Incorporating Directional Uncertainties into Polynomial Chaos Expansions for Astronautics Problems

Abstract

Modern astronautics problems require methods of tractable uncertainty quantification for problems with data in a variety of mathematical spaces. Polynomial Chaos Expansions (PCEs) enable tractable uncertainty propagation, sensitivity analysis, and provide a surrogate model to efficiently solve stochastic optimization problems. Existing PCE methods are mostly isolated to bases defined via tensor products of univariate functions over subdomains of the real line or integers. The goal of this work is to incorporate random vectors on the n-dimensional sphere, thereby extending the use of PCEs to problems that include spacecraft attitude uncertainty. Random inputs with probability densities on the n-sphere are generally correlated. When directional random inputs are independent, products of univariate functions fail to produce an orthogonal basis. Basis functions must preserve the periodic response of the system produced by the underlying structure of the domain. This paper presents an approach to generating an orthogonal basis with respect to a density function on the n-sphere by combining hyperspherical harmonics with an orthogonalization procedure based on the raw moments of the harmonic functions. For highly concentrated densities on the unit sphere, the spherical cap harmonics provide a more numerically stable solution while preserving periodicity. Numeric errors in the proposed procedure are presented for multiple cases. Performance of the PCEs is quantified when propagating uncertainty for a highly eccentric orbit with a random translational maneuver error, and a second case based on rigid-body dynamics with the attitude state parameterized as a quaternion.

Appendices

Appendix A Derivation of Basis for Symmetric Densities

Combining Eqs. (49), (33), and (10) for the Kent distribution with $n=2$ and $\beta =0$,

$$\begin{aligned} \left<\phi _{i}\phi _{j}\right>&= \int _{0}^{2\pi }\int _{0}^{\pi } Y_{\mu _0\mu _1}^{(\ell )}(\theta _1,\theta _2)Y_{\eta _0\eta _1}^{(k)}(\theta _1,\theta _2) \rho (\theta _1)\rho (\theta _2)\,d\theta _2\,d\theta _1, \end{aligned}$$

(A1)

$$\begin{aligned}&\quad= \dfrac{1}{2\pi }\int _{0}^{2\pi }\int _{0}^{\pi } Y_{\mu _0\mu _1}^{(\ell )}(\theta _1,\theta _2)Y_{\eta _0\eta _1}^{(k)}(\theta _1,\theta _2) \rho (\theta _2)\,d\theta _2\,d\theta _1, \end{aligned}$$

(A2)

with multi-indices $i=(\mu _0,\mu _1,\ell )$ and $j=(\eta _0,\eta _1,k)$. Focusing on the case $\ell = k = 0$, then

$$\begin{aligned} \int _{0}^{2\pi }\int _{0}^{\pi }&Y_{\mu _0\mu _1}^{(0)}(\theta _1,\theta _2)Y_{\eta _0\eta _1}^{(0)}(\theta _1,\theta _2) \rho (\theta _2)\,d\theta _2\,d\theta _1 \nonumber \\&= \int _{0}^{2\pi }\int _{0}^{\pi } P_{\mu _0,\mu _1}(\sin (\theta _2))\cos (\mu _1\theta _1) P_{\eta _0,\eta _1}(\sin (\theta _2))\cos (\eta _1\theta _1) \nonumber \\&\quad \times \rho (\theta _2)\,d\theta _2\,d\theta _1, \end{aligned}$$

(A3)

$$\begin{aligned}&= \int _{0}^{2\pi } \cos (\mu _1\theta _1) \cos (\eta _1\theta _1) \,d\theta _1\nonumber \\&\quad\times \int _{0}^{\pi } P_{\mu _0,\mu _1}(\sin (\theta _2)) P_{\eta _0,\eta _1}(\sin (\theta _2)) \rho (\theta _2)\,d\theta _2, \end{aligned}$$

(A4)

$$\begin{aligned}&= \pi \delta _{\mu _1\eta _1}\int _{0}^{\pi } P_{\mu _0,\mu _1}(\sin (\theta _2)) P_{\eta _0,\eta _1}(\sin (\theta _2)) \rho (\theta _2)\,d\theta _2. \end{aligned}$$

(A5)

