# Propagation architecture

The core of a numerical integration is the successive evaluation of the state derivative $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},s;\mathbf{p})$$. Here, $$\mathbf{x}$$ is the propagated state. For a typical propagation, this is the translational state of a single body, but it may be a combination of any types of dynamics for any number of bodies (see here), and may also include the variational equations associated with these dynamics. The variable $$s$$ is the independent variable of the differential equation governing the state (typically but not necessarily the time $$t$$), and the vector $$\mathbf{p}$$ denotes a set of parameters which are held constant during the propagation, but which do influence the solution of the equations of motion.

## Propagator pre-processing

Before starting the propagation, several steps are taken to initialize the variables/objects required for the propagation. Below, we provide an overview of the steps that are taken for the propagation of a single arc (typical Tudat propagation):

• It is checked whether the propagator settings are feasible (e.g. no body A propagated w.r.t. B, and body B propagated w.r.t. body A, etc.).

• Several objects are created that are used during/after the propagation

• An EnvironmentUpdater object (in C++; not exposed to Python), which decides which environment models need to be updated for each function evaluation (A single function evaluation and how)

• A ReferenceFrameManager object (in C++; not yet exposed to Python), which allows translations between ephemerides with different origins to be performed

• If the set_integrated_result to the SingleArcOutputSettings is set to true: A set of IntegratedStateProcessor objects (in C++; not exposed to Python) which is used to post-process the propagation results, and make any required transformations to reset the ephemerides of the propagated bodies.

• A DynamicsStateDerivativeModel (in C++; not exposed to Python) object that handles the calculation of a single state derivative evaluation (see A single function evaluation)

• A PropagationTerminationCondition (in C++; not exposed to Python) object that checks whether the propagation is terminated for a given time/state pair

• A set of functions that extract the dependent variables from the environment is created (if any)

• The initial state is converted from processed formulation (in which it has to be provided) to propagated formulation (see Processed vs. Propagated State Elements)

## A single function evaluation

The top-level calculation of a single state derivative evaluation $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},s;\mathbf{p})$$ is handled by the DynamicsStateDerivativeModel class (in C++; not exposed to Python). In summary, a single function evaluation entails the following steps:

• Check that the independent variable or state are neither infinity nor NaN. If either one is, an exception is thrown.

• The current state of the environment from the previous time step (translational state; orientation, altitude, etc. of bodies) are cleared, ensuring that they are recomputed for the current time step. (see Interacting with the environment during propagation for the properties of bodies that are set during a time step and how to retrieve them).

• The propagated state vector $$\mathbf{x}$$ is split into its constituent parts (e.g. translational, rotational, etc. states of separate bodies that are propagated).

• The separate propagated states, in ‘propagated coordinates’, are converted to ‘processed coordinates’ (see Processed vs. Propagated State Elements). For instance: if Kepler elements of the Moon w.r.t the Earth are propagated, this step converts those Kepler elements (propagated state) to Cartesian elements (processed state)

• Each propagated state, in processed coordinates, is set as the current state of the Body object. Consequently, when the state attribute of a propagated body is called during the propagation (see Interacting with the environment during propagation) the state from the state vector, converted to Cartesian elements (if needed) is retrieved. If the propagation origin is different from the global frame origin (see Translational states), a frame translation is applied before updating the body’s current state. For instance, if the Kepler elements of the Moon w.r.t. Earth are propagated (propagation origin: Earth), and the global frame origin is the SSB, the state attribute of the Moon will be the Cartesian state w.r.t. the SSB.

• The time-dependent properties of the environment are updated to the current time and propagated state. Only those time/state-dependent models that are needed for either the dynamics or the dependent variables are updated during each time step. For instance, if Jupiter is a body in the environment, but Jupiter’s state plays no role in either the dynamics or in the dependent variables that are saved, its state is not updated at each time step.

• Each state derivative model (acceleration, torque, etc.) required for the calculation of the state derivative is evaluated. If variational equations are required, the state derivative partials are evaluated

• The derivative of each propagated state $$\mathbf{x}_{i}$$ is evaluated from the separate state derivatives (e.g. accelerations are used to compute derivative of Kepler elements, if propagating Kepler elements), and concatenated into the complete state derivative vector $$\dot{\mathbf{x}}$$

## A single time step

Depending on the integrator that is used, a single time step may require one or several function evaluations of the state derivative function $$\mathbf{f}$$. The full propagation loop, which succesively calls the numerical integrator to advance the state, is in the integrateEquationsFromIntegrator function (in C++; not exposed to Python). The steps for a single time step are the following:

• Check that the independent variable or state are neither infinity nor NaN. If either one is, the propagation is tagged as being unsuccesful (nan_or_inf_detected_in_state from PropagationTerminationReason) and the results up until the current point are returned.

