This article considers the problem of online trajectory design under time-varying environments. We formulate the general trajectory optimization problem within the framework of time-varying constrained convex optimization and propose a novel version of online gradient ascent algorithm (OGA) for such problems. Respecting the online nature, we carefully select the step size of OGA at each iteration so that the iterates stay feasible. Importantly, the proposed algorithm allows noisy gradients, expanding the range of practical applicability. In contrast to the most available literature, we present the offline sublinear regret of OGA up to the path length variations of the offline optimal solution, the cumulative gradient, and the error in the gradient variations. Furthermore, we establish a lower-bound on the offline dynamic regret, which defines the optimality of any trajectory. To show the efficacy of the proposed algorithm, we consider two practical problems of interest. First, a device to device (D2D) communications setting, where the goal is to design a user trajectory while maximizing its connectivity to the internet. Second, planning energy-efficient trajectories for unmanned surface vehicles (USV) under strong disturbances in ocean environments. Different from the state-of-the-art trajectory planning algorithms that entail planning and re-planning the full trajectory using the forecast data at each time instant, the proposed algorithm is entirely online and relies mostly on the ocean velocity measurements at the vehicle location. The detailed simulation results demonstrate the significance of the proposed algorithm on both synthetic and real data sets. Video result is available at https://tinyurl.com/y3ahmhsf.

中文翻译：

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针对时变环境使用不精确梯度反馈的在线轨迹优化

本文考虑时变环境下的在线轨迹设计问题。我们在时变约束凸优化的框架内制定了一般轨迹优化问题，并针对此类问题提出了一种新版本的在线梯度上升算法（OGA）。尊重在线性质，我们在每次迭代时仔细选择 OGA 的步长，以便迭代保持可行。重要的是，所提出的算法允许噪声梯度，扩大了实际适用范围。与大多数可用的文献相比，我们提出了 OGA 的离线次线性遗憾，直到离线最优解的路径长度变化、累积梯度和梯度变化的误差。此外，我们建立了离线动态后悔的下限，它定义了任何轨迹的最优性。为了显示所提出算法的有效性，我们考虑两个感兴趣的实际问题。首先是设备到设备 (D2D) 通信设置，其目标是设计用户轨迹，同时最大限度地提高其与互联网的连接性。其次，在海洋环境的强烈干扰下为无人水面舰艇（USV）规划节能轨迹。与需要使用每个时刻的预测数据规划和重新规划完整轨迹的最先进的轨迹规划算法不同，所提出的算法完全在线并且主要依赖于车辆位置的海洋速度测量. 详细的仿真结果证明了该算法在合成数据集和真实数据集上的重要性。