Similarly, for $\ell =k=1$,

$$\begin{aligned} \int _{0}^{2\pi }\int _{0}^{\pi }&Y_{\mu _0\mu _1}^{(1)}(\theta _1,\theta _2)Y_{\eta _0\eta _1}^{(1)}(\theta _1,\theta _2) \rho (\theta _2)\,d\theta _2\,d\theta _1 \end{aligned}$$

(A6)

$$\begin{aligned}&= \pi \delta _{\mu _1\eta _1}\int _{0}^{\pi } P_{\mu _0,\mu _1}(\sin (\theta _2)) P_{\eta _0,\eta _1}(\sin (\theta _2)) \rho (\theta _2)\,d\theta _2, \end{aligned}$$

(A7)

and for the case of $\ell \ne k$,

$$\begin{aligned} \int _{0}^{2\pi }\int _{0}^{\pi }&Y_{\mu _0\mu _1}^{(0)}(\theta _1,\theta _2)Y_{\eta _0\eta _1}^{(1)}(\theta _1,\theta _2) \rho (\theta _2)\,d\theta _2\,d\theta _1 = 0, \end{aligned}$$

(A8)

$$\begin{aligned}&\int _{0}^{2\pi }\int _{0}^{\pi } Y_{\mu _0\mu _1}^{(1)}(\theta _1,\theta _2)Y_{\eta _0\eta _1}^{(0)}(\theta _1,\theta _2) \rho (\theta _2)\,d\theta _2\,d\theta _1 = 0. \end{aligned}$$

(A9)

Hence,

$$\begin{aligned} \int _{0}^{2\pi }\int _{0}^{\pi }&Y_{\mu _0\mu _1}^{(\ell )}(\theta _1,\theta _2)Y_{\eta _0\eta _1}^{(k)}(\theta _1,\theta _2) \rho (\theta _2)\,d\theta _2\,d\theta _1 \nonumber \\&= \pi \delta _{\ell k}\delta _{\mu _1\eta _1}\int _{0}^{\pi } P_{\mu _0,\mu _1}(\sin (\theta _2)) P_{\eta _0,\mu _1}(\sin (\theta _2))\rho (\theta _2)\,d\theta _2. \end{aligned}$$

(A10)

Substituting Eq. (A10) and the definition of $\rho (\theta _2)$ into (A2),

$$\begin{aligned} \left<\phi _{i}\phi _{j}\right>\propto \delta _{\ell k}\delta _{\mu _1\eta _1}\int ^{\pi }_{0} P_{\mu _0,\mu _1}(\sin (\theta _2))P_{\eta _0,\mu _1}(\sin (\theta _2))e^{\kappa \cos (\theta _2)}\,d\theta _2. \end{aligned}$$

(A11)

Appendix B Description of Linear Post-maneuver Solution

This appendix briefly describes the method used to approximate the propagated deviation in the trajectory using traditional linearization-based methods. Due to the definition of the inputs $\theta _1$ and $\theta _2$, namely the mean vector corresponding to the singularity at the pole in the canonical basis, the problem is ill-defined when examining deviations in the random inputs in a linearized approach. Instead, we model the variations in the maneuver direction by angles $\alpha$ and $\beta$ relative to the nominal maneuver direction, and

$$\begin{aligned} \varvec{x}(\theta _1,\theta _2) = \varvec{x}(\alpha ,\beta ) = \dfrac{1}{\sqrt{\tan ^2(\alpha ) + \tan ^2(\beta ) + 1}}\begin{bmatrix}\tan \left( \alpha \right) \\ \tan \left( \beta \right) \\ 1 \end{bmatrix}. \end{aligned}$$

(B12)

Angles $\theta _1$ and $\theta _2$ may be transformed to $\alpha$ and $\beta$ through $\varvec{x}$.