• Advance the time and state from $$(t_{i},\mathbf{x}_{i})$$ to $$(t_{i+1},\mathbf{x}_{i+1})$$ by calling the performIntegrationStep function of the selected numerical integrator (which may involve one or more function evaluations $$\mathbf{f}$$). The time step that is taken may be fixed, or may be adjusted by the integrator, depending on the selected integration algorithm.

• If an exception is thrown during the propagation, the propagation is tagged as being unsuccesful (runtime_error_caught_in_propagation from PropagationTerminationReason) and the results up until the current point are returned.

• If needed, the state $$\mathbf{x}_{i+1}$$ is corrected to account for matters such as normalization conditions. Possible corrections are:

• If the propagated state involves one or more quaternions $$\mathbf{q}$$ representing a rotation, these are renormalized as $$\mathbf{q}\rightarrow \mathbf{q}/|\mathbf{q}|$$ to ensure that the norm of the quaternion is reset to unity

• If the state contains a shadow parameter (modified Rodrigues parameters; exponential map), it is checked whether the element set has to switched to the shadow elements. Note that this will cause a discontuity in the state history between $$\mathbf{x}_{i}$$ and $$\mathbf{x}_{i+1}$$, but not a discontinuity in the ‘processed’ (for translational dynamics: Cartesian) state.

• If a termination condition was reached during one of the sub-stages of the time step, the propagation is stopped, and the results returned. Note that this only happens if the assess_termination_on_minor_steps input to one of the integrator setting functions in integrator is set to true (false by default)

• If output is to be saved at the current time step (default: saved every time step):

• The pair $$(t_{i+1},\mathbf{x}_{i+1})$$ is added to the propagated state history

• If any dependent variables are to be saved, the environment is updated to the current time/state $$(t_{i+1},\mathbf{x}_{i+1})$$, see A single function evaluation, and the dependent variables are extracted.

• It is checked whether the $$(t_{i+1},\mathbf{x}_{i+1})$$ pair meets the termination conditions. If the termination conditions are exceeded, and the terminate_exactly_on_final_condition input to the termination condition settings is set to false (see propagator), the propagation is finished, and the results are returned. If this variable is set to true:

• In either case, the PropagationTerminationReason is set to termination_condition_reached, and the state and dependent variable history is returned.

## Propagator post-processing

After the propagation is finished, the following post-processing steps are performed before returning the simulation to the user:

• The propagated states are converted to processed states. After the propagation, the time histories of both may be extracted from the unprocessed_state_history and state_history attributes, respectively

• If the set_integrated_result to the SingleArcOutputSettings is set to true, the propagated states (in processed formulation) are used to reset the environment of the propagated body/bodies. For the different state types, this means:

• Translational dynamics: the propagated translational state of the body is used to create an interpolator (lagrange_interpolation(), number_of_points =6), which is used to update the tabulated() ephemeris of the body. If needed, a translation from the propagation origin to the ephemeris origin is applied (see Frame origin). NOTE: this is only possible if the body has a tabulated ephemeris alreacy, or no ephemeris. In the latter case a tabulated ephemeris is created, with ephemeris origin equal to the propagation origin. In case you want to use a non-tabulated ephemeris for the propagated body, you can use the tabulated_from_existing() function to override existing body settings (see Overriding existing settings objects). When doing so, the behaviour of the non-tabulated ephemeris will be emulated by a non-tabulated ephemeris.

• Rotational dynamics: the propagated rotational state of the body is used to create an interpolator (lagrange_interpolation(), number_of_points = 6), which is used to create a tabulated rotation model (not yet exposed to Python). At present, this option is only possible if the propagated body starts out with no rotation model. An update to allow the same flexibility as for the translational dynamics (see above) is planned

• Mass dynamics: the propagated mass of the body is used to create an interpolator (lagrange_interpolation(), number_of_points =6), which is used to update the mass function of the body.

• If the clear_numerical_solutions to the SingleArcOutputSettings is set to true, the state (processed and unprocessed) and dependent variable history are deleted, after having reset the environment (if set_integrated_result was set to true; see above). In this case, the ephemerides are reset with the propagated dynamics, but the results of the propagation cannot be extracted from the unprocessed_state_history, state_history and dependent_variable_history` attributes. Note that the dependent variable history will be lost entirely in this case.