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This paper presents a trajectory planning algorithm for a robot operating in dynamic human environments. Environments such as pedestrian streets, hospital corridors, train stations or airports. We formulate the problem as planning a minimal cost trajectory through a potential field, defined from the perceived position and motion of persons in the environment. A Rapidly-exploring Random Tree (RRT) algorithm is proposed as a solution to the planning problem, and a new method for selecting the best trajectory in the RRT, according to the cost of traversing a potential field, is presented. The RRT expansion is enhanced to account for the kinodynamic robot constraints by using a robot motion model and a controller to add a reachable vertex to the tree. Instead of executing a whole trajectory, when planned, the algorithm uses a Model Predictive Control (MPC) approach, where only a short segment of the trajectory is executed while a new iteration of the RRT is computed. The planning algorithm is demonstrated in a simulated pedestrian street environment.