ArticlePDF Available

Extending propagation with user-defined equations, applications to optimization and partial derivatives computation

Authors:
  • CS SI, France, Toulouse

Abstract

This paper presents a new way to deal with transition matrices handling in variational equations. While working simultaneously on several problems dealing with orbit propagation: low-thrust trajectories and orbit determination, we implemented a feature allowing to add user equations to a propagator in order to solve the first problem. Then it appeared that this feature could be reused to deal with the second one. Indeed, the orbit determination problem is based on variational equations involving transition matrices, which can also be considered as additional parameters, propagated at the same time as the original state vector. This method allows a very modular implementation of both problems, with looser coupling in the equations. It has been successfully integrated in the Orekit open-source library.
Pommier-Maurussane, V. et al. Extending propagation with user-defined equations
Presented at the 22nd International Symposium on Space Flight Dynamics, São José dos Campos, Brazil, February, 2011
Journal of Aerospace Engineering, Sciences and Applications, Sep. - Dec. 2011, Vol. III, No 3
45
EXTENDING PROPAGATION WITH USER-DEFINED EQUATIONS,
APPLICATIONS TO OPTIMIZATION AND PARTIAL DERIVATIVES COMPUTATION
Véronique Pommier-Maurussane
CS Communication&Systems, Parc de la Plaine, 5 rue Brindejoncs des Moulinais,
BP 15872, 31506 Toulouse cedex 5, Veronique.Pommier@c-s.fr
Luc Maisonobe
CS Communication&Systems, Parc de la Plaine, 5 rue Brindejoncs des Moulinais,
BP 15872, 31506 Toulouse cedex 5, Luc.Maisonobe@c-s.fr
Pascal Parraud
CS Communication&Systems, Parc de la Plaine, 5 rue Brindejoncs des Moulinais,
BP 15872, 31506 Toulouse cedex 5, Pascal.Parraud@c-s.fr
Abstract: This paper presents a new way to deal with transition matrices handling in variational equations. While
working simultaneously on several problems dealing with orbit propagation: low-thrust trajectories and orbit
determination, we implemented a feature allowing to add user equations to a propagator in order to solve the first
problem. Then it appeared that this feature could be reused to deal with the second one. Indeed, the orbit determination
problem is based on variational equations involving transition matrices, which can also be considered as additional
parameters, propagated at the same time as the original state vector. This method allows a very modular
implementation of both problems, with looser coupling in the equations. It has been successfully integrated in the
Orekit open-source library.
Keywords: Jacobians, propagation, optimization, customization, Orekit.
1 Introducing the problem
High accuracy orbit propagation often involves solving Initial Value Problems (IVP) using the motion equations with
various force models. However this is not sufficient to solve some specific problems like low thrust trajectory
optimization with boundary constraints which involve both solving an optimal control problem (the optimization part)
and solving a Two Point Boundary Value Problems (TPBVP, the boundary constraints part). This is also not sufficient
for orbit determination which involves computing the derivatives of the orbital state vector throughout the propagation
time with respect to initial state and to models parameters.
For the first problem type, the optimal control problem, in addition to position-velocity parameters, extra parameters
have to be integrated: the dual parameters of the Pontryagin principle. These parameters have their own differential
equations that should be added to the classical equations of motion.
For the second problem type, the Two Points Boundary Value Problem, initial parameters need to be adjusted in order to
have the final state vector reach the desired boundary condition. This is done by solving a non-linear optimization
problem. The best algorithms dealing with such problems require computation of a Jacobian matrix (from worst to best
algorithm: steepest descent, conjugate gradient, Gauss-Newton, and Levenberg-Marquardt). This matrix represents the
final state derivative with respect to the initial state.
For the third problem type, the orbit determination problem, initial parameters and force models parameters need to be
adjusted as well, in order to have small residuals for all measurements. This is done by least squares problems solving
or Kalman filtering. These algorithms also need Jacobian matrices. They use the current state derivative with respect to
the initial date and the current state derivative with respect to the force models parameters at each measurement time.
In the last two cases, Jacobians are computed either only at final states or during all propagation. These matrices are
sometimes called transition matrices: Ψy(t) and Ψp(t). They make the connection between the propagated state and both
initial state and force models parameters. They are time-dependent matrices following orbit propagation: dy(t)/dy0 et
dy(t)/dp, where y stands for the state vector, y0 this state vector initial value at t0 and p a parameters vector (for example
drag coefficients).
Pommier-Maurussane, V. et al. Extending propagation with user-defined equations
Journal of Aerospace Engineering, Sciences and Applications, Sep. - Dec. 2011, Vol. III, No 3
47
2 Straightforward implementation
The spacecraft trajectory is computed by integrating the Ordinary Differential Equations (ODE) defined by:
   
dτyτ,f=tyyt,f=
dttdy
(1)
The common way to solve the TPBVP is to compute the Jacobian of the final state Ψy(t) and Ψp(t) and use them in an
optimization algorithm to minimize
   
tyty ˆ
where
 
ty
ˆ
is the expected final state.
The final state Jacobians are defined as:
   
