models are not perfect, they are simplifications of what is really happening and thus have some uncertainty. We carry with the estimate uncertainty, but its good stuff to know. Through the orbit estimation process we can understand just where we are inside the uncertainty football or rugby ball shaped volume of “it’s in there somewhere.”. Furthermore, when we want to know the position and velocity of an object in some future time, we must use this imperfect math to advance (propagate) the state and covariance estimate forward. Uncertainty will build up. But we can also advance the covariance matrix and have a good idea just how uncertain the estimate is. It works. So next time you see your colleagues who estimate orbits. Here’s a photo of some space patriots at Vandenberg), be nice to them, they got a tough job but do it with skill, knowledge and dedication and their pesky companion tagging along call uncertainty.

THOSE WHO HELPED ME: I want to give a shout out here to Brent Skrehart of the National Space Defense Center, Richard Osedacz who long ago was an AF Academy astro student and now my astro mentor and TS Kelso who is the mastermind of Celestrak for their help in telling this story of uncertainty in orbit estimate.