Sundman-Transformed Differential Dynamic Programming with Modified Equinoctial Elements

Abstract

Previous efforts addressed the challenge of low-thrust many-revolution trajectory optimization by applying a Sundman transformation to change the independent variable of the spacecraft equations of motion to an orbit anomaly and performing the optimization with differential dynamic programming (DDP). The approach may be enhanced by representing the spacecraft state with orbital elements rather than position and velocity coordinates. This paper shows how the modified equinoctial element state representation enters the DDP algorithm. Example transfers from geostationary transfer orbit (GTO) to geosynchronous orbit (GEO) demonstrate how gains in computational efficiency are possible with minimal impact to solution quality. Those gains are leveraged to compute the Pareto front of time versus delivered mass for a benchmark orbit transfer from the literature.

Appendix: The Dynamics Matrix and Tensor for the Augmented Modified Element State Vector

In order to compute the STMs for the augmented MEE state vector, the dynamics matrix and tensor must be obtained for each term in Eq. 14. Derivatives of b are available directly, but the preceding terms require careful application of the chain rule. First, the thrust term is restated.

$$ A\boldsymbol{\Delta}_{\text{Thrust}} =\frac{ A\boldsymbol{u}T_{a}}{m} $$

(27)

The thrust contribution to the dynamics matrix is the first derivative of Eq. 27 with respect to the augmented MEE state vector.

$$ \frac{\partial (A\boldsymbol{u}T_{a}/m)}{\partial\boldsymbol{X}_{MEE}} = \frac{A\boldsymbol{u}}{m}\frac{\partial T_{a}}{\partial \boldsymbol{X}_{MEE}} + \frac{A}{m}\frac{\partial\boldsymbol{u}}{\partial\boldsymbol{X}_{MEE}}T_{a} + A\frac{\partial}{\partial \boldsymbol{X}_{MEE}}\left( \frac{1}{m}\right)\boldsymbol{u}T_{a} + \frac{\partial A}{\partial \boldsymbol{X}_{MEE}}\frac{\boldsymbol{u}T_{a}}{m} $$

(28)

Power models frequently use the Cartesian inertial state, so it is convenient to obtain the derivative of the thrust available with respect to the X_IJK and then make the transformation to the X_MEE sensitivity.

$$ \begin{array}{@{}rcl@{}} \frac{\partial T_{a}}{\partial \boldsymbol{X}_{MEE}} &=& \frac{\partial T_{a}}{\partial \boldsymbol{X}_{IJK}}\frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}} \end{array} $$

(29a)

$$ \begin{array}{@{}rcl@{}} {\left( \frac{\partial^{2} T_{a}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)}^{i,a} &=& \left( \frac{\partial T_{a}}{\partial \boldsymbol{X}_{IJK}}\right)^{\gamma_{1}}\left( \frac{\partial^{2} \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{\gamma_{1},ia}\\ &&+ \left( \frac{\partial^{2} T_{a}}{\partial \boldsymbol{X}^{2}_{IJK}}\right)^{\gamma_{1},\gamma_{2}}\left( \frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{1},i}\left( \frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{2},a} \end{array} $$

(29b)

Tensor notation has been introduced with the convention that superscripts are indices, and repeated indices of γ are summed over.

With the second derivative of the thrust available now defined with respect to the X_MEE, the dynamics tensor contributions from thrust can be obtained.

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2} (A\boldsymbol{u}T_{a}/m)}{\partial\boldsymbol{X}^{2}_{MEE}} &=& t_{11} + t_{12} + t_{13} + t_{14}\\ &&+ t_{21} + t_{22} + t_{23} + t_{24}\\ &&+ t_{31} + t_{32} + t_{33} + t_{34}\\ &&+ t_{41} + t_{42} + t_{43} + t_{44} \end{array} $$

(30a)

$$ \begin{array}{@{}rcl@{}} t_{11}^{i,ab} &=& \left( \frac{A}{m}\right)^{i,\gamma_{1}}\boldsymbol{u}^{\gamma_{1}}\left( \frac{\partial^{2} T_{a}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{a,b} \end{array} $$

(30b)

$$ \begin{array}{@{}rcl@{}} t_{12}^{i,ab} &=& \left( \frac{A}{m}\right)^{i,\gamma_{1}}\left( \frac{\partial\boldsymbol{u}}{\partial\boldsymbol{X}_{MEE}}\right)^{\gamma_{1},a}\left( \frac{\partial T_{a}}{\partial \boldsymbol{X}_{MEE}}\right)^{b} \end{array} $$

