Robust Cislunar Trajectory Optimization in the Presence of Stochastic Errors

Abstract

The focus of this research is the optimization of impulsive maneuvers in the circular restricted three body problem with stochastic error sources. Two and three impulse transfer trajectories with corrective maneuvers are designed to minimize an upper statistical bound for total $\Delta V$, including the sum of nominal impulsive $\Delta V$ plus the 3$\sigma$ upper bound for corrections. Error sources include an initial state dispersion, maneuver execution error, and random white disturbances modeled as process noise. Direct optimization is performed via nonlinear programming. At each step in the nonlinear program, the optimal location and number of trajectory correction maneuvers (TCM) is determined, simultaneously satisfying a set of position dispersion constraints along the trajectory. Analytical gradients of the nominal maneuver magnitude and trace of the TCM covariance are provided to the nonlinear program to avoid finite differencing and associated numerical errors. Partial derivatives of the TCM covariance require the propagation of second-order state transition tensors. This work incorporates process noise into the optimization problem by propagating the effect of the accumulated process noise covariance. As a result, populating analytical gradients requires the partial derivative of the accumulated process noise covariance, called the accumulated process noise covariance state sensitivity tensor, which is propagated similar to state transition tensors. Robust trajectory scenarios analyzed include a three impulse trajectory from low-Earth orbit (LEO) to a near-rectilinear halo orbit (NRHO), and a two impulse NRHO rendezvous.

Appendices

Appendix A: Analytical Gradients

The design method in this paper computes, propagates required values, and provides analytical gradients of the cost function and constraints with respect to the problem parameters, $\varvec{S}$, to the NLP. Many stochastic cost terms involve STMs, which require the propagation of second-order STTs to compute the gradient analytically. Similar to propagating second-order STTs, the gradient of QBM terms requires propagating a tensor that characterizes the state sensitivity of the accumulated process noise covariance, $\frac{\partial {\bar{Q}}\left( t_{1},t_{0}\right) }{\partial \varvec{x}_{0,i}}$. This will be referred to as the Q bar tensor, or QBT. This section provides an overview of two gradient formulations: first, a sequential method for assembling the gradient of multiple TCMs and second, the propagation and manipulation of the QBT.

1.1 A.1 Multiple TCM Cost Analytical Gradient

The partial derivative of the deterministic cost, deterministic constraints, a single TCM RSS magnitude, and the target position dispersion covariance constraint are derived in more detail in [20]. This paper incorporates multiple TCMs, maneuver execution error, and process noise. A portion of the derivation for a single TCM is repeated here.

The approach used to calculate the gradient of multiple TCMs starts with the gradient of a single TCM. The partial derivative of the RSS of a single TCM’s covariance, $\sigma _{\delta V_{r}}$, with respect to a segment parameter vector $\varvec{s}_{i}$ is an application of matrix differentiation rules.

$$\begin{aligned} \frac{\partial \sigma _{\delta V_{r,1}}}{\partial \varvec{s}_{i}}=\left[ \begin{array}{cc} \frac{\partial \sigma _{\delta V_{r,1}}}{\partial \varvec{x}_{0,i}}&\frac{\partial \sigma _{\delta V_{r,1}}}{\partial \Delta t_{i}}\end{array}\right] \end{aligned}$$

(75)

Showing the partial derivative of $\sigma _{\delta V_{r}}^{2}$ with respect to a segment initial state makes the following derivation slightly cleaner. Maneuver execution error and process noise are also assumed to be zero for this derivation and will be subsequently included.

$$\begin{aligned} \left[ \frac{\partial \sigma _{\delta V_{r,1}}^{2}}{\partial \varvec{x}_{0,i}}\right] _{y}&=\frac{\left[ \partial \text {tr}\left( T_{1}P_{c_{1}}^{-}T_{1}^{\top }\right) \right] }{\left[ \partial \varvec{x}_{0,i}\right] _{y}} \end{aligned}$$

