on the accuracy of state-transition matrices

On the Accuracy of State

Transition Matrices

Etienne Pellegrini

Ryan P. Russell

8/13/15

Summary

• Introduction and motivations

• Three methods for partials computation

- Variational equations

- Complex Step Derivative approximation

- Finite differences

• Testing framework

• The caveats of variable-step integration

• Timings and accuracy

• Conclusions

2 Etienne Pellegrini - AAS/AIAA Astrodynamics Specialists Conference - 8/13/15 - Vail, CO

Introduction and Motivation

• Partials are at the center of gradient-based optimization and

root-solving algorithms

• Accurate partials are crucial for sensitive problems

• Papers usually focus on the accuracy of the integrators with

respect to the physical trajectory

Here, we’re interested in the accuracy of the partials

with respect to the approximated trajectory

• Three common methods:



- Automatic differentiation / Complex-step derivative


State Transition Matrices

• Defined by:

• Such that:

• Used to predict the change in 𝑿 due to a change

in 𝑿0

• Can be used for optimization, root-solving,

covariance propagation, differential correction, etc…


Summary









• Conclusions


The variational equations

• Propagate the variational equations:

• Simple example: Euler integration

Identical!


Finite differences methods

• Applicable in any case, easy to implement

• Forward differences

• Central differences

• Generic expression


Finite differences methods

• Matrix form:

• Solve: Vandermonde

System

• Second order:


Complex Step Derivatives

• Complex step derivative: Lyness & Moler (1967),

Squire (1998), Martins (2001)

No subtraction! h can be taken as small as we

wan

Martins demonstrated that CSD was linked to AD

(with extra computations and different type)


Multicomplex step derivatives

• Lantoine et al. (2012) use multicomplex numbers:

• Necessitates complex and bicomplex implementation

of the functions: MCXLib (dll) and generize (Python

preprocessor) allow automatic implementation


Summary









• Conclusions


Example orbits

• 3 Body orbits

PO1: 𝜆𝑚𝑎𝑥 = 2.47 PO2: 𝜆𝑚𝑎𝑥 = 2.8 108


Example orbits

PO3: 𝜆𝑚𝑎𝑥 = 17.63


Example orbits

• 2 Body orbits

TB1: 𝜆𝑚𝑎𝑥 = 1 TB2: 𝜆𝑚𝑎𝑥 = 8.98 103


Linesearch application and fixed path

• Linesearch application:

- Propagate trajectory and STMs for a reference 𝑋0

- Propagate linesearch states: along [1,1,1,1,1,1] vector

- Predict change:

- Compute error:

• Fixed path:

- Propagate state using variable-step integration

- Record step sizes

- Reuse same step sizes for subsequent integrations


Summary









• Conclusions


State vs. State + variational equations

• Propagating variational equations alongside state different

integration path.

TB2, Variable-step, 1E-6 Difference in State and STM



TB2, Variable-step, 1E-9 Difference in State and STM



• Yields incorrect STMs

• Recommendation: Fixed-step, or compute error on State only

𝜖 = 10−6 𝜖 = 10−9


Partials of the variable-step integration

• Remember the Euler integration of the variational

equations:

• Now, variable-step:

Extra term


Partials of the variable-step integration

Variational

Forward

diff

CSD

Central

diff

PO3, 𝜖 = 10−3


Discontinuity: the number of steps

• Variation in step sizes sometimes yields a different number of

steps Discontinuity in the integration function

• Can not be captured by any of the methods

PO3 PO3, perturbed


Discontinuity: the number of steps

• Finite differences: degraded by the discontinuity

PO3 PO3, perturbed


Summary









• Conclusions


Accuracy of the methods

• First- and second-order approximations, TB2, 𝜖 = 10−3

• Only ‘coherent’ scenarios are presented

Variable-step integration Fixed path integration


Accuracy and timings

• RMS over 𝛿𝑋 ∈ [10−12, 10−7]


Accuracy and timings

PO1 PO3

TB1


PO2

• Most sensitive. Nature changes with tolerance


Conclusions

• Three main recommendations for the accurate propagation of

State Transition Matrices using variable-step integration:

- Do not mix state only and state + variational equations

- Variable-step integrators can lead to discontinuities of the

integration function. When possible, prefer fixed-step

integration, with regularized EOMs

- The variational equations DO NOT capture the change of the

integration path. CSD and finite differences should be

preferred for low-fidelity propagations

• Timings and Accuracy

- CSD is best or tied for best in all cases. Takes longer than

variational equations, especially for fixed-step.

- Forward differences achieves poor accuracy. Central is better,

generic even better, but have compute time penalties


on the accuracy of state-transition matrices

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