This paper presents a derivative-free method for computing approximate solutions to the uncertain Lambert problem (ULP) and the reachability set problem (RSP) while utilizing higher-order sensitivity matrices. These sensitivities are analogous to the coefficients of a Taylor series expansion of the deterministic solution to the ULP and RSP, and are computed in a derivative-free and computationally tractable manner. The coefficients are computed by minimizing least squared error over the domain of the input probability density function (PDF), and represent the nonlinear mapping of the input PDF to the output PDF. A non-product quadrature method known as the conjugate unscented transform is used to compute the multidimensional expectation values necessary to determine these coefficients with the minimal number of full model propagations. Numerical simulations for both the ULP and the RSP are provided to validate the developed methodology and illustrate potential applications. The benefits and limitations of the presented method are discussed.

中文翻译：

---

不确定朗伯问题和可达集问题概率解的高阶灵敏度矩阵方法

本文提出了一种利用高阶灵敏度矩阵计算不确定朗伯问题 (ULP) 和可达性集问题 (RSP) 的近似解的无导数方法。这些灵敏度类似于 ULP 和 RSP 确定性解的泰勒级数展开的系数，并以无导数和计算上易于处理的方式计算。通过最小化输入概率密度函数 (PDF) 域上的最小二乘误差来计算系数，并表示输入 PDF 到输出 PDF 的非线性映射。使用称为共轭无迹变换的非乘积正交方法来计算确定这些系数所需的多维期望值，并使用最少的完整模型传播次数。提供了 ULP 和 RSP 的数值模拟来验证开发的方法并说明潜在的应用。讨论了所提出方法的优点和局限性。