We study distributed planning for multi-robot systems to provide optimal service to cooperative tasks that are distributed over space and time. Each task requires service by sufficiently many robots at the specified location within the specified time window. Tasks arrive over episodes and the robots try to maximize the total value of service in each episode by planning their own trajectories based on the specifications of incoming tasks. Robots are required to start and end each episode at their assigned stations in the environment. We present a game theoretic solution to this problem by mapping it to a game, where the action of each robot is its trajectory in an episode, and using a suitable learning algorithm to obtain optimal joint plans in a distributed manner. We present a systematic way to design minimal action sets (subsets of feasible trajectories) for robots based on the specifications of incoming tasks to facilitate fast learning. We then provide the performance guarantees for the cases where all the robots follow a best response or noisy best response algorithm to iteratively plan their trajectories. While the best response algorithm leads to a Nash equilibrium, the noisy best response algorithm leads to globally optimal joint plans with high probability. We show that the proposed game can in general have arbitrarily poor Nash equilibria, which makes the noisy best response algorithm preferable unless the task specifications are known to have some special structure. We also describe a family of special cases where all the equilibria are guaranteed to have bounded suboptimality. Simulations and experimental results are provided to demonstrate the proposed approach.

我们研究了多机器人系统的分布式规划，以便为分布在空间和时间上的协作任务提供最佳服务。每项任务都需要在指定时间窗口内在指定位置由足够多的机器人提供服务。任务在剧集中到达，机器人试图通过根据传入任务的规格规划自己的轨迹来最大化每个剧集中的服务总价值。机器人需要在环境中指定的站点开始和结束每一集。我们通过将这个问题映射到一个游戏来提出这个问题的博弈论解决方案，其中每个机器人的动作是它在一个情节中的轨迹，并使用合适的学习算法以分布式方式获得最佳关节计划。我们提出了一种基于传入任务的规范为机器人设计最小动作集（可行轨迹的子集）的系统方法，以促进快速学习。然后，我们为所有机器人遵循最佳响应或嘈杂的最佳响应算法以迭代地规划其轨迹的情况提供性能保证。虽然最佳响应算法导致纳什均衡，但嘈杂的最佳响应算法以高概率导致全局最优联合计划。我们表明，所提出的博弈通常可以具有任意差的纳什均衡，这使得噪声最佳响应算法更可取，除非已知任务规范具有某些特殊结构。我们还描述了一系列特殊情况，其中所有均衡都保证具有有界次优性。