(76)

$$\begin{aligned}&=\text {tr}_{bg}(\left[ \frac{\partial T_{1}}{\partial \varvec{x}_{0,i}}\right] _{bd,y}\left[ P_{c_{1}}^{-}\right] _{dw}\left[ T_{1}\right] _{gw}+\left[ T_{1}\right] _{bp}\left[ \frac{\partial P_{c_{1}}^{-}}{\partial \varvec{x}_{0,i}}\right] _{pq,y}\left[ T_{1}\right] _{gq}\nonumber \\&\quad +\left[ T_{1}\right] _{bd}\left[ P_{c_{1}}^{-}\right] _{dw}\left[ \frac{\partial T_{1}}{\partial \varvec{x}_{0,i}}\right] _{gw,y}) \end{aligned}$$

(77)

where index notation is used, combined with brackets to avoid confusion when indices are adjacent to other subscripts. The subscripts on the trace operator ($\text {tr}_{bg}$) identify the indices that the trace is performed on, resulting in the corresponding matrix indices collapsing to a scalar. Additionally, when segment i is relevant to the STM terms inside $T_{1}$ or $P_{c_{1}}^{-}$, their partial derivative requires propagated STT terms:

$$\begin{aligned} \frac{\left[ \partial P_{c_{1}}^{-}\right] _{pq}}{\left[ \partial \varvec{x}_{0,i}\right] _{y}}&=\frac{\partial \Phi \left( t_{c_{1}},t_{0}\right) _{pu}\left[ P_{0}\right] _{uv}\Phi \left( t_{c_{1}},t_{0}\right) _{qv}}{\left[ \partial \varvec{x}_{0,i}\right] _{y}} \end{aligned}$$

(78)

$$\begin{aligned}&=\Phi \left( t_{c_{1}},t_{0}\right) _{pu,y}\left[ P_{0}\right] _{uv}\Phi \left( t_{c_{1}},t_{0}\right) _{qv}+\Phi \left( t_{c_{1}},t_{0}\right) _{pu}\left[ P_{0}\right] _{uv}\Phi \left( t_{c_{1}},t_{0}\right) _{qv,y} \end{aligned}$$

(79)

$$\begin{aligned} \frac{\left[ \partial T_{1}\right] _{bd}}{\left[ \partial \varvec{x}_{0,i}\right] _{y}}&=\frac{\left[ \partial \left( \left[ \begin{array}{cc} -\Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right) \Phi _{rr}\left( t_{n_{1}},t_{c_{1}}\right)&\quad -I_{3\times 3}\end{array}\right] \right) \right] _{bd}}{\left[ \partial \varvec{x}_{0,i}\right] _{y}} \end{aligned}$$

(80)

The rr and rv subscripts identify top-left and top-right $3\times 3$ submatrices of the STM terms. When applied to STT terms, the result is a $3\times 3\times 6$ tensor, with no crop** occurring in the third dimension.

$$\begin{aligned} \frac{\partial \left( \Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right) _{bz}\Phi _{rr}\left( t_{n_{1}},t_{c_{1}}\right) _{zd}\right) }{\left[ \partial \varvec{x}_{0,i}\right] _{y}}&=\frac{\partial \Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right) _{bz}}{\left[ \partial \varvec{x}_{0,i}\right] _{y}}\Phi _{rr}\left( t_{n_{1}},t_{c_{1}}\right) _{zd}\nonumber \\&\quad +\Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right) _{bz}\Phi _{rr}\left( t_{n_{1}},t_{c_{1}}\right) _{zd,y} \end{aligned}$$

(81)

An application of the matrix inverse derivative property and STT terms are required to evaluate the partial derivative of $\Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right)$:

$$\begin{aligned} \frac{\partial \Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right) _{bz}}{\left[ \partial \varvec{x}_{0,i}\right] _{y}}=-\Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right) _{bo}\Phi _{rv}\left( t_{n_{1}},t_{c_{1}}\right) _{oe,y}\Phi _{rv}^{-1}\left( t_{n_{1}},t_{c_{1}}\right) _{ez} \end{aligned}$$

(82)

The same fundamental steps apply when evaluating the partial derivative with respect to $\Delta t_{i}$ but the results are simplified by the fact that tensors are not required. Equation 11 is applied for STM time sensitivity rather than utilizing STTs for STM state sensitivity.

Some creative manipulation of STM endpoints is required to obtain the appropriate sensitivity in many cases. As segments are propagated independently of one another each NLP iteration, from the beginning of each segment for the corresponding duration, care must be taken to only incorporate the sensitivity of modifications to the appropriate time indices for the corresponding segment, and not the entire STM sensitivity between endpoints (when the time between corrections and/or a target might span multiple segments).