(30c)

$$ \begin{array}{@{}rcl@{}} t_{13}^{i,ab} &=& A^{i,\gamma_{1}}\boldsymbol{u}^{\gamma_{1}}\left[\frac{\partial}{\partial\boldsymbol{X}_{MEE}}\left( \frac{1}{m}\right)\right]^{a}\left( \frac{\partial T_{a}}{\partial\boldsymbol{X}_{MEE}}\right)^{b} \end{array} $$

(30d)

$$ \begin{array}{@{}rcl@{}} t_{14}^{i,ab} &=& \left( \frac{\partial A}{\partial \boldsymbol{X}_{MEE}}\right)^{i,\gamma_{1}a}\left( \frac{\boldsymbol{u}}{m}\right)^{\gamma_{1}}\left( \frac{\partial T_{a}}{\partial\boldsymbol{X}_{MEE}}\right)^{b} \end{array} $$

(30e)

$$ \begin{array}{@{}rcl@{}} t_{22}^{i,ab} &=& \left( \frac{A}{m}\right)^{i,\gamma_{1}}\left( \frac{\partial^{2}\boldsymbol{u}}{\partial\boldsymbol{X}^{2}_{MEE}}\right)^{\gamma_{1},ab}T_{a} \end{array} $$

(30f)

$$ \begin{array}{@{}rcl@{}} t_{23}^{i,ab} &=& A^{i,\gamma_{1}}\left[\frac{\partial}{\partial\boldsymbol{X}_{MEE}}\left( \frac{1}{m}\right)\right]^{a}\left( \frac{\partial\boldsymbol{u}}{\partial\boldsymbol{X}_{MEE}}\right)^{\gamma_{1},b}T_{a} \end{array} $$

(30g)

$$ \begin{array}{@{}rcl@{}} t_{24}^{i,ab} &=& \left( \frac{\partial A}{\partial \boldsymbol{X}_{MEE}}\right)^{i,\gamma_{1}a}\left( \frac{\partial\boldsymbol{u}}{\partial\boldsymbol{X}_{MEE}}\right)^{\gamma_{1},b}\left( \frac{T_{a}}{m}\right) \end{array} $$

(30h)

$$ \begin{array}{@{}rcl@{}} t_{33}^{i,ab} &=& A^{i,\gamma_{1}}\boldsymbol{u}^{\gamma_{1}}\left[\frac{\partial^{2}}{\partial\boldsymbol{X}^{2}_{MEE}}\left( \frac{1}{m}\right)\right]^{a,b}T_{a} \end{array} $$

(30i)

$$ \begin{array}{@{}rcl@{}} t_{34}^{i,ab} &=& \left( \frac{\partial A}{\partial \boldsymbol{X}_{MEE}}\right)^{i,\gamma_{1}a}\boldsymbol{u}^{\gamma_{1}}\left[\frac{\partial}{\partial\boldsymbol{X}_{MEE}}\left( \frac{1}{m}\right)\right]^{b}T_{a} \end{array} $$

(30j)

$$ \begin{array}{@{}rcl@{}} t_{44}^{i,ab} &=& \left( \frac{\partial^{2} A}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{i,\gamma_{1}ab}\boldsymbol{u}^{\gamma_{1}}\left( \frac{T_{a}}{m}\right) \end{array} $$

(30k)

$$ \begin{array}{@{}rcl@{}} t_{21}^{i,ab} &=& t_{12}^{i,ba}, \quad t_{31}^{i,ab} = t_{13}^{i,ba}, \quad t_{32}^{i,ab} = t_{23}^{i,ba} \\ t_{41}^{i,ab} &=& t_{14}^{i,ba}, \quad t_{42}^{i,ab} = t_{24}^{i,ba}, \quad t_{43}^{i,ab} = t_{34}^{i,ba} \end{array} $$

(30l)

Proceeding to the mass flow rate term,

$$ \frac{\partial (\dot{\boldsymbol{m}}T_a)}{\partial\boldsymbol{X}_{MEE}} = \dot{\boldsymbol{m}}\frac{\partial T_a}{\partial \boldsymbol{X}_{MEE}} + \frac{\partial \dot{\boldsymbol{m}}}{\partial \boldsymbol{X}_{MEE}}T_, $$

(31a)

$$ \begin{array}{@{}rcl@{}} \left( \frac{\partial^{2} (\dot{{\boldsymbol{m}}}T_{a})}{\partial{X}^{2}_{MEE}}\right)^{i,ab} &=& \dot{{\boldsymbol{m}}}^i\left( \frac{\partial^{2}T_{a}}{\partial{X}^{2}_{MEE}}\right)^{a,b} + \left( \frac{\partial \dot{{\boldsymbol{m}}}}{\partial \boldsymbol{X}_{MEE}}\right)^{i,a}\left( \frac{\partial T_a}{\partial \boldsymbol{X}_{MEE}}\right)^{b}\\ &&+ \left( \frac{\partial \dot{{\boldsymbol{m}}}}{\partial \boldsymbol{X}_{MEE}}\right)^{i,b}\left( \frac{\partial T_a}{\partial \boldsymbol{X}_{MEE}}\right)^{a}, \end{array} $$

(31b)

where the second derivatives of $\dot {\boldsymbol {m}}$ are zero and ignored in presentation.

