Current Federal Aviation Administration regulations require that passing aircraft must either meet a specified horizontal or vertical separation distance. However, solving for optimal avoidance trajectories with conditional inequality path constraints is problematic for gradient-based numerical nonlinear programming solvers since conditional constraints typically possess non-differentiable points. Further, the literature is silent on robust treatment of approximation methods to implement conditional inequality path constraints for gradient-based numerical nonlinear programming solvers. This paper proposes two efficient methods to enforce conditional inequality path constraints in the optimal control problem formulation and compares and contrasts these approaches on representative airborne avoidance scenarios. The first approach is based on a minimum area enclosing superellipse function and the second is based on use of sigmoid functions. These proposed methods are not only robust, but also conservative, that is, their construction is such that the approximate feasible region is a subset of the true feasible region. Further, these methods admit analytically derived bounds for the over-estimation of the infeasible region with a demonstrated maximum error of no greater than 0.3% using the superellipse method, which is less than the resolution of typical sensors used to calculate aircraft position or altitude. However, the superellipse method is not practical in all cases to enforce conditional inequality path constraints that may arise in the nonlinear airborne collision avoidance problem. Smith et al., J Aeronaut Aerospace Eng 2017, 6:3 DOI: 10.4172/2168-9792.1000195 Therefore, this paper also highlights by example when the use of sigmoid functions are more appropriate.
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机译：当前的美国联邦航空管理局法规要求经过的飞机必须满足指定的水平或垂直间隔距离。但是，由于条件约束通常具有不可微的点，因此对于基于梯度的数值非线性规划求解器，用条件不等式路径约束求解最佳回避轨迹是有问题的。此外，文献没有提及对逼近方法的鲁棒处理，以实现基于梯度的数值非线性规划求解器的条件不等式路径约束。本文提出了两种在最优控制问题制定中强制执行条件不等式路径约束的有效方法，并将这些方法在代表性的空中避让场景中进行了比较和对比。第一种方法是基于包含超椭圆函数的最小面积，第二种方法是基于使用S型函数。这些提出的方法不仅鲁棒，而且保守，也就是说，它们的构造使得近似可行区域是真实可行区域的子集。此外，这些方法使用超椭圆方法允许分析得出的边界用于过高估计不可行区域，并证明最大误差不大于0.3％，该误差小于用于计算飞机位置或高度的典型传感器的分辨率。但是，超椭圆法并非在所有情况下都适用于强制实施可能发生在非线性机载避免碰撞问题中的条件不等式路径约束。 Smith et al。，J Aeronaut Aerospace Eng 2017，6：3 DOI：10.4172 / 2168-9792.1000195因此，本文还通过示例突出显示了当使用S型函数更合适时。
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