When multiple TCMs are performed along a nominal trajectory, each at $t_{c_{k}}$ with execution error covariance $R_{TCM}$, the total TCM RSS $\sigma _{\delta V}$ gradient requires gradients of each individual $\sigma _{\delta V_{r,k}}$ plus the gradient of $\sigma _{\delta V_{v}}$. Modifications to TCMs have impacts to the dispersion covariance that affect the gradient of subsequent TCMs. The sequential nature of the following formulation maintains the sensitivity of each TCM to modifications in elements that have an impact on the preceding dispersion covariance. Said differently, each TCM affects the dispersion covariance at all future TCMs. The following sequential covariance sensitivity formulation simplifies this operation.

The first TCM, $k=1$, mirrors the single TCM scenario with the addition of $R_{TCM}$. The RSS of the first TCM, $\sigma _{\delta V_{r,1}}$, is

$$\begin{aligned} \sigma _{\delta V_{r,1}}=\sqrt{\text {tr}\left( T_{1}P_{c_{1}}^{-}T_{1}^{\top }+R_{TCM}\right) } \end{aligned}$$

(83)

The post-TCM covariance is defined by Eq. 47. To simplify notation, the gradient of $\sigma _{\delta V_{r,1}}$ with respect to a segment initial state will be represented by the function ${\mathcal {D}}$, where ${\mathcal {D}}$ and the partials being passed represent the application of the single TCM gradient derivation in Eqs. 75 through 82:

$$\begin{aligned} \frac{\partial \sigma _{\delta V_{r,1}}}{\partial \varvec{x}_{0,i}}={\mathcal {D}}\left( T_{1},P_{c_{1}}^{-},R_{TCM},\frac{\partial T_{1}}{\partial \varvec{x}_{0,i}},\frac{\partial P_{c_{1}}^{-}}{\partial \varvec{x}_{0,i}}\right) \end{aligned}$$

(84)

The second TCM ($k=2$) now incorporates the effects of TCM 1 to the next covariance update $P_{c_{2}}^{-}$; assuming the next covariance update is another TCM:

$$\begin{aligned} \sigma _{\delta V_{r,2}}=\sqrt{\text {tr}\left( T_{2}P_{c_{2}}^{-}T_{2}^{\top }+R_{TCM}\right) } \end{aligned}$$

(85)

$$\begin{aligned} P_{c_{2}}^{-}&=\Phi \left( t_{c_{2}},t_{c_{1}}\right) \left( \left( I+N_{1}\right) P_{c_{1}}^{-}\left( I+N_{1}\right) ^{\top }+GR_{TCM}G^{\top }\right) \Phi \left( t_{c_{2}},t_{c_{1}}\right) {}^{\top } \end{aligned}$$

(86)

The state sensitivity of the dispersion before TCM 2, $\frac{\partial P_{c_{2}}^{-}}{\partial \varvec{x}_{0,i}}$, can be expressed as another function ${\mathcal {J}}$, shortening another lengthy application of matrix differentiation. Evaluating Eq. 87 is similar to Eq. 75 with the addition of the partial derivative of the TCM execution error term, which is nonzero due to being multiplied by STMs.

$$\begin{aligned} \frac{\partial P_{c_{2}}^{-}}{\partial \varvec{x}_{0,i}}={\mathcal {J}}\left( T_{1},\frac{\partial T_{1}}{\partial \varvec{x}_{0,i}},P_{c_{1}}^{-},\frac{\partial P_{c_{1}}^{-}}{\partial \varvec{x}_{0,i}},\Phi \left( t_{c_{2}},t_{c_{1}}\right) ,\frac{\partial \Phi \left( t_{c_{2}},t_{c_{1}}\right) }{\partial \varvec{x}_{0,i}},R_{TCM}\right) \end{aligned}$$

(87)

Now $\frac{\partial P_{c_{2}}^{-}}{\partial \varvec{x}_{0,i}}$ is expressed as a function of gradients at TCM 1 and the dynamics between TCMs 1 and 2. Calculating the dispersion covariance sensitivities sequentially enables subsequent TCMs and their gradients to be expressed as a function of values at the previous dispersion covariance update.

