HYBRID LINEAR-NONLINEAR INITIAL ORBIT DETERMINATION WITH SINGLE ITERATION REFINEMENT FOR RELATIVE MOTION

Application of Volterra theory to the Keplerian circular relative motion initial orbit determination problem has been considered recently. A series of azimuth-elevation angular measurements are coupled through the observation geometry with an analytic second-order three-dimensional solution for relative motion. One recently explored solution strategy to the resulting nonlinear measurement equations reformulates the problem as an equivalent set of linear equations with constraints solved by matrix decomposition and computation of an unknown scale factor. Two strengths of this technique include 1) improved observability (compared to zero observability when using a linear dynamics solution) and 2) sound computational numerics (eigen computations). One deficiency of this technique is the requirement for additional measurements. In the three-dimensional case, only six measurements are needed to directly solve the nonlinear formulation, while twenty-five measurements are necessary in the reformulated equivalent linear problem. Similarly, in the two-dimensional case, requirements are four vs. fourteen. In this paper, a hybrid solution technique is considered where a linear motion solution and only six measurements are used to obtain an initial estimate of the relative (unsealed) state vector. This state vector is then inserted into the unknown scale factor computation process that uses a nonlinear motion solution. Although this hybrid technique tends to improve the state estimation result beyond the purely linear approach, accuracy is still lacking. A single Newton-Raphson iteration refinement step using the nonlinear measurement equations, inserted between the linear and nonlinear state computations, has been found to restore much of the missing accuracy. The purpose of this investigation is to examine a simple modification to the existing strategy to retain initial state estimation accuracy with fewer measurements.