The effects of perturbations are included by considering the summation of their first and second derivatives in an inertial frame.

$$ \begin{array}{@{}rcl@{}} \frac{\partial \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}_{IJK}} &=& \sum\limits_{i=0}^{n_{p}}\frac{\partial \boldsymbol{\delta}_{p_{i}}}{\partial \boldsymbol{X}_{IJK}} \end{array} $$

(32a)

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2} \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}^{2}_{IJK}} &=& \sum\limits_{i=0}^{n_{p}}\frac{\partial^{2} \boldsymbol{\delta}_{p_{i}}}{\partial \boldsymbol{X}^{2}_{IJK}} \end{array} $$

(32b)

As previously obtained for the thrust available, the perturbation sensitivities must be found with respect to the augmented MEE state vector.

$$ \begin{array}{@{}rcl@{}} \frac{\partial \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}_{MEE}} &=& \frac{\partial \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}_{IJK}}\frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}} \end{array} $$

(33a)

$$ \begin{array}{@{}rcl@{}} \left( \frac{\partial^{2} \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{i,ab} &=& \left( \frac{\partial \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}_{IJK}}\right)^{i,\gamma_{1}}\left( \frac{\partial^{2} \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{\gamma_{1},ab} \\&&+ \left( \frac{\partial^{2} \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}^{2}_{IJK}}\right)^{i,\gamma_{1}\gamma_{2}}\left( \frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{1},a}\left( \frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{2},b} \end{array} $$

(33b)

Derivatives of the rotation matrix Q^T are also required. Those too are first found inertially.

$$ \begin{array}{@{}rcl@{}} \left( \frac{\partial Q^{T}}{\partial \boldsymbol{X}_{IJK}}\right)^{i,ab} &= \frac{\partial (Q^{T})^{i,a}}{\partial \boldsymbol{X}^{b}_{IJK}} \end{array} $$

(34a)

$$ \begin{array}{@{}rcl@{}} \left( \frac{\partial^{2} Q^{T}}{\partial \boldsymbol{X}^{2}_{IJK}}\right)^{i,abc} &= \frac{\partial (Q^{T})^{i,a}}{\partial \boldsymbol{X}^{b}_{IJK}\partial \boldsymbol{X}^{c}_{IJK}} \end{array} $$

(34b)

Next, the derivatives of Q^T are found with respect to the augmented MEE state vector.

$$ \begin{array}{@{}rcl@{}} \left( \frac{\partial Q^{T}}{\partial \boldsymbol{X}_{MEE}}\right)^{i,ab} &=& \left( \frac{\partial Q^{T}}{\partial \boldsymbol{X}_{IJK}}\right)^{i,a\gamma_{1}}\left( \frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{1},b} \end{array} $$

(35a)

$$ \begin{array}{@{}rcl@{}} \left( \frac{\partial^{2} Q^{T}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{i,abc} &=& \left( \frac{\partial Q^{T}}{\partial \boldsymbol{X}_{IJK}}\right)^{i,a\gamma_{1}}\left( \frac{\partial^{2} \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{\gamma_{1},bc}\\ &&+ \left( \frac{\partial^{2} Q^{T}}{\partial \boldsymbol{X}^{2}_{IJK}}\right)^{i,a\gamma_{1}\gamma_{2}}\left( \frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{1},b}\left( \frac{\partial \boldsymbol{X}_{IJK}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{2},c} \end{array} $$

(35b)

The necessary terms have been defined to assemble the dynamics matrix and tensor contributions from perturbations.

$$ \begin{array}{@{}rcl@{}} \left( \frac{\partial (AQ^{T}\boldsymbol{\delta}_{p})}{\partial\boldsymbol{X}_{MEE}}\right)^{i,a} &=& \left( AQ^{T}\frac{\partial \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}_{MEE}}\right)^{i,a}\\&&+ A^{i,\gamma_{1}}\left( \frac{\partial Q^{T}}{\!\partial\boldsymbol{X}_{MEE}\!}\right)^{\gamma_{1},\gamma_{2}a}\boldsymbol{\delta}_{p}^{\gamma_{2}} + \left( \frac{\partial A}{\!\partial \boldsymbol{X}_{MEE}}\!\right)^{i,\gamma_{1}a}(Q^{T}\boldsymbol{\delta}_{p})^{\gamma_{1}} \end{array} $$