Let $\varvec{s}(t) = \begin{bmatrix} \varvec{r}(t)^T&\varvec{v}(t)^T\end{bmatrix}^T$ be the translation state vector at time t. A maneuver performed at $t_m$ is modeled as impulsive with velocity change $\Delta \varvec{v}$ as in Eq. (56) for a given $\varvec{x}$. The variables $\varvec{s}_m^+$ and $\varvec{s}_m^-$ denote the post- and pre-maneuver state, respectively, and

$$\begin{aligned} \varvec{s}_m^+ = \varvec{s}_m^- + \Delta \varvec{s}_m, \quad \quad \Delta \varvec{s}_m = \begin{bmatrix} \varvec{0}_3 \\ \Delta \varvec{v}(\varvec{x})\end{bmatrix}. \end{aligned}$$

The start and end epochs of the scenario are $t_0$ and $t_f$ with $t_0< t_m < t_f$. As in Eq. (56), the maneuver is assumed to be independent of $\varvec{s}(t_m)$. Group the initial state and maneuver parameters into the vector

$$\begin{aligned} \varvec{\lambda }_0 = \begin{bmatrix}\varvec{s}(t_0)^T&{\alpha }&\beta \end{bmatrix}^T. \end{aligned}$$

Let $\varvec{s}^*(t_f)$ be the propagated state given initial conditions and maneuver parameters $\varvec{\lambda }_0^*$. We wish to determine the approximate solution $\varvec{s}(t_f)$ that corresponds to the initial conditions $\varvec{\lambda }_0 = \varvec{\lambda }^*_0 + \Delta \varvec{\lambda }_0$, where $\Delta \varvec{\lambda }_0$ is a deviation relative to a nominal vector $\varvec{\lambda }^*_0$. We will generate the solution

$$\begin{aligned} \varvec{s}(t_f) \approx \varvec{s}^*(t_f) + \Delta \varvec{s}(t_f),\quad \quad \Delta \varvec{s}(t_f) = \left[ \dfrac{\partial \varvec{s}(t_f)}{\partial \varvec{\lambda _0}}\right] _{\varvec{\lambda }^*_0} \Delta \varvec{\lambda }_0. \end{aligned}$$

Hence,

$$\begin{aligned} \dfrac{\partial \varvec{s}(t_f)}{\partial \varvec{\lambda }_0} =\,&\dfrac{\partial \varvec{s}(t_f)}{\partial \varvec{s}_m^+}\dfrac{\partial \varvec{s}_m^- + \Delta \varvec{s}_m}{\partial \varvec{\lambda }_0},\nonumber \\ =\,&\varvec{\Phi }_{f,m}\left( \dfrac{\partial \varvec{s}_m^-}{\partial \varvec{\lambda }_0} + \dfrac{\partial \Delta \varvec{s}_m}{\partial \varvec{\lambda }_0}\right) ,\nonumber \\ =\,&\varvec{\Phi }_{f,m}\left( \begin{bmatrix}\varvec{\Phi }_{m,0}&\varvec{0}_{6\times 2}\end{bmatrix} + \begin{bmatrix} \varvec{0}_{6\times 6}&\dfrac{\partial \Delta \varvec{s}_m}{\partial \alpha }&\dfrac{\partial \Delta \varvec{s}_m}{\partial \beta }\end{bmatrix} \right) ,\nonumber \\ =&\varvec{\Phi }_{f,m}\left( \begin{bmatrix}\varvec{\Phi }_{m,0}&\dfrac{\partial \Delta \varvec{s}_m}{\partial \alpha }&\dfrac{\partial \Delta \varvec{s}_m}{\partial \beta }\end{bmatrix} \right) , \end{aligned}$$

(B13)

where $\varvec{\Phi }_{i,j}$ is the state transition matrix from time $t_j$ to time $t_i$ with reference trajectory $\varvec{\lambda }^*_0$. The partials with respect to $\alpha$ and $\beta$ are then found by differentiating Eq. (B12).

The accuracy of this approach based on linearization is compared to the PCE-based surrogate using the same samples. The evaluated partials Eq. (B13) and the final reference trajectory $\varvec{s}^*(t_f)$ are generated once given the initial mean of the prior PDF with $\alpha ^*$ and $\beta ^*$ equal to zero radians. For each random input vector, $\varvec{s}(t_0,\varvec{\xi }) - \varvec{s}^*(t_0)$, $\alpha (\varvec{\xi })$, and $\beta (\varvec{\xi })$ are computed to yield $\Delta \varvec{\lambda }_0(\varvec{\xi })$. Finally, the RMS error is based on the position elements of the vector

$$\begin{aligned} \varvec{s}^*(t_f) + \Delta \varvec{s}(t_f,\varvec{\xi }) - \varvec{s}(t_f,\varvec{\xi }) \end{aligned}$$

(B14)

over all points considered in the analysis.