For corrected nominal maneuvers, the gradient of Eqs. 53 and 55 follows a similar process. The only new term is now the partial derivative of the unit vector in the direction of the nominal maneuver, ${\hat{i}}_{\Delta \varvec{V}}$.

$$\begin{aligned} \frac{\partial {\hat{i}}_{\Delta \varvec{V}}}{\partial \varvec{x}_{0,i}}=\frac{\partial {\hat{i}}_{\Delta \varvec{V}}}{\partial \varvec{\Delta V}}\frac{\partial \varvec{\Delta V}}{\partial \varvec{x}_{0,i}} \end{aligned}$$

(88)

where

$$\begin{aligned} \frac{\partial {\hat{i}}_{\Delta V}}{\partial \varvec{\Delta V}}=\frac{I_{3\times 3}}{\Vert \varvec{\Delta V}\Vert }-\frac{\varvec{\Delta V}\varvec{\Delta V}^{\top }}{\Vert \varvec{\Delta V}\Vert ^{3}} \end{aligned}$$

(89)

Suppose the nominal maneuver occurs at the mth trajectory node. The corresponding $\Delta \varvec{V}$ is only sensitive to modifications in segments $m-1$ and m. Therefore, the only nonzero gradients are with respect to $\varvec{x}_{0,m-1}$, which is shown below, and $\varvec{x}_{0,m}$, which is trivial.

$$\begin{aligned} \frac{\partial \varvec{\Delta V}}{\partial \varvec{x}_{0,m-1}}=\frac{\partial \left( \varvec{v}_{0,m}-\varvec{v}_{f,m-1}\right) }{\partial \varvec{x}_{0,m-1}}=\left[ \begin{array}{cc} -\Phi _{m-1,vr}&-\Phi _{m-1,vv}\end{array}\right] \end{aligned}$$

(90)

where $\Phi _{m-1}$ corresponds to the STM from the beginning to the end of segment $m-1$ and subscripts vr and vv are its corresponding bottom-left and bottom-right $3 \times 3$ sub-matrices.

The gradient of the total stochastic cost is the sum of each individual gradient.

1.2 A.2 Accumulated Process Noise Covariance Gradient

When including the effect of process noise and evaluating $\frac{\partial P_{c_{1}}^{-}}{\partial {\bar{x}}_{0,i}}$ in Eq. 58, $\frac{\partial {\bar{Q}}\left( t_{c_{1}},t_{0}\right) }{\partial \varvec{x}_{0,i}}$, the QBT, becomes a required term. These QBM state sensitivities are obtained via numerical integration of Eq. 91, with a brief derivation following. The first step involves taking the partial derivative of $\dot{{\bar{Q}}}$ (Eq. 57) with respect to the initial state. Applying the product rule results in Eq. 91 which is numerically integrated to obtain the QBT, similar to an STT, with initial conditions $\frac{\partial {\bar{Q}}(t_{0},t_{0})}{\partial \varvec{x}(t_{0})}=0_{6\times 6\times 6}$:

$$\begin{aligned} \left[ \frac{\partial \dot{{\bar{Q}}}(t,t_{0})}{\partial \varvec{x}(t_{0})}\right] _{ijk}&=\frac{\partial F(t)_{i,m}}{\partial \varvec{x}(t_{0})_{k}}{\bar{Q}}(t,t_{0})_{mj}+F(t)_{i,m}\frac{\partial {\bar{Q}}(t,t_{0})_{mj}}{\partial \varvec{x}(t_{0})_{k}}\nonumber \\&\quad +\frac{\partial F(t)_{jm}}{\partial \varvec{x}(t_{0})_{k}}{\bar{Q}}(t,t_{0})_{im}+F(t)_{j,m}\frac{\partial {\bar{Q}}(t,t_{0})_{im}}{\partial \varvec{x}(t_{0})_{k}} \end{aligned}$$

(91)

where $F\left( t\right) _{i,j}$ is the system jacobian. A chain rule application utilizes the second derivative of the system dynamics with respect to the state, $F\left( t\right) _{i,jk}$:

$$\begin{aligned} \frac{\partial F(t)_{ij}}{\partial \varvec{x}(t_{0})_{k}}=\frac{\partial F(t)_{ij}}{\partial \varvec{x}(t)_{m}}\frac{\partial \varvec{x}(t)_{m}}{\partial \varvec{x}(t_{0})_{k}}=F(t)_{i,jm}\Phi (t,t_{0})_{m,k} \end{aligned}$$

