3.5. Propagator Settings: Conventional vs. Propagated Coordinates

In the description of some of the objects in this part of the wiki about the simulation set-up, you may have noticed the use of two names to describe the states of an object. These two names are conventional and propagated, and they describe two slightly different concepts. In this part of the Wiki, you will get to know in what they differ and how that may affect your application.

Conventional Coordinates

These coordinates are mainly used to describe the state of the object(s) you are propagating outsite of integration. This means that you will find conventional coordinates in these scenarios:

  • To describe the initial conditions of an object
  • To update the acceleration model of an object
  • To update the environment model of an object; this also means that the states extracted from the Body are expressed in the conventional coordinates
  • As an output to the getEquationsOfMotionNumericalSolution function of the DynamicsSimulator object

For sake of completeness, below you will find the conventional coordinates for each propagator type which supports a multiple propagated coordinates:

  • Translational Motion: Cartesian coordinates
  • Rotational Motion: Quaternions
Propagated Coordinates

The propagated coordinates, on the other hand, are mainly used to describe the state of an object inside of the integration environment. Thus, you can expect to see these elements here:

  • To describe the equations of motion
  • To describe the state and state derivative during integration
  • As an output to the getEquationsOfMotionNumericalSolutionRaw function of the DynamicsSimulator object

As a user, you will generally only interact with the conventional coordinates, but you will have the choice over which propagated coordinate to use for propagation/integration. Note that setting a type of coordinates for propagation does not necessarily also constrain which propagator is used. For instance, Cowell and Encke propagation, both use Cartesian coordinates as the propagated states. The same is also true vice-versa: Gauss’ version of Lagrange’s planetary equations can be used with both Keplerian and modified equinoctial elements.

In the remaining part of this page, you will find a description of where you need to keep a closer look at the difference between these two types of coordinates and which coordinates are available in Tudat.

3.5.1. Important To Keep In Mind

Since the conventional coordinates are used to update the environment and accelerations of the bodies, but the propagated coordinates are the ones used in propagation, you can see that whenever the conventional and propagated coordinates differ, there is a need to convert between the two at every time step (or even multiple times, if the time step is divided in multiple steps for integration). Thus, this leads to a set of extra stages to be perfomed during propagation, which may in turn lead to a longer computation time. This convertion is also necessary when outputting the conventional state at the end of propagation.


The fact that using a different set of propagated coordinates may lead to a longer computation time is not always true. As a matter of fact, the default translational propagator (i.e., cowell) is considerably slower and less accurate than other propagators, in certain situations. Check out Comparison of Propagator Types to get an idea of the difference in performance among the various translational propagators offered by Tudat.

Another fact to consider, is that sometimes there may be a difference between the size of the conventional and propagates states. For instance, a Cartesian state is expressed with 6 elements, but the USM7 state with 7. This may lead to some confusion when extracting the results, so keep this in mind. In the next section, you can find the size of each propagated type used in Tudat.

3.5.2. Propagated Coordinates in Tudat

For the following state types, one can choose between the listed propagated coordinates. You can also find a similar list in Propagator Settings: Basics, where each propagator available is listed.

Translational Motion
  • Cartesian coordinates (with cowell and encke propagators); state size: 6
  • Keplerian elements (with gauss_keplerian propagator); state size: 6
  • Modified equinoctial elements (with gauss_modified_equinoctial propagator); state size: 6
  • Unified state model with quaternions, or USM7 (with unified_state_model_quaternions propagator); state size: 7
  • Unified state model with modified Rodrigues parameters, or USM6 (with unified_state_model_modified_rodrigues_parameters propagator); state size: 7
  • Unified state model with exponential map, or USMEM (with unified_state_model_exponential_map propagator); state size: 7
Rotational Motion
  • Quaternions (with quaternions propagator); state size: 7
  • Modified Rodrigues parameters, or MRPs (with modified_rodrigues_parameters propagator); state size: 7
  • Exponential map (with exponential_map propagator); state size: 7

Any other state type that has not been mentioned, is only described by one coordinate type, and thus their conventional and propagated coordinates always match.