This paper describes a trajectory planning algorithm for mobile robot navigation in crowded environments; the aim is to solve the problem of planning a valid path through moving people. The proposed solution relies on an algorithm based on the Informed Optimal Rapidly-exploring Random Tree (InformedRRT*), where the planner continuously computes a valid path to navigate in crowded environments. While the robot executes the trajectory of the current path, this re-planning method always allows a feasible and optimal solution to be obtained. Compared to other state-of-the-art algorithms, this solution does not compute the entire path each time an obstacle is detected, instead it evaluating the current solution validity, i.e., the presence of moving obstacles on the current path; in this case the algorithm tries to repair the current solution. Only if the current path is completely unacceptable is a new path computed from scratch. Thanks to its reactivity, our solution always guarantees a valid path that brings the robot to the desired goal position. This dynamic approach is validated in a real case scenario where a mobile robot moves through a human crowd in a safe and reliable way.