## Tracking systems, their mathematical models and their errors: Theory, Part 1F. O. Vonbun, Werner D. Kahn, Goddard Space Flight Center, United States. National Aeronautics and Space Administration This paper treats the RMS-errors associated with the position and velocity of a satellite or spacecraft when tracked by all types of present day tracking systems. These errors are based on uncertainties in measurements made with the systems as well as those associated with their location. The present paper (Part I) is principally a theoretical treatment which establishes the mathematical models necessary to solve for the errors in satellite position and velocity. It is presumed throughout the paper that these errors are to be determined for discrete points in the satellite's orbit. This approach enables one to calculate the error propagation during short time intervals (order of seconds). This is of particular interest for instance for evaluation of a guidance system during a short burning phase. A least square solution of non-simultaneous observations would diverge (matrices involved become ill-conditioned). This condition imposes a constraint on the method of solution which is, that either one tracking system can measure both position and/or velocity, or that several tracking systems observe the satellite simultaneously to produce the equivalent effect. Both these alternatives are considered in Part I. |

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A(pxl Angle and Angular Angular Rate Measuring angular rate systems axis directed Axoi bias errors calculate Cartesian coordinate system consistent with Equations Dl through D4 Equation b-11 Equation D-20 Equation D-8 Equations for Tracking error analysis Error Equations Figure fully determined identity matrix inertial coordinate system interferometer Ionosphere least squares method MATHEMATICAL MODELS matrix form matrix notation Method of Least number of measurements number of observations number of unknowns obtained orthogonal matrix over-determined over-determined system polynomial posi position and velocity position errors Q(kxk quantity radar radar system range and range Range Measuring System range rate systems Reference represents Right ascension RMS-errors satellite orbit satellite position vector Semimajor axis simultaneous observations spacecraft system measures system X tion and velocity total number tracking station transformation types of tracking uncertainty in range unit position vector unknown parameters values variational equation relating velocity vector Y-system