Skip to main content

Orbital rendezvous performance comparison of differential geometric and ZEM/ZEV feedback guidance algorithms


In this paper, the performance of two distinct classes of feedback guidance algorithms is evaluated for a spacecraft rendezvous problem utilizing a continuous low-thrust propulsion system. They are the DG (Differential Geometric) and ZEM/ZEV (Zero-Effort-Miss/Zero- Effort-Velocity) feedback guidance algorithms. Even though these two guidance algorithms do not attempt to minimize the onboard fuel consumption or ΔV directly, the ΔV requirement is used as a measure of their orbital rendezvous performance for various initial conditions and a wide range of the rendezvous time (within less than one orbital period of the target vehicle). For the DG guidance, the effects of its guidance parameter and terminal time on the closed-loop performance are evaluated by numerical simulations. For the ZEM/ZEV guidance, its nearfuel- optimality is further demonstrated for a rapid, short-range orbital rendezvous, in comparison with the corresponding open-loop optimal solutions. Furthermore, the poor ΔV performance of the ZEM/ZEV guidance for a slow, long-range orbital rendezvous is remedied by simply adding an initial drift phase. The ZEM/ZEV feedback guidance algorithm and its appropriate variants are then shown to be a simple practical solution to a non-impulsive rendezvous problem, in comparison with the DG guidance as well as the open-loop optimal guidance.

This is a preview of subscription content, access via your institution.


  1. [1]

    Goodman, J. L. History of space shuttle rendezvous and proximity operations. Journal of Spacecraft and Rockets, 2006, 43(5): 944–959.

    Article  Google Scholar 

  2. [2]

    Barbee, B., Carpenter, J. R., Heatwole, S., Markley, F. L., Moreau, M., Naasz, B. J., Van Eepoel, J. A guidance and navigation strategy for rendezvous and proximity operations with a noncooperative spacecraft in geosynchronous orbit. The Journal of the Astronautical Sciences, 2011, 58(3): 389–408.

    Article  Google Scholar 

  3. [3]

    Wen, C., Gurfil, P. Guidance, navigation and control for autonomous R-bar proximity operations for geostationary satellites. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2017, 231(3): 452–473.

    Article  Google Scholar 

  4. [4]

    Luo, Y.-Z., Sun, Z.-J. Safe rendezvous scenario design for geostationary satellites with collocation constraints. Astrodynamics, 2017, 1(1): 71–83.

    Article  Google Scholar 

  5. [5]

    Zhang, S., Han, C., Sun, X. New solution for rendezvous between geosynchronous satellites using low thrust. Journal of Guidance, Control, and Dynamics, 2018, 41(6): 1396–1405.

    Article  Google Scholar 

  6. [6]

    Wang, W., Chen, L., Li, K., Lei, Y. One active debris removal control system design and error analysis. Acta Astronautica, 2016, 128: 499–512.

    Article  Google Scholar 

  7. [7]

    Wang, W., Song, X., Li, K., Chen, L. A novel guidance scheme for close range operation in active debris removal. Journal of Space Safety Engineering, 2018, 5(1): 22–33.

    Article  Google Scholar 

  8. [8]

    Chiou, Y.-C., Kuo, C.-Y. Geometric approach to three-dimensional missile guidance problem. Journal of Guidance, Control, and Dynamics, 1998, 21(2): 335–341.

    Article  Google Scholar 

  9. [9]

    Kuo, C.-Y., Chlou, Y.-C. Geometric analysis of missile guidance command. IEEE Proceedings: Control Theory and Applications, 2000, 147(2): 205–211.

    Google Scholar 

  10. [10]

    Kuo, C.-Y., Soetanto, D., Chiou, Y.-C. Geometric analysis of flight control command for tactical missile guidance. IEEE Transactions on Control Systems Technology, 2001, 9(2): 234–243.

    Article  Google Scholar 

  11. [11]

    Li, C., Jing, W., Wang, H., Qi, Z. iterative solution to differential geometric guidance problem. Aircraft Engineering and Aerospace Technology, 2006, 78(5): 415–425.

    Article  Google Scholar 

  12. [12]

    Li, C., Jing, W. Analysis of 3D geometric guidance problem. Transactions of the Japan Society for Aeronautical and Space Sciences, 2008, 51(172): 124–129.

    Article  Google Scholar 

  13. [13]

    Li, C., Jing, W., Qi, Z., Wang, H. A novel approach to the 2D differential geometric guidance problem. Transactions of the Japan Society for Aeronautical and Space Sciences, 2007, 50(167): 34–40.

    Article  Google Scholar 

  14. [14]

    Li, C.-Y., Jing, W.-X. Fuzzy PID controller for 2D differential geometric guidance and control problem. IET Control Theory & Applications, 2007, 1(3): 564–571.

    Article  Google Scholar 

  15. [15]

    Li, C., Jing, W., Wang, H., Qi, Z. Gain-varying guidance algorithm using differential geometric guidance command. IEEE Transactions on Aerospace and Electronic Systems, 2010, 46(2): 725–736.

    Article  Google Scholar 

  16. [16]

    Dhananjay, N., Ghose, D., Bhat, M. S. Capturability of a geometric gudiance law in relative velocity space. IEEE Transactions on Control Systems Technology, 2009, 17(1): 111–122.

    Article  Google Scholar 

  17. [17]

    Li, K., Chen, L., Bai, X. Differential geometric modeling of guidance problem for interceptors. Science China Technological Sciences, 2011, 54(9): 2283–2295.