(36)

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2} (AQ^{T}\boldsymbol{\delta}_{p})}{\partial\boldsymbol{X}^{2}_{MEE}} &=& t_{11} + t_{12} + t_{13}\\ &&+ t_{21} + t_{22} + t_{23}\\ &&+ t_{31} + t_{32} + t_{33} \end{array} $$

(37a)

$$ \begin{array}{@{}rcl@{}} t_{11}^{i,ab} &=&A^{i,\gamma_{1}}(Q^{T})^{\gamma_{1},\gamma_{2}}\left( \frac{\partial^{2}\boldsymbol{\delta}_{p}}{\boldsymbol{X}^{2}_{MEE}}\right)^{\gamma_{2},ab} \end{array} $$

(37b)

$$ \begin{array}{@{}rcl@{}} t_{12}^{i,ab} &=& A^{i,\gamma_{1}}\left( \frac{\partial Q^{T}}{\partial\boldsymbol{X}_{MEE}}\right)^{\gamma_{1},\gamma_{2}a}\left( \frac{\partial \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{2},b} \end{array} $$

(37c)

$$ \begin{array}{@{}rcl@{}} t_{13}^{i,ab} &=& \left( \frac{\partial A}{\partial \boldsymbol{X}_{MEE}}\right)^{i,\gamma_{1}a}(Q^{T})^{\gamma_{1},\gamma_{2}}\left( \frac{\partial \boldsymbol{\delta}_{p}}{\partial \boldsymbol{X}_{MEE}}\right)^{\gamma_{2},b} \end{array} $$

(37d)

$$ \begin{array}{@{}rcl@{}} t_{22}^{i,ab} &=& A^{i,\gamma_{1}}\left( \frac{\partial^{2} Q^{T}}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{\gamma_{1},\gamma_{2}ab} \boldsymbol{\delta}_{p}^{\gamma_{2}} \end{array} $$

(37e)

$$ \begin{array}{@{}rcl@{}} t_{23}^{i,ab} &=& \left( \frac{\partial A}{\partial \boldsymbol{X}_{MEE}}\right)^{i,\gamma_{1}a}\left( \frac{\partial Q^{T}}{\partial\boldsymbol{X}_{MEE}}\right)^{\gamma_{1},\gamma_{2}b}\boldsymbol{\delta}_{p}^{\gamma_{2}} \end{array} $$

(37f)

$$ \begin{array}{@{}rcl@{}} t_{33}^{i,ab} &=& \left( \frac{\partial^{2} A}{\partial \boldsymbol{X}^{2}_{MEE}}\right)^{i,\gamma_{1}ab}(Q^{T})^{\gamma_{1},\gamma_{2}}\boldsymbol{\delta}^{\gamma_{2}} \end{array} $$

(37g)

$$ \begin{array}{@{}rcl@{}} t_{21}^{i,ab} &=& t_{12}^{i,ba}, \quad t_{31}^{i,ab} = t_{13}^{i,ba}, \quad t_{32}^{i,ab} = t_{23}^{i,ba} \end{array} $$

(37h)

The complete assembly of the dynamics matrix is the summation of Eqs. 28, 31a and 33a and derivatives of b.

$$ \frac{\partial\dot{\boldsymbol{X}}_{MEE}}{\partial\boldsymbol{X}_{MEE}} = \frac{\partial (A\boldsymbol{u}T_{a}/m)}{\partial\boldsymbol{X}_{MEE}} + \frac{\partial (\dot{\boldsymbol{m}}T_{a})}{\partial\boldsymbol{X}_{MEE}} + \frac{\partial (AQ^{T}\boldsymbol{\delta}_{p})}{\partial\boldsymbol{X}_{MEE}} + \frac{\partial \boldsymbol{b}}{\partial \boldsymbol{X}_{MEE}} $$

(38)

Finally, the complete assembly of the dynamics tensor is the summation of Eqs. 30a, 31b and 33b and second derivatives of b.

$$ \frac{\partial^{2}\dot{\boldsymbol{X}}_{MEE}}{\partial\boldsymbol{X}^{2}_{MEE}} = \frac{\partial^{2} (A\boldsymbol{u}T_{a}/m)}{\partial\boldsymbol{X}^{2}_{MEE}} + \frac{\partial^{2} (\dot{\boldsymbol{m}}T_{a})}{\partial\boldsymbol{X}^{2}_{MEE}} + \frac{\partial^{2} (AQ^{T}\boldsymbol{\delta}_{p})}{\partial\boldsymbol{X}^{2}_{MEE}} + \frac{\partial^{2} \boldsymbol{b}}{\partial \boldsymbol{X}^{2}_{MEE}} $$

(39)