To integrate the equations of motion for a launch vehicle trajectory to the orbit, which is the best coordinate system to consider of the following two:

  • Rectangular with Earth-centered inertial frame or
  • Relative spherical coordinates?

Is there any other alternative coordinate systems, other than these two to effectively compute the flight parameters with numerical integration?


2 Answers 2


It seems that rectangular coordinates in Earth-centered inertial frame is a viable choice. The relative spherical coordinates may be susceptible to singularity during the lift-off (i.e initial conditions such as velocity and flight path angle are 0 m/s and 90 degrees, respectively).

From page 42 of Ezgi Civek Coskun's 2014 Ph.D. Thesis Multistage Launch Vehicle Design with Thrust Profile and Trajectory Optimization:

Knowing the fact that Newton’s laws are valid only with respect to the inertial frame, it is the easiest and the fastest way to define and integrate the equations of motion in Earthcentered-inertial (ECI) frame. And also, rectangular coordinates are preferred since they offer a simpler formulation and eliminate singularity problems at lift-off (zero initial velocity and 90° flight path angle).

While on the other hand, the relative spherical coordinates give a better insight and understanding about the vehicle’s motion, so they are used as the output coordinates to present the results to the user. Furthermore, since the final state of the launch vehicle’s ascent trajectory is defined with respect to the orbital frame, it is also required to compute the orbital elements all along the trajectory in order to assess whether the terminal boundary constraints are satisfied or not. Different coordinate systems and the related transformations between them are presented in Appendix B.1 and Appendix B.2, respectively.

Although the rotational dynamics are not modeled in the equations of motion, the attitude of the launch vehicle can be defined based on certain simplifying assumptions, and thus trajectory control variables can be physically interpreted easily. The attitude, in general, describes the orientation of a body-fixed reference frame with respect to an external reference frame. Euler angles or aerodynamic angles are often used to specify the attitude of the launch vehicle during flight.

  • 1
    $\begingroup$ +1 Great answer and source! Apologies for not including the correct character set for the thesis author's name. My attempt at conversion with Python yielded Ezgİ Cİvek CoŞkun and I'm not confident this is correct. $\endgroup$
    – uhoh
    Commented Apr 5, 2019 at 1:32
  • 1
    $\begingroup$ This is what we used in the Shuttle Mission Simulator. $\endgroup$ Commented Apr 5, 2019 at 12:50

Most rocket companies use several coordinates systems for launch vehicles during ascent.

Generally a local ground fixed reference frame centered at the launch pad is used along with an Earth Centered Inertial frame (ECI) and Earth Centered Earth Fixed Frame (ECEF).

  • $\begingroup$ Could you please explain what is a local ground fixed reference frame centered at the launch pad? Is it equivalent to an inertial frame fixed on the launch pad at the instant of the launch or a local vertical and local horizontal moving along with the vehicle? If it is the former, for a 3 DoF system, how do you initialize the inertial attitude angles? $\endgroup$
    – Harish
    Commented Jul 16, 2020 at 12:33
  • $\begingroup$ I think that's the Downrange-Crossrange-Altitude (DCA) frame. github.com/nasa/Coordinate-systems-class-library for a list of frames and conversion routines. $\endgroup$
    – lamont
    Commented Aug 30, 2023 at 3:57

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