The latest issue of The College Mathematics Journal contains a good article by Donald Teets, titled “Push-To Telescope Mathematics” (see references below), describing the alignment process used by computerized telescopes to determine their orientation, and thus how to point to selected stars. This is a beautiful problem, in part because of just how little information is needed for its solution. Just power up the telescope, point it at any two known stars, and after some simple computation, the telescope knows how to find any other star in its database.

But some care must be taken; I think the solution in the article overlooks an important practical consideration, which I’ll get to shortly. Fortunately, there is an equally beautiful solution to a natural generalization of this problem, that I suspect is probably closer to what is actually implemented in the software of these computerized telescopes.

The basic problem is this: when the telescope is first powered up, it does not know its orientation relative to the stars. That is, it does not know the relationship between its “body frame” azimuth and altitude (which it can sense and/or control with motors) and the corresponding “reference frame” right ascension and declination.

We can point the telescope at a known star , and tell the telescope that it is pointing at that star. This gives some information about the telescope’s orientation: a particular direction in the body frame corresponds to a known direction in the reference frame. From now on, no matter where the telescope slews, it always knows how to return to . [1]

However, knowing the direction to just one star is not enough to completely determine orientation. Intuitively, this makes sense: for example, when the telescope is pointed at , it may still be “rolled” through any angle about its axis without changing its direction. Slewing the telescope again to point at one more known star will do the trick.

(Actually, this problem is interesting in that one measurement is not enough, but two is too many. Three parameters are needed to specify the telescope’s orientation– think yaw, pitch, and roll– and each star direction measurement provides two parameters. So with two stars, the telescope’s orientation is over-determined. We’ll come back to this shortly.)

At this point, we have the following mathematical problem: given 3-D unit (column) vectors and indicating the position of star in the inertial reference frame and sensor body frame, respectively, what is the 3×3 rotation matrix such that ?

The CMJ article solves the problem as follows: let be the 3×3 matrix with columns given by

and be the 3×3 matrix

Then ; solving, we have . Simple, right?

This works fine… as long as our measurements of direction in the body and/or reference frames are exact. If there are any errors– and there almost certainly are– then in general the matrix may not even represent a proper rotation (i.e., it may not be orthogonal with determinant 1).

Fortunately, there are two well-documented incremental improvements to this approach. The first is the TRIAD algorithm, whose only difference is essentially to ensure that the input matrices are themselves rotations, even if the measurements are inaccurate:

Note that the TRIAD algorithm effectively treats the first measurement as “perfect” (i.e., exactly), and accounts for error in the second measurement.

A more realistic approach would be to assume that all of the measurements contain errors, and to try to determine the “best” rotation that minimizes the effect of those errors in a least-squares sense. It turns out that we can do just that, with a very elegant algorithm described by Markley (see references below) that looks only slightly more sophisticated than what we have already seen. As a bonus, this algorithm can accept more than just two measurements:

Given n measurements of unit vector pairs in the reference and body frames, let be the 3xn matrix with columns corresponding to the reference frame vectors, and be the 3xn matrix of body frame vectors. Then the rotation matrix is computed using the singular value decomposition of :

Finally, it is worth emphasizing that this is really just one of several algorithms that solve the same least-squares problem. Shuster provides a good history and survey of these algorithms in the paper referenced below, including his quaternion-based QUEST algorithm.

References:

Markley, F. L., Attitude Determination using Vector Observations and the Singular Value Decomposition, The Journal of the Astronautical Sciences, 36(3) (July-September 1988): 245-258. [PDF] Shuster, M. D., The Quest for Better Attitudes, The Journal of the Astronautical Sciences, 54(3-4) (July-December 2006): 657–683. [PDF] Teets, D., Push-To Telescope Mathematics. The College Mathematics Journal, 43(3) (May 2012): 227-231. [JSTOR]

[1] This is not quite true. I am glossing over the fact that the telescope “body” frame is rotating with the Earth. So after some time has passed, the telescope does not even have enough information to know how to rotate back to the star . But as will be shown, we can handle this problem, with a second star… and an onboard clock to measure the time difference between the two star measurements.