Multi Agents’ Multi Targets Mission Under Uncertainty Using Probability Navigation Function Shlomi Hacohen 1 Shraga Shoval 2 and Nir Shvalb 2 Abstract— In this paper we consider the problem of co- operative control of a swarm of autonomous heterogeneous mobile agents that are required to intercept a group of moving targets while avoiding contacts with dynamic obstacles. Traditionally these type of problems are solved by decomposing the solution into several sub problems: targets assignments, coordinated interception control, motion planning and motion control. In this paper we present a simultaneous solution to these problems based on the Probabilistic Navigation Function (PNF). The proposed solution considers uncertainties in the targets and obstacles locations. such that the locations and geometries of the targets and obstacles are given by Gaussian probability distributions. These probabilities are convoluted with the agents’, obstacles’ and targets’ geometries to provide a Global Probability Navigation Function - ϕ . The PNF provides an analytic solution, and guarantees a simultaneous interception of all targets while limiting the risk of the agents to a given value. The complexity of the solution is linear with the number of targets and agents, and therefore is not limited to small problems. Although the solution provided by the PNF is not optimal, it provides simple and efficient solution, making it suitable for a large range of real time applications. I. I NTRODUCTION The complexity of many environments and missions may require capabilities that are not feasible for a single agent. Additionally, time and process constraints may require the use of multiple agents working simultaneously on different aspects of the mission in order to successfully accomplish the objectives in time. To achieve the level of autonomy and flexibility in a large variety of scenarios, teams of agents must be assembled, so they can work together to accom- plish the specific missions that no individual agent alone can. The difficulties in designing a cooperative team are significant [1]; how to formulate, decompose, and allocate tasks among a group of heterogeneous intelligent agents, how to enable agents to communicate and interact, how to ensure that agents act coherently in their actions and many more. Previous research in distributed heterogeneous agents cooperation includes [2], who proposes a three- layered control architecture for a robotic swarm that consists of a planner level, a control level, and a functional level. Caloud et al., [3] propose an architecture that includes a task planner, a task allocator, a motion planner, and an execution monitor. Asama et al., [4] describe an architecture called ACTRESS that utilizes a negotiation framework to allow agents to recruit help when needed. Kumar et al. [5] use a 1 Department of Mecanical Engineering Ariel University, Israel 2 Department of Industrial and Management Engineering Ariel University, Israel hierarchical division of authority to perform cooperative fire- fighting. Cooperation can be defined as how the components of a system work together to achieve the global emerging properties. In cooperative teams, individual members that appear to be independent, work together to create a highly complex performance, greater-than-the-sum-of-its-parts. The behavior of a team is often hard to predict as the number of interactions between the team members increases combina- torically. The interactions, or interfaces between the members are often the most susceptible elements, as the members do not necessarily ”speak the same language”. As a result, the performance of the team can be lower-than-the-sum-of-its- parts. The traditional assignment problem of agents to tasks is considered to be one of the fundamental combinatorial optimization problems. The mathematical formulation of the traditional assignment problem is: Given two sets, Agents A and Tasks T , and a weight function C : A × T → R . Find a bijection function f : A → T such that the cost function ∑ a∈A C (a, f (a)) is minimized or maximized. The Hungarian algorithm [6] is one of the first algorithms suggested for solving the linear assignment problem within time bounded by a polynomial expression of the number of agents. There have been several more studies on efficient so- lutions to the assignment problem. Most algorithms proposed for the assignment problem are for static environments, e.g., the graph matching algorithm [7], network simplex algorithm [8], distributed auction algorithm [9], genetic algorithm [10], agent based algorithm [11], dynamic Tabu search algorithm [12], and an adaptive task assignment algorithm in a dynamic environment [13]. In this paper we consider a scenario where a team of autonomous mobile agents are required to intercept a group of dynamic targets while avoiding collisions with obstacles that are scattered in the environment. Due to imperfect information (e.g. limited capabilities of sensors, noisy en- vironment etc.), the locations of the agents, the targets and the obstacles are given by a probabilistic distribution function as illustrated in 1. In the scenario shown, there are 2 agents denoted by A 1 and A 2 , 3 targets denoted by T 1 , T 2 and T 3 , and 2 obstacles (O 1 and O 2 ). The shaded areas around the objects represent their Probability Density Function (PDF) such that a darker shade represents higher probability for the objects to be at a certain location. Notice that the targets and the obstacles may have different geometries, resulting in different PDFs. 2017 13th IEEE International Conference on Control & Automation (ICCA) July 3-6, 2017. Ohrid, Macedonia 978-1-5386-2679-5/17/$31.00 ©2017 IEEE 845