双积分器的旅行推销员问题
Traveing Salesperson Problems for a double integrator
摘要：
本文针对速度有界、控制输入有界的双积分器提出了一些新的路径规划策略。首先，我们研究了旅行商问题（TSP）的以下版本：给定$\real^d$中的一组点，找到双积分器在点集上的最快行程。我们首先给出在最坏的情况下完成这样一次旅行所需时间的渐近界。然后，我们研究了一个随机版本的TSP的双积分点随机抽样均匀分布在一个紧凑的环境中的$\real^2$和$\real^3$。我们提出了新的算法，在一个恒定的因素内执行的最佳策略，具有很高的概率。最后，我们研究了一个动态TSP：给定一个随机过程，产生的目标，是否有一个政策，保证未访问的目标的数量不会随着时间的推移发散？如果存在这种稳定的政策，那么达到目标的最低等待时间是多少？我们提出了新的稳定滚动时域算法，其性能是在一个恒定的因素，从最佳的高概率，在$\real^2$以及$\real^3$。我们还认为，这些算法给一个特定的非完整车辆，Dubins车辆相同的性能。
英文摘要：
In this paper we propose some novel path planning strategies for a double integrator with bounded velocity and bounded control inputs. First, we study the following version of the Traveling Salesperson Problem (TSP): given a set of points in $\real^d$, find the fastest tour over the point set for a double integrator. We first give asymptotic bounds on the time taken to complete such a tour in the worst-case. Then, we study a stochastic version of the TSP for double integrator where the points are randomly sampled from a uniform distribution in a compact environment in $\real^2$ and $\real^3$. We propose novel algorithms that perform within a constant factor of the optimal strategy with high probability. Lastly, we study a dynamic TSP: given a stochastic process that generates targets, is there a policy which guarantees that the number of unvisited targets does not diverge over time? If such stable policies exist, what is the minimum wait for a target? We propose novel stabilizing receding-horizon algorithms whose performances are within a constant factor from the optimum with high probability, in $\real^2$ as well as $\real^3$. We also argue that these algorithms give identical performances for a particular nonholonomic vehicle, Dubins vehicle.
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