 
   
pty
=tψ
ty ty
=tψ
p
y
0
(2)
The first way to compute these matrices is by finite differences. A first central trajectory y(t) is computed by a simple
solver from an initial state y0(t), then this initial state is slightly shifted to get several close trajectories. The final states
of all those trajectories are combined together to compute the Jacobian matrix Ψy(t) and Ψp(t) by finite differences. In
the Ordinary Differential Equations (ODE) world, this is known as external differentiation.
It is well-known since more than 20 years (Hairer, Wanner and Nørsett 1987) that numerical stability of external
differentiation methods is very poor when using modern adaptive step size integration methods. This is due to the noise
introduced when different initial states lead to different conditional branches used in the step size control.
The following figure1 shows this behavior. The smooth lines are the exact derivatives and the noisy curves are the
derivatives computed by external differentiation.
Figure 1. External differentiation errors
1 This figure is inspired by Hairer, Wanner and Nørsett own example
Pommier-Maurussane, V. et al. Extending propagation with user-defined equations
Journal of Aerospace Engineering, Sciences and Applications, Sep. - Dec. 2011, Vol. III, No 3
48
The common way to solve the orbit determination problem is to compute the differential equations that govern the
evolution of the Jacobian matrices at the same time as the main problem is solved, thus preserving consistency. These
equations are called variational equations y(t)/dt and p(t)/dt in which Ψy(t) and Ψp(t) are the transition matrices
defined previously. The variational equations are defined as follows:
(3)
where Ψy(t) is the matrix dy(t)/dy0 and Ψp(t) the matrix dy(t)/dp.
y
J=
y
f
is the partial derivatives of the time derivative f with respect to the state y;
p
J=
p
f
is the partial derivatives of the time derivative f with respect to the force models parameters p.
These equations show that the state global Jacobian time derivatives is linked to the local Jacobian of the state time
derivatives.
The Jacobian matrices Jy(t) and Jp(t) can be computed gradually throughout the propagation.
The transition matrices initial values are the identity matrix for dy(t)/dy0 at t0 and the null matrix for dy(t)/dp at t0. From
these initial values, the variational equations compute the values at any time. However, if one considers that the user
propagates section by section from t0 to t1, then from t1 to t2 and so on, he may want to have all his matrices computed
with respect to the initial time t0.
This is typically what happens for orbit restitution: the initial time corresponds to the expected orbit adjustment time. It
is thus necessary to have an option for setting the transition matrices to any initial value. In the case defined previously,
the user will start each section by setting the matrices to the current value of the matrices obtained at the end of previous
section.
When these Jacobian matrices cannot be computed explicitly, finite differences are used, which is then called internal
differentiation.
This method has been implemented almost everywhere, it works well but has some drawbacks and is clearly not
extensible. One of the problems is that the differentiation process is fully embedded in the core propagation equations.
Switching from one force model to another or changing the ODE solver thus implies numerous adaptations and
validations. This is even more difficult for the optimal control problem as the equations for dual parameters are
complex, problem-dependent and can almost never be differentiated analytically. That's why common propagation
solvers cannot be used for low thrust trajectories with boundary constraints, and very specific tools are usually
developed.
3 A bypass resolution
3.1 Additional equations handling
The low-thrust trajectory problem can be solved by adding a set of extra parameters, the dual parameters, to the state
vector. Via the corresponding differential equations, these parameters are then included in the propagation process. To
allow that feature, it is only necessary to make a few changes in the ODE solver. These changes include handling an
array of equations and extending the state vector to take these parameters into account. This method is quite easy to
implement and to validate.
Pommier-Maurussane, V. et al. Extending propagation with user-defined equations
Journal of Aerospace Engineering, Sciences and Applications, Sep. - Dec. 2011, Vol. III, No 3
49
Figure 2. State vector extension
As shown in the previous diagram, the additional equations for additional parameters evolution
 
zy,t,g=
dt
dz
can
depend both on the original state vector y(t) and on the additional parameters z(t). On the contrary, the original
equations for state vector evolution
only depend on the state vector y(t) itself and have no relations with the
additional parameters.
In the orbit determination problem case, adding equations is not merely extending the size of an array and providing one
function to compute its derivatives. There is a fly in the ointment as far as adaptive step size is concerned. Adaptive step
size is done by estimating a local error on the state vector. If the error exceeds a predefined threshold, the step is
rejected, a smaller step size is computed and used for another attempt on the current step. As adding equations extends
the state vector size, the extra parameters are included in the error estimation. It is often very difficult to specify a
threshold for these parameters, as they have almost no physical meaning. From a propagation standpoint, there is also
no real need for including these parameters into error estimation, which should be based on original position-velocity
state only.
Some modern solvers provide continuous output models between steps using dedicated interpolators. In that case, the
additional equations must be provided to the integrator in order to propagate the whole state vector. This allows for
example to compute residuals in orbit determination, we will see how in next section.
3.2 Application to the transition matrices problem
While working on both variational equations and low-thrust trajectories problem, it appeared that somehow, they were
similar problems and could be solved using the same mechanism. Basically, they fit in a single frame which consists in
adding parameters to the state vector, and adding to the ODE solver the associated differential equations to propagate
them. In that case, the additional parameters are the elements of the Ψy(t) and Ψp(t) matrices, and the corresponding
equations are the variational equations defined previously :
Figure 3. Using extension mechanism for variational equations
That mechanism is very interesting for orbit determination, when applied to residuals estimation and model parameters
estimation. Orbit determination is an iterative process, each iteration being a propagation initialized from the current
estimated orbit. During propagation, at each integration step the local model from the ODE solver is used to compute
Solver
y(t0)
z(t0)
y(t)
z(t)
Position-velocity
original state vector
Dual parameters
 