    MATH  Article  Google Scholar 

  18. [18]

    Li, K., Chen, L., Tang, G. Improved differential geometric guidance commands for endoatmospheric interception of high-speed targets. Science China Technological Sciences, 2013, 56(2): 518–528.

    Article  Google Scholar 

  19. [19]

    Li, K., Chen, L., Tang, G. Algebraic solution of differential geometric guidance command and time delay control. Science China Technological Sciences, 2015, 58(3): 565–573.

    Article  Google Scholar 

  20. [20]

    Li, K., Su, W., Chen, L. Performance analysis of threedimensional differential geometric guidance law against low-speed maneuvering targets. Astrodynamics, 2018, 2(3): 233–247.

    Article  Google Scholar 

  21. [21]

    Ariff, O., Zbikowski, R., Tsourdos, A., White, B. A. Differential geometric guidance based on the involute of the target’s trajectory. Journal of Guidance, Control, and Dynamics, 2005, 28(5): 990–996.

    Article  Google Scholar 

  22. [22]

    White, B. A., Zbikowski, R., Tsourdos, A. Direct intercept guidance using differential geometric concepts. IEEE Transactions on Aerospace and Electronic Systems, 2007, 43(3): 899–919.

    Article  Google Scholar 

  23. [23]

    Meng, Y., Chen, Q., Ni, Q. A new geometric guidance approach to spacecraft near-distance rendezvous problem. Acta Astronautica, 2016, 129: 374–383.

    Article  Google Scholar 

  24. [24]

    Ebrahimi, B., Bahrami, M., Roshanian, J. Optimal sliding-mode guidance with terminal velocity constraint for fixed-interval propulsive maneuvers. Acta Astronautica, 2008, 62(10–11): 556–562.

    Article  Google Scholar 

  25. [25]

    Furfaro, R., Selnick, S., Cupples, M. Nonlinear sliding guidance algorithms for precision lunar landing. In: Proceedings of the 21st AAS/AIAA Space Flight Mechanics Meeting, 2011: AAS 2011–167.

    Google Scholar 

  26. [26]

    Guo, Y., Hawkins, M., Wie, B. Optimal feedback guidance algorithms for planetary landing and asteroid intercept. In: Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, 2011: AAS 2011–588.

    Google Scholar 

  27. [27]

    Guo, Y., Hawkins, M., Wie, B. Applications of generalized zero-effort-miss/zero-effort-velocity feedback guidance algorithm. Journal of Guidance, Control, and Dynamics, 2013, 36(3): 810–820.

    Article  Google Scholar 

  28. [28]

    Guo, Y., Hawkins, M., Wie, B. Waypoint-optimized zero-effort-miss/zero-effort-velocity feedback guidance for Mars landing. Journal of Guidance, Control, and Dynamics, 2013, 36(3): 799–809.

    Article  Google Scholar 

  29. [29]

    Ahn, J., Guo, Y., Wie, B. Precision ZEM/ZEV feedback guidance algorithm utilizing Vinti’s analytic solution of perturbed Kepler problem. In: Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, 2016: AAS 16–345.

    Google Scholar 

  30. [30]

    Ahn, J., Wang, P., Guo, Y., Wie, B. Optimal terminaltime determination for the ZEM/ZEV feedback guidance law. Astrodynamics, 2019. (to be published)

    Google Scholar 

  31. [31]

    Wie, B. Two-phase ZEM/ZEV guidance for Mars and Lunar powered descent & landing with hazard avoidance and retargeting. Keynote Talk in the 4th IAA Conference on Dynamics and Control of Space Systems, 2018.

    Google Scholar 

  32. [32]

    Wie, B., Zimmerman, B., Lyzhoft, J., Vardaxis, G. Planetary defense mission concepts for disruptiong/pulverizing harzadous asteroids with short wanrning time. Astrodynamics, 2017, 1(1): 3–21.

    Article  Google Scholar 

Download references


This work is supported by the National Natural Science Foundation of China (Grant Nos. 61673135 and 61603114).

Author information



Corresponding author

Correspondence to Bong Wie.

Additional information

Pengyu Wang received his B.S. degree in automation from Harbin Engineering University, China, in 2015, and M.S. degree in control science and engineering from Harbin Institute of Technology, China, in 2017. He is now a Ph.D. candidate at Harbin Institute of Technology focusing on the development and application of control theories in aerospace problems, including Mars pinpoint landing, spacecraft rendezvous, and missile impact-time guidance, etc.

Yanning Guo received his Ph.D. degree in control science and engineering from Harbin Institute of Technology, China, in 2012, and was a visiting scholar at Iowa State University in 2010–2011. Currently, he is an associate professor at Harbin Institute of Technology, and specializes in optimal control, sliding-mode control, as well as visual navigation and localization.

Bong Wie is a professor of aerospace engineering at Iowa State University. He is the founding director of the Asteroid Deflection Research Center established in 2008 at Iowa State University. He received his M.S. and Ph.D. degrees in aeronautics and astronautics from Stanford University in 1978 and 1981, respectively. In 2006, the AIAA (American Institute of Aeronautics and Astronautics) presented Prof. Wie with the Mechanics and Control of Flight Award for his innovative research on advanced control of complex spacecraft such as agile imaging satellites, solar sails, and large space structures.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, P., Guo, Y. & Wie, B. Orbital rendezvous performance comparison of differential geometric and ZEM/ZEV feedback guidance algorithms. Astrodyn 3, 79–92 (2019).

Download citation


  • differential geometric guidance
  • ZEM/ZEV feedback guidance
  • orbital rendezvous
  • initial drift phase