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Reputation plays an important role in the collaboration in people's daily life. In many cases, task allocation in the human world is based on someone's reputation, which is gained from the evaluation of the completion of historical tasks. In the collaboration system of the heterogeneous multi-robot, reputation is introduced to solve the task allocation problem. A detailed formal model based on reputation for heterogeneous multi-robot collaboration is given, including the framework, reputation matrix, reputation attenuation curve, new robot member reward characteristics and robot alliance reward characteristics. The reputation in the collaboration system is divided to three categories: direct reputation from one robot to the other, overall reputation of a robot in the collaboration system and the robot where the group's reputation. Task attempts to be assigned to the robot with relatively high reputation, which can greatly improve the success rate of implementation of its mandate, thereby reducing the time of the system task recovery and redistribution. Simulation results show that the model can be used in a multi-robot task allocation system, and has good efficiency.