(92)

Similar to the QBM, some manipulation of the QBT is also required to obtain the appropriate QBT endpoints and its effect at the appropriate time. Finding the equation for sequential QBT combination (i.e., assembling $\frac{\partial {\bar{Q}}_{t_{2}}\left( t_{2},t_{0}\right) }{\partial \varvec{x}(t_{0})}$ from $\frac{\partial {\bar{Q}}_{t_{1}}\left( t_{1},t_{0}\right) }{\partial \varvec{x}(t_{0})}$ and $\frac{\partial {\bar{Q}}_{t_{2}}\left( t_{2},t_{1}\right) }{\partial \varvec{x}(t_{1})}$) involves taking the partial derivative of Eq. 59, which equals the following after two chain rule applications and term collection:

$$\begin{aligned} \frac{\partial {\bar{Q}}_{t_{2}}\left( t_{2},t_{0}\right) }{\partial \varvec{x}(t_{0})}&=W_{ij,m}\Phi \left( t_{1},t_{0}\right) _{m,k}+\Phi \left( t_{2},t_{1}\right) _{i,m}\frac{\partial {\bar{Q}}_{t_{1}}\left( t_{1},t_{0}\right) _{mn}}{\partial \varvec{x}(t_{0})_{k}}\Phi \left( t_{2},t_{1}\right) _{j,n} \end{aligned}$$

(93)

where

$$\begin{aligned} W_{ij,m}&=\frac{\partial {\bar{Q}}_{t_{2}}\left( t_{2},t_{1}\right) _{ij}}{\partial \varvec{x}(t_{1})_{m}}+\Phi _{II}\left( t_{2},t_{1}\right) _{i,nm}{\bar{Q}}_{t_{1}}\left( t_{1},t_{0}\right) _{np}\Phi \left( t_{2},t_{1}\right) _{j,p}\nonumber \\&\quad +\Phi \left( t_{2},t_{1}\right) _{i,n}{\bar{Q}}_{t_{1}}\left( t_{1},t_{0}\right) _{np}\Phi _{II}\left( t_{2},t_{1}\right) _{j,pm} \end{aligned}$$

(94)

Subtracting a contribution involves taking the partial derivative of Eq. 61, performing a product rule and two chain rule applications, and collecting terms:

$$\begin{aligned} \frac{\partial {\bar{Q}}_{t_{2}}\left( t_{2},t_{1}\right) _{ij}}{\partial \varvec{x}(t_{1})_{k}}&=\left( \frac{\partial {\bar{Q}}_{t_{2}}\left( t_{2},t_{0}\right) _{ij}}{\partial \varvec{x}(t_{0})_{m}}-\Phi \left( t_{2},t_{1}\right) _{i,n}\frac{\partial {\bar{Q}}_{t_{1}}\left( t_{1},t_{0}\right) _{np}}{\partial \varvec{x}(t_{0})_{m}}\Phi \left( t_{2},t_{1}\right) _{j,p}\right) \Phi \left( t_{0},t_{1}\right) _{m,k}\nonumber \\&\quad -\Phi _{II}\left( t_{2},t_{1}\right) _{i,nm}{\bar{Q}}_{t_{1}}\left( t_{1},t_{0}\right) _{np}\Phi \left( t_{2},t_{1}\right) _{j,p}\nonumber \\&\quad -\Phi \left( t_{2},t_{1}\right) _{i,n}{\bar{Q}}_{t_{1}}\left( t_{1},t_{0}\right) _{np}\Phi _{II}\left( t_{2},t_{1}\right) _{j,pm} \end{aligned}$$

(95)

Appendix B: Parameter Values Used

• Earth gravitational parameter: $\mu _{E}=398600.4415\text { km}^{3}/\text {s}^{2}$.

• Moon gravitational parameter: $\mu _{M}=4902.8\text { km}^{3}/\text {s}^{2}$.

• Circular lunar orbit radius: 384,400 km.

Appendix C: Robust LEO to PLF to NRHO Insertion Trajectory Parameters

See Table 10.

Table 10 Robust NRI with flexible initial orbital plane trajectory states and durations in the CR3BP