# Reachable domain for spacecraft with ellipsoidal Delta-V distribution

## Abstract

Conventional reachable domain (RD) problem with an admissible velocity increment, Δv, in an isotropic distribution, was extended to the general case with Δv in an anisotropic ellipsoidal distribution. Such an extension enables RD to describe the effect of initial velocity uncertainty because a Gaussian form of velocity uncertainty can be regarded as possible velocity deviations that are confined within an error ellipsoid. To specify RD in space, the boundary surface of RD, also known as the envelope, should be determined. In this study, the envelope is divided into two parts: inner and outer envelopes. Thus, the problem of solving the RD envelope is formulated into an optimization problem. The inner and outer reachable boundaries that are closest to and farthest away from the center of the Earth, respectively, were found in each direction. An optimal control policy is then formulated by using the necessary condition for an optimum; that is, the first-order derivative of the performance function with respect to the control variable becomes zero. Mathematical properties regarding the optimal control policy is discussed. Finally, an algorithm to solve the RD envelope is proposed. In general, the proposed algorithm does not require any iteration, and therefore benefits from quick computation. Numerical examples, including two coplanar cases and two 3D cases, are provided, which demonstrate that the proposed algorithm works efficiently.

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11702293).

## Author information

Authors

### Corresponding author

Correspondence to Changxuan Wen.

Changxuan Wen received his B.S. and Ph.D. degrees from Beihang University, China, in 2009 and 2016, respectively, and was a visiting scholar of Technion- Israel Institute of Technology, Israel, in 2014 and 2015. Since June 2015, he has been an associate research fellow at Technology and Engineering Center of Space Utilization, Chinese Academy of Sciences. His academic interests include flight dynamics, satellite conjunction analysis, geostationary servicing, reachable domain, and guidance and navigation. Email: wenchangxuan@gmail.com

Yang Gao received his B.S. and M.S. degrees from Beihang University and Chinese Academy of Sciences in 1997 and 2000 respectively. He received his Ph.D. degree, in 2003 from the University of Missouri, Columbia, USA. From 2004 to 2005, he was a postdoctoral research fellow from the University of Missouri. Since June 2005, he has been an associate research fellow at Academy of Opto-Electronic, Chinese Academy of Sciences. Currently, he is a research professor at Technology and Engineering Center of Space Utilization, Chinese Academy of Sciences. His academic interests include orbit determination, spacecraft dynamics and control, guidance and navigation, trajectory optimization, and space mission design. Email: gaoyang@csu.ac.cn

Chao Peng received his B.S. degree from the Taiyuan University of Technology in 2008 and his M.S. degree from the Chinese Academy of Sciences in 2012. Since June 2012, he has been a junior engineer at the Technology and Engineering Center of Space Utilization, Chinese Academy of Sciences. His academic interests include spacecraft dynamics and control, trajectory optimization, and space mission design. Email: pc309@csu.ac.cn

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Wen, C., Peng, C. & Gao, Y. Reachable domain for spacecraft with ellipsoidal Delta-V distribution. Astrodyn 2, 265–288 (2018). https://doi.org/10.1007/s42064-018-0025-x