My education experiment: deconstructing Satellite Laser Ranging (part 1)

Imagine a spacecraft the size of your car travelling at an altitude of 600 km with a velocity of 7.6 km/s. Now try to determine where the spacecraft is and how fast it is flying, also called its dynamic state. There is no speedometer (no wheels turning) or a pitot tube like in an aircraft (no air to measure). So, we, as astrodynamicists (yes, I think that is a word), have different techniques to accurately determine the state of the spacecraft. Highly accurate measurements of distances (also called range) between the satellite and ground stations are used, as well as the changes in these distances (called range-rate). If you follow this blog a little bit, you know that we work on building a ground station at the TU Delft that measures these range-rates with respect to cubesats. With some complicated, but very elegant mathematics, these measurements are transformed into very precise orbit determinations of satellites. Some of my colleagues can tell you within one cm accurate where their satellite is. In my course, I have tried to device an experiment to show the basics of this technique to my students and hopefully show its powerful application. 

One of the most precise instruments to determine the range of a satellite (the distance between your ground station and the spacecraft) is the SLR system (Satellite Laser Ranging). The system shoots a laser pulls towards the satellite. The satellite needs to have a retro-reflector, which is a mirror that will reflect light back in the same direction as it was coming from: two mirrors with a perfect 90 degree angle between them will do the trick. The laser light will be reflected back and the SLR system records the arrival time of the laser pulls. Subtracting the starttime of the laser pulse will result in the travel time of the laser light. Multiplying this travel time with the speed of light and dived it by two, will give you the range of the satellite w.r.t. the ground station. The technology of SLR is so advanced, that they can determine the range sometimes up to one mm accurate. But how do you determine a state vector in 3 dimensions-3 position and 3 velocity coordinates-with only one range measurement? My experiment will try to convince you that it is possible.

Of course, I can’t simulate a satellite flying by, so I have the second best thing: a student on a skateboard holding a board with a sketch of a retro-reflector (sorry, I thought that was funny). Back to the experiment, I deconstructed the problem to a 2D situations. Our “satellite" will move in a straight line (aka orbit) and is only affected by the rolling resistance (aka air drag) after a push by me (initial state vector). The “satellite” will move and during that motion, two “ground stations” will try to measure the range between them and the “satellite”. I positioned the “satellite" in front of the class and the two ground stations of the corners in the back of the class. Both “ground stations” have a laser ranging device, bought at the local hardware store.

The students need to measure the range every two seconds, which will result in 4 measurements during the whole experiment. Before the experiment starts I ask the students to determine their position within the room. Also, I have marked 4 positions for the “satellite” that simulates the motion of our satellite. I have learned from experience that pushing a student on a skateboard might not be a good idea and also making 4 measurements within 6 seconds is a very challenging task for the students. This is how the setup of the experiment looks like:

The path of the satellite is shown in red and both ground stations are denoted by the green and blue dot. I used data from a classroom session of last year. The experiment started and laser light was travelling through the class room. Students and teacher liked this! The data that was gathered was the following:

All measurements are in meters and the colours denoted the ground station. The precision of the laser ranging device is around the mm, and the students tried there best to determine the range as accurate as possible, but there are always measurement errors. So, how can we now determine position (and maybe even the velocity of the satellite). 

I show two approaches to my students: the kinematic and the dynamic method. The kinematic method can be best explain with the intersecting circle example. For every position (start with t = 0), draw a circle around the station with the radius equal to the station’s observation. So, for station 1 this is 9.686 meters and for station 2 this is 11.226 meters. The circles intersect each other in two points, one point inside the class-room and one point outside the class-room. Let's eliminate the latter, as the experiment was done inside the class-room, and in real-life, we know that satellites will not fly inside the Earth. That point has an x- and y-coordinate, both can be determined by using the equation for the radius of a circle:

The coordinates of the stations are known, which means we have two equations (one for station 1 and one for station 2) and 2 unknowns (the x and y coordinate of point 1). This is a determined system and with some elementary math, or a calculator, we can determine the position of point 1. Repeating this for t = 2 sec, t = 4 sec, and t = 6 sec, will result in the trajectory of our satellite.

The black dashed line shows our computed trajectory for which we used the eight observations to calculated it. The computed trajectory is pretty close to the theoretical one (we had some good observers), but the trajectory wiggles a bit, which should not be ad our satellite moved in a straight path. This of course illustrates the observation errors. So, with the kinematic approach we are able to determine the trajectory of our satellite.

However, it is quite unpractical, because two ranging stations need to track the satellite simultaneously and in the end we only acquired 4 locations. In between these observations, we have to rely on interpolations, which can be very unreliable in some cases. Furthermore, we have no solution for the velocity of the satellite (we could use numerical differentiation to obtain 3 velocity estimates, but we want more). It would be better if we could only use one tracking station (reduces the cost) and construct a model of the trajectory, such that we also have a reasonable idea of the position and velocity of the satellite. This is where the dynamic approach makes its entry.

See part 2 (coming soon)