Analytical guidance for spacecraft relative motion under constant thrust using relative orbit elements

• Spacecraft relative motion analytical guidance with on/off continuous control.
• Use of relative orbit elements (ROEs) to provide a geometrical visualization.
• Significant advantage and simplicity over Cartesian coordinates.
• Comparison against Cartesian coordinates for the input-shaping input case.
• Solutions applicable to low-thrust thrusters.

Proximity control of modern nano-spacecraft often relies on low and discrete thrust engines that are characterized by low consumption, and generate on-off force profiles. New guidance solutions must take into account the nature of this type of orbital engines. This paper introduces novel analytical guidance solutions for spacecraft relative motion considering continuous, on-off thrust, and using relative orbit elements as a geometrical representation of the dynamics. The solutions provide the relative state vector at any given time, accommodating any thrust magnitude along the three directions of the relative frame, as well as generic activation times and durations. Relative orbit elements geometrically interpret key aspects of the relative motion, including for example, the relative ellipse size, and the evolution of its center in time. The new solutions provide the guidance designer with a direct visualization of the thrust effects on the relative motion geometry, offering new possibilities for analytical guidance in the presence of continuous thrust engines, such as low thrust engines on nano-spacecraft. The paper presents the analytical solutions, and tests their effectiveness using a sample thrust profile based on input-shaping, previously developed by one of the authors using classical Cartesian coordinates. The use of relative orbit elements shows substantial benefits and added simplicity with respect to Cartesian-based approaches, holding the promise for straightforward onboard spacecraft implementation. The software developed for this research will be available open source1, to be used by spacecraft guidance designers as trajectory design tool.