The Initial Orbit Determination (IOD) Problem With Range, Range-Rate and Angles

The Initial Orbit Determination (IOD) problem has been tacked since Laplace, Euler and Gauss
in the service of determining the orbits of planets, asteroids, and comets. The measurements that
established the orbit characteristics were angles over long arcs.1 The Space Age brought two new
measurement types to bear: range and range-rate.2, 3 Initially these measurements were provided by
radars through skin-tracking. Soon, however, transponders allowed better measurement accuracy
and availability. To date, there are three types of IOD problems: angles only, range and angles, and
range and range-rate.

The Orion program exposed the need for an additional capability in the case of an extended
loss of communication: when the measurements included antenna angles (azimuth and elevation or
right-ascension and declination), range and range-rate. This paper contains the details of this new
methodology. We will begin with state determination with range, range-rate and angle measurements.

First the state estimate, given a sequence of these four time-synchronized measurements, will
be analytically developed. Next the covariance matrix associated with this state estimate will be
analytically developed. Finally, these will be verified by Monte Carlo simulations.