Trajectory Linearisation of Relative Motion to Elliptical Orbit Target Using Tschauner–Hempel Equation

Abstract This paper presents a trajectory approximation method for relative motion around targets in elliptical orbits. It employs the Tschauner–Hempel (TH) equation, derived from the three-body problem, with a State Transition Matrix (STM) dependent on the true anomaly. To reduce the computational cost, Yamanaka-Ankersen (YA) STM is used to derive an STM solution. This is expressed with sinusoidal functions which are approximated using true anomaly-dependent coefficients. Additionally, a conversion between true anomaly and time intervals is derived. The chaser’s state is expressed in terms of the true anomaly interval and linearised with an eccentricity approximation. This is converted into a time-based linear solution. The performance analysis of the true anomaly linearised solution shows that it is 3 times faster than the YA STM for $$e = 0.01$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.01</mml:mn> </mml:mrow> </mml:math> . Furthermore, it is more accurate than CW STM and maintains a similar computational speed.