zy,t,g=
dt
dz
 
yt,f=
dt
dy
y(t)
z(t)
Ψy(t) , Ψp(t)
Position-velocity
Variational equations
 
py
py Ψy,t,g=
dt
dΨ
/
/
 
yt,f=
dt
dy
Pommier-Maurussane, V. et al. Extending propagation with user-defined equations
Journal of Aerospace Engineering, Sciences and Applications, Sep. - Dec. 2011, Vol. III, No 3
50
both current state and partial derivatives at the current measurement time. These are used to update the normal
equations, which are used by the upper optimizer to adjust the orbit estimation. Computing the partial derivatives Ψy(t)
by additional equations mechanism allow looser coupling between the core propagator and the other components of the
orbit determination system. Adding estimated model parameters is also very simple due to the modular structure of the
equations handling.
During orbit propagation, the evolution of the state vector is given by differential equations coming from force models.
Variational equations need local Jacobians of the state vector time derivatives (Jy and Jp matrices), which are computed
from the force models Jacobians. If analytical equations are available, these Jacobians can be computed directly,
otherwise they can still be computed by finite differences, for the original state vector as well as for additional
parameters. Corresponding steps can either be user-specified or computed automatically. During a single propagation,
both cases can occur, as some force models are more complex than others.
All the matrices involved are computed with respect to Cartesian parameters, even if propagation is done in equinoctial
parameters. In that case, a post-processing conversion is required.
4 Conclusion
We have shown that low thrust trajectory problem could be handled by a classical propagator using a simple feature: the
additional equations, which can also be applied to the partial derivatives computation in the orbit determination
problem.
This method has been fully implemented in version 5.1 of the Orekit open-source library which release is scheduled in
early 2011.
5 References
Hairer, Wanner, Nørsett, Solving Ordinary Differential Equations I Non-Stiff Problems, Springer Verlag, 1987
Pommier-Maurussane, V., Maisonobe, L., Orekit: an Open-source Library for Operational Flight Dynamics
Applications, Proceedings 4th International Conference on Astrodynamics Tools and Techniques - 4th ICATT,
Madrid, Spain, 2010.
Vallado, D., A., Fundamentals of Astrodynamics and Applications, Space Technology Library, 2007
... Orekit is a open source software library for space flight dynamics. 10,11 It includes the ability to model standard astrodynamical elements such as orbits, time systems and reference frames. The algorithms necessary to make use of these elements, such as frame conversions, time conversion and various orbit propagation methods, are included as well. ...
Conference Paper
Full-text available
Open source software tools have been gaining acceptance in the astrodynamics community for some applications, though heritage tools still dominate precision orbit determination and propagation. This paper examines recent tide modeling improvements in the open source Orbit Extrapolation Toolkit (Orekit) and compares it with the US Naval Research Laboratory's (NRL) heritage Orbit Covariance Estimation And ANalysis (OCEAN) system. First, the two tools are compared directly against each other by propagating a given state vector for Stella, a geodetic satellite sensitive to tidal variations in the geopotential. Second, orbits were fit to International Laser Ranging Service (ILRS) laser ranging data using OCEAN and orbit determination software built around Orekit so that a more useful comparison could be made. Five days of data were used to solve for orbital parameters using OCEAN and Orekit. This solution orbit is then propagated forward 25 days and compared to subsequent five day orbit solutions. This comparison between predicted and fitted orbit solutions is used as a metric to compare the quality of each piece of software's dynamic modeling capability. Results from the direct orbit propagation comparison indicate the RSS of postion difference between the OCEAN and Orekit propagated orbit grow to only 7 meters over 25 days. It is also seen that the difference between OCEAN's and Orekit's implementation of Earth tides are less than 3% of the total tidal effect. The results of the orbit determination analysis show that the Orekit orbit solution comparison is at worst on the same order of magnitude in accuracy as the OCEAN orbit solution comparison, and at best more accuate than the OCEAN orbit solution comparison. While OCEAN produces a more accurate orbit prediction than Orekit in the majority of the cases studied, more testing is need to understand the origin of the difference.
ResearchGate has not been able to resolve any references for this publication.