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Gibbs Method for Orbit Determination
The Gibbs Method for orbit determination uses three specific position vectors (R1, R2, and R3) measured at three specific times (t1, t2, and t3) to determine the orbital elements of that object or spacecraft. Figure 1 demonstrates a typical measurement of such vectors. The Gibbs Method fundamentally acts as a technique taking these three position vectors at three unique times to then estimate the velocity of the spacecraft at the second observed position. This algorithm acts in the Earth Centered Inertial reference frame. Converting the estimated spacecraft state vector at a given time produces a set of Keplerian elements that can be propagated forward in time. Once solved for the velocity of the second position, you can then find all the orbital elements for the spacecraft: h – specific angular momentum, i- inclination, Ω - right ascension of the ascending node, e – eccentricity, ω – argument of perigee, and θ - true anomaly. See Figure 2 which shows these orbital elements and corresponding state vector. Solving the Gibbs Method relies on knowing the Gauss formulation. The Gibbs problem is formed under the assumptions of supposing we know three non-zero, coplanar position vectors, which represent three time-sequential vectors of a satellite or object in its orbit. . These assumptions are necessary to achieve a solution. Requiring the vectors to be non-zero simply prevents divide-by-zero operations, as well as needless analysis of orbits that don’t exist. The sequential requirement is very important because we consider the vectors sequential while forming the solution and take several cross products based on the given order. Changing from a sequential order will give erroneous results. Finally we require the vectors to be coplanar, but we can relax this somewhat in practice. Because actual observations aren’t perfect, we may obtain vectors which deviate by a degree or two from each other (in inclination).

Figure 1: Three position vectors in ECI frame at three sequential times.

Figure 2: Shows the orbital elements in which the state vector represents.

History
Suppose that from the observations of a space object at the three successive times t1, t2, and t3 (t1 < t2 < t3) we have obtained the geocentric position vectors r1, r2, and r3. The problem is to determine the velocities v1, v2, and v3 at t1, t2, and t3 assuming that the object is in a two-body orbit. The solution using purely vector analysis is due to J.W. Gibbs (1839–1903), an American scholar who is known primarily for his contributions to thermodynamics. Josiah Willard Gibbs was an American scientist who made important theoretical contributions to physics, chemistry, and mathematics. He was a professor of mathematical physics at Yale College where he inaugurated the new subject — three dimensional vector analysis. He had printed for private distribution to his students a small pamphlet on the “Elements of Vector Analysis” in 1881 and 1884. Gibbs’ pamphlet became widely known and was finally incorporated in the book “Vector Analysis” by J.W. Gibbs and E.B. Wilson and published in 1901.

Formulation of Gibbs Method Algorithm to find orbital parameters
Given three different times t1, t2, t3, three position vectors are measured respectively r1, r2, r3. Using this information all orbital elements can be found using this algorithm:

Start with the Laplace Vector and eccentricity of orbit: v x h = μ(r/r + e)

Where h is the angular momentum and e is the eccentricity vector. To isolate the velocity, take the cross product of this equation with the angular momentum:

h x (v x h)= μ(h x r/r + h x e)

Which then simplifies too:

v = μ(h x r/r + h x e)

Rewrite in perifocal frame:

v = μ/h(w x r/r +eq)

We can then rewrite and define new variables N, D, and S.

v = (μ/ND)^1/2 (D x r/r + S)

where N = r1(r2 x r3) + r2(r3 x r1) + r3(r1 x r2)

D = r1 x r2 + r2 x r3 + r3 x r1

S = r1(r2 - r3) + r2(r3-r1) + r3(r1-r2)

where r1, r2, and r3 are position vectors 1, 2, and 3, and r1, r2, r3 are the magnitudes of these vectors.

Using this algorithm, you can then obtain the velocity of the second position vector, from the state vector we can find all orbital elements.

Variation of Gibbs Method
One of the main questions that typically arises from the Gibbs method is how to make these calculations when the three different position vectors are too close to one another. For those types of occasions where the position vectors are very closely spaced, solutions from the Gibbs method are unreliable at best. One solution is the Herrick-Gibbs Method which tries to find the middle velocity vector given three sequential vectors (r1, r2, and r3) and their observation times (t1, t2, and t3). The Herrick-Gibbs method is a variation of the original Gibbs method. The basic premise uses a Taylor-series expansion (thus the requirement for the times of each observation) to obtain an expression for the middle velocity vector. Because this method is only approximate, the Herrick-Gibbs method is not as robust as the Gibbs method, and has more limited application.

Figure 5 Orbit Geometry for the Herrick-Gibbs Method. Orbit determination using this method works well with